On aspects of characterising and calibrating the

interferometric gravitational wave detector,

GEO 600

Martin R. Hewitson, M. Sci.

Department of Physics and Astronomy

University of Glasgow

Presented as a thesis for the degree of Ph.D.

in the University of Glasgow, University Avenue,

Glasgow G12 8QQ.

c© M. R. Hewitson 6th May 2004

i

“The only thing to do with good advice is pass it on. It is never any use

to oneself.” — Oscar Wilde.

Summary

Gravitational waves are small disturbances, or strains, in the fabric of space-time. The

detection of these waves has been a major goal of modern physics since they were pre-

dicted as a consequence of Einstein’s General Theory of Relativity. Large-scale astro-

physical events, such as colliding neutron stars or supernovae, are predicted to release

energy in the form of gravitational waves. However, even with such cataclysmic events,

the strain amplitudes of the gravitational waves expected to be seen at the Earth are in-

credibly small: of the order 1 part in 1021 or less at audio frequencies. Because of these

extremely small amplitudes, the search for gravitational waves remains one of the most

challenging goals of modern physics.

One of the most promising types of detector currently under development are laser in-

terferometric detectors. The detectors are based on the Michelson interferometer and

measure phase differences between two beams of coherent light which have been made

to travel over orthogonal paths. These detectors are sensitive over a wide range of fre-

quencies, typically from around 50 Hz up to a few thousand Hertz. Currently a number

of such detectors exist in the world: the LIGO interferometers, GEO 600, VIRGO, and

TAMA. GEO 600 is a German-British interferometric gravitational wave detector being

constructed near Hannover, Germany, which aims to use advanced optical techniques—

power-recycling and signal-recycling—to reach its design sensitivity. At the end of 2003,

the last major hardware installations of the detector were completed; an intense period of

optimisation now follows to ready the instrument for long term observations. One of the

key aspects to commissioning and using GEO 600 to make astrophysical observations is

the main thrust of this thesis: the calibration of the instrument.

iii

In the past, calibrated data from interferometric gravitational wave detectors has typi-

cally been produced using frequency-domain calibration methods. This involves deter-

mining the transfer function of the detector from differential arm-length changes to the

detector output at a specific time. The reciprocal of this transfer function is used in the

frequency domain to convert spectra of the detector’s electrical output to strain; this is

the process of calibration.

Typical methods for the measurement of the detector transfer function renders the de-

tector effectively blind to gravitational wave signals. Because of this, it is generally

measured infrequently (once per day for example). If it is to be used to calibrate the

output of the detector from other times, it must be either assumed constant between suc-

cessive measurements, or it must be adjusted based on some other measured parameter

of the detector. Since it is difficult to reconstruct a calibrated time-series by using this

frequency-domain calibration approach, the calibration information is usually supplied,

along with the uncalibrated detector output, to any analysis groups. Due to this, analysis

methods are restricted to applying time-domain search methods to un-calibrated data;

the results of the search must then be calibrated. Frequency-domain search methods

can perform the calibration pre- or post-analysis. Each of these possible methods relies

heavily upon the fact that every group that receives the uncalibrated data and the cal-

ibration information must perform the calibration properly. As the number of groups

who individually perform the calibration increases, so does the possibility that errors are

introduced during the calibration process.

One way to avoid such problems is to approach the calibration of interferometric gravita-

tional wave detectors in the time-domain. By injecting calibration lines continually, we

can, in principle, determine the transfer function of the instrument from differential arm-

length changes to detector output on-line. If we then take the reciprocal of this transfer

function and apply it to the output of the detector using suitable time-domain filters, we

can, in real-time, reconstruct the external strain signal experienced by the detector.

This thesis starts by detailing the data recording system of GEO 600: an essential part of

producing a calibrated data set. The full data acquisition system, including all hardware

and software aspects, is described in detail. Comprehensive tests of the stability and

iv

timing accuracy of the system show that it has a typical duty cycle of greater than 99%

with an absolute timing accuracy (measured against GPS) of the order 15µs.

The thesis then goes on to describe the design and implementation of a time-domain

calibration method, based on the use of time-domain filters, for the power-recycled con-

figuration of GEO 600. This time-domain method is then extended to deal with the more

complicated case of calibrating the dual-recycled configuration of GEO 600.

The time-domain calibration method was applied to two long data-taking (science) runs.

The method proved successful in recovering (in real-time) a calibrated strain time-series

suitable for use in astrophysical searches. The accuracy of the calibration process was

shown to be good to 10% or less across the detection band of the detector. In principle,

the time-domain method presents no restrictions in the achievable calibration accuracy;

most of the uncertainty in the calibration process is shown to arise from the actuator used

to inject the calibration signals. The recovered strain series was shown to be equivalent

to a frequency-domain calibration at the level of a few percent. A number of ways

are presented in which the initial calibration pipeline can be improved to increase the

calibration accuracy. The production and subsequent distribution of a calibrated time-

series allows for a single point of control over the validity and quality of the calibrated

data.

The techniques developed in this thesis are currently being adopted by the LIGO inter-

ferometers to perform time-domain calibration of their three long-baseline detectors. In

addition, a data storage system is currently being developed by the author, together with

the LIGO calibration team, to allow all the information used in the time-domain cali-

bration process to be captured in a concise and coherent form that is consistent across

multiple detectors in the LSC.

Acknowledgements

I would like to offer my thanks and appreciation for the help and guidance of my super-

visors, Harry Ward and Graham Woan; their interest and enthusiasm in the work of this

thesis were invaluable motivators. I would also like to thank Ken Strain for his consul-

tation and expertise on technical matters to do with gravitational wave detectors, and in

particular, GEO 600. David Robertson provided significant support and assistance with

the various electronics aspects of this thesis, as well as giving sound advice on many

other matters; for this, he deserves my thanks. I would also like to thank Karsten Kotter

for his help and efforts regarding the data acquisition work of chapter two.

The GEO team comprises a great many people, too numerous to list here. While my

appreciation and thanks go out to all of them, I would like to specifically thank Hartmut

Grote, Joshua Smith, Uta Weiland, Harald Luck and Benno Willke for their help at the

GEO site, both in the work carried out there, and in making the environment a pleasing

place to work.

I would like to thank Gerhard Heinzel for his great expertise on many aspects of digital

signal processing, as well as for his general guidance on the overall calibration scheme

presented in this thesis.

As a member of The Institute for Gravitational Research at Glasgow University, I must

extend a big thank you to Jim Hough, head of the group, and all the other members of

the group for creating a stimulating and enjoyable environment in which to carry out this

work.

Let me also offer my thanks to Rejean Dupuis for the many interesting talks about all

vi

manner of things while sharing the office with me during the three years of this research.

Joshua Smith deserves my greatest thanks for having the desire and stamina to proof-

read this thesis.

I would like to acknowledge that this work would not have been possible without the

support and funding of PPARC together with the University of Glasgow.

And finally, let me say a big thank you to all my family and friends who have supported

me throughout this endeavour, and in particular to Annette who, with seemingly infinite

patience, has got me through this.

Preface

The contents of this thesis form an account of the work carried out by the author be-

tween October 2000 and October 2003. The main thrust of the work is in calibrating

the interferometric gravitational wave detector GEO 600. In addition, other work is pre-

sented that was in some cases necessary for the main work, and in some cases, a natural

extension of the scope of the calibration work.

In chapter1, the field of gravitational wave research is briefly described with a partic-

ular focus on interferometric gravitational wave observatories. An overview is given

of the design, construction, and current status of one such gravitational wave detector,

GEO 600. In addition, an outline mathematical derivation (in the framework of General

Relativity) is given describing what an interferometric gravitational wave detector, and

in particular, GEO, actually measures. The final part of this chapter focuses on setting

the context for the remainder of the work. Various aspects of the search for gravita-

tional waves, from the point of view of current data analysis methods, are used to set the

accuracy with which the calibration and data acquisition at GEO should be performed.

This work is derived in part from published literature, and in part from the authors un-

derstanding of the GEO 600 detector, mainly from the point of view of calibrating the

instrument.

The initial design of the data acquisition system used in GEO was based on that im-

plemented for the LIGO detectors. This design, and the purchase of the commercial

hardware elements, was completed prior to the start of this thesis work. The poor initial

performance of the system led to a redesign of the some of the timing hardware and a

complete redesign and implementation of the software layer of the data acquisition sys-

viii

tem. The redesign of the hardware timing layer and the entire software layer form the

main part of chapter2. Also presented are detailed experiments regarding the timing

performance of the final system. Some discussion is also given over to the storage of the

data and associated information. A small working team comprising the author, Karsten

Kotter, Harry Ward, and David Robertson was set up to work on the data acquisition

scheme. The design and writing of the software part of the system was carried out in the

most part by the author, with help and input from the rest of the data acquisition team.

The work on the hardware aspects of the system was carried out by the team as a whole.

In chapter3, the principles of calibrating GEO 600 are presented. A model of the de-

tector is developed to include all aspects of the instrument that relate any differential

arm-length changes (and hence strains) to the detector output signal. The chapter goes

on to lay the foundations of a time-domain calibration scheme that was used to calibrate

the instrument during an extended science run. Various validation techniques are dis-

cussed and employed to arrive at an estimation of the accuracy of the final calibration

scheme. Various characterisation investigations that arise naturally in the course of cal-

ibrating the instrument are also presented. The work presented in this chapter was done

mostly by the author with input from various members of the GEO team.

After GEO was run in power-recycled mode for some time, the instrument was up-

graded to be a dual-recycled interferometer. This involved the installation of an extra

mirror—the signal-recycling mirror—and the development of a new locking scheme for

this detector configuration. The calibration of the dual-recycled GEO is much more

complicated than for the power-recycled configuration. However, the method presented

in chapter3 for calibrating the power-recycled GEO was extended to deal with the dual-

recycling calibration. Chapter4 presents the principles behind calibrating the dual-

recycled GEO, including some discussion about the fundamental differences between

the two interferometer configurations. The chapter also sets the scene for using these

principles in the development of a time-domain calibration pipeline. The detailed de-

velopment of the pipeline and its application to simulated data is given in chapter5.

Chapter6 extends the pipeline to calibrating GEO for another extended science run. The

major part of the development and implementation of the dual-recycled GEO calibration

pipeline is the work of the author with help coming from other members of the GEO

ix

team.

The work of this thesis relies heavily on various digital signal processing methods.

One of the most used aspects of this field is time-domain digital filtering. AppendixA

presents a brief introduction to some of the digital filtering techniques used throughout

the work of this thesis. The discussion is by no means exhaustive, nor does it provide a

thorough introduction to the field of digital filtering. It does, however, summarise some

of the results and methods used in the calibration work. Most of the discussion presented

here is derived from published material; the analytic calculations of filter coefficients for

particular filter types using the bilinear transform are the work of the author.

Appendix B gives some pseudo-code listings of some aspects of the data acquisition

software.

Contents

Summary ii

Acknowledgements v

Preface vii

1 Introduction 1

1.1 GEO 600 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Design configuration. . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Signal extraction. . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.3 Sensitivity of GEO. . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Measuring gravitational waves. . . . . . . . . . . . . . . . . . . . . . 12

1.2.1 Application to calibrating GEO. . . . . . . . . . . . . . . . . 14

1.3 Calibration and DAQ requirements set from astrophysical searches. . . 15

1.3.1 Signals from continuous wave sources. . . . . . . . . . . . . . 16

1.3.2 Signals from the stochastic gravitational wave background. . . 16

1.3.3 Signals from inspiralling binary objects. . . . . . . . . . . . . 17

CONTENTS xi

1.3.4 Signals from burst type signals. . . . . . . . . . . . . . . . . . 18

1.3.5 GEO as part of a detector network. . . . . . . . . . . . . . . . 19

1.3.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2 Data acquisition and storage 22

2.1 System requirements. . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Hardware components. . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.1 The data collecting units. . . . . . . . . . . . . . . . . . . . . 24

2.2.2 Data archiving system. . . . . . . . . . . . . . . . . . . . . . 34

2.2.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3 Software design and the data flow model. . . . . . . . . . . . . . . . . 35

2.3.1 System configuration interface. . . . . . . . . . . . . . . . . . 37

2.3.2 Data flow model . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.3 DCU software . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3.4 Data archiving software and signal processing. . . . . . . . . . 46

2.3.5 System administration and configuration. . . . . . . . . . . . 53

2.4 System timing accuracy and stability. . . . . . . . . . . . . . . . . . . 54

2.4.1 Gross timing accuracy. . . . . . . . . . . . . . . . . . . . . . 54

2.4.2 Systematic timing offsets. . . . . . . . . . . . . . . . . . . . . 55

2.4.3 Fine timing accuracy. . . . . . . . . . . . . . . . . . . . . . . 56

CONTENTS xii

2.4.4 Long-term timing stability. . . . . . . . . . . . . . . . . . . . 61

2.4.5 Long-term timing stability during S1 science run. . . . . . . . 63

2.4.6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.5 System diagnostics and alarm system. . . . . . . . . . . . . . . . . . . 69

2.6 Environmental signal calibration and de-whitening. . . . . . . . . . . 70

2.6.1 User input system. . . . . . . . . . . . . . . . . . . . . . . . . 71

2.6.2 Calibration information storage. . . . . . . . . . . . . . . . . 73

2.6.3 An example calibration procedure. . . . . . . . . . . . . . . . 75

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3 Calibrating the power-recycled GEO 77

3.1 Principles of calibrating the power-recycled GEO. . . . . . . . . . . . 78

3.2 The Michelson locking scheme. . . . . . . . . . . . . . . . . . . . . . 80

3.2.1 The Michelson control servo. . . . . . . . . . . . . . . . . . . 80

3.3 Inducing a known mirror displacement. . . . . . . . . . . . . . . . . . 83

3.3.1 The electro-static drives (ESDs). . . . . . . . . . . . . . . . . 84

3.3.2 The injected calibration signal for S1. . . . . . . . . . . . . . 86

3.4 Measuring the optical gain of the detector. . . . . . . . . . . . . . . . 88

3.5 Recording the signals. . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.6 A model of the Michelson locking servo. . . . . . . . . . . . . . . . . 95

CONTENTS xiii

3.6.1 Relative gain. . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.6.2 Feedback paths. . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.6.3 Closed-loop transfer function. . . . . . . . . . . . . . . . . . 102

3.7 Real-time calibration. . . . . . . . . . . . . . . . . . . . . . . . . . .103

3.7.1 Deriving the calibration equation. . . . . . . . . . . . . . . . . 104

3.7.2 Up-sampling the optical and relative gains. . . . . . . . . . . . 107

3.7.3 Issues of signal dynamic range and numerical precision. . . . . 109

3.7.4 Suppressing the calibration lines. . . . . . . . . . . . . . . . . 112

3.7.5 Sign convention of GEO. . . . . . . . . . . . . . . . . . . . . 115

3.8 Application to the S1 science run. . . . . . . . . . . . . . . . . . . . . 116

3.8.1 Stability of the input calibration lines. . . . . . . . . . . . . . 116

3.8.2 Results and characterisation of the calibration process. . . . . 118

3.8.3 Validating the calibration scheme. . . . . . . . . . . . . . . . 123

3.8.4 Calibration accuracy. . . . . . . . . . . . . . . . . . . . . . . 128

3.8.5 Calibration bandwidth. . . . . . . . . . . . . . . . . . . . . . 129

3.9 A summary of the calibration procedure. . . . . . . . . . . . . . . . . 130

3.10 Future requirements and improvements. . . . . . . . . . . . . . . . . 132

4 Calibration of the dual-recycled GEO 600 : principles 133

4.1 Improving techniques from the power-recycled GEO calibration scheme134

CONTENTS xiv

4.1.1 Improvements carried forward. . . . . . . . . . . . . . . . . . 134

4.1.2 Modelling the optical response of the DRMI. . . . . . . . . . 135

4.1.3 Obtaining the optimal signal-to-noise ratio of the calibration. . 140

4.2 Time-domain calibration of DR GEO. . . . . . . . . . . . . . . . . . 142

4.2.1 System identification process. . . . . . . . . . . . . . . . . . . 144

4.2.2 Correcting for the optical response. . . . . . . . . . . . . . . . 146

4.2.3 Correcting for the length-control loop. . . . . . . . . . . . . . 147

4.2.4 Calculating the strain signal. . . . . . . . . . . . . . . . . . . 148

4.3 Summary of the procedure. . . . . . . . . . . . . . . . . . . . . . . . 148

5 Calibration of the dual-recycled GEO 600 — Simulations and software 152

5.1 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . .153

5.1.1 Simulation outputs. . . . . . . . . . . . . . . . . . . . . . . . 154

5.2 Calibration software. . . . . . . . . . . . . . . . . . . . . . . . . . . .156

5.2.1 System identification. . . . . . . . . . . . . . . . . . . . . . . 156

5.2.2 Optical correction. . . . . . . . . . . . . . . . . . . . . . . . . 157

5.2.3 Calculation of the strain signal. . . . . . . . . . . . . . . . . . 161

5.2.4 Output of software. . . . . . . . . . . . . . . . . . . . . . . . 162

5.3 Simulation experiments. . . . . . . . . . . . . . . . . . . . . . . . . .166

5.3.1 Experiment 1 – Timing aspects of the pipeline. . . . . . . . . . 166

CONTENTS xv

5.3.2 Experiment 2 – Performance of the optical filter. . . . . . . . . 167

5.3.3 Experiment 3 – Overall optical gain correction. . . . . . . . . 169

5.3.4 Experiment 4 – System identification performance with fixed

optical gain . . . . . . . . . . . . . . . . . . . . . . . . . . . .170

5.3.5 Experiment 5 – System identification performance with varying

optical gain . . . . . . . . . . . . . . . . . . . . . . . . . . . .172

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .174

6 Calibration of the dual-recycled GEO 600 — application to the S3 sciencerun 176

6.1 Experimental setup and detector details. . . . . . . . . . . . . . . . . 177

6.1.1 Optical response of the detector. . . . . . . . . . . . . . . . . 177

6.1.2 Length-control servo electronics. . . . . . . . . . . . . . . . . 180

6.1.3 Closed-loop transfer function of the Michelson servo. . . . . . 183

6.1.4 Demodulation phase for high-power diode. . . . . . . . . . . . 184

6.1.5 Injected calibration lines. . . . . . . . . . . . . . . . . . . . . 187

6.1.6 Recording of relevant signals. . . . . . . . . . . . . . . . . . . 193

6.2 Calibration software. . . . . . . . . . . . . . . . . . . . . . . . . . . .193

6.2.1 System identification. . . . . . . . . . . . . . . . . . . . . . . 194

6.2.2 Optical correction. . . . . . . . . . . . . . . . . . . . . . . . . 195

6.2.3 Length-control servo correction. . . . . . . . . . . . . . . . . 197

6.3 Results and validation. . . . . . . . . . . . . . . . . . . . . . . . . . .199

CONTENTS xvi

6.3.1 Frequency-domain calibration comparison. . . . . . . . . . . . 199

6.3.2 Recovered parameters. . . . . . . . . . . . . . . . . . . . . . 201

6.3.3 Calibration suppression performance. . . . . . . . . . . . . . 206

6.3.4 Calibration accuracy. . . . . . . . . . . . . . . . . . . . . . . 208

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .214

A Digital filters 217

A.1 IIR filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .217

A.1.1 The Bilinear transform. . . . . . . . . . . . . . . . . . . . . . 218

A.1.2 Changes of Gain. . . . . . . . . . . . . . . . . . . . . . . . . 220

A.1.3 Cascading systems. . . . . . . . . . . . . . . . . . . . . . . . 221

A.2 Measured transfer functions of the DAQ anti-alias filters. . . . . . . . 223

A.3 A design for IIR Chebychev/Butterworth filters. . . . . . . . . . . . . 226

B Code listings 233

B.1 DCU code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .233

B.1.1 Bounded buffer. . . . . . . . . . . . . . . . . . . . . . . . . .234

B.1.2 VMIC interrupt handler . . . . . . . . . . . . . . . . . . . . . 235

B.1.3 Bounded buffer resynchronisation. . . . . . . . . . . . . . . . 237

B.1.4 DAQ checking software. . . . . . . . . . . . . . . . . . . . . 239

Glossary of terms 242

List of Figures

1.1 A schematic of GEO.. . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 The duty cycle of the detector during S1.. . . . . . . . . . . . . . . . . 8

1.3 Schematic of the SR cavity.. . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Design sensitivity of GEO 600 in broad-band and narrow-band modes.12

1.5 Timing accuracy for time of arrival of astronomical signals.. . . . . . . 19

2.1 A DCU crate layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 The specifications of the signals required to drive the ICS110b in slave

mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 A simplified schematic of the internal workings of the ICS110b ADC card.27

2.4 Amplitude spectral densities of a sinusoidal signal recorded by the DAQ

system with different preamplifier gain settings.. . . . . . . . . . . . . 28

2.5 Measured DC offsets of an ICS board.. . . . . . . . . . . . . . . . . . 30

2.6 A plot of the timing signals from the Glasgow timing card at the start of

the acquisition process.. . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.7 An overview of the data acquisition system.. . . . . . . . . . . . . . . 36

LIST OF FIGURES xviii

2.8 A schematic of the flow of data through the various software elements

of the DAQ system.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.9 A schematic of the operation of the bounded buffer model.. . . . . . . 42

2.10 A flow diagram of the part of the data archiving software that receives

and processes the data.. . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.11 A measured transfer function of the 4096 Hz IIR anti-alias filter that is

applied in the data archiving software.. . . . . . . . . . . . . . . . . . 51

2.12 The file format used for the initial storage of the data.. . . . . . . . . . 52

2.13 A test of the gross timing accuracy of the DAQ system.. . . . . . . . . 55

2.14 A plot of the injected ramp signal alongside the 1 PPS signal from the

DAQ GPS receiver.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.15 A example of an injected ramp as it appears in the data.. . . . . . . . . 57

2.16 HP33120A generated ramps of different frequencies.. . . . . . . . . . 59

2.17 The dependency of measured offset scatter on injected ramp frequency.59

2.18 A plot of mean measured offset versus injected ramp frequency.. . . . 61

2.19 A plot of standard deviation of the measured offsets versus injected ramp

frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.20 A 24 hour time series if measured offsets.. . . . . . . . . . . . . . . . 62

2.21 A histogram of 24 hours of measured offsets.. . . . . . . . . . . . . . 62

2.22 The behaviour of the timing of the DAQ system across system reboots.. 63

2.23 The injected calibration signal from the S1 science run.. . . . . . . . . 64

2.24 Results of the S1 timing tests.. . . . . . . . . . . . . . . . . . . . . . . 65

LIST OF FIGURES xix

2.25 Bad states of the S1 timing tests.. . . . . . . . . . . . . . . . . . . . . 65

2.26 Timing artifacts of the S1 timing tests.. . . . . . . . . . . . . . . . . . 66

2.27 The offset drift from the GPS servo.. . . . . . . . . . . . . . . . . . . 68

2.28 An example system calibration function.. . . . . . . . . . . . . . . . . 76

2.29 An example de-whitening function.. . . . . . . . . . . . . . . . . . . 76

3.1 The optical layout of the power-recycled GEO.. . . . . . . . . . . . . 78

3.2 A schematic of the Michelson servo.. . . . . . . . . . . . . . . . . . . 80

3.3 Simulated outputs from a low-noise PRMI lock.. . . . . . . . . . . . . 81

3.4 Simulated outputs from a high-noise PRMI lock.. . . . . . . . . . . . 81

3.5 The model ESD response compared to the design ESD response.. . . . 84

3.6 The calibrated response of the ESDs.. . . . . . . . . . . . . . . . . . . 86

3.7 The calibration lines shown in the injected calibration signal, the quad-

rant photodiode error-point, and the high-power error-point signals.. . 87

3.8 The whitening filters for the three detector error-point signals.. . . . . 92

3.9 The de-whitening filters for the three detector error-point signals.. . . . 92

3.10 The difference between the whitening and de-whitening filters used in

the calibration process.. . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.11 A ‘map’ of the signal recovery for the calibration process.. . . . . . . . 94

3.12 A loop diagram of the Michelson longitudinal control servo.. . . . . . 95

3.13 The Michelson servo gain control circuit.. . . . . . . . . . . . . . . . 98

LIST OF FIGURES xx

3.14 The gain curves of the two Michelson gain knobs.. . . . . . . . . . . . 98

3.15 The measured transfer function of the slow path electronics.. . . . . . 99

3.16 The measured transfer function of the fast path electronics.. . . . . . . 99

3.17 Bode plots of the Michelson servo feedback paths.. . . . . . . . . . . . 101

3.18 The modelled Michelson servo closed-loop transfer function for various

optical gain settings. . . . . . . . . . . . . . . . . . . . . . . . . . . .103

3.19 The measured Michelson servo closed-loop transfer function.. . . . . . 103

3.20 A system diagram of the time-domain calibration function.. . . . . . . 104

3.21 The transfer functions of the three terms of the calibration function.. . 106

3.22 Measurements of the individual terms of the calibration function.. . . . 107

3.23 Explanation of the interpolation of the optical gain.. . . . . . . . . . . 109

3.24 Discrete and continuous estimation of the optical and relative gains.. . 110

3.25 The effect of discrete and continuous optical gain updating.. . . . . . . 110

3.26 The percentage error introduced as a function of effective dynamic range.111

3.27 The process of simulating the high-power error-point at the calibration

line frequencies.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112

3.28 Snap-shot amplitude spectral densities of the simulated and recorded

high-power error-point signals.. . . . . . . . . . . . . . . . . . . . . . 113

3.29 Calibrated strain spectra with and without calibration lines suppressed.. 114

3.30 The contamination introduced by the process of suppressing the calibra-

tion lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115

LIST OF FIGURES xxi

3.31 Time evolution of the three injected calibration peaks used to determine

the optical gain during S1.. . . . . . . . . . . . . . . . . . . . . . . . 117

3.32 Comparison of the injected 732 Hz calibration peak with temperature.. 117

3.33 The time evolution of the estimated optical and relative gains for the

entire S1 data run.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .119

3.34 A comparison of the estimated optical gain with temperature on the laser

bench. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .120

3.35 Low frequency spectra of the optical gain and the Michelson differential

longitudinal feedback.. . . . . . . . . . . . . . . . . . . . . . . . . . .120

3.36 A closer view of the optical gain showing a signal with a 20 minute period.121

3.37 A low-frequency power spectral density of the optical gain showing a

peak around 19.5 minutes.. . . . . . . . . . . . . . . . . . . . . . . . 121

3.38 High resolution spectra of the calibration lines.. . . . . . . . . . . . . 122

3.39 732 Hz calibration line in the calibrated strain channel with and without

line suppression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122

3.40 High resolution spectrum of the 732 Hz calibration line in the recorded

high-power error-point signal.. . . . . . . . . . . . . . . . . . . . . . 123

3.41 A chain of dependence that leads to a valid calibration.. . . . . . . . . 124

3.42 Deviations of ESD from the 1/ f 2 model.. . . . . . . . . . . . . . . . . 125

3.43 Comparing the time- and frequency-domain models of the Michelson

servo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125

3.44 The errors on the optical and relative gain estimations.. . . . . . . . . 127

3.45 An amplitude spectral density comparison of the 1220 Hz peak.. . . . . 129

LIST OF FIGURES xxii

4.1 Signal-recycling cavity response for 600 Hz de-tuned.. . . . . . . . . . 136

4.2 MI demodulation phase curves for two signal frequencies.. . . . . . . 137

4.3 Signal-recycling cavity response for 600 Hz de-tuned.. . . . . . . . . . 138

4.4 A pole/zero fit to a particular signal-recycling optical response.. . . . . 139

4.5 A model of the length control servo for the dual-recycled GEO.. . . . . 143

4.6 A schematic of the tasks involved in the dual-recycled GEO calibration

scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143

4.7 The data processing pipeline for calibrating the dual-recycled GEO.. . 151

5.1 A block diagram of the open-loop simulation.. . . . . . . . . . . . . . 153

5.2 Typical spectra and time-series of the main simulation output signals.. 155

5.3 A sample time-series of the recovered optical gain estimates together

with the up-sampled version of the estimates.. . . . . . . . . . . . . . 158

5.4 Spectral content of the up-sampled optical gain estimates.. . . . . . . . 158

5.5 The magnitude difference of an S-domain filter and an IIR filter designed

with the bilinear transform.. . . . . . . . . . . . . . . . . . . . . . . . 159

5.6 A Bode plot of an ideal S-domain filter and a bilinear IIR filter.. . . . . 160

5.7 The difference between the responses of an S-domain and bilinear IIR

filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .160

5.8 An example of the inverse optical IIR filter.. . . . . . . . . . . . . . . 161

5.9 The differences between an inverse optical IIR filter and a perfect filter.161

LIST OF FIGURES xxiii

5.10 A snap-shot spectrum of the simulated error-point spectrum compared

with the up-sampled version.. . . . . . . . . . . . . . . . . . . . . . . 162

5.11 The magnitude and phase differences between the simulated error-point

and the up-sampled version.. . . . . . . . . . . . . . . . . . . . . . . 162

5.12 A bitwise representation of the quality channel.. . . . . . . . . . . . . 165

5.13 A schematic of the core calibration pipeline for experiment 1.. . . . . . 167

5.14 Time-series of the input and output to the simulation of experiment 1.. 167

5.15 The time-domain amplitude differences between the input and output

signals from experiment 1.. . . . . . . . . . . . . . . . . . . . . . . . 167

5.16 Snap-shot spectra of the underlying differential displacement and the

calibration output signals from experiment 3.. . . . . . . . . . . . . . 168

5.17 The magnitude and phase differences between the spectra of figure5.16. 168

5.18 The magnitude and phase difference between the IIR optical filter used

in the simulations and an ideal optical response.. . . . . . . . . . . . . 169

5.19 Snap-shot spectra of the underlying differential displacement and the

calibration output signals from experiment 3.. . . . . . . . . . . . . . 170

5.20 The magnitude and phase differences between the spectra of figure5.16. 170

5.21 The overall optical gain estimates plotted together with the underlying

overall optical gain.. . . . . . . . . . . . . . . . . . . . . . . . . . . .171

5.22 Time-series of the four recovered parameters and theχ2 channel from

experiment 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .172

5.23 Amplitude spectra of the time-series shown in figure5.22. . . . . . . . 172

5.24 Histograms of the recovered parameters from experiment 4.. . . . . . . 173

LIST OF FIGURES xxiv

5.25 A histogram of the recoveredχ2 values from experiment 4.. . . . . . . 173

5.26 Time-series of the four recovered parameters and theχ2 channel from

experiment 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .174

5.27 Amplitude spectra of the time-series shown in figure5.26. . . . . . . . 174

5.28 Histograms of the recovered parameters from experiment 5.. . . . . . . 175

5.29 A histogram of the recoveredχ2 values from experiment 5.. . . . . . . 175

6.1 A schematic of the length control servo and differential displacement

sensing setup of GEO for S3.. . . . . . . . . . . . . . . . . . . . . . . 178

6.2 Measured optical transfer functions of GEO for different detunings of

the signal-recycling mirror.. . . . . . . . . . . . . . . . . . . . . . . . 179

6.3 The optical transfer function measured for S3 II together with a pole/zero

fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .180

6.4 Model fits to the fast path electronics for S3 II.. . . . . . . . . . . . . . 182

6.5 Magnitude and phase differences between the measured and modelled

fast path electronics for S3 II.. . . . . . . . . . . . . . . . . . . . . . . 182

6.6 Model fits to the fast path electronics for S3 II.. . . . . . . . . . . . . . 183

6.7 Magnitude and phase differences between the measured and modelled

fast path electronics for S3 II.. . . . . . . . . . . . . . . . . . . . . . . 183

6.8 A measurement of the closed-loop transfer function of the Michelson

servo for S3 II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .184

6.9 The magnitude and phase differences between the measured and mod-

elled closed-loop transfer function of the Michelson servo for S3 II.. . 184

LIST OF FIGURES xxv

6.10 A plot of demodulation phase for the high-power diode against the am-

plitude of a 10 Hz calibration line.. . . . . . . . . . . . . . . . . . . . 185

6.11 A plot of demodulation phase for the high-power diode against the am-

plitude of a 1200 Hz calibration line.. . . . . . . . . . . . . . . . . . . 186

6.12 A schematic overview of the calibration signal generator.. . . . . . . . 188

6.13 A snap-shot spectrum of the high-power error-point signal during S3 I

showing the injected calibration lines.. . . . . . . . . . . . . . . . . . 191

6.14 A snap-shot spectrum of the injected calibration signal during S3 I.. . . 191

6.15 A snap-shot spectrum of the high-power error-point signal during S3 II

showing the injected calibration lines.. . . . . . . . . . . . . . . . . . 191

6.16 A snap-shot spectrum of the injected calibration signal during S3 II.. . 191

6.17 A comparison between the IIR dewhitening filter used for the high-

power error-pointPand an analogue model.. . . . . . . . . . . . . . . 194

6.18 Time evolution of the inverse optical filter coefficients (f 11p). . . . . . 197

6.19 Time evolution of the inverse optical filter coefficients (f 12p). . . . . . 197

6.20 Spectral density plots of the simulated and measured fast feedback signals.198

6.21 Differences between the fast path electronics model and IIR filters.. . . 199

6.22 A comparison of the time- and frequency-domain calibration methods

around 100 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .201

6.23 A comparison of the time- and frequency-domain calibration methods

around 1230 Hz.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .202

6.24 The recovered optical parameters from the S3 I science run.. . . . . . . 203

LIST OF FIGURES xxvi

6.25 The recovered optical parameters from the S3 II science run.. . . . . . 205

6.26 Histograms of the optical parameters recovered in calibrating the S3 II

data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .206

6.27 The time evolution of the peak amplitude of the 609 Hz line in calibrated

and un-calibrated data.. . . . . . . . . . . . . . . . . . . . . . . . . .207

6.28 The spectral content of measurements of the peak amplitude of the 609 Hz

calibration line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .207

6.29 The time evolution of the peak amplitude of the 1011 Hz line in cali-

brated and un-calibrated data.. . . . . . . . . . . . . . . . . . . . . . . 208

6.30 The spectral content of measurements of the peak amplitude of the 1011 Hz

calibration line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .208

6.31 The systematic magnitude error introduced by incorrectly estimating the

pole frequency of the optical response.. . . . . . . . . . . . . . . . . . 212

6.32 The systematic magnitude error introduced by incorrectly estimating the

pole Q of the optical response.. . . . . . . . . . . . . . . . . . . . . . 212

6.33 The systematic magnitude error introduced by incorrectly estimating the

zero frequency of the optical response.. . . . . . . . . . . . . . . . . . 213

A.1 A measured transfer function of the 8192 Hz DAQ IIR anti-alias filter.. 224

A.2 A measured transfer function of the 2048 Hz DAQ IIR anti-alias filter.. 224

A.3 A measured transfer function of the 1024 Hz DAQ IIR anti-alias filter.. 225

A.4 A measured transfer function of the 512 Hz DAQ IIR anti-alias filter.. . 225

List of Tables

2.1 Hardware components of the GEO 600 DAQ system.. . . . . . . . . . 25

2.2 Signals required to drive the ICS110b in slave mode.. . . . . . . . . . 26

2.3 A summary of the command functionality of the Glasgow timing card.. 32

2.4 A summary of the error states of the Glasgow timing card.. . . . . . . 34

2.5 The fields contained within each data packet on the DCU.. . . . . . . . 39

2.6 The fields of the data structure that stores information about each DCU

channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.7 The DAQ software message structure.. . . . . . . . . . . . . . . . . . 45

2.8 The command set of the data archiving server.. . . . . . . . . . . . . . 47

2.9 A summary of the possible sample rates that can be configured using the

DAQ user interface.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.10 A table of the injected ramp frequencies with the measured offsets.. . . 60

2.11 A table of the injected ramp frequencies with the variance of the mea-

sured ramps.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.1 Typical values for the calibration peak measurements and resulting opti-

cal gain during S1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

LIST OF TABLES xxviii

3.2 A summary of the recorded channels used in the calibration process.. . 91

3.3 Calibration line correction factors for the recording process.. . . . . . 94

3.4 Poles and zeros of the feedback path electronics.. . . . . . . . . . . . . 99

3.5 The Michelson servo gain parameters.. . . . . . . . . . . . . . . . . . 102

3.6 A summary table of the observed systematic and random errors in the

calibration process.. . . . . . . . . . . . . . . . . . . . . . . . . . . .128

3.7 A summary table of the input/output data streams and parameters.. . . 131

4.1 A summary table of the input/output data streams and parameters.. . . 150

5.1 The output signals from the open-loop simulation of GEO.. . . . . . . 154

5.2 A table of the output signals of the dual-recycled IFO calibration software.163

6.1 Table of injected calibration lines for S3 I.. . . . . . . . . . . . . . . . 189

6.2 Table of injected calibration lines for S3 II.. . . . . . . . . . . . . . . . 190

6.3 The recorded signals relevant for the calibration process for the S3 I data

run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192

6.4 A summary of the systematic and random errors of the dual-recycling

calibration process.. . . . . . . . . . . . . . . . . . . . . . . . . . . .210

B.1 Table of symbols and operators used in the pseudo code listings.. . . . 233

Chapter 1

Introduction

The existence of Gravitational Waves (GW) was predicted in 1916 by Einstein’s General

Theory of Relativity (GR). Gravitational waves are small disturbances in the fabric of

space-time and their magnitudes are typically expressed as dimensionless strains. Cur-

rent astrophysical models of the expected sources of gravitational waves predict strain

amplitudes at the Earth of below 10−21/√

Hz. These extremely small amplitude signals

make the search for gravitational waves one of the most challenging goals in modern

physics.

Although gravitational waves have not been directly measured so far, indirect evidence

for their existence was gained by the work of Hulse and Taylor [Taylor]. Timing mea-

surements of pulsar PSR 1913+16, a pulsar in a binary system, showed that the orbit

of the pulsar was decaying. This rate of decay of the orbit can be predicted by GR if

the loss of orbital energy is associated with the emission of gravitational radiation. The

observations carried out over many years were an excellent fit to the predicted effect,

providing the first clear evidence for the existence of gravitational waves.

So far, two main detection techniques have been employed in the direct search for grav-

itational waves: resonant mass detectors, and, more recently, interferometric detectors.

Resonant mass detectors aim to detect gravitational waves in a narrow band of frequen-

cies centred around the fundamental resonance of a large mass. A passing gravitational

wave would excite that resonance and give rise to a signal. A number of resonant

2

mass detectors are currently operational in the world, including, [Allegro], [Auriga],

and [Niobe].

Interferometric gravitational wave detectors are based on the well known Michelson

interferometer. When a gravitational wave passes through such an instrument, the length

of the two arms is differentially changed. This differential length change results in a

phase change in the light in the two arms. By allowing the light in the two arms to

interfere, a signal can be extracted that is proportional to the amount of phase change

induced by the passing gravitational wave, and hence is proportional to the amplitude of

the gravitational wave.

GEO 600 (or simply, GEO) is one of a number of such interferometric gravitational wave

detectors that are currently entering the final stages of commissioning and optimisation

(see [GEO 600] for an overview discussion of GEO). The [LIGO] project in the USA

has three detectors—two with 4 km armlength and one with 2 km armlength. In Japan,

the [TAMA ] project has a 300 m armlength detector, and in Italy, the [VIRGO] project is

constructing a 3 km armlength detector. Once all of these detectors are running at their

design sensitivity, they will form an international detector network that will search for

gravitational waves from a variety of astrophysical sources.

As these interferometers near the end of their commissioning, the astrophysical impor-

tance of the data they produce increases significantly. The recording and subsequent

calibration of the detector outputs is a primary concern if the data is to be used suc-

cessfully for astrophysical searches. The search from gravitational waves is typically

carried out under four astrophysical source categories: continuous wave emissions (e.g.,

from Pulsars), short duration burst sources (e.g., from supernovae or other violent cos-

mic events), from the inspiralling of compact binary systems (e.g., neutron star binary

systems), and from the stochastic gravitational wave background. Each of these searches

sets requirements on the timing accuracy and stability with which data must be recorded,

as well as on the accuracy of any calibration that is carried out.

1.1 GEO 600 3

1.1 GEO 600

The GEO 600 project (known hereafter as GEO) is a German-British collaboration

whose aim is to produce a large-scale, highly advanced, interferometric gravitational

wave detector. As well as demonstrating various key advanced interferometry con-

cepts, the detector, when fully commissioned, aims to run in a continuous data taking

mode, providing high quality gravitational wave data for the various astrophysical search

pipelines being developed throughout the world. In addition, GEO will also become part

of a world-wide network of Gravitational wave detectors that will carry out coincidence

searches and be able to determine directional information for sources.

The construction of the detector began in 1995 at a site near Hannover, Germany, and

is currently entering the final stages of installation and commissioning. The design of

GEO is based on the Michelson interferometer with the addition of two mirrors. All the

main mirrors that make up the interferometer are suspended from triple pendulums. For

optimal sensitivity, the detector is operated with a dark fringe at the output port, with the

result that during normal ‘locked’ operation all the input light leaves the interferometer

via the input port. Because of this, the Michelson can be viewed as a highly reflective

mirror when it is held at a dark fringe. Placing a mirror at the input port forms a Fabry-

Perot cavity between this mirror and the Michelson ‘mirror’ with the result that, when

on resonance, the circulating light power in the interferometer is greatly increased. This

cavity is referred to as the power-recycling (PR) cavity. The PR cavity is kept on reso-

nance using a Pound-Drever-Hall locking scheme [PoundDreverHall]. By implementing

such a cavity, the amount of light that leaves the output port of the interferometer due to

any deviation from the dark fringe is increased. This in turn can reduce the shot noise

limit to the sensitivity of the detector.

Placing a further mirror—a signal-recycling (SR) mirror—at the output port forms an-

other Fabry-Perot cavity, the signal-recycling cavity, between the Michelson ‘mirror’

and the SR mirror. As the name suggests, the signal-recycling cavity causes any signals

leaving the interferometer to be partially reflected back into it, where they can interact

for a longer time with the gravitational wave that is producing the signal, resulting in

an increase in the signal strength. The signal-recycling cavity is kept on resonance, for

1.1 GEO 600 4

signals of a particular frequency, using a Schnupp modulation scheme [Heinzel99]. This

results in another enhancement of the signal strength (although at the expense of band-

width), which can further reduce the shot noise limit to the detector’s sensitivity. The

microscopic positioning of the SR mirror allows the peak sensitivity of the detector to

be tuned in frequency; by increasing the reflectivity of the SR mirror (for example, by

changing the mirror) the Q of the cavity resonance can be increased making the detector

more sensitive over a narrow band of frequency. The Michelson, together with these ad-

ditional two mirrors, is referred to as a dual-recycled (DR) interferometer. The locking

scheme for the Michelson is outlined in section1.1.2and a more detailed description of

the locking scheme for the dual-recycled GEO 600 is given in [Heinzel02].

1.1.1 Design configuration

master

laser

slave

laser

first

mode-cleaner

second

mode-cleaner

inboard mirror (MCn)

far mirror (MFn)

inboard mirror (MCe)

600m

3.9m

North

(main test mass)

(main test mass)

far mirror (MFe)

power

recycling

recyclingmirror (MPR)

signal

mirror (MSR)

beamsplitter (BS)

output

photodiode

Figure 1.1:A schematic layout ofGEO 600showing the full optical system.

In reality, the inboard mirrors are suspended higher than the other mirrors

so that the arms are folded in a vertical plane. They are shown folded here

in a horizontal plane to simplify the illustration.

The design configuration of GEO incorporates many subsystems. Detailed descriptions

of the design of GEO exist, (see [Willke02] and [GEO 600] for examples), and so it is

not covered here in any depth. However, it is useful to consider the current status of

1.1 GEO 600 5

GEO and for this we need a brief overview of the detector as a whole. Figure1.1shows

a schematic of the full detector. A brief description of each subsection follows.

The laser and mode-cleaner system

GEO uses an injection locked master/slave laser system that is capable of delivering up

to around 12 W of laser light at 1064 nm. The laser light is spatially filtered through

two sequential mode-cleaners. The mode-cleaners are triangular ring cavities, each with

a round-trip length of around 8 m. All of the mode-cleaner mirrors are suspended by

double pendulums (see [Gossler02] for a description of the mode-cleaners).

Up to the current time, GEO has been operating in a ‘low-power’ mode achieved by

attenuating the laser light to 2 W at the input to the first mode-cleaner. In this mode, we

have about 1 W of light after the mode-cleaners to inject into the main interferometer.

Mirror suspensions

In order to isolate the instrument from seismic noise, all mirrors within GEO are sus-

pended from pendulums hung from support structures. These suspensions provide vari-

ous degrees of mechanical isolation from the ground. The main optics of the core instru-

ment (the beamsplitter, the two far mirrors, and the two inboard mirrors) are suspended

as triple pendulums [GEOpendulum]; all other optics (the mode-cleaner mirrors, the PR

mirror, and the SR mirror) are suspended as double pendulums. The main test optics are

constructed as a monolithic fused silica stage designed to reduce the losses of the system

that could manifest as access thermal noise. All pendulums are suspended from an up-

per structure via two wires hung from spring blades. The upper structure is mounted on

three stacks that provide a degree of vertical isolation from the ground. The suspension

and control of the main test masses is discussed further in section3.2 in the context of

calibration.

1.1 GEO 600 6

Detector control and alignment

GEO has four suspended cavities in total: the two mode-cleaner ring cavities, the power-

recycling cavity, and the signal-recycling cavity. Each of these cavities has to have both

its alignment and length controlled in order to keep the detector at its operating point.

A combination of digital and analogue control loops using various actuators is used to

achieve this. All of the 25 pendulums are locally damped in, for the case of the dou-

ble pendulums, four degrees of freedom, and for the case of the triple pendulums, six

degrees of freedom. The local controls consist of analogue servos that feedback to the

upper masses of the pendulums using coil/magnet actuators. The sensors and actuators

for these servos are co-located on the suspension systems. The control servos are ‘su-

pervised’ by a digital system that allows monitoring of the signals in the system and

control of the gains and offsets. The local controls also provide the means to do manual

alignment of the optics prior to locking the detector. The design and implementation of

the local control system used in GEO is described in [Casey99].

GEO employs a hierarchical locking scheme for the length control of the four cavities.

The first step is to stabilise the frequency of the laser light to the length of the first mode-

cleaner by feeding back to the master laser. A second servo loop that applies feedback

signals to one mirror of the first mode-cleaner is then used to lock the length of the first

mode-cleaner (and hence laser) to the length of the second mode-cleaner. This three ele-

ment system is then locked to the length of the power-recycling cavity such that, in a fully

locked state, the master laser frequency follows the length of power-recycling cavity to

keep the light resonant. With the laser light resonant in the PR cavity, the Michelson is

then locked by applying differential length control signals to the two inboard mirrors.

The core interferometer has 1200 m long arms that are folded to fit physically into a

600 m space. The arms of the interferometer are aligned approximately North and East.

More details of the locking scheme can be found in [Freise02], [Freise03] and [Grote03].

1.1 GEO 600 7

Detector development

During August 2002, the power-recycled Michelson (without the SR mirror installed)

took part in a world-wide coincident science run. During this run the interferometer

was still using test optics suspended by steel wires for the two inboard mirrors and the

beamsplitter. For eighteen days GEO took data with an overall duty cycle of over 98%

(see figure1.2). In chapter2, the data acquisition system used in GEO is described in

detail; this system was in place and operational for the entire run, recording around 47

detector and environmental signals, as well as hundreds of control signals. The data rate

at the time was 577 kbytes per second resulting in a total data volume of approximately

800 GB. In addition, a short-messaging-system (SMS) alert system was in place to mon-

itor various aspects of the instrument and report any problems that may occur. The full

DAQ system, along with the SMS alert system, is described in detail in chapter2. The

on-line calibration scheme described in chapter3 was implemented prior to the S1 run

and was tested throughout. The results of this experiment are also presented in chapter

3. The run provided an excellent opportunity to consider various matters of detector

characterisation and data analysis.

Following the success of the S1 run, work began on the installation of the final optics.

The two final, high-quality, inboard mirrors, as well as the final beamsplitter, were sus-

pended as monolithic suspensions. Around the same time, a signal-recycling mirror

was installed and work began on locking the dual-recycled interferometer. After some

initial success (locks of a few minutes duration), various issues became apparent that

were believed to be causing significant difficulty in locking of the dual-recycled inter-

ferometer. These needed to be addressed before continuing the DR work. A significant

amount of work was directed towards implementing a seismic isolation feed-forward

system to reduce the motion of the main mirrors in the band from 0.1 to a few Hz. This

work, together with the development of a new locking scheme allowed the dual-recycled

interferometer to be locked stably for periods of many hours. A short period of opti-

misation followed to ready the instrument for another science run. In November 2003,

GEO joined the LIGO detectors in a coincidence science run, termed S3. After one week

of running with a duty cycle around 95%, GEO was taken off-line for a further period

1.1 GEO 600 8

0

0

2

2 4

4 6

6 8

8

10

10

12

12

14

14

16

16

18

18

80

90

100

GEO600 duty cycle during S1 run

Duty

Cycl

e(%

)

Overall Duty Cycle: 98.73%

Overall Lock Time: 396.82 hours

Overall Time Outside Maintenance Periods: 401.92 hours

Time (days)S1 start

DAQ Problem Periods

Maintenance Periods

Lock Times

Figure 1.2:The duty cycle of GEO for the entire S1 science run. DAQ prob-

lem times are highlighted as well as the times set aside as maintenance pe-

riods.

of optimisation. After significant improvements that brought the sensitivity of GEO to

within a factor of 4 of one of the LIGO interferometers, GEO rejoined for the last two

weeks the S3 science run (30th December 2003 to 14th January 2004). In this second

part of S3, GEO achieved a duty cycle of around 97%. The DAQ system continued to

operate reliably for this science run and achieved a duty cycle of 100% for the 3 weeks

of GEO-participation.

1.1.2 Signal extraction

In order to calibrate GEO, it is necessary to have a good picture of how a signal that is

proportional to gravitational wave strain is extracted. It can be shown (see section1.2)

that the effect of a monochromatic gravitational wave passing through an interferometer

can be thought of as producing a periodic differential change in the arm-lengths of the

1.1 GEO 600 9

detector. This in turn is equivalent to applying a phase modulation to the carrier light

resonant in the interferometer (see [Heinzel99] for further discussion). The result is that

modulation sidebands are imposed on the carrier light. In order to see how this signal is

extracted, we must first look at the locking method of the Michelson interferometer.

Locking the Michelson

The Michelson interferometer in GEO is locked using a Schnupp modulation (or frontal

modulation) scheme. Here, phase modulation sidebands are imposed on the carrier light

before the light enters the interferometer. This is done using an electro-optic-modulator

(EOM) driven at the so called ‘Michelson frequency’,fMI . The frequency is chosen so

that the sidebands become resonant in the power-recycling cavity. The carrier light, at

frequencyfc, along with the two sidebands atfc − fMI and fc + fMI , propagates in both

arms and recombines at the beamsplitter.

A deliberate asymmetry in the length of the two arms means that even on a dark fringe,

a small amount of control-sideband light leaves the output port of the interferometer.

When detected by a photodiode, this gives a signal that has a component at the modula-

tion frequency which is proportional to the amount of deviation. Furthermore, the signal

obtained after demodulation is bipolar and so from this we get a control signal which,

when appropriately filtered, can be applied differentially to the actuators of the end mir-

rors in order to keep the Michelson to the dark fringe. This control signal is referred to

throughout this work as the detector error-point signal.

The effect of a gravitational wave is to modulate the length of each arm. Due to the

quadrupole nature of gravitational waves, the two arms are modulated in anti-phase.

The result is that modulation sidebands (signal sidebands) are imposed on the carrier

light, and on the control sidebands. Because of the anti-phase modulation, the signal

sidebands in the two arms have the opposite sign and so leave the interferometer after

constructively interfering at the beamsplitter. After detecting the beats between the var-

ious light fields and demodulating at the Michelson frequency, we get a signal that is

proportional to the gravitational wave signal strength (see [Heinzel99]). For a simple

1.1 GEO 600 10

Michelson with power-recycling, the phase,φdemod, with which the demodulation is per-

formed can be optimised to give the maximum signal strength. If we also demodulate

with a phase 90away from the optimal demodulation, we get a signal that should con-

tain no Michelson control signal or gravitational wave signal. These two demodulation

quadratures are referred to throughout this text as theP (or in-phase) demodulation and

Q (or out-of-phase) demodulation respectively.

Including signal-recycling

−ωsig +ωsig

ω0

beam-splitter

mirror

outputsignal

signalsidebands

power-recycling

signal-recyclingmirror

controlsidebands

carrier

Figure 1.3:A schematic view of the signal-recycling cavity which together

with the power-recycling cavity, forms a coupled system where the signal

interaction time of the signal sidebands and the gravitational wave is in-

creased.

The introduction of a signal-recycling mirror, as already mentioned, has the effect of

resonantly enhancing any signal in a certain range of frequencies. This can be thought of

as the signal sidebands passing through a Fabry-Perot type cavity, with the combination

of the two end mirrors forming one mirror of the cavity (the ‘Michelson mirror’), and

1.1 GEO 600 11

the SR mirror forming the other. Figure1.3 shows a schematic of this setup. In this

example, the response of the SR cavity is tuned to the so called ‘broad-band’ case where

the peak response is at the carrier frequency. Careful consideration of the light fields

propagating through the interferometer shows that the signals will (for general tunings

of the SR mirror) appear spread between two modulation quadratures,PandQ , (see

[Heinzel99] for a full treatment). The spreading of the two signals betweenPandQ is

discussed further in chapter4 in the context of calibration.

1.1.3 Sensitivity of GEO

When fully operational, GEO should be one of the most sensitive displacement measur-

ing instruments in the world. The design sensitivity is of the order 10−19m/√

Hz over

the band from 50 Hz to 6 kHz. The sensitivity is limited in principle by various technical

1 and fundamental2 noise sources that are present in the system. These noise sources

appear as an apparent strain.

At low frequencies (up to 40 Hz), the sensitivity of the detector will be dominated by

seismic noise coupling from ground motion to the main test masses via the suspen-

sion system. Around 40 Hz, the sensitivity becomes dominated by the thermorefractive

noise [Thermorefractive] present in the beamsplitter. At higher frequencies, say above

1 kHz, the sensitivity is expected to be dominated by shot noise present in the detection

of the signal.

Figure1.4 shows design sensitivity curves for two configurations of GEO: one with a

low reflectivity SR mirror (around 95%) resulting in a large bandwidth of operation, the

other shows GEO configured with a higher reflectivity SR mirror (around 99%) giving

a smaller bandwidth. The termbroad-bandis often used in the literature to refer to a

signal-recycling cavity that is tuned such the carrier light is resonant in the SR cavity,

1Here, technical noise sources are those that arise from deviations in the performance of the physi-

cal subsystems from the expected optimal/theoretical performance. The process of commissioning the

detector aims to reduce this noise to a minimum.2Fundamental noise sources are those noise sources that are anticipated to be present due to the design

of the detector.

1.2 Measuring gravitational waves 12

resulting in a peak sensitivity for signals at zero frequency. The tuning of SR response

away from zero frequency is referred to as ade-tunedstate.

total noisethermorefractive noiseinternal thermal noisesuspension noiseshot noiseseismic noise

10 310 2

Frequency (Hz)10 3

Frequency (Hz)10 2

10−20

10−18

10−22

10−24

Appar

ent

Str

ain

(1/√

Hz)

Figure 1.4:The apparent strain spectral densities resulting from the presence

of different noise sources in the detector. The shot noise, and hence the total

noise, is shown for GEO operating in a large bandwidth mode (left plot),

i.e., with a low reflectivity SR mirror, and in a low bandwidth mode (right

plot), i.e., with a higher reflectivity SR mirror.

1.2 Measuring gravitational waves

In order to correctly calibrate the output of GEO, it is important to be clear about how the

output of the detector is related to gravitational wave strain. We have seen above that the

output of the detector is in general proportional to differential length changes of the two

arms in a frequency dependent way. To relate differential arm length changes to strain,

we can consider a simplified case where we are observing the effect of gravitational

waves far from any source. Here, the effect of gravitational waves on the space-time

metric,gµν , can be approximated as a small disturbance on an otherwise flat Minkowski

space-time,ηµν . This is expressed as

gµν = ηµν + hµν (1.1)

1.2 Measuring gravitational waves 13

where the perturbations,hµν , are given as two polarisations of plane waves by

hµν =

0 0 0 0

0 h+ h× 0

0 h× h+ 0

0 0 0 0

(1.2)

If we consider an optimal orientation of a gravitational wave with respect to our inter-

ferometer such thath×(t) = 0 andh+(t) = h(t), then we can rewrite the metric as

gµν =

−1 0 0 0

0 1+ h(t) 0 0

0 0 1− h(t) 0

0 0 0 1

. (1.3)

We can now consider the flight path of a photon from the beam splitter to one of the

end mirrors. We call this path,x, and calculate the proper time taken for the photon to

traverse this path as

0 = −(cdt)2+ (1 + h(t))(dx)2

+ (1 − h(t))(dy)2+ (dz)2, (1.4)

with dy = dz = 0. So,

(cdt)2= (1 + h(t))(dx)2, (1.5)

⇒ cdt =

√1 + h(t)dx. (1.6)

For |h| 1, we can approximate the square root term to get

cdt =

(1 +

h

2

)dx. (1.7)

Now we can integrate over the path the photon takes to get the total flight time, making

the assumption that the gravitational wave is low frequency and hence constant over the

flight time. Then we see that

c∫ t

t0dt =

∫ L

0dx

(1 +

h

2

)−

∫ 0

Ldx

(1 +

h

2

), (1.8)

⇒ c1tx = 2L

(1 +

h

2

), (1.9)

⇒ c1tx = 2L + Lh. (1.10)

1.2 Measuring gravitational waves 14

We can then do the same for a photon traversing the other arm, so with dx= dz = 0 we

get

c1ty = 2L − Lh. (1.11)

The total difference in time of arrival of these two photons is then given by

c1t = c(1tx − 1ty), (1.12)

⇒ c1t = 2Lh, (1.13)

wherec1t = 21Lx + 21L y = 41L and1L is the physical length change of an indi-

vidual arm. We see then that the effect of a gravitational wave,h, on our interferometer

is given by

h(t) =21L(t)

L. (1.14)

1.2.1 Application to calibrating GEO

In GEO, the arm length,L, is 1200 m. The output of the instrument is proportional to

the differential arm length change. If each arm changes length by1L metres, then the

detector output is proportional to 21L metres.

In its most general sense, calibration information is determined by injecting a signal

differentially into two actuators that adjusts the length of both arms. The result of this is

to extend one arm by1L metres and to shorten the other arm by1L metres. The signal,

vout, measured at the output of the detector can be then expressed as

vout[V] = 21L.G[m][V/m], (1.15)

whereG is the, potentially time and frequency dependent, optical gain of the detector in

units of Volts per metre. In order to determine the strain,h, measured by the detector,

we must convert the measured output,vout Volts, into a differential arm length change,

21L metres, by

21L(t) =vout(t, f )

G(t, f ), (1.16)

1.3 Calibration and DAQ requirements set from astrophysical searches 15

and then convert this differential arm length change, 21L(t) metres, into dimensionless

strain by

h(t) =21L(t) [m]

1200 [m]. (1.17)

1.3 Calibration and DAQ requirements set from astro-

physical searches

The work of this thesis is concerned with acquiring data from the detector and then

calibrating the main detector output to a strain signal. Various requirements are set on

the timing accuracy of the data acquisition system and on the accuracy with which any

calibration is performed. Trying to meet these requirements plays a large part in deter-

mining how both the data acquisition system and the calibration scheme were developed

and implemented. From private communications with various members of the analysis

teams, the calibration accuracy required is in the range 1-10% in order to be able to ex-

tract certain astrophysical results. In order to see what motivates such a statement, it is

useful to consider the needs of the data analysis groups that aim to extract astrophysical

results from the output of gravitational wave detectors like GEO. Most of the arguments

for calibration and timing accuracy are given in qualitative terms since in most cases,

quantifying the effect of calibration and timing errors is extremely complicated. Any

references to such quantitative discussions are given when appropriate. Some simple

examples are given that lead to requirements of DAQ timing accuracy.

As mentioned earlier, astrophysical searches for gravitational waves fall broadly into

four categories. Within the LIGO Scientific Community ([LSC]), four search groups are

organised as follows:

Continuous waves group: the search for signals from continuous wave sources, for

example, from Pulsars.

Burst group: the search for burst type signals from violent astrophysical

events, for example, signals from supernovae.

1.3 Calibration and DAQ requirements set from astrophysical searches 16

Stochastic group: the search for the gravitational stochastic background—gravitational

waves left over from the time of the Big Bang.

Inspiral group: the search for signals from inspiralling compact objects, for

example, coalescing neutron star-neutron star systems.

1.3.1 Signals from continuous wave sources

The requirements of the Continuous Wave (CW) search group are in some ways the most

strict. The pulsar analysis aims to extract a known signal waveform from the output

of the detector. For the first generation of detectors, astrophysical predictions for the

strength of known Pulsars places signals at the level of, or well below, the noise of the

detectors. In order to recover the signal from the noise of the detector, a coherent search

must be performed over long data sets (weeks or even months in length). Details of

the Pulsar search methods are given in [PulsarSearch]. For a signal to remain coherent

as it appears in the recorded detector output, and ultimately the calibrated strain signal,

the timing accuracy of the data acquisition system must be high. In addition, the phase

coherence of the calibration process must also be high.

With regards calibration of the main detector output signal, the pulsar search requires

that the phase of calibrated strain signal remains coherent at each frequency,i.e., that no

jumps in phase occur. The absolute amplitude calibration, while not affecting the ability

to detect a signal, will affect the pulsar parameters recovered from any search. In the

first pulsar searches, where the instrument sensitivity is at level that makes detections

very improbable, the searches aim to set upper limits on the strain amplitudes (and other

parameters) of known pulsars [PulsarSearch]. These upper limits are directly affected by

any uncertainty in the amplitude calibration of the detector.

1.3.2 Signals from the stochastic gravitational wave background

The search for a stochastic gravitational wave background aims to correlate the noise

between two (or more) detectors. In order to do this, the sampling of the data at the

1.3 Calibration and DAQ requirements set from astrophysical searches 17

two sites must remain coherent. In addition, in order to determine the source position of

any observed correlations, accurate time-stamping of the data is necessary. The absolute

amplitude calibration is unimportant in this search, but again, the phase coherence of the

calibration process must be high. Details of the Stochastic background search are given

in [StochasticSearch].

1.3.3 Signals from inspiralling binary objects

The search for signals from inspiralling binary compact objects (for example, binary

neutron star systems), again uses matched-filtering methods to look for signals in the

detector output signal. Large banks of signal templates are compared to sections of

data and a measure of confidence is formed. Based on this measure, candidate events

can be highlighted. If the confidence is sufficiently high, then a detection may be

claimed. A description of the current LSC search for inspiralling binary objects is given

in [BinarySearch].

The waveforms of inspiral events typically spend a few seconds at frequencies and am-

plitudes where we would hope to see them in the detection band of the instruments. The

predicted event rates for such signals is very low (a few per year) and so a high duty

cycle of both the instrument and the DAQ system is essential if we are to maximise our

chances of a detection.

The calibration of the detector can affect the detection of inspiral events quite signif-

icantly. In the case of calibrating the power-recycled IFO (where the instrument’s re-

sponse to gravitational waves is flat in frequency), an error in the calibration will only

affect the apparent strain amplitude with which a signal is detected (assuming that the

calibration remains approximately constant over the time-scale of the inspiral event).

For the calibration of the dual-recycled IFO where the sensitivity of the instrument is

frequency dependent, an error in the calibration could lead to a deformation of the wave-

form as it appears in the calibrated strain signal. This could mean that the signal matches

a waveform template that has the wrong astrophysical parameters, or worse, that the sig-

nal is not matched to any template.

1.3 Calibration and DAQ requirements set from astrophysical searches 18

As with any signal that is short lived, early candidate events will rely on coincident

detections between multiple interferometers before any event is claimed as a detection.

It is therefore essential that these coincident detections appear at the same time (taking

into account the direction of the source) in the strain signals of the individual detectors.

A discussion of the coincident detection of binary neutron stars in the context of deter-

mining the Hubble constant is given in [Schutz]. This provides another argument for the

overall timing and calibration accuracy. Here, Schutz makes the case that, using 10 or

more coincident binary neutron star detections, the Hubble constant can be determined

to about 3%. In order to do this, the position of each source on the sky must be deter-

mined to within 3. This requires the time-of-arrival of each event to be measured to

within 400µs and the amplitude of each event to be determined to within 3%. In order

to determine the time-of-arrival of an event to better than 400µs, it is essential that the

timing accuracy of the DAQ system recording the data is much less than this. Similarly,

to determine the strain amplitude of the event to better than 3%, we need to produce a

calibrated strain output from the detector that is accurate to better than 3%.

1.3.4 Signals from burst type signals

In some sense, the search for burst type signals is the most challenging of the four as-

trophysical searches since few, if any, model astrophysical waveforms exist. If we are to

be able to infer any details of the astrophysical process that produced any detected burst

event, we will want accurate calibration across the frequency bandwidth of the signal. In

addition, since the search for bursts aims to highlight anomalous time/frequency events

in the detector output, then, as with the inspiral search, it will rely strongly on coincident

detection between multiple IFOs to be able to claim a detection; the same timing and

duty-cycle requirements on the DAQ system also apply here then. Details of the search

for burst type events are given in [BurstSearch].

1.3 Calibration and DAQ requirements set from astrophysical searches 19

1.3.5 GEO as part of a detector network

As mentioned in the above discussions of inspiral and burst searches, we can look at

the timing requirements from the point of view of coincident detection and direction

determination of a source. Multiple coincident detections not only provide confidence

in the detection itself, but can also be used to gain information about the direction of the

source of the signal.

source

detector 4

detector 3

detector 2

detector 1

dS

α

α

Figure 1.5:A diagram of the effect of timing inaccuracies in the

data of one detector when determining the position of a source

on the sky. The normal to the set of parallel planes that are

consistent with the recorded arrival times at each detector gives

the direction of the source in the sky. An effective change in

position of one of the detectors due to timing errors leads to a

slight change in the set of planes that fit the observations and

hence to the apparent position of the source in the sky.

1.3 Calibration and DAQ requirements set from astrophysical searches 20

Using four or more detectors around the world, detection of the same event can lead to a

direction for the source if we can accurately determine the time of arrival of the event at

each detector. This sets natural requirements on the timing accuracy of the DAQ system

of each detector. In order to be able to determine the position of a source, each detector

must be able to measure the time of arrival of the signal to much better than the light

travel time between the detectors. The maximum possible light travel time between two

fictitious detectors corresponds to a separation of around 12000 km (the diameter of the

earth),i.e., to about 40 ms. This is of course a rather generous upper limit and in practise

the separation of the detectors will be much less for most source positions. Let us say

that we want each detector to be able to measure the time of arrival with an accuracy of

about 1% of this,i.e., around 0.4 ms. We can look at the effect of this on determining the

position of a source by considering the simplified situation shown in figure1.5. We have

three detectors situated on the equator with a source situated directly above the North

pole. These three detectors by themselves are only enough to measure the direction in

the sky to within±180. Adding a fourth detector out of the plane of the equator allows

the source direction to be properly determined. Let us assume that only one detector

has timing errors in its DAQ system. These timing errors translate directly to an error

in the measured time of arrival of the signal which in turn translates into an error in the

position of the detector (shown as dS in the diagram). The result is an angular error ,α,

in the position on the sky given by:

α ≈dS

6 × 106, (1.18)

where dS= cdt, 6×106 is the radius of the Earth, and dt is the error in measuring the time

of arrival. The angular error is then justα ≈ 50dt. So for a timing error of dt= 0.4 ms,

we get an uncertainty of the position of the source of about 20 milli-radians; if we get

timing accuracies for the detectors to the 1µs level then we get an angular positional

accuracy (in this rather special case) of around 50 micro-radians for high signal-to-noise

events.

1.3 Calibration and DAQ requirements set from astrophysical searches 21

1.3.6 Summary

From the above discussions and simple arguments, we see that we need to aim for a tim-

ing accuracy in the data acquisition system of GEO of the order of a few microseconds.

We also see that the accuracy with which the instrument is calibrated needs to be of the

order 10% or better.

Chapter 2

Data acquisition and storage

The data acquisition system (DAQ system, DAQs) is one of the most important subsys-

tems in GEO. Without the accurate and reliable recording of the detector output signals,

it would be almost impossible to perform any astrophysical research using GEO.

At the beginning of this research work, there existed a DAQ system based upon the

system used in the [LIGO] project. The GEO implementation of this system proved to

be too unreliable in various aspects of the hardware as well as in the supporting software.

There was a clear need for a redesign of the software system and for some improvements

in the hardware components.

The choice of hardware for the GEO DAQ system was mainly governed by the fact that

the LIGO project (having done a considerable amount of research), was already using a

similar hardware setup. The base sampling rate of the system was chosen to be a power

of 2 that sufficiently spans the target gravitational wave band (up to a few kHz) while not

producing excessive data rates. The data rate is a particularly important consideration

since data needs to be archived essentially indefinitely. Due to the need to store not only

the detector output that contains the gravitational wave information, but also many other

signals,e.g., environmental monitors and detector control signals, the amount of data

can grow quite large. With these considerations in mind, a maximum sample rate of

16384 Hz was chosen in the LIGO project and was adopted in the GEO project in order

to make data exchange and combined data analysis more simple.

2.1 System requirements 23

To ensure that the DAQ system met the requirements discussed in section1.3.6, the

software system was completely redesigned and implemented, while most of the existing

hardware was kept and a new timing circuit was added to improve the reliability and

diagnostic abilities of the system. This chapter presents an overview of the entire DAQ

system and focuses in detail on some of the hardware components and the software

system. The reliability and timing stability of the new DAQ system are also discussed in

detail.

2.1 System requirements

As we have seen from section1.3.6, the data acquisition system requirements are funda-

mentally very strict. We require a system that:

• operates continually for long periods (∼100% uptime for longer than one year),

• can record many channels simultaneously,

• has good sample resolution (≥16-bit),

• can sample signals for the full gravitational wave band of interest (sample rate

≥12 kHz),

• accurately and robustly time stamps the data.

Achieving long uptimes requires not only robust hardware but also good software design

that is not only stable in itself but which also can recover quickly from any external dis-

turbances,e.g., power cuts or network failures. For this reason a considerable amount of

effort was focused on designing a software system that not only fulfils the requirements

in terms of system robustness, but is also user-friendly.

The choice of hardware in the data acquisition system is mainly governed by the need to

use high resolution ADCs. Very few commercial solutions exist (this may indeed be the

2.2 Hardware components 24

only one at the time of writing), that have high sampling resolution (16 effective bits or

higher) and synchronous sampling over many channels1.

After choosing the appropriate hardware and designing a suitable software system, the

timing accuracy of the system must be assessed. In principle there are two different

requirements on the timing accuracy of the system: the offset of the system from absolute

time, and the fluctuation of the timing accuracy about this absolute offset.

2.2 Hardware components

The hardware components of the DAQ system can be conveniently split into two sepa-

rate entities: the data collecting units, and the data archiving system; each of these use

separate software applications that are described in section2.3. These two subsystems

will be described in this section.

2.2.1 The data collecting units

The hardware part of the data acquisition system is based on a VME rack design. Each

hardware element, with the exception of the timing card, is a commercial product. All

hardware is implemented as VME cards that slot into a VME crate. The VME crate and

its associated hardware and software is called aDCU (Data Collecting Unit) within the

GEO project. GEO uses three such DCUs, one in the central station and one in each

end station, each comprising at least 4 VME-type cards: one processor card, one GPS

receiver, a timing card, and one ADC (analogue to digital converter) card. The central

station has an additional ADC card that operates at a slower sampling rate (512 Hz)

and is used to record signals from environmental monitors and other slowly varying

signals. The two ADC cards used within GEO, the 16384 Hz card and the 512 Hz card,

are referred to respectively as afastADC card and aslowADC card. Figure2.1shows a

schematic of the layout of one DCU VME crate. The details of the hardware components

1Synchronous sampling is desirable particularly when coincidence analysis is performed on two or

more channels.

2.2 Hardware components 25

inputssignal

inputssignal

to GPS antenna

1 PPS

4 MHz

VM

IC31

13A

GPS

rece

iver

Tim

ing

card

ICS11

0B

Pro

cess

orC

ard

VMIC clock signal

Figure 2.1:A schematic layout of one DCU VME crate.

are summarised in table2.1.

Hardware componentManufacturer Model

Processor card Artesyn [Artesyn] Baja4700E

Fast ADC card ICS [ICS] 32 and 16 channel 110b

Slow ADC card VMIC [ VMIC] 64 channel 3113A

GPS receiver jxi2, inc [Brandywine] VME-SYNCCLOCK32

Timing card Glasgow University Version 1.0

Table 2.1:Hardware components of theGEO 600DAQ system.

The ICS ADC card

The main ADC cards used in GEO are those supplied by [ICS]. The model used is

the ICS110b in both 16 channel and 32 channel configurations. We also use the op-

tional signal processing daughter board that provides additional anti-alias filtering, a

programmable gain stage, and a modest amount of input protection. The ICS110b uses

16 stereo, sigma-delta, ADC chips (8 in the 16 channel version) to give simultaneous

2.2 Hardware components 26

sampling of up to 32 channels with sample rates up to 100 kHz.

The ICS ADC boards have various modes of operation that allow them to run stand-

alone or in conjunction with other ICS boards in the same crate. In GEO, the cards are

run in a so-called ‘slave mode’ which allows the commencement of sampling, as well as

the sampling instances, to be externally defined. In this mode, the card expects timing

signals to be supplied by another ICS110b which runs in the so-called ‘master mode’.

The signals needed to drive the board in slave mode are summarised in table2.2. Figure

2.2 shows the specification of the timing signals needed to drive the ICS board in this

mode.

clock−

clock+

frame tc

SyncAcquire

Ts

TH

Tc

Tc2 ≥ TH ≥ 2.5 ns

Tc2 ≥ Ts ≥ 10 ns

Requirements:

Figure 2.2:The specifications of the signals required to drive the ICS110b in

slave mode. These specifications were supplied by ICS on request. Samples

are acquired on everyframe tc pulse as long as the acquire sync line is

held low.

Signal Rate

+Clock +4 MHz (222) TTL

-Clock -4 MHz (222) TTL

frame tc 16384 Hz low duty cycle TTL

acquire sync TTL level control signal

Table 2.2:Signals required to drive the ICS110b in slave mode.

It is useful to have an overview of how the ICS board operates in order to understand the

2.2 Hardware components 27

design of the DAQ system. Figure2.3 shows a block diagram of the internal operation

of the ICS110b card2. Data are clocked out of the ADC chips as serial bits and are then

arranged into 24-bit words by the serial-to-parallel converter before being inserted into

the FIFO. The number of bits used to store each sample of data depends on the software

configuration of the card; the two possible modes are calledpacked(24-bit, 4 byte) and

unpacked(16-bit, 2 byte).

ADC

ADC

ADC

ADC

parallelto

serial

parallelto

serial

FIFO32k word

FIFO32k word

VM

EB

us

controllogic

acquire sync

Ch 3

Ch 1

Ch 2

Ch 0

clock+

clock−frame tc

Figure 2.3:A simplified schematic of the internal workings of the ICS110b

ADC card. The timing signals and the signals to be sampled are input via

front-panel connectors; the digitised data are read out over the VME bus.

The ICS boards are also fitted with optional daughter boards that provide some level of

input protection, extra anti-alias filtering, and a programmable preamplifier stage. Tests

were done on the noise performance of the ICS card with this programmable gain stage.

It was found that the programmable preamplifiers add noise if they are used to amplify

or suppress the input signal more than a certain amount. These tests also showed that

although the ADC board can in principle sample data at 24 bit resolution, the input noise

2The diagram shown is a slightly simplified version of the diagram given in the user manual of the

ICS110b card.

2.2 Hardware components 28

102

Frequency (Hz)

10110−10

10−8

ASD

(V/√

Hz)

10−6

10−4

10−2

100

navs:16384nfft:

ndata: 983040

6016384

1.50fs:enbw:

−20 dB preamp gain0 dB preamp gain+20 dB preamp gain16-bit digitisation noise24-bit digitisation noise

Figure 2.4:Amplitude spectral densities of a sinusoidal signal recorded by

the DAQ system in unpacked (24-bit) mode with different preamplifier gain

settings. The theoretical spectral densities for 16- and 24-bit digitised sig-

nals are shown for comparison.

of the daughter board is around the same level as 16 bit quantisation noise. For these

reasons, the ICS ADC boards in GEO are only used in packed mode (16 bit quantisation)

and the preamplifier gains are only used in the range from -8 dB to 8 dB.

Figure2.4 shows measured amplitude spectral densities of a sinusoidal signal recorded

by the DAQ system in unpacked (24-bit) mode for different preamplifier gain settings.

A sinusoidal signal was chosen so that a sensible fraction of the input range was used—

in this case, around 25%. The amplitude of the signal ‘seen’ by the ADC chips was

therefore 1 V for each preamplifier gain setting. For comparison, the quantisation noise

from 16-bit and 24-bit digitisation are shown. Quantisation noise can be characterised as

normally distributed random noise with mean zero and standard deviationσquantwhere,

σquant=1

√12

LSB. [DSPGuide] (2.1)

2.2 Hardware components 29

The theoretical noise signal(s) can then be calculated as:

nN(t) =1

√12

4 [V]

2Nnnorm(t), (2.2)

whereN is the number of bits in the digitisation process (e.g., 16 or 24), andnnorm(t) is

normally distributed random noise with mean zero, and unit standard deviation3.

Since we do not want to add digitisation noise to signals when we record them, signals

will typically need to be whitened using analogue filters before being sampled by the

DAQ system. We can see from figure2.4 that we must confine all signals to about 5

orders of magnitude in dynamic range before recording them.

When analysing data from the DAQ system, one also has to be aware that each channel

of the ICS boards has a DC offset that both varies in time and between channels. Figure

2.5shows a sample of the offsets from the ICS card installed in the north station (thot).

The offsets are sampled once per minute for 24 hours. A daily drift, most probably due

to temperature variations, can clearly be seen. For signals with amplitudes around a few

millivolts, this effect could be significant.

The VMIC ADC board

The additional ADC board that is used in the central station DCU is supplied by [VMIC].

This board samples up to 64 channels asynchronously with 12-bit resolution and sample

rates up to 512 Hz per channel. The board can be configured to accept an external signal

that instigates a complete scan of all the channels. This external signal is supplied via

the VME backplane. When the scan is complete, the data from all the channels can be

read out via the VME bus.

Generating the timing signals (The Glasgow timing card)

The timing signals for both the ICS and VMIC cards are generated from a custom built

timing card (referred to as the Glasgow timing card). The timing card in essence simu-

3The input range of the ICS board is−2 V to +2 V.

2.2 Hardware components 30

Channel 6

Channel 5

Channel 4

Channel 3

Channel 2

Channel 1

0 5 10 15 20 25

0.011

0.012

0.013

0.014

0.015

Time (Hours)

Am

plitu

de

(V)

Figure 2.5:The measured DC offsets of the first 6 channels of the ICS ADC

board installed in the north station (thot). A 24 hour variation, most proba-

bly from temperature changes, can be clearly be seen.

lates the ICS110b in master mode but with some additional features. Using the signals

from the GPS receiver, the timing card generates the appropriate signals for driving the

two types of ADC card. The card is designed around an [ALTERA] programmable logic

chip (PLC).

The GPS receiver generates two signals: a 4194304 Hz(222)4 TTL signal, and a pulse

signal that defines the transition boundary of one GPS second to the next. The pulse

signal is referred to as a 1 PPS signal (one Pulse Per Second). The 1 PPS signal is a short

pulse, around 200µs in length, that repeats every second. The signal is nominally low

(0 V) but goes high (+5 V) about 200µs before the second boundary and falls to zero

on the second boundary. The two signals from the GPS unit are fed to the timing card

circuit via front panel BNC connectors. The timing card passes the unaltered 4 MHz

signal through to the ICS card (-Clock ) and also replicates the signal with the opposite

sign to give the+Clock signal. In addition, the combination of the 4 MHz signal and

the 1 PPS signal are used to derive the 16384 Hzframe tc signal which is aligned

4Since the 222 Hz signal is the only signal referred to in this section of text that is close to 4 MHz it

will be commonly referred to as simply a 4 MHz signal for convenience.

2.2 Hardware components 31

such that the firstframe tc pulse is as close as possible to the GPS second boundary.

From figure2.6, it can be seen that theframe tc signal is displaced from the 1 PPS

falling edge by about half a cycle of the 4 MHz clock. The ‘acquire sync’ line is just a

software controllable logic line that is used to start and stop the acquisition on the ICS

ADC board.

For the VMIC card, a 512 Hz clock signal, also derived from the 4 MHz GPS clock, is

directed to the VME bus where it is routed to the backplane connector of the VMIC card.

Only external control of the clock signal is available for the VMIC; all other control, for

example, starting acquisition, must be done in software.

The timing card can accommodate a certain amount of software control via the VME

bus. The software that controls the DCU must be able to interact with the timing card in

order to start and stop the acquisition process as well as to check for any errors that may

occur on the card.

Experience with an earlier version of a different design of timing card built by LIGO

showed that the cable that feeds the timing signals to the ICS card was susceptible to

environmental interference since it is an unshielded ribbon cable. This interference,

combined with the low voltage logic levels (MECL) used on the LIGO timing card,

occasionally caused rogue timing pulses which in turn caused misalignments of the data

in the ICS card’s internal FIFO buffer. The result was that the signal recorded from one

channel would appear in the data stream of another—the so-called ‘channel hopping’

problem. In the original DAQ system, these errors proved difficult to catch without

looking at the data continually and therefore impossible to recover from on a suitable

time-scale. In order to recover quickly from such an error one must first detect it as

soon as it occurs. To do this, the Glasgow timing card was designed with the ability of

reading back the signals as they appear on the line that delivers them to the ICS card.

The timing pulses are then counted and checked to be the correct number: 222 for the

fast clock lines and 214 for the frame tc line. If either of these lines exhibit extra

pulses in a particular second, an error flag is raised on the timing card. These error flags

are checked in the controlling software once per second. The control software can then

flush the FIFO on the ICS card and reset the timing card. Acquisition then continues

2.2 Hardware components 32

on the next 1 PPS boundary. With this method, the down-time resulting from this type

of error can be kept to a minimum. This system was tested by deliberately introducing

extra pulses onto theclock andframe tc lines. The extra pulses were caught and the

appropriate action was taken by the control software. Although the timing card logic can

identify extra pulses on the clock lines, it cannot currently detect missing pulses. It is

possible, though as yet unobserved, that such missing pulses could cause similar effects

as those described above for extra pulses. To catch such effects, a separate software

application was developed that continually monitors the data being recorded by the data

acquisition system, checking that it conforms to a user configured set of requirements

(see section2.5for details).

Since the installation of the Glasgow timing cards, no occurrence of channel hopping

has been observed. This is surely due to the fact that the Glasgow built system uses only

TTL logic levels and not the MECL levels used on the LIGO cards, resulting in a system

that is less susceptible to interference. Table2.3summarises the commands that control

the timing card. The possible error states that can be read back from the timing card are

summarised in table2.4.

Command Function Software Flag Control register

Initialise Check for the presence

of the timing card on

the VME bus.

TIMING CARDINIT 0

Arm Arms the timing card to

start on the next 1 PPS

falling edge.

TIMING CARDARM 1

Reset Resets the state of the

card to the power-up

state.

TIMING CARDRESET 2

Acquire Sync Raise the acquire sync

line.

TIMING CARDAS 4

Table 2.3:A summary of the command functionality of the Glasgow timing

card. All software flags have the prefixICS110b which was omitted to

reduce the size of the table.

2.2 Hardware components 33

−0.5

Acquire Sync

frame tc

1PPS

clock+

clock−

4MHz

Acquisition time (µ s)

0−1 10.5

GPS secondboundary

Figure 2.6:A plot of the measured timing signals from the Glasgow timing

card at the start of the acquisition process. The acquisition begins at t= 0,

about 290 ns after the GPS second boundary. The signals from the GPS card

are also shown as a reference.

2.2 Hardware components 34

Command Function Software Flag Control register

4M Error A flag that indicates the

presence of an error on

the 4 MHz timing line.

TIMING CARD4M ERROR 32

Frametc Error A flag that indicates the

presence of an error on

theframe tc line.

TIMING CARDFTC ERROR 1

Table 2.4:A summary of the error states of the Glasgow timing card. These

flags can be read from the card via the VME bus allowing the controlling

software to check for errors. All software flags have the prefixICS110b

which was omitted to reduce the size of the table.

Figure2.6shows the signals produced by the Glasgow timing card at the start of acqui-

sition. It can be seen that these signals conform to the specification given by ICS for

driving the ICS110b in slave mode.

2.2.2 Data archiving system

The on-board FIFO of the ICS card can store around 1/16 seconds of data. The accu-

mulating and subsequent storing of data is done on two levels: in the short term, up to

20 seconds of data can be stored on the VME processor card in chunks of one second

duration; in the long term, data is archived on large hard-disk arrays by a data archiving

computer, both in an proprietary GEO format (for up to 3 days) and in a more interna-

tionally accepted frame format (indefinitely) [FrameLib][FrameSpec].

Data are collected from the ICS card 16 times per second via the VME bus and are

stored in one second long packets in an internal data buffer. When a full second of

data is available, it is sent via Ethernet to the data archiving computer to be archived

to disk. The data archiving computer is a Sun Enterprise Server which has a 170 GB

RAID array attached—enough to store around 3 days of data. The data are converted in

close-to-real-time into frame format and then archived indefinitely on further large disk

arrays.

2.3 Software design and the data flow model 35

Control data archival

In addition to the storage and archival of the signals recorded via the ICS and VMIC

ADC cards, it is also necessary to archive the many control channels that are present

in GEO. The main interferometer subsystems of GEO (such as mirror offsets and local

control gains) are supervised and controlled by an extensive [LabVIEW] system which

comprises some 800 or more channels. This data is very important for detector diagnos-

tic purposes and so it is desirable to archive it along with the other recorded data.

A LabVIEW VI (virtual instrument) was written to gather the state of all the control

channels once per second, package them in an appropriate way, and send the data to the

data archival computer for storage on disk. This VI is calledControl-to-DAQ.vi .

From the point of view of the data collecting computer, the LabVIEW Control-to-DAQ

VI looks just like another hardware DCU. Section2.3.4on page52describes the storing

of the control data in more detail.

2.2.3 Overview

Figure 2.7 shows an overview of the hardware components that make up the central

elements of the DAQ system for GEO5. The data rates shown are the upper limits that are

comfortably realisable with the current hardware. In principle, the data rate to disk can

be reduced considerably by decimating the data in the data archiving software, leading

to reduced storage requirements.

2.3 Software design and the data flow model

The software that controls the main aspects of the DAQ system is best discussed under

the three broad headings of data collection, data archival, and system administration. The

data collection software is a client-based application that connects to the server-based

5The central elements are those that are located at the detector site. Other elements of the DAQ

system—frame making computer, backup tape system, etc—are located at the AEI in Hannover.

2.3 Software design and the data flow model 36

1 MB/s

500 kB/s

500 kB/s

5 kB/s

2M

B/s

≤1 MB/s

data archiving

RAIDarray

and signalprocessing

(morgan)

32-bit floating point

DCU(alchemist)

32/64 channels@ 16384 Hz/512Hz

16-bit/12-bit

DCU

DCU

DCU

(pandora)

16 channels@ 16384 Hz

16-bit

(thot)

16 channels@ 16384 Hz

16-bit

@ 1Hz' 800 channels

(control-to-DAQ.vi)

100 Mbit switch

backuptape drive

Figure 2.7:An overview of the data acquisition system including the approx-

imate data rates that are present on the network. Each DCU and computer

has been named to allow easier identification within the project; these names

are also shown.

2.3 Software design and the data flow model 37

data archival application. This allows multiple DCUs that run the same client software

to connect to one data collecting server application. This design makes the system very

expandable, being limited only by the I/O capabilities of the computer hardware running

the applications. In GEO, three standard DCU client applications and one special client

application (thecontrol-to-DAQ.vi Virtual Instrument), can communicate data to

the one data archival server application.

Before considering the details of each of these software tasks, it is useful to look at an

overview of the data flow model that steered the design and writing of the software and

at the system interface that allows easy configuration of individual channels, as well as

other administrative aspects such as data storage times.

2.3.1 System configuration interface

The configuration of the entire DAQ system is held in a database. This includes informa-

tion about each DCU (location, which ADC cards are installed,etc.), the configuration of

each channel (sample rate, sample resolution, gain,etc.), as well as other system param-

eters, for example calibration information. Each channel of the data acquisition system

can be configured individually using a simple web interface. When changes are made

to a channel, the web interface communicates those changes first to the data archiving

software and then, if the changes were carried out successfully (for example a gain was

successfully changed), to the database. Using this kind of system, a complete record

of the DAQ configuration is kept and can be viewed for any time in the past. This is

essential for correct interpretation and characterisation of stored data, particularly if the

system configuration changes frequently as in engineering periods.

2.3.2 Data flow model

Figure2.8shows a schematic of the flow of data within the DAQ system software. The

different data types are indicated and the software sub-tasks are shown. Data are quan-

tised within this data flow model into data packets, the definition of which follows.

2.3 Software design and the data flow model 38

Signal data

Configuration data

Log messages

User signals

ICS Buffer (20 data packets)

VMIC Buffer (20 data packets)

message cue (100)

collectdata

VMIC senddata

sendmessages servermain

login/config

DC

UD

ata

colle

ctor

userreceivecontroldata

updateconfig

receivedata

receivemessages

diskstore

interface

database

dataconfig

Con

trol

dat

a

data (disk)config

control-to-DAQ.vi

VME bus

ICScollectdata

Con

figura

tion

Figure 2.8: A schematic of the flow of data through the various software

elements of the DAQ system. The individual software sub-tasks are shown as

grey boxes. The dotted grey lines separate different physical locations of the

software.

2.3 Software design and the data flow model 39

On the DCU, each data packet contains one second of data from all the channels and the

time-stamp of that second. The DCU software has an internal bounded buffer capable

of storing up to 20 such data packets6 for both the ICS and VMIC data. The use of such

buffers means that the system can cope with short term (less than 20 s) network problems

without loss of data. When the data packets are received by the data archiver, they are

further segmented into data packets of the same duration that describe only one channel

for one second of time. At this level, more information about each channel is added to

the data packets. Tables2.5and2.6summarise the contents of the DCU data packet and

the data archiver packet respectively. In the case of the DCU that contains both the ICS

and VMIC ADC cards, an extra bounded buffer is implemented to store up to 20 packets

of data from the VMIC data.

Field name Description Data type

gps time The time stamp for this packet. long int

gps nsecs The nanosecond at which the time long int

time stamp was applied.

num channels The number of channels in this packet. long int

nbytes The number of bytes of data in this packet.long int

data The data vector for this packet. char *

Table 2.5:The fields contained within each data packet on the DCU. Each

packet contains one second of signal data. In the case of the ICS card, the

number of bytes of data in each data packet depends on the configuration

of the card; the user interface to the DAQ system allows the ICS card to be

configured to record 16 bit or 24 bit data. In addition the number of chan-

nels to record can also be configured. Within the software source code, this

structure is type defined asdata chunk .

The different data packets arise in this way because all the channels on the DCU have the

same format (sample rate, sample resolution,etc.), which is defined by the configuration

of the ADC cards. Since this information is stored in the system database, we don’t need

to carry it with the data to the data archival software; it can retrieve the information itself

6The number of chunks that can be stored is restricted by the amount of RAM on the processor board.

2.3 Software design and the data flow model 40

Field name Description Data type

record Record this channel to disk? (0 or 1) short int

sw lp Apply software low-pass filter? (0 or 1) short int

name The signal name. char*

card number The internal GEO reference number. short int

number The channel number within the ADC card. short int

sample rate in The sample rate of data from DCU. short int

sample rate out The sample rate of data to record to disk. short int

sample res in The sample resolution of data from DCU. short int

sample res out The sample resolution of data to record to disk.short int

gain The gain setting of this channel on the ADC

card.

float

inputrange The input voltage range of the ADC card. float

fc The cut-off frequency of the software low-pass

to be applied.

double

in history The filter history from the previous second of

input data.

long int*

out history The filter history from the previous second of

output data.

double*

a Filter coefficients. double*

b Filter coefficients. double*

Table 2.6:The fields of the data structure that stores information about each channel for

each second of data received from a DCU. This information is stored on disk along with

the signal data. Storing all of this information makes it possible in principle to track the

history of any particular segment of data. At the current time, not all of the information

is carried over into the frame file format; in particular, the filter coefficients are not

carried over but since the sample rate is, the coefficients used can be reconstructed in

retrospect. In addition, some of the information (gain, output sample resolution, and

output sample rate), is combined into one variable, theslope. This allows conversion

from recorded ADC units of a signal to the actual voltage that was connected to the ADC

card.

2.3 Software design and the data flow model 41

directly from the database. This is not so when the time comes to write the data to disk.

When the data collector receives the data, it performs any downsampling and resolution

changes that have been configured for each individual channel. It is then essential that

each data packet carries with it a full description of each channel contained within. Each

recorded signal in GEO is given a name that conforms to the channel naming convention

of LIGO. The general format is a prefix code that specifies the detector (G1: in the

case of GEO) followed by three further tags separated by underscores, describing the

signal. For example, G1:LSCMID EP represents an error-point channel that is part

of the MIchelson Differential control which in turn belongs to the Length Sensing and

Control subsystem of GEO.

2.3.3 DCU software

The DCU code is essentially a client application that is capable of connecting to the

server based data archiving software (see section2.3.4). It is implemented as a real-time

application that runs on the [VxWorks] real-time operating system. Within this operating

system, one must define tasks and assign a priority to each. The operating system then

uses the processor time to ensure that, as much as is possible, each task runs in real-time

according to its priority. Each grey box within the DCU area of figure2.8represents one

such task.

Before looking briefly at the details of each of these tasks, it is first useful to understand

the operation of the bounded buffers that are implemented in the GEO DAQ system.

Bounded buffer design

A buffer in this context, is simply an area of RAM set aside for the storage of data. The

bounded buffer system is a simple design that allows one task to write data to one place

in the buffer (the head) while another task reads data from a different place (the tail).

The buffer is maintained such that at least one data packet lies between the head and

the tail of the buffer. After a data packet is written to the head position of the buffer,

2.3 Software design and the data flow model 42

the head pointer is advanced one place such that when the next packet is written to the

buffer it does not overwrite the previous packet; similarly, when a data segment is read

from the buffer tail, the tail pointer is advanced one place, rendering the previous data

packet obsolete and ready for overwriting. When either the head or tail reaches the

end of the buffer, the place pointer wraps around to the beginning of the buffer. This

means that if no data packets are read from the buffer but are continually written to

the buffer, data would start to be lost after the head first wraps around—in the case of

current implementation in GEO, after 20 seconds. Figure2.9 shows a schematic of the

bounded buffer operation. The process of reading from, and writing to, a bounded buffer

is protected by a semaphore system. The semaphore system is simply a way to allow

separate tasks running on the VxWorks system to communicate the fact that they are in

the process of changing a global variable. In this way, a global variable, in this case the

length control variable of the bounded buffer, cannot be changed by more than one task

at a time. Pseudo code for the buffer operation is shown in appendixB.1.1.

...... 134N-3N-2N-1N 2

tail pointerhead pointer

current buffer length

‘Write to head’

operation

‘Read from tail’

operation

Figure 2.9:A schematic of the operation of the bounded buffer model that

is used to manage the data in the DCUs of the DAQ system. The maximum

buffer length is N whereas the current buffer length is the distance between

the head and the tail pointers. In the current GEO DCU code, N= 20.

The main task

The main task of the DCU code has responsibility for getting the whole acquisition

process going. It first checks for the existence of the necessary hardware cards: the

2.3 Software design and the data flow model 43

timing and GPS cards must be available as well as at least one of the two types of

ADC card. If the checks are successful, it then contacts the data archiving software and

receives the configuration details for the installed ADC card(s). Having received all the

necessary configuration details (number of channels, sampling resolution,etc.), it creates

and initialises the required bounded buffers. It then creates all the other sub-tasks and

waits, potentially forever, for all the other tasks to finish before rebooting the system.

(Here the assumption is that if all the other tasks have stopped executing then something

must be wrong and a reboot is the chosen course of action.)

The ICS ADC task

Upon starting, the ‘ICS collect data task’7 resets the timing card and the ICS FIFO buffer

before arming the timing card in preparation for the next 1 PPS trigger from the GPS

card. Having performed this setup procedure, the code enters an infinite loop in which

one second segments of data are collected and written to the ICS bounded buffer. Within

this infinite loop, various checks are made on the condition of the timing hardware (e.g.,

GPS synchronisation). If an unrecoverable error is discovered (for example, the number

of lost seconds exceeds a certain threshold), then the task exits the infinite loop and

quits; at the same time, arun flag is set to false to inform all the other tasks that they too

should exit. The main task would then reboot the system in an attempt to recover normal

operation.

Within the infinite loop, a sub-function is called (precisely once per second during nor-

mal operation), to gather one second of data. Since the ICS on-board FIFO buffer is only

large enough to contain about 1/16 seconds of data, this sub-function8 collects data 16

times each second and writes it to the appropriate place in the bounded buffer. The read-

ing of each sixteenth of a second data segment is instigated when the ICS FIFO reaches

half capacity, at which time a flag within the status register of the ICS card is raised,

signalling to the software that data are ready.

7Within the DCU code, the ICS collect data task is namedcollect data(void) and is located in

collect data.c .8The sub-function responsible for gathering one second of data is called

ics110b adc read one second() and is located inics110b adc.c .

2.3 Software design and the data flow model 44

The VMIC ADC task

The VMIC ADC board has a somewhat different operation mode from that of the ICS

ADC board. Because the VMIC board does not not have any on-board storage capacity,

the samples for each channel must be read out before the next sample is acquired. When

the VMIC board receives an external clock pulse, it takes a scan of the all the channels

and then raises an interrupt on the VME bus. This interrupt is caught by an interrupt

handler in the DCU code, which then collects one sample for each channel and writes

it to the correct place in the VMIC bounded buffer. The data are time stamped via the

GPS card when the first sample of a one second data packet is collected (approximately

1/512 s after the 1 PPS edge). Pseudo code for the interrupt handler can be seen in

appendixB.1.2.

The send data task

The send data task9 is responsible for sending the data packets to the data archiving

computer. As long as there are data packets in the bounded buffers, the task tries to send

that data as quickly as possible.

The send data task is also responsible for monitoring the network link to the data col-

lecting computer. If the send data task cannot send any data (for example, if the network

goes down for a short time), it initialises a reconnection procedure. This reconnection

procedure consists of continually trying to resume the network connection to the data

archiving computer. If this is achieved, the DCU resumes communication with the data

archiving software (the server) and acquisition continues.

In addition, in the case when an ICS and a VMIC card are installed, the send data task

monitors the states of the two bounded buffers to ensure that when it sends data packets

from these two bounded buffers, the packets have the same time stamp,i.e., that they

represent data from the same instant of time. If this is not the case, an attempt is made

to resynchronise the buffers. SectionB.1.3contains pseudo code that shows the resyn-

9The code is found in the source fileclient.c of the DCU code.

2.3 Software design and the data flow model 45

chronisation procedure. This asynchronisation can occur if the timing signals to either

ADC card are somehow disturbed. Although other procedures are in place to correct

the timing signals, it is equally important to correct for the effects on the data buffers as

well.

The server task

The server task is a small server routine that allows clients to connect to the DCU and

instigate a limited number of actions. The main purpose of the server is to receive noti-

fication that a user wishes to reconfigure the gain of one or more of the programmable

preamplifier gains on the ICS board. In practice, only the data archiving software con-

tacts this server to apply changes to the preamplifier gains. Figure2.8shows (in red) the

path of this user communication data.

The send message task

Whenever anything out of the ordinary occurs in the DCU software, it is essential that

that information is made available to the people at the detector and to any data analysts

that may use the data in the future. For this reason, a message handling system was

designed. At any point in the code where a message needs to be communicated to the

outside world, it is placed on amessage queue. The message queue is essentially another

bounded buffer that contains places for 100 messages. The contents of a message is

shown in table2.7. The send message task is responsible for sending any messages

Field name Description Data type

gps time The time stamp for this message.long int

message The text to communicate. char[]

Table 2.7:The DAQ software message structure.

that are in the message queue to the data archiving computer. The received messages

are then recorded to log files (see section2.3.5 for more details). Having a message

queue is especially useful if a problem occurs with the network connection: in that case,

2.3 Software design and the data flow model 46

error messages can still be raised by the DCU tasks and then communicated to the data

archiving computer once the network is restored.

2.3.4 Data archiving software and signal processing

The data archiving software (which, due to historical reasons, is calleddatacollector ),

is a server application that accepts connections from DCU-like clients. In addition, a user

interface client is used to configure various aspects of the DAQ system. The commands

available to connected clients are summarised in table2.8.

The server is a so-calledforking server. This means that when a new client connects

to the server, the server processforks (essentially creates a copy of itself). From then

on, the copied process handles the commands from newly connected client. In fact,

when a DCU connects to the data archiving server, two new process are created: one

that handles the flow of channel data, and one that handles messages. This is necessary

since the DCU code runs the data sending and message sending tasks in parallel. Hence,

there will exist 2N + 1 datacollector processes on the data archiving computer,

whereN is the number of connected DCUs. In this way, data from each DCU is handled

independently from the all the other DCUs, creating a very robust solution, since if one

DCU fails, all the others carry on as normal.

On the ICS board, data are acquired and stored in the internal FIFO such that the first

sample for each channel is contiguous, the second sample from each channel follows,

and so on. This is not the most natural arrangement of data, especially if one wishes

to analyse or process individual channels in an efficient manner. Therefore, the first

things that is done when the data archiving software receives the data is to rearrange

the data such that all the samples for each channel are contiguous. After this, each

channel is processed separately, depending on the user-set configuration. For example, a

channel may be resampled to a lower sample rate (with or without low pass filtering) and

then reduced in resolution. Up to now, all channels from all DCUs have been recorded

and sent to the data archiving process. At this point, the user configuration instructs

the data archiver which channels should be archived to disk. With a combination of

2.3 Software design and the data flow model 47

Command Action

QUIT Quit the current connection.

LOGIN Allows a DCU to login and receive its configu-

ration information. (This does not apply to the

control-to-DAQ.vi DCU.)

CREATE FRAME This is the main command for the DCU clients.

A DCU can send one second of data to the data

archiving computer using this command. Any

necessary signal processing is done before the

data are written to disk in the GEO internal for-

mat.

RECEIVE CONTROL DATAReceive one second of LabVIEW control data

from thecontrol-to-DAQ.vi DCU.

ERROR MESSAGE Receive an error message from a DCU and

record it to disk in the appropriate log file.

RECEIVE CONFIG This command receives a new channel configu-

ration from the user interface. If the new con-

figuration includes a change of the preampli-

fier setting on a DCU, the command contacts

the server of the appropriate DCU and instructs

the change of gain. The user interface client is

informed if the changes were successfully im-

plemented, after which, the database record is

updated.

Table 2.8:The command set of the data archiving server. Only commands

relevant to the receiving and archiving of data are shown. A number of other

commands exist that allow various monitoring tasks to be configured.

2.3 Software design and the data flow model 48

downsampling, resolution reduction, and switching off channels, the data rates can be

kept to acceptable levels. Figure2.10shows the path of the data through the command

that is called when a client sends aCREATE FRAMEcommand.

Signal processing

The ADC chips (CS5396/97) on board the ICS110b have an internal digital filter that

gives a flat (±0.004 dB) band-pass of 40% of the output sample rate (16384 Hz in the

case of GEO). Since some signals simply do not contain information at these frequen-

cies, it is desirable to down-sample the data. This not only saves storage space but also

reduces overheads in data transfer and processing. In some cases, even though a signal

has frequency content up to 8192 Hz10, we may only be interested in the data at lower

frequencies; again it is desirable to decimate this data. Before doing so, it is necessary

to filter away the unwanted high frequency components so as to avoid aliasing effects.

For these reasons, a modest amount of signal processing capability has been built into

the data archiving software, namely:

• Apply software anti-alias filter prior to downsampling.

• Down-sample data.

• Lower the sample resolution from 16-bit to 8-bit.

If it is configured for a particular channel, software anti-alias filtering will be performed

first. The software anti-alias filters are 6-pole IIR Butterworth11 filters that are matched

to the Nyquist frequency of the downsampled data. For example, if the output sample

rate of a particular channel is configured to be 4096 Hz, the normalised corner frequency,

fc, of the anti-alias filter will be set to a fraction of the output Nyquist rate given by:

fc = 0.3 ×fsout

fsin

= 0.3 ×4096

16384= 0.075. (2.3)

10The Nyquist rate of the raw 16384 Hz data from the ICS110b.11Due to the potentially large number of channels that must be filtered in real-time, we can only use a

low-order anti-alias filter. With more computing resources, this could be increased.

2.3 Software design and the data flow model 49

Is

minute?this a new

timeCheck storage

Delete dataexceedingstorage time

Yes

Newchannel Yes

config?

YesRecordchannel?

Apply

filter?

YesFilter Data

Read and

Yes

reorder data

CREATE FRAME

RecalculateIIR filters

anti-alias

All channelsprocessed?

Makefilename

to diskWrite data

Updatelog files

No

No

No

No No

Yes

No

Downsampleand changeresolution

if neccessary

timefrom lastdata > 1?

from DCUGPS time

Read

Do nextchannel

Write logmessageto indicatelost data

Figure 2.10:A flow diagram of the part of the data archiving software that

receives and processes the data from a DCU before writing it to disk.

2.3 Software design and the data flow model 50

In other words, the corner frequency is set to 60% of the output Nyquist frequency.

This was chosen to give enough suppression at the output Nyquist frequency while not

degrading the signal too much in the pass band. The possible output sample rates are

summarised in table2.9.

fsin Hz fsout Hz

16384 16384, 8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16

512 512, 256, 128, 64, 32, 16, 8, 4, 2, 1

Table 2.9:A summary of the possible sample rates that can be configured

using the DAQ user interface. A software low-pass filter can also be applied

if desired.

By recording the same white noise source in two channels of the DAQ system, the trans-

fer function of each possible anti-alias filter was measured. For this experiment, the

downsampling section of code was deactivated such that the filtered and unfiltered noise

appeared in the data files with the same (16384 Hz) sample rate; in normal operation,

if a low-pass filter is selected in the configuration interface, decimation is automatically

performed.

An example of the measured transfer functions is shown in figure2.11. More of the

measured transfer functions are given in appendixA.2. The IIR filter coefficients are

calculated according to the formula laid out in appendixA.3.

Decimation is done by simply dropping unwanted samples. So, for decimating from an

input sample frequency of 16384 Hz to an output sample frequency of 8192 Hz, every

second sample is dropped. The system only allows decimation by multiples of two.

Further savings in storage space can be achieved by reducing the sampling resolution.

This is particularly sensible for signals that represent discrete digital levels such as the

interferometer lock indicator. The reduction in sampling resolution from either 16-bit

(ICS) or 12-bit (VMIC) to 8-bit is done by dividing the data by 28 and 24 respectively.

The data samples can then be stored as single bytes whereas without any resolution

change, both the ICS and the VMIC data samples are stored as two bytes.

2.3 Software design and the data flow model 51

measured filter responsemodel filter response

Frequency (Hz)

Frequency (Hz)

150

100

50

0

−50

−100

−150

101 102 103

101 102 103

Mag

nit

ude

(dB

)P

has

e(

)

10 0

10−2

10−4

Figure 2.11:A measured transfer function of the IIR anti-alias filter that is

applied in the data archiving software. In this case, the filter is used when

decimating data from 16384 Hz to 4096 Hz. Here the corner frequency can

be calculated from equation2.3 to be 1228.8 Hz. The model response of the

filter is overlaid for comparison.

Data format and organisation

The data from each DCU’sdatacollector process is written to disk in a separate

file (raw data files). Figure2.12shows a schematic of the GEO internal file structure.

During the conversion to the frame format (handled by a separate application), data from

all DCUs is collected into one frame file. In GEO, each raw data file contains data from

one DCU spanning one second of time. The frame format is flexible in its contents:

GEO-type frame files contain data from multiple DCUs spanning one minute of real

time.

The raw data files are organised in a directory structure related to the GPS time of the

data. Figure2.12also shows the details of the GEO directory structure. Having an or-

ganised structure like this makes dealing with large numbers of files (upwards of 345600

2.3 Software design and the data flow model 52

length of DCU name (2 bytes) DCU name

Channel 0 descriptor (∼1MB)

Channel N descriptor (∼1MB)

Num bytes of data (4 bytes)

Data for all channels

The structure of a GEO

An example file name including the fulldirectory structure.

raw data file.

base dir/data/2001/day232/hour11/min03/GEO alchemsit 683305200.grd

GPS time (4 bytes) Num Channels (4 bytes)

Figure 2.12:The file format used for the initial storage of the data. The

format allows for fast reading and writing which is important for keeping

down the I/O load of the data archiving computer. Fast reading allows for

prompt viewing of the data on-site. Also shown is an example raw data

filename complete with the full directory structure. All GEO raw data files

are written following this directory structure.

per day) much easier.

Receiving the control data

As well as receiving the data from the VME based DCUs (CREATE FRAMEcommand),

the data archival software also receives data from thecontrol-to-DAQ.vi DCU—

so calledcontrol data.

The LabVIEW control system within GEO uses a hierarchical channel naming conven-

tion organised as follows:

Physical System:SubSystem:Control .

Because of the large number of channels (>800) and low sample rate (1 Hz), it would be

extremely inefficient to record each channel as a separate entity as with the VME based

2.3 Software design and the data flow model 53

DCUs. For this reason, channels are grouped together based on physical systems such

that a channel name is now a long string describing all the channels contained within

in this one ‘super-channel’. The associated data vector is organised such that the data

sample from each channel is in the same position as the channel appears in the super-

channel name. In order to make the resulting super-channel name parsable, it is defined

as follows:

Top level#Sublevel 1#N1#Ch1#...#Ch N#Sublevel 2#N2#Ch1#...

whereN1 is the number of channels within the first sublevel and so on. An example of

one such super-channel will no doubt make this clearer. Here are the first few elements

of the channel name that describes the auto-alignment control of the first mode-cleaner.

AAMC1#ROT#18#DENOM#ERRP-A#ERRP-B#FBACK-A#FBACK-B#SCAN#ERR-X#...

In this example theTop level physical system is calledAAMC1and it contains multi-

ple sub-levels; the first sub-level that is shown is for rotation signals (ROT), which itself

has 18 sub-sub-levels:DENOM, ERRP, etc..

Since this data requires no reordering, no decimation, or any other signal processing, it

is more convenient for it to communicate with the data archiving process via a different

command:RECEIVE CONTROL DATA. The data archival process treats each super-

channel as a normal DCU channel and records it to disk using the file format described

in figure2.12.

2.3.5 System administration and configuration

Due to the importance of the DAQ system, it is of course essential to maintain knowledge

of the status of the system at all times. This knowledge comes in two forms: firstly the

state of the system from the point of view of user configuration, including what hardware

is installed and running at any one time. This information must be archived so that if

data are analysed some time after recording, the analyst can determine the status of the

system at that time. The second set of information to consider is the actual state of the

2.4 System timing accuracy and stability 54

running system, particularly any errors and warning messages that may be generated.

The DAQ system implemented in GEO uses two separate systems to deal with these

two points. User configuration and hardware status are stored in a database, with all

user interaction being done via a web-based interface. The run-time status of the system

(error messages,etc.), are stored in log files that are kept with the raw data. Although

the raw data is lost after a few days, the log files are kept indefinitely, allowing anyone

in the future to to check the state of the running system at any time in the past.

2.4 System timing accuracy and stability

Having designed and built a data acquisition system to a particular set of requirements, it

is essential to check whether the running system meets the original design specification,

particularly on long time scales. Extensive tests were done on the timing accuracy and

stability of the system. These tests address the issue on a number of different levels:

gross absolute timing accuracy, fine absolute timing accuracy, and long term timing sta-

bility. In addition, checks were done to ensure that the timing stability is constant over

the various possible error states of the system. This section will present these experi-

ments in detail.

In the following sections, times will be given in either UTC format (e.g., 2003-03-26

14:20:03) or GPS format (e.g., 714150013).

2.4.1 Gross timing accuracy

The question of whether or not the DAQ time stamping is correct to the one or two

second level is the first, and indeed, the simplest question to answer. One can simply

unplug (or switch off), a signal at a known time, determined from some external timing

reference, and then look in the data to see if the recorded signal behaves as expected.

This test was done using the [DCF77] time signal as a reference. A signal was unplugged

and the time recorded by eye (accurate to about half a second). The signal was then

2.4 System timing accuracy and stability 55

reinstated and the time recorded again. Figure2.13shows the results of this experiment.

The times during which the signal was unplugged (between 14:21:00 and 14:21:01) and

replugged (between 14:21:30 and 14:21:31), as measured against the DCF77 reference,

are shown by a horizontal bar.

Signal unplugged

amplitu

de

(V)

amplitu

de

(V)

−2−2

−2−2

Signal replugged

−1.5

−1.5 −1

−1

−1

−1

−0.5

−0.5

0

1

2

0

1

2

3

30

0

0.5

0.5 1

1 1.5

1.5

2

2

2.5

2.5

time from 2003-04-23 14:20:58 (s)

time from 2003-04-23 14:21:28 (s)

Figure 2.13:A test of the gross timing accuracy of the DAQ system. Here we

show that the time stamping of the data is at least accurate to the one second

level.

2.4.2 Systematic timing offsets

Before looking in detail at the fine timing performance of the system we should first

consider any known systematic timing offsets present in the system.

Due to the design of the ICS ADC cards, a systematic offset in the time-stamping of

the data arises due to the ‘gate delay’ in the ADC chips and other circuitry on the cards.

Since this delay is constant it can, once determined, be corrected for (at least at the

level of one sample). When the system was in development, this delay was determined

2.4 System timing accuracy and stability 56

by injecting a signal at a known time (triggered from an external GPS clock) and then

observing the position of the signal in the data. With this method the gate delay of the

ICS board was determined to be between 37 and 38 samples. When the data collecting

code first collects data from the ICS board, this gate delay is removed by reading 37

samples from the ICS internal FIFO and dumping them. The recorded data are now

aligned with GPS time to within one sample (<60µs). Any remaining gate delay cannot

be corrected for in this way and would require much more sophisticated techniques to

deal with it if required. The following section deals with measuring this residual offset.

2.4.3 Fine timing accuracy

Having tested the gross timing accuracy and having considered any systematic timing

offsets, we can look in detail at the fine absolute time accuracy (on time-scales of

nanoseconds). One way to do this is to inject a particular signal at a known time and

then look to see when this signal appears in the data (as with determining the gate de-

lay). Since the timing resolution of the data is only about 60µs (inverse of the sample

rate), one needs a signal that can be easily interpolated between the recorded samples

such that it can be monitored at a sub-sample level.

One such suitable signal is a linear ramp voltage. A straight line fit can be made to the

recorded ramp voltage, allowing the origin of the ramp to be determined to sub-sample

resolution. If in addition one knows with good accuracy when the ramp signal actually

had its origin, one can determine any time shifts that arise due to residual delays in the

data acquisition process. This was essentially the method used to determine the accuracy

of the DAQ time-stamping at a sub-sample level; it is described in the following text.

The injected signal

For convenience, an HP33120A signal generator triggered from a Rubidium clock12 was

used to inject a repeating ramp signal once per second for an extended period.

12The Rubidium clock provides an independent 1 PPS signal.

2.4 System timing accuracy and stability 57

The two time references in this experiment are the 1 PPS edges of the DAQ GPS receiver

and the Rubidium clock that triggers the ramp. The relative offset of these two references

must be determined before trying to determine any offsets in the DAQ system itself. By

measuring the two 1 PPS edges on an analogue oscilloscope, we determined that an

absolute offset of around 8.2µs exists between the two (the Rubidium 1 PPS leads the

DAQ GPS 1 PPS).

Figure2.14shows one such injected ramp signal plotted with the 1 PPS signal from the

DAQ system. The signals were recorded on an analogue oscilloscope. They-axes are

scaled differently so that the slope of the ramp signal is visible.

DAQ 1 PPS signal

Injected ramp signal

Am

plitu

de

(V)

Am

plitu

de

(mV

)

Time (µs)

00 0

1

2

3

5

4

1

3

5

4

2

−50−100 50 100

Figure 2.14:A plot of the injected ramp

signal alongside the 1 PPS signal from

the DAQ GPS receiver. The signals

are averaged to show the features more

clearly. The digitisation of the synthe-

sised ramp generated by the HP33120A

can be clearly seen. The frequency of this

injected ramp was 10 Hz.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Am

plitu

de

(V)

0

1

5

2

3

4fitting segment

Time (s)

−5

−4

−3

−2

−1

datafitzero crossing

Figure 2.15: A example of an injected

ramp as it appears in the data. The best fit

line is shown and the data segment used

for the fit is indicated by the horizontal

bar.

Analysing the data

The analysis of the data is in principle very simple: for each second of data containing

the ramp signal, fit a straight line to a section of the recorded ramp and determine the

point from which the ramp originated. From the fit we get a dependence between time,

2.4 System timing accuracy and stability 58

t , and ramp voltage,v, of:

v = mt + c, (2.4)

wherem andc are the parameters that are determined by the fitting process.

From equation2.4 we wish to determine the value oft for which v corresponds to the

start of the ramp. To do this we must define this starting ramp voltage,v0. It is not

sufficient to determine the value oft for whichv is zero since DC offsets in the recording

system or on the ramp signal itself would bias the result. Instead we take the average

of the data from the end of the second of data, where the ramp signal is constant at its

nominal ‘zero’ level—this gives a good measure of the ‘zero’ signal level when the ramp

is not present. We then determine the time of the start of the ramp,t0, as:

t0 = (v0 − c)/m. (2.5)

Figure2.15shows an example of one of the injected ramps as it appears in the data. Also

shown is the straight line fit that was made to the indicated segment of data.

An extra complication arises due to the fact that the HP33120A does not produce an

analogue ramp signal but instead digitally synthesises the signal and outputs a voltage

via a DAC. Because of this, the point at which the ramp appears to start (the first non-zero

sample sent to the DAC), depends on the frequency one chooses for the ramp. Figure

2.16shows clearly this effect.

The ramp can be characterised by its frequency or, if the peak voltage is kept constant,

by its length. For increasing length (decreasing frequency), the position of the first non-

zero output sample increases in distance from the origin of the ramp. In order to remove

this systematic effect from the results, we injected ramps of different frequencies and

determined the offset of the ramp in the data as a function of the ramp length. Once the

relationship between ramp length and measured offset is determined, we can extrapolate

down to a ramp of zero length (essentially a step function) to get the best measure of

the timing offset introduced by the DAQ system. With the same experiment, we can

determine how the standard deviation of the offset measurements varies with the ramp

frequency. This can be considered analytically if we assume that the scatter in the mea-

sured offsets comes from the fact that there is amplitude noise on the ramp signal. Figure

2.4 System timing accuracy and stability 59

0

100Hz

200Hz

time (µs)

0.001

0.002

0.003

0.005

0.004

0.006

0.01

0.007

0.008

0.009A

mplitu

de

(v)

0 5 1510 20 25 30

50Hz20Hz

Figure 2.16:HP33120A generated ramps

of different frequencies. The resolution

of this plot is below the sample resolu-

tion of the DAQ system so it is difficult

to interpret how these ramps will appear

in the data. However, placing a straight

line through the centre of the steps of each

ramp highlights quite well the relation-

ship between ramp origin and ramp fre-

quency.

δT

α

δT

δAα

δA

Figure 2.17: A conceptual explanation

of why the scatter in the measured off-

set measurements depends on the length

of the injected ramp. (See text for further

discussion.)

2.17shows a pictorial explanation of this. We can see that amplitude noise,δA, leads to

noise in the measured offset,δT . The inset figure shows that these two values are related

to the slope,α, of the ramp signal by:

tanα =δA

δT. (2.6)

The slope is in turn related to the length,L, of the injected ramp by:

tanα =A

L. (2.7)

So we see immediately that the relationship between the scatter in the measured offsets,

δT , and the ramp length,L, is given by:

δT =δA

AL , (2.8)

whereA is the amplitude of the ramp (which was fixed in these experiments). If we then

assume that the amplitude noise does not vary for ramps of different frequency, then we

see a linear dependence ofδT on ramp length.

2.4 System timing accuracy and stability 60

Results of the fine timing analysis

Table2.10shows the frequencies of the injected ramps along with the mean of the mea-

sured offsets for each ramp frequency. Plotting the data (figure2.18) clearly shows the

linear dependence between measured offset and ramp length.

Ramp freq. (Hz) Offset (µs)

1.1 182.48

1.5 136.43

2 103.93

3 72.43

8 32.03

20 17.07

40 12.16

Table 2.10:A table of the injected ramp

frequencies with the measured offsets.

Each measured offset is computed from

the mean of the fits from about 1200 in-

jected ramps.

Ramp freq. (Hz) σ of offsets (s)

10 6.21× 10−7

10 5.08× 10−7

20 3.23× 10−7

50 1.91× 10−7

100 1.0 × 10−7

200 7.7 × 10−8

200 6.23× 10−8

30 2.81× 10−7

12 5.61× 10−7

15 3.79× 10−7

400 7.61× 10−8

Table 2.11:A table of the injected ramp

frequencies with the variance of the mea-

sured ramps. Each variance was com-

puted from one minute of data,i.e., 60 in-

jected ramps.

A straight line fit was made to the data and the resulting relationship between ramp

frequency and measured offset was determined. The systematic offset of 8.2µs between

the two reference 1 PPS signals was removed before making the linear fit. The resulting

relationship is given by:

t0 =192.66

Rf+ 15.89µs, (2.9)

wheret0 is the absolute time offset of the DAQ system measured against the Rubidium

clock, andRf is the frequency of the injected ramp. We can see immediately that for a

ramp of infinite frequency (a step function), that the offset is 15.89µs.

2.4 System timing accuracy and stability 61

Figure2.19shows a plot of the data from table2.11—the scatter of the measured offsets

for injected ramps of different frequency. The resulting relationship for the standard

deviation,σoffset as a function of ramp frequency,Rf, is given by:

σoffset =5.23

Rf+ 0.063µs. (2.10)

The linear fit is not as good for this data, essentially due to the small number of points

used to compute the standard deviations. Nevertheless, the trend is clear and an estimate

for the variance of the absolute DAQ offset can be given as around 60 ns.

t0 = 192.66× ramp length + 15.89µs

Ramp length (1/Hz)

0 0.2 0.4 0.6 0.8 1

Mea

nm

easu

red

offse

t(µ

s)

mean measured offset

linear fit

20

40

60

80

100

120

140

160

180

200

Figure 2.18:A plot of mean measured off-

set versus injected ramp frequency. A lin-

ear fit to the data is shown along with the

equation of the best fit line. The equation

shown has units ofµs.

σ = 5.23× ramp length + 0.063µs

std of measured offsets

linear fit

0.120.10.080.060.040.020

Ramp length (1/Hz)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Sta

ndar

ddev

iation

ofm

easu

red

offse

ts(µ

s)

Figure 2.19:A plot of standard deviation

of the measured offsets versus injected

ramp frequency. A linear fit to the data is

shown along with the equation of the best

fit line. The equation shown has units of

µs.

2.4.4 Long-term timing stability

It is not only important to determine that the timing accuracy is good at one particular in-

stant, but also to determine the long-term performance of the DAQ timing. This includes

considering the timing stability across system reboots.

To investigate the long-term timing stability, a ramp of one particular frequency (10 Hz)

was injected for about 24 hours. Figure2.20shows a plot of the measured offsets as a

2.4 System timing accuracy and stability 62

function of time while figure2.21shows a histogram of these measured offsets. We can

see that the mean of the measured offsets agrees with equation2.9.

0 5 10 15 20 2532

33

39

38

37

36

35

34

offse

t(µ

s)

40

time (H)

Figure 2.20:A 24 hour time series of mea-

sured offsets for a 10 Hz injected ramp.

µ = 35.47

σ = 0.48

4000

3500

3000

2500

2000

1500

1000

500

033.533 34 34.5 35 35.5 36 36.5 37 37.5 38

offset (µs)

count

Figure 2.21:A histogram made from mea-

sured offsets from 24 hours of ramp injec-

tions. Overlaid is a fitted Gaussian distri-

bution with the mean and variance shown.

The fitted mean and variance agree well

with equations2.9and2.10.

The behaviour of the system across system reboots was also investigated. Here we make

a distinction between the two possible types of reboot: a hardware reboot, which involves

a full power recycle of the crate, and a software reboot which only reboots the VxWorks

operating system and restarts the acquisition software.

Again using an injected ramp of 10 Hz, the behaviour of the system was investigated

across both hardware and software reboots. Figure2.22 shows the results of this ex-

periment. It can clearly be seen that the timing offset after a software reboot is about

2300µs. This corresponds almost exactly to the 37 sample gate delay of the ICS board.

After some investigation, the reason for this was found to be a bug in the ICS driver code.

After a software reboot, the part of the driver code that allows reading from the internal

FIFO does not work properly the first time it is called. Since the first action of the DCU

code is to remove the gate delay by reading 37 samples, it is clear that the result should

be that the gate delay is not removed. A software patch was implemented to deal with

this so that now the problem doesn’t occur.

2.4 System timing accuracy and stability 63

0 10 20 30 40 50 60

time (min)

3000

2500

2000

1500

1000

500

0

∆T

(µs)

har

dw

are

reboot

soft

war

ere

boot

soft

war

ere

boot

soft

war

ere

boot

har

dw

are

reboot

timing offset with respect to Rb reference clock

Figure 2.22:The timing behaviour of the DAQ system across both hardware

and software reboots. It can be seen that the software reboot has an adverse

effect on the timing accuracy due to incorrect resetting of the ICS FIFO

reading algorithm.

2.4.5 Long-term timing stability during S1 science run

During 2002, GEO took part in the first coincidence science run undertaken by large-

scale interferometric gravitational wave detectors throughout the world. Since the timing

tests of the DAQ were not completed prior to the run, it was essential to validate the

timing accuracy of the data taken during the run. (See section3.8 for details about the

S1 science run.)

During the S1 science run, no ramp signal of the kind described in the previous sections

was recorded, mainly due to limited time and available channels. Because of this, we

were unable to determine the absolute timing accuracy of the S1 data. However, since

the hardware and software used in S1 was identical to that used in the tests described

above, one can reasonably assume that the absolute timing accuracy of the S1 data was

the same. What is left, however, is to determine whether there were any departures from

2.4 System timing accuracy and stability 64

4 62 310

amplitu

de

(V)

5

2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

time (ms)

linear fit

Fitting interval

Figure 2.23:A closeup of calibration signal that was injected during the S1

science run. The signal proved useful in confirming the DAQ timing stability

for the entire S1 data set. A linear fit was made to the data points contained

within the indicated time interval; typically this was only about 4 samples.

An example linear fit is shown.

this nominal timing accuracy, perhaps due to the reboot effects described above or to

some other unknown effects.

To do this retrospectively required us to find a predictive signal that could be used in

much the same way as the injected ramp signal discussed above. For the entire S1 data

run, a calibration signal was injected into the length control actuators of the Michelson

locking servo. This calibration signal was also directly recorded in the DAQ system (see

section3.8for details). Figure2.23shows a closeup of the calibration signal in the time-

domain. A section of the data was chosen where the signal was approximately linear.

For the entire S1 data set, we were able to make a linear fit to this same section of data

in each second of the recorded signal. This provides an accurate record of the timing

accuracy of the DCU which recorded this signal. Fortunately, the DCU that recorded the

calibration signal also recorded all the main interferometer output signals; data from the

2.4 System timing accuracy and stability 65

other DCUs remain unvalidated.

Results

In order to properly interpret the results which follow, it is first necessary to look in detail

at some artifacts that arise from this non-optimal measurement process. In addition, a

systematic effect related to the GPS receiver was identified in this process.

Figure2.24shows the results for the full extent of the S1 data; two things immediately

stand out as being unusual: two short periods where the offset was considerably larger

than the majority of the time, and a general linear drift in the measured offsets. First

‘Bad State’

‘Good State’

300250200150100500 350 400

time (H)

5000

4500

4000

3500

3000

2500

2000

offse

t(µ

s)

Figure 2.24: The measured offsets from

the S1 timing test.

‘Bad State’

‘Good State’

2500

2000

1500

1000

384383.5 384.50 0.5 1

2500

2000

1500

1000

500

0 0

500

offse

t(µ

s)

offse

t(µ

s)

time (H) time (H)

Figure 2.25:A close up of the bad states

found from the S1 timing test.

the two periods of excess deviation. Figure2.25shows a closer view of the two ‘bad

states’. Using the error log files of the DAQ system, the start of these two periods can be

correlated with two software reboots that occurred13.

The linear drift can be explained entirely as an artifact of the measurement process. The

calibration signal that was used for this purpose was derived from a 244 Hz square wave

signal output from an HP33120A signal generator. This signal was then differentiated

electronically to give a forest of peaks in the frequency domain at the odd harmonics of

244 Hz. As was explained earlier, these generators digitally synthesise the output signal

13The DAQ system was rebooted in order to try and recover from a source of excess noise believed to

come from the ICS daughter boards.

2.4 System timing accuracy and stability 66

and output an analogue voltage via a DAC. Because of this, its frequency will not be

exactly 244 Hz but instead some tiny fraction of a Hz away from it. The result is that the

waveform shown in figure2.23will appear to drift slowly in time relative to the DAQ

1 PPS. In the context of this experiment, this drift exhibits itself as a slow drift in the

measured offset of the data segment we chose. By measuring the slope of this drift we

can determine that the frequency of the square wave output from the HP33120A was not

244 Hz but actually 243.999999938 Hz.

200

200

time (H)

time (H)

hardware reboothardware reboot hardware reboot

lower limit of fitting window

upper limit of fitting window

‘bad state’

‘bad state’

250 300 350 4000 50 100 150

400350300250150100500

45

40

sam

ple

35

30

10

−10

0

5

−5offse

t(µ

s)

Figure 2.26:The timing artifacts from the S1 timing data analysis. The upper

graph shows measured timing offsets (with the mean offset removed), when

the DAQ system was operating normally. The steps that can be seen are due

to the changes in which samples are used in fitting. As the slowly drifting

calibration signal moves through the fitting window, different samples have

to be chosen to fit to. The lower graph reflects the times when the fitting

algorithm changed which samples were used to fit to. The vertical lines in

the upper graph (labelled ‘Hardware Reboot’) are times when the system

was rebooted and the locking servo of the DAQ GPS receiver was not fully

settled.

Having considered and explained these gross deviations from the nominal timing accu-

2.4 System timing accuracy and stability 67

racy we can remove them both from the data (ignore reboot periods, and remove the

slow trend) and look in more detail at the measured offsets. The top plot of figure2.26

shows the results after doing just that. At this level of detail, three more artifacts be-

come visible: on three occasions the timing offsets appear to describe a vertical straight

line, at regular intervals the timing offsets make small jumps and also seem to follow a

curve-like structure.

The curve-like structure can once again be entirely explained by the fact that the signal

we use to measure the offset, drifts with respect to the DAQ 1 PPS. As the signal drifts

through the window in which the fit is performed, the slight non-linearity of the signal

in this region gives rise to these curve-like structures.

The steps in the measured offsets are due to the combined effect of the measurement

process and the slow drift. After some time, one of the samples to which the linear fit is

performed will drift out of the right side of the fitting window. Now the fitting routine

uses one less sample in the fitting process and a step appears in the measured offset

values. When the next sample drifts in from the left, the fitting routine gets one more

sample to fit to. Again a step appears in the measured offsets. The bottom plot of figure

2.26 shows the times in the analysis when the upper and lower samples in the fitting

window were changed to best follow the waveform; the correlation between the samples

used for fitting and the steps in the measured offset is clear.

The small vertical spike-like structures (marked ‘Hardware Reboot’ in figure2.26) need

closer inspection to get a clue as to their origin. Figure2.27shows a closeup view of

two of these events. When one checks these times in the log files of the DAQ system,

one finds exact correlation with hardware reboots of the system. When the power to the

GPS receiver is cycled, as it is when the system is hardware rebooted, it takes some time

for the PPL (Phase Locked Loop) that locks the on-board quartz oscillator to the GPS

1 PPS signal to settle. This is what we see in the two plots of figure2.27. The excess

noise on the measured timing offsets that can be seen on the left of the hardware reboots

is in fact the reason for the reboots. On three occasions during S1, excess noise appeared

on the signals recorded on the central station DCU (the one recording the calibration

signal). This excess noise is believed to originate in the daughter boards that are fitted

2.4 System timing accuracy and stability 68

to the ICS ADC boards, but the reason for it is as yet unknown. These results confirm

hardware reboot∆

T(µ

s)

6

5

4

3

2

1

0

−1277 277.5 278 278.5

time (H)

138 138.5 139 139.5

time (H)

∆T

(µs)

6

5

4

3

2

1

0

−1

hardware reboot

measured offsets measured offsets

Figure 2.27:The offset drift of the DAQ GPS servo as it reacquires lock. The

times on the x-axis are given relative to the start of the S1 run.

that the relative timing accuracy of the DAQ system for the duration of the S1 was to

specification with the exception of the following times:

• 2002-08-23 15:00:00 to 2002-08-23 15:32:00,

• 2002-09-08 15:01:00 to 2002-09-08 15:06:00.

2.4.6 Conclusions

From the above experiments, we conclude that the time stamping of the GEO 600 DAQ

system contains a systematic offset of 15µs and does not differ from this in normal

operation by much more than around 60 ns. The experiments revealed a failure in the

ICS110b driver code that has now been corrected for such that the accuracy of the timing

stamping is no longer affected by software reboots of the system. Deviations from the

nominal timing accuracy of up to a few microseconds, lasting for around 30 minutes,

2.5 System diagnostics and alarm system 69

can be observed after a hardware reboot of the system. A more condensed presentation

of the timing tests of the GEO 600 DAQ system are presented in [DAQtiming].

2.5 System diagnostics and alarm system

Since it is absolutely essential that the DAQ system remains active for the entire time

that the detector is running in normal operation, it is useful to have an independent

system that checks the DAQ system to see that it is functioning correctly and reports

to an operator if it is not. To do this, a separate software application was written to

monitor particular recorded channels to make sure that the recorded data conforms to

certain criteria. This application reads the GEO raw data files a few seconds after they

are written and performs a user defined set of checks. The reporting to the operator(s)

is done via a file relay system. This DAQ checking software continually updates the

time stamp and contents of some remote files (on the machine where the final frame files

are written) which are then checked by another application which sends SMS (short-

message-system) messages to a set of chosen operators in case of an error. An error

condition is raised if either the time stamp of the remote files does not update (perhaps

because of a network failure), or if the contents of the remote files (ultimately an error

message) is not the default content.

This system can be, and is, used to monitor many channels of the interferometer. For

channels such as lock indicators it is possible to monitor the state of the interferometer.

Using a ramp signal like the one discussed in section2.4, the software also allows the

timing accuracy of the DAQ system to be constantly monitored.

The DAQ checking application is configured by a configuration file that can be changed

at any time without restarting the application. This allows for new checks to be initiated

during long data taking runs without stopping the existing checks. AppendixB.1.4shows

the README for the application.

An early version of this diagnostic alarm system was used during the S1 science run

with effect that on one or two occasions, alarms were raised which allowed operators to

2.6 Environmental signal calibration and de-whitening 70

attend to the problems in good time such that the duty cycle of the DAQ system was over

98%. Figure1.2shows the performance of the DAQ system (and the whole detector) for

the 18 days of the science run.

The problems caught by this monitor were previously unseen events. In one instance,

all channels of the DAQ system became contaminated with 50 Hz noise. The monitor

program was calculating the RMS of the injected calibration signal and checking that it

was between quite tight limits. When the 50 Hz contamination was introduced, the RMS

increased beyond the specified limit and an alarm was raised. The cause of this error

state is as yet unknown but normal operation can be resumed by cycling the power to the

ADC board. The monitor system can catch this problem quickly with the result that the

duty cycle of the system is only marginally affected. The other error state that occurred

was a floating point exception on one of the DCUs. Again, the source of the problem is

unknown but because of the monitor system, the DAQ system was quickly restarted.

2.6 Environmental signal calibration and de-whitening

As has already been mentioned, gravitational wave detectors will, by design, record a

large number of environmental and system signals. These signals will play a central role

in characterising the behaviour of the detector as well as being used for the generation

of vetoes in astrophysical searches. In order to draw physically meaningful information

from any of the recorded signals, it is important that the signal is understood in terms of

its physical properties,i.e., what it actually measures. In calibrating the recorded signals,

we attach physical meaning to what was before, just a recorded voltage.

The calibration of any one signal is done using information gathered by direct measure-

ment or by well understood models of a particular system. In some cases, the calibration

information is supplied by the manufacturer of the instrument producing the signal; in

other cases, the signal will need to be characterised by measurements or modelling. In

either case, this calibration information needs to be stored in an accessible manner so

that data can be correctly interpreted in any subsequent analysis.

2.6 Environmental signal calibration and de-whitening 71

This section describes in detail the scheme developed to store the calibration data in

such a way that it remains tied to the signal data it is meant to calibrate. In order to reach

the desired goal of having calibration information directly to hand when one analyses

a particular signal, we must consider two questions: ‘How do we input the calibration

information in a user-friendly manner?’ and ‘How do we store this information with the

signal data streams?’ The answer to each of these questions will be the subject of this

section.

2.6.1 User input system

As the DAQ system configuration and state is already stored in a database system, it was

a natural extension of the database to include the calibration information for each signal.

This is particularly useful as it allows the calibration information to evolve to follow

changes in the signal description,e.g., the fitting of a new sensor or preamplifier. In this

way, the history of the calibration information can be tracked so that data covering long

time scales (perhaps years) can be correctly interpreted.

What information do we store?

It was decided that the main process of calibrating a signal be split into two separate

steps: de-whitening of the signal, and calibration to physical units. This is a very natural

distinction to make since the whitening filter electronics are often developed separately

from the sensor itself. We call these two calibration functions ade-whiteningfunction

and asystem calibrationfunction respectively. Before either of these steps are performed

however, the recorded signal must be converted from ADC counts to a Voltage as it

appeared on the input cable to the DAQ system. This conversion is taken as read in the

following discussion although it is included in the details about storing the calibration

information.

Another segmentation of the information arises naturally in the frequency domain. Here

we take the view that a particular de-whitening function or system calibration function

2.6 Environmental signal calibration and de-whitening 72

may be split into multiple sub-functions, each covering a distinct but non-overlapping

range of frequencies. A direct consequence of this system is that the full calibration of a

signal in a particular range of frequencies requires knowledge of both the de-whitening

function and the system calibration function for at least that range of frequencies.

Users of the system are then able to enter both types of calibration function using a web-

based interface to the database. Both types of calibration function are described by a set

of poles and zeros and a gain such that the transfer function of the calibration function

is given by:

T(s) = k(s − z1)(s − z2)...

(s − p1)(s − p2)...(2.11)

wheres = i ω, z1...zN are the positions of the zeros,p1...pM are the positions of the

poles, andk is the gain.

Each entry is then characterised by the following flags:

Function type: A de-whitening function or a system calibration function.

Start Frequency: The lower frequency of the valid range of the function.

Stop Frequency: The upper frequency of the valid range of the function.

Units: The units of the output of the function per Volt,e.g., m/V.

Pole/Zero Units: The units the zeros and poles will be entered in (radian or Hertz).

Number of poles: The number,M , of poles in the function.

Number of zeros: The number,N, of zeros in the function.

A list of poles: A pole position for each of theM poles.

A list of zeros: A zero position for each of theN poles.

A gain: The gain,k, of the function.

Offset: Any inherent offset in the system.

2.6 Environmental signal calibration and de-whitening 73

The output of the function is then given by:

aout(t) = FT Ain(ω).T(ω) + offset, (2.12)

whereaout(t) is the calibrated/de-whitened signal,FT represents a Fourier transform

operation,Ain(ω) is the recorded signal,T(ω) is the calibration/de-whitening function

evaluated fors = i ω, and offset is any DC offset inherent to the sensor/system.

2.6.2 Calibration information storage

Having entered calibration functions for the signals, we now have to ensure that this in-

formation is carried with the recorded signals so that independent analysts can correctly

interpret the signals without asking experimentalists at the site and without interrogating

the database. To achieve this it was decided that the information necessary to calibrate a

signal would be stored, with the signal data, in the frame files14. In addition, this means

that the process of calibration can be, in some cases, automated. (Care should be taken

when automating any calibration process since numerical problems can easily arise.)

Since no predefined storage capacity for comprehensive calibration information exists

within the frame file format, the various storage structures that do exist were used.

The conversion factor from ADC counts to Volts is stored for each channel in the frame

files in a field calledslope in the channel description structure. The slope is computed

in the following way:

slope =inputrange

2sample res out × 10gain /20, (2.13)

where the variables stated refer to those described in table2.6.

For the other calibration information, we decided to use a vector structure within the

frame files. When building frame files one can specify additional data vectors that can be

attached to individual channels. This is precisely the format that was required for storing

the calibration information. One drawback of this method is that the vector structures14Reminder: frame files are the final data storage format of the data. See [FrameLib] and [FrameSpec]

for details.

2.6 Environmental signal calibration and de-whitening 74

referred to here are static for the duration of the frame file. Since one frame file in the

GEO format spans 60 seconds of time, this means that when calibration information

changes for a particular channel, there will arise a period of uncertainty of up to one

minute. Since the calibration information rarely changes, this is an acceptable penalty

for the convenience of having the calibration information stored with the data.

In order to use these vector structures for this purpose, we must first manipulate the

calibration information into a suitable vector format. We use two vectors to do this: one

vector that contains a string description of the calibration information, and one vector

that contains the numerical data pertaining to the functions in the string description.

These two vectors are referred to as thecomment andcode vectors respectively.

The comment vector is formulated as follows:

comment = “mSysTF#nWhtTF#user sys1#datesys1#unit in sys1#

unit out sys1#......#unit in sysM#datesysM#unit in sysM#

unit out sysM#user wht1#datewht1#unit in wht1#

unit out wht1#...#unit in whtN#datewhtN#unit in whtN#

unit out whtN#”,

wheremSysTFspecifies the number of system calibration functions,nWhtTFspecifies

the number of de-whitening functions,user sysm specifies which user entered this func-

tion, datesysm specifies the date this function was entered,unit in sysm specifies the

units of the input to this function, andunit out sysm specifies the units of the output of

this function.

The associateddata vector is formulated as:

data = [nSys, nWht, 〈sys1〉, ..., 〈sysM〉, 〈wht1〉, ..., 〈whtN〉],

wherenSysis the number of system response functions,nWhtis the number of whitening

filter response functions, and each sub-vector,〈sysm〉 and〈whtn〉, has the form:

〈sysn〉 = [G, offset, numpoles, numzeros, start f , stop f , <(pole1), =(pole1), ...,

<(poleN), =(poleN), <(zero1), =(zero1), ...,<(zeroM), =(zeroM)].

2.7 Summary 75

2.6.3 An example calibration procedure

An example of the calibration of a particular channel will serve to clarify the above

definitions.

Let us look at the channel that contains the feedback signal to the length control actuator

of the second mode-cleaner from the power-recycling locking servo (named G1:LSCMIC FP-

MMC2B). If one looks at the calibration vectors contained within the frame files for this

signal, one sees the following:

comment = “001#001

#hrg#2002-01-17 12:17:15#V#Hertz

#hrg#2002-08-26 11:19:10#V#V#”,

and for the data vector:

data = [1, 1,

1.428× 1011, 0, 2, 0, 0, 16384, −0.134, 3.8744, −0.134, −3.8744,

3.882× 10−9, 0, 0, 2, 0, 8192, −20230, 0, −20230, 0].

For this channel we can see that there is one de-whitening calibration function and one

system calibration function defined, both of which were entered by the userhrg, the first

on 2002-01-17 at 12:17:15, the second on 2002-08-26 at 11:19:10. The de-whitening

function has no poles and two zeros and converts Volts to Volts, while the system cali-

bration function has two poles and no zeros and converts Volts to Hertz. The magnitude

and phase responses of these two calibration functions can be plotted from this data and

are shown in figures2.28and2.29.

2.7 Summary

The data acquisition system as described in this chapter has proved to be sufficiently

reliable, with typical duty cycles during data taking periods of 99% and higher. Sustain-

able data rates of around 1 Mb/s are achievable with the system, the largest bottle neck

2.7 Summary 76

100

Frequency (Hz)

−45

−90

101−180

−135

Phas

e(

)

140

160

180

200

220

240

Mag

nit

ude

(dB

)

0

Figure 2.28: A Bode plot of the system

calibration function for the feedback sig-

nal to the second Mode-cleaner from the

power-recycling locking servo. See the

text for details. The transfer is from Volts

to Hertz.

104103

40

30

20

10

180

135

90

45

0

0

Mag

nit

ude

(dB

)P

has

e(

)

Frequency (Hz)

Figure 2.29: A Bode plot of the de-

whitening function for the feedback sig-

nal to the second Mode-cleaner from the

power-recycling locking servo. See the

text for details. The transfer is from Volts

to Volts.

being the Ethernet radio link between the GEO site and the frame making computer in

Hannover (limited to about 30 Mbit/s). The GEO DAQ system is summarised in [DAQ].

Although a number of unexplained error states still exist within the system, thedaqchk

monitoring software allows quick recovery from any occurrences of these errors. Be-

cause of these errors, in the long term, the system could become the limiting factor in

the duty cycle of the detector. As the reliability of the detector grows, so must that of the

data acquisition system. If the problems that have so far been identified cannot be elimi-

nated, then it may become necessary to move to a new system if one becomes available,

or alternatively, develop a new system.

Chapter 3

Calibrating the power-recycled GEO

During the summer of 2002, GEO took part in the first world-wide coincident science

run (S1) of the LIGO Scientific Collaboration (LSC) for large-scale interferometric grav-

itational wave detectors. GEO, along with the three LIGO detectors in the USA and, for

part of the time, with the TAMA detector in Japan, ran continuously inscience mode1

for eighteen days and collected data from many environmental and system channels as

well as the main detector output signals.

For this science run, GEO was configured as a power-recycled Michelson interferometer

with full autoalignment [GEOAA]. Prior to the run, a calibration scheme was developed

with the aim of producing an on-line calibrated strain channel for the detector. The

calibration scheme was implemented and ran for the whole of the science run, providing

an excellent opportunity to study the stability and performance of the scheme over an

extended period. Extensive study of the calibrated detector output after the science run

highlighted a few areas where the scheme could be improved.

This chapter presents the design and implementation of the calibration scheme for the

power-recycled GEO that was used during S1. Details of the performance of the scheme

1The term science mode, at least from the point of view of GEO, refers to the running operation of the

detector. In a science run, as opposed to an engineering run, no unscheduled maintenance is permitted and

the detector is operated remotely as much as possible. By minimising external disturbances, we ensure

the highest possible data quality and the best chance of extracting meaningful science results.

3.1 Principles of calibrating the power-recycled GEO 78

as applied to the S1 data set are also presented along with a summary of the calibration

accuracy achieved. A summary of the method is given at the end of this chapter in

section3.9.

3.1 Principles of calibrating the power-recycled GEO

The power-recycled configuration of GEO consists of a core Michelson interferometer

with an additional mirror—the power-recycling (PR) mirror—placed at the input port

of the Michelson. Figure3.1 shows the optical layout of the core instrument of the

power-recycled interferometer, complete with the names of the individual mirrors.

optical layoutPower-recycled GEO

east arm

north arm

(MFn)

(MFe)

(MCn)

input port

(MCe)

beam splitter

(BS)

power recycling

mirror

(MPR)

high-power

output photodiode

output photodiode

quadrant

Figure 3.1:The optical layout of the core instrument of the power-recycled

GEO. The internal naming convention of the individual mirrors is shown.

Although in reality the arms of GEO are folded in a vertical plane, they are

shown here folded in a horizontal plane for simplicity.

In the power-recycled configuration, the transfer function from displacement ofMCe or

3.1 Principles of calibrating the power-recycled GEO 79

MCn (or both) to the demodulated detector output signal is, in the absence of any control

servos, flat in frequency up to the first null corresponding to the inverse light travel time

of the arm (∼ 125 kHz). Since this is well above the design detection band (50 Hz to

∼ few kHz), we assume it to be flat for the remainder of the discussion. In principle then,

to calibrate the detector output, we need only determine the gain factor that converts the

volts recorded at the detector output to mirror displacement (this gain is referred to as

the optical or detector gain throughout this text). This can be done by inducing a known

mirror displacement and then observing the signal in the detector output.

The Michelson interferometer is held at its operating point (on a dark fringe) by a con-

trol servo (see section3.2 for details), which introduces a frequency dependence into

the transfer function from mirror displacement to detector output signal for frequencies

below the unity gain point of the control servo. This frequency dependence must be de-

termined and corrected for in order to calibrate the detector output down to the lower cut

off frequency of the detection band. Having determined the frequency dependent optical

gain at all frequencies, we can convert the recorded output signal into a displacement

signal; this signal represents the differential arm-length changes of the detector. As we

have seen from section1.2, this differential displacement or length change, 21L(t), can

then be converted to a strain signal,h(t), by

h(t) = 21L(t)

L, (3.1)

whereL is 1200 m for GEO.

Another important aspect of the power-recycled configuration is that the readout scheme

for GEO is such that the output signal can be confined to a single quadrature of the

demodulated output signal. This is in contrast to the Dual-recycled interferometer con-

figuration (see section4), where the signal will be split between the two demodulated

quadratures of the output photodiode signal, which may then need to be combined to

produce a single calibrated detector output that has an optimal signal-to-noise ratio. De-

tails aside, this means that for the case of the power-recycled interferometer, we need

only calibrate one recorded signal.

3.2 The Michelson locking scheme 80

beamsplitter

−1−1

HT

HTmixer

Light fields from

demodulationsignal

photodiode

output

quadrant

Intermediate

MassUpper

Mass

To farmirror

electronicsFrom ESD

electronicsFrom IMD

tank

MCe

stack

TestMass

2 stacks omitted for clarity

ESDIMD

to MCn

(MFe)

Figure 3.2:A schematic of the split-feedback Michelson servo. The suspen-

sion setup of MCe is shown to indicate where the two feedback signals are

applied in the pendulum chain; the suspension of MCn is nominally identi-

cal.

3.2 The Michelson locking scheme

GEO relies on a complex control system that uses many servo loops to keep the detector

at its nominal operating point. The full details of the control system are comprehensively

covered in [Freise03] and [Grote03]; only the parts important to the calibration process

are discussed here, namely, the Michelson differential length-control servo.

3.2.1 The Michelson control servo

The Michelson control servo senses, and minimises, deviations from the optimal dark

fringe operating point. In order to keep GEO at its operating point, the servo must com-

pensate for disturbances from DC to 50 Hz or more. This range of frequencies represents

a large range in amplitude of mirror displacement (upwards of 60 dB) that must be con-

trolled. In order to obtain the necessary noise performance and dynamic range over this

bandwidth, the servo is designed to use a split feedback system. A simplified schematic

of this split feedback system is shown in figure3.2along with a schematic of the pendu-

3.2 The Michelson locking scheme 81

IMD feedback

ESD feedback

Error-point

enbw:

navs:nfft:

fs:

ndata: 1000004000

2520000.75

107

105

104

103

102

101

100

101 102 103

Am

plitu

de

(arb

.unit

s)

Frequency (Hz)

106

effective DAQ noise

Figure 3.3:Simulated output signals from

a low-noise PRMI locking servo. This

represents a detector with high sensitivity.

An arbitrary effective DAQ noise level is

shown to compare with figure3.4. Three

sinusoidal signals, injected at the point

where gravitational wave signals would

enter the loop, are shown in order to give

a convenient measure of the sensitivity of

the detector. The parameters for the servo

loop are nominally the same as those of

the MI loop used in the S1 science run.

IMD feedback

ESD feedback

Error-point

enbw:

navs:nfft:

fs:

ndata: 1000004000

2520000.75

106

105

104

103

102

101

101 102 103

Frequency (Hz)

107

Am

plitu

de

(arb

.unit

s)

100

effective DAQ noise

Figure 3.4:Simulated output signals from

a high-noise PRMI locking servo. This

represents a detector with low sensitiv-

ity. The parameters are the same as those

used to generate figure3.3. The noise

level is increased at various points in the

model to simulate a detector with lower

sensitivity than that in figure3.3.

lum and actuator system that is used to control the length of the Michelson. BothMCe

andMCn pendulums are the same design; the servo controls the differential length of

the Michelson by feeding back the same signal to these two suspensions with opposite

signs. The unity gain frequency of the servo is nominally around 100 Hz.

Low frequency feedback signals (< 10 Hz) are applied to the intermediate mass drive

(IMD) using a coil and magnet actuator while high frequency signals (> 10 Hz) are

applied directly to the test mass using an electro-static drive (ESD). In both cases, forces

are applied from a second, reaction pendulum chain suspended directly behind the main

suspension chain.

Due to the presence of violin mode resonances of the main pendulum suspensions, there

is a limit to the achievable gain and bandwidth of the feedback signals that can be ap-

3.2 The Michelson locking scheme 82

plied with the Michelson servo. Because of this, we cannot extract signal information for

the whole detection band from the feedback signal and must instead use the error-point

signal of the servo to derive the calibrated output signal. During the S1 science run,

because the detector noise was still far above the target noise levels, it was possible to

extract a calibrated strain signal from the error-point signal alone without degrading the

signal-to-noise ratio of any gravitational wave signals that may have been present. In the

future, when the detector is operating at its target sensitivity, it may be no longer possi-

ble to extract the gravitational wave signal for the full band from the error-point alone;

instead we will need to use both the feedback and error-point signals to recover the

gravitational wave information across the full detection band with good signal-to-noise

ratio. This requirement comes from the fact that within the bandwidth of the Michel-

son servo, the error-point signal is kept, by design, as small as possible. If within this

bandwidth the signal is not sufficiently large compared to any noise in the subsequent

measurement electronics (DAQ system), then the gravitational wave information will

be degraded. Within the bandwidth of the servo any gravitational wave information is

contained within the feedback signal with a good signal-to-noise, and is huge compared

to any subsequent measurement noise. Thus it can be recorded and used to extract the

gravitational wave information with good fidelity. Outside the bandwidth of the Michel-

son servo, gravitational wave signals are not suppressed by the control loop and can be

made suitably large during the readout process to ensure good recording of the signal.

Figures3.3and3.4show simulated feedback and error-point signals for low- and high-

sensitivity detectors generated using a model of the Michelson locking servo; the model

(described in more detail in section3.6on page95) can be used to demonstrate the effect

discussed above. Three sinusoidal disturbances are injected at the point in the model

where a gravitational wave signal would enter (labelledexternal displacementin figure

3.12). These peaks serve as a measure of the sensitivity of the detector. Amplitudes

are arbitrary and equal for the three signals. The change in sensitivity is modelled by

changing the noise level at various points in the model. Noise is added in the model in

the two feedback electronic subsystems, and in the detector (optical) gain subsystem (to

simulate laser noise). All noise sources are broadband white noise. The effective DAQ

noise is fixed at an arbitrary level so that it contributes only a very small amount of noise

3.3 Inducing a known mirror displacement 83

to the signal from the low-sensitivity detector. From this we can see that all potential

gravitational wave signals can be recovered with optimal signal-to-noise ratio from the

(high-noise) error-point signal. In the low-noise detector however, a better signal-to-

noise ratio can be achieved at lower frequency if the signal is measured in the feedback

instead of the error-point.

An extra complication arises from the Michelson servo because the error signal for the

servo was derived from the same output photodiode that was used to derive alignment

error signals. The photodiode is in fact a quadrant diode, the sum of the quadrants being

used to derive the error signal for the longitudinal Michelson lock, while the individual

quadrants are used with a differential wavefront sensing method to determine any mirror

misalignments in the system. The quadrant diode saturates with the high light levels

present before, and during, lock acquisition, and so the light leaving the detector must

be attenuated before it impinges on the quadrant diode. This has the obvious effect of

considerably reducing the sensitivity of the detector since the detected signal-to-noise

ratio is reduced due to the presence of shot noise. To avoid this, a second, high-power,

photodiode was used in S1 to sense the output light, thus lowering the shot-noise in the

detected signal. This high-power photodiode is not part of the Michelson loop and has

different gain to the low-power quadrant diode. Since it is this high-power photodiode

signal that we want to use to derive the calibrated strain channel, we must also determine,

and correct for, the relative gain between the two photodiodes.

3.3 Inducing a known mirror displacement

The process of calibrating GEO centres on the following basic premise: introduce a

differential arm length change of known amplitude into the Michelson and observe the

signal in the detector output. In the case of the power-recycled Michelson it is in prin-

ciple only necessary to do this at one frequency since the frequency dependence of the

system is known. However, determining the calibration at more frequencies serves to

increase confidence and minimise the error in the process. Since the bandwidth of the

Michelson servo only covers a small part of the detection band, it is possible to do the

3.3 Inducing a known mirror displacement 84

plot a plot c

plot b plot d

Frequency (Hz)Frequency (Hz)

102

Frequency (Hz)

0.8Hz response

5Hz response

102100

Phas

e(

)A

mplitu

de

Phas

ediff

eren

ce(

)A

mplitu

de

rati

o

10 5

10 0

10−5

10−10

0

−50

−100

−150

−200100 102 102 103

103

1.01

1.005

1

0.995

0.99

2.5

2

1.5

1

0.5

0

Frequency (Hz)

Figure 3.5:Plot (a) shows the magnitude response of the ESD with the design

0.8 Hz pole and the ESD model with a complex pole at 5 Hz scaled to match

at higher frequencies. In both cases the Q of the complex pole was chosen to

be 2. Plot (b) shows the phase response of the two systems. Plot (c) shows

the ratio of the magnitude responses over the lower part of the detection

band (50 Hz to 1 kHz) while plot (d) shows the phase difference of the two

systems.

calibration at frequencies outside the servo bandwidth,i.e., at high frequencies (above

say 200 Hz). The electro-static actuators (ESDs) of GEO provide a convenient way to

induce a known displacement. How well we know what displacement is induced by

these actuators is then dependent on their calibration.

3.3.1 The electro-static drives (ESDs)

The electro-static drives in GEO are used as the main high-frequency feedback actuators

in the Michelson length control loop. The drives use electrodes deposited on the surface

of a reaction mass to apply forces to the test mass via electric fields. The dielectric

3.3 Inducing a known mirror displacement 85

properties of the test mass cause it to be pulled into the electro-static field generated

between the electrodes. Since the ESDs can only pull on the test mass, they are biased

with a DC force so that bipolar displacements can be achieved.

The strength of each ESD depends quite strongly2 on the distance between the electrodes

and the test mass. The induced displacement of the test mass can be modelled as a simple

pendulum system, such that for frequencies below the pendulum resonance, the induced

displacement is proportional to the force applied, and for frequencies well above the

pendulum resonance, the induced displacement is inversely proportional to the square of

the frequency. The calibration of the ESDs was done at two frequencies using two in-

dependent methods: at DC using a fringe counting technique, and at a higher frequency

against the master laser piezo actuator. The 1/ f 2 model was then assumed. In all of the

calibration scheme, the response of the pendulums (and thus the ESDs) was modelled

as a 5 Hz resonance with a quality factor of 2 and with the DC gain set so that the re-

sponse of the system matched the ‘real’, expected, pendulum responses (around 0.8 Hz3)

above 10 Hz. This was done to simplify the design of stable Infinite Impulse Response

(IIR) filters used to do the calibration. Above 50 Hz (the lower cut off frequency of the

detection band), the deviation of the model response from the ‘real’ response is below

1%. Figure3.5 show the modelled ESD response against the (design) ‘real’ pendulum

response. The differences between the two responses are also shown.

If we want to use the ESDs as the basis for the calibration of the S1 data, we need to be

confident that the force applied does not vary over time. This confidence was gained by

repeatedly calibrating the drives before and after the science run, where the calibration

factor for the two ESDs was observed to remain effectively constant for the entire S1

data taking period. The calibration methods for the electro-static drives is discussed in

detail in section3.8.3. Figure3.6 shows the calibrated response of both electro-static

drives assuming the 0.8 Hz resonance and using the measured DC calibration factors.

Full details of the electro-static drive design and implementation are given in [Grote03].

2The dependence of the force,F , on the separation,d, of the test and reaction masses was shown by

Strain [Strain02] to be of the formF ∝ d1.5.3Design models of the pendulum show the main longitudinal pendulum mode to be around 0.8 Hz.

3.3 Inducing a known mirror displacement 86

10−8

10−10

10−12

10−14

10−16

10 210 110 0 10 3

Frequency (Hz)

Am

plitu

de

(m/V

)

MCe ESD (100 nm/V @ DC)

MCn ESD (98 nm/V @ DC)

Figure 3.6:The calibrated responses of the two electro-static drives around

the time of the S1 run. A 0.8 Hz longitudinal pendulum resonance is as-

sumed.

3.3.2 The injected calibration signal for S1

The purpose of the injection signal is to produce a small number of calibration peaks

(3 or more), that will appear in the recorded detector error-point with a sufficiently high

signal-to-noise ratio (at least 20 or above measured in one second of data to get calibra-

tion accuracy better than 5%). The frequencies at which the calibration peaks should be

injected are only constrained by the unity gain point of the Michelson servo (we want cal-

ibration lines above it) and by the presence of any other lines in the error-point spectrum.

We also want the calibration lines to be stable in frequency, for example, by locking the

oscillating source to a GPS frequency standard; doing this confines the spectral peaks to

particular bins for the duration of any subsequent analysis.

For the S1 run, these requirements were most conveniently met by using the square-wave

output of an HP33120A signal generator locked to a 10 MHz GPS frequency standard.

This square-wave signal is filtered through two analogue differentiator stages to produces

3.3 Inducing a known mirror displacement 87

calibration peaks

ndata:

nfft:

navs:

fs:

enbw:

1638400

16384

100

16384

1.50

plot a

10 4

10 4

10 410 3

10 3

10 3

Frequency (Hz)

10 0

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10 2

Am

plit

ude

(VR

MS)

10−3

10−4

10−5

10−6

10−7

10 2

Am

plit

ude

(VR

MS)

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10 2

Am

plit

ude

(VR

MS)

plot b

plot c

Figure 3.7:Snap-shot spectra of the injected calibration signal (plot a), the

quadrant photodiode error-point signal (plot b), and the error-point signal

from the high-power photodiode (plot c). All signals show the calibration

lines. The signals are shown as they were recorded,i.e., not de-whitened.

3.4 Measuring the optical gain of the detector 88

a forest of spectral peaks at the odd harmonics which rise in amplitude proportional to

f ; the high frequency content of the signal was suppressed with a low-pass filter so as to

avoid saturating the electro-static drives. The fundamental frequency was set to 244 Hz

so that the first 3 odd harmonics did not overlap with any other spectral features. This

fundamental frequency was deemed high enough to avoid any significant suppression by

the loop gain of the Michelson servo. As we shall see later in this text (page112), this

turned out not to be the case and so only the higher harmonics were used in the cali-

bration process. The fundamental and first 3 harmonics appear in the high-power error-

point (after being filtered by the response of the ESD) with an approximately uniform

signal-to-noise ratios. Plot (a) of figure3.7shows a spectral snapshot of the calibration

signal. The calibration lines can also be seen in snap-shots of the quadrant (plot b) and

high-power photodiode (plot c) error-point signals.

3.4 Measuring the optical gain of the detector

Having successfully injected a series of calibration peaks, we can proceed to observing

these in the detector output and using them to determine the optical gain of the detector.

For each calibration peak we can calculate the optical gain,Goptf , by:

Goptf [V/m] =|Af|

|af||εf|

[V]

[V][m /V], (3.2)

where |A f | is the amplitude of a calibration peak measured in the high-power error-

point , |a f | is the amplitude of the calibration peak as applied to the electro-static drive,

and |ε f | is the amplitude response of the electro-static drive. The subscriptf denotes

evaluation at a particular frequency,i.e., at a particular calibration peak. Typical values

for these quantities during S1 are shown in table3.1.

Since we make an estimation of the optical gain for each calibration peak, we can use

this information to generate a more reliable estimation of the optical gain by combining

the individual measurements based on their signal-to-noise ratios.

We can consider each calibration line separately in the time domain as the sum of a

sinusoidal signal of peak amplitudeA and frequencyf with some Gaussian random

3.4 Measuring the optical gain of the detector 89

Peak f (Hz) |a f | (V) |A f | (V) |ε f | (m/V) Gopt (V/m)

244 0.092 3.72× 10−2 0.19× 10−11 1.5 × 1010

732 0.216 0.92× 10−2 0.02× 10−11 1.3 × 1010

1220 0.235 0.36× 10−2 0.0077× 10−11 1.2 × 1010

1708 0.176 0.16× 10−2 0.0039× 10−11 1.35× 1010

Table 3.1:The values shown are typical of the values measured during S1.

The peaks are measured in the calibration signal and the high-power in-

phase error-point signal respectively. The optical gain values are calculated

from equation3.2. Before calculating the optical gain, the values tabulated

in the other columns must be corrected in order to remove any effects of

the recording process. Details of the correction procedures are discussed in

section3.5and are summarised in table3.3and figure3.11.

noise of varianceσ 2n such that the resulting time series,v(t), is given by

v(t) = Asin(2π f t) + n(t). (3.3)

We then make a measurement,P [Vrms2], of a particular calibration peak in a power

spectrum of the error-point signal. This measurement is made up partly from the under-

lying signal power,A2/2 [Vrms2], and partly from the noise power,η [Vrms2], in the

measurement bin. We can relate the mean value ofη (taken by averaging spectral bins

that contain only contributions from the noise source) to the underlying noise source by

〈η〉 =ENBWσ 2

n

fs/2

[Vrms2

], (3.4)

where ENBW is the equivalent noise bandwidth of the power spectrum, andfs is the

sampling frequency of the data. On average, the measured peak is then given by

〈P〉 =A2

2+ 〈η〉. (3.5)

And so an estimate,A2, of the underlying peak signal power can be obtained from one

power spectrum by,

A2= 2(P − 〈η〉). (3.6)

with a variance,σ 2A, that can be estimated by the signal-to-noise ratio of the peak being

measured.

3.5 Recording the signals 90

From the considerations above, we can see that the amplitude,A f of each calibration

peak can be estimated as

A f =

√2(Pf − 〈η〉) ± σA. (3.7)

We can now make independent estimates of the optical gain as shown in equation3.2

and associate a variance,σ 2Goptf

, with each estimate by

σ 2Gopt f

= G2optf

[σ 2

A

|A f |2

+

σ 2ε f

|ε f |2

], (3.8)

whereσε f represents the error in the calibration of the electro-static drive response. Here

we assume negligible error in the measurement of the input calibration peak amplitudes.

We can now weight these individual measurements together to get an improved estimate

of the optical gain as follows:

Gopt =

∑f

Goptf

σ 2Gopt f∑

f

1

σ 2Gopt f

, (3.9)

which has an error given by

σ 2Gopt

=1∑

f

1

σ 2Gopt f

. (3.10)

The optical gain was measured once per second throughout the S1 science run using

the method described above. In addition to this weighted average, the optical gain mea-

surements were smoothed using a Hanning window of length 60 seconds. This was

done to reduce the effect of large transients. In doing this, we concentrate on calibrating

only slowly changing fluctuations in the optical gain. Results of the measurements are

presented in section3.8.

3.5 Recording the signals

Since various signals are needed to recover a calibrated strain channel, it is important to

look in detail at the recording of each of these channels. Table3.2 shows a list of the

3.5 Recording the signals 91

Channel Description ADC Pre-amp Whitening/Attenuation

name Channel Gain (dB)

EP-P HP The error-point signal

demodulated in-phase

at the high-power

photodiode.

23 0 whitening filter hpwp

(figure3.8)

EP-Q HP The error-point signal

demodulated out-of-

phase at the high-power

photodiode.

24 0 whitening filter hpwq

(figure3.8)

EP-P The error-point signal

demodulated in phase

at the quadrant photodi-

ode.

4 -8 whitening filter qwep

(figure 3.8), -11.5 dB ,

×2 diff. transmitter

CAL The calibration signal

applied differentially to

the electro-static drives.

2 -8 -12.7 dB, ×2 differen-

tial transmitter

Table 3.2:The channels used in the calibration process along with details of

the recording process. These details need to be accounted for when doing the

calibration. All channels were recorded at 16384 Hz with 16-bit sampling

resolution. All channel names have the prefixG1:LSC MID which was

omitted here for clarity.

channels used in the calibration process. The details of the recording of each signal are

shown, in particular, which input channel of the ICS ADC board was used, what gain, if

any, was applied, and the details of any whitening filters or additional attenuation4.

In the case of the calibration signal and the quadrant photodiode error-point, a factor of

two needs to be included to account for the differential transmitter that sends the signal to

the DAQ system. In the case of the high-power photodiode signals, the differential trans-

4A passive attenuation patch panel is used in the central station at GEO. This serves two purposes: as a

possible 10 dB attenuation for each channel; and as a convenient way to convert signals arriving on single

pole LEMO connectors to a double pole connector needed for the input to DAQ system.

3.5 Recording the signals 92

mission is included in the whitening filter transfer function. In all cases, the whitening

filters were simple transient differentiator filters giving some low frequency suppression

and some high frequency gain, thus maximising the use of the dynamic range of the

DAQ system. Figure3.8 shows Bode plots of the three whitening filters mentioned in

table3.2. Figure3.9shows the de-whitening IIR filters that were designed to correct for

the whitening process.

20

0

−20

−40

−60

−80

200

150

100

50

010 1 10 2 10 3

10 0 10 1 10 2 10 3

10 0

Mag

nit

ude

(dB

)P

has

e(

)

Frequency (Hz)

high power in-phase

high power out-of-phase

quadrant in-phase

Figure 3.8:The whitening filters applied

before recording the three detector error-

point signals. Although the out-of-phase

high-power error-point signal was not

used directly in the calibration scheme,

the calibration lines were monitored in it

as a diagnostic.

high power in-phase

high power out-of-phase

quadrant in-phase

10 0 10 1 10 2 10 3

80

60

40

20

0

−20

0

−50

−100

−150

−200

Phas

e(

)

Frequency (Hz)

10 0 10 1 10 2 10 3

Mag

nit

ude

(dB

)

Figure 3.9: The de-whitening filters de-

signed to correct the three detector error-

point signals for the whitening process

applied during recording the signals. The

filters were implemented as IIR filters.

An important point in the calibration procedure is how well we can correct for the

whitening process during the calibration procedure. The de-whitening filters were de-

signed as IIR filters. Infinite Impulse Response filters are defined in theZ-domain

whereas the whitening filters built with analogue electronics are defined in theS-domain.

Since each of these domains cannot be exactly mapped on to the other (see [DSPGuide]

for details), the designed IIR filters can only approximate the equivalentS-domain filter.

Hence, the IIR filters that were designed do not exactly correct for the whitening pro-

cess. The differences between the two filter types are shown in figure3.10. We can see

that a slight mismatch in the filters for the high-power photodiode processing leads to a

3.5 Recording the signals 93

systematic gain difference of about 0.3 dB (3.5%).

high power in-phase

high power out-of-phase

quadrant in-phase

10 0 10 1 10 2 10 3

Frequency (Hz)

0.5

0

−0.5

−0.5

0

0.5

Phas

ediff

eren

ce(

)M

agnit

ude

diff

eren

ce(d

B)

Figure 3.10:A plot of the difference between the whitening and de-whitening

filters used for the three detector error-point signals. The difference is cal-

culated as the complex product of the whitening and de-whitening filter re-

sponses. In principle, if the de-whitening filter could be perfectly designed,

these differences would be zero.

From these details we can immediately tabulate the factors necessary to correct the cali-

bration peak amplitudes shown in figure3.7 for the recording process. Using these cor-

rection factors, the values of the optical gain shown in table3.1 can be calculated from

equation3.2and the other (corrected) tabulated values in table3.1. Table3.3shows the

correction factors for each calibration peak. The pre-amplifier gains shown in table3.2

are included in the process of converting ADC counts to Volts and are thus not included

in these factors.

As can be seen, many factors need to be accounted for during the calibration process. To

make this process clearer, figure3.11shows a ‘map’ of the part of the calibration process

that corrects for the recording of the signals.

3.5 Recording the signals 94

Signal Factor

244 Hz 732 Hz 1220 Hz 1708 Hz

EP-P HP 0.154 0.134 0.132 0.130

EP-Q HP 0.114 0.099 0.098 0.097

EP-P 0.188 0.188 0.188 0.188

CAL 2.17 2.17 2.17 2.17

Table 3.3:Correction factors for each calibration line. These factors correct

the peak heights shown in figure3.7 for the recording process, allowing the

optical gain values shown in table3.1to be calculated by hand.

Detector

DAQ

Sof

twar

eP

roce

sses

Whiteningfilter

Whitening

filter

quad-pd output

signalcalibration

signal generator

−12.7 dB

−11.5 dB

+12.7 dB

×0.5

+11.5 dB

×0.5

dewhitening filter dewhitening filter

RecoveredCalibration

SignalRecovered

high-power-pd output

calibration input

×2 Diff.Trans

×2 Diff.Trans

quad photodiodeerror-point

signal

Recoveredhp-photodiode

error-pointsignal

Figure 3.11:A ‘map’ of the signal recovery required before performing the

calibration process. The processes outlined in red are the software steps

necessary to recover a correct representation of the analogue signals shown

in the rest of the diagram. The correction of the out-of-phase error-point

signal is omitted.

3.6 A model of the Michelson locking servo 95

3.6 A model of the Michelson locking servo

Once the recorded signals have had the recording process corrected for, we can in prin-

ciple calculate the optical gain of the detector and from there calibrate the high-power

photodiode output to a strain signal, at least for frequencies above the unity gain point

of the Michelson servo. For the lower frequencies, we have to correct for the loop gain

of the Michelson servo. This can be done using a model of the Michelson loop.

H1[m] [V] [V][V]

Detector

IMD electronics

[V][V]

Hrel

H2

[V]H4

ESD electronics

[V][V]

[m]

[m]

[V]H3

H5

IMD

ESD

calibrationinput

Relative gain

photodiode

output

Quadrant

‘fast path’

High-power

photodiode

output

displacementexternal

signalse.g., h(t) induceddisplacement ‘slow path’

Φ

Figure 3.12:A loop diagram of the Michelson longitudinal locking servo.

The diagram shows the two feedback paths, each split into an electronics

stage and an actuator stage. Each (blue) box represents a gain stage of the

servo. The signal transfer of each stage is shown by the physical units at the

input and output.

A loop diagram of the Michelson servo is shown in figure3.12. Each (blue) box in

the diagram represents a, possibly frequency dependent, gain stage of the servo. For

example, the box labelleddetectoris flat in frequency and has gain equal to the optical

gain discussed in section3.4, whereas the box labelledESDincludes the 1/ f 2 frequency

response of the pendulum and the calibration factor of the ESD(s). Each element (box) of

figure3.12needs to be modelled in order to create the full model. Section3.4has already

presented details of the measurement ofH1, we will now look at the other elements,

3.6 A model of the Michelson locking servo 96

H2...H5, andHrel.

3.6.1 Relative gain

The two photodiodes (high-power and quadrant) both have a flat frequency response over

the range of frequencies that concerns the calibration. Although in reality the detector

has two outputs, one from each photodiode, it is useful to model the system as a single

output system that is then split into two paths, with one of the paths including a flat gain

stage that represents the relative gain between the two photodiode/readout systems. This

relative gain is labelledHrel in figure3.12. The relative gain arises firstly from the fact

that the two photodiodes have a different light level falling on them, and secondly from

the fact that the two photodiodes are of different construction and so produce different

photo-current for a given light level. Both of these factors lead to an effective gain

between the two diodes that could vary in time and must therefore be tracked.

The relative gain is calculated by comparing the size of the calibration peaks in the

quadrant and high-power photodiodes such that for each calibration peak, we calculate

the dimensionless relative gain as

Hrel f =|Q f |

|A f |

[V]

[V], (3.11)

where|Q f | is the amplitude of the calibration peak at frequencyf as measured in the

quadrant photodiode signal, and|A f | is the magnitude of the calibration peak as mea-

sured in the high-power photodiode signal. Both signals have the recording process cor-

rected for before this gain is calculated. The variance of each of these estimates, which

can be calculated by considering the variances of each peak measurement, is given by

σ 2Hrel f

= H2rel f

σ 2q f

|q f |2

+

σ 2hpf

|hpf |2

. (3.12)

Since the transfer function between the two photodiodes is flat in the frequency band

we are interested in, we can combine the individual estimates of the relative gain for

each calibration peak frequency to get an improved estimate. This is done in much the

same way as was done for the optical gain (section3.4). The weighted average of these

3.6 A model of the Michelson locking servo 97

individual relative gain measurements is then given by

Hrel =

∑f

Hrel f

σ 2Hrel f∑

f

1

σ 2Hrel) f

, (3.13)

which has an error given by

σ 2Hrel

=1∑

f

1

σ 2Grel f

. (3.14)

As with the optical gain, the relative gain was calculated once per second and then

smoothed with a 60 second Hanning window. Results of the measurements are presented

in section3.8.

3.6.2 Feedback paths

Each of the feedback paths (as in reality) is split into two parts in the model: an electronic

part and an actuator. The path that includes theintermediate mass drive(IMD), labelled

as the ‘slow path’ in figure3.12, applies low frequency signals (DC–a few Hz); the path

that includes theelectro-static drive(ESD), labelled as ‘fast path’ in figure3.12, applies

feedback signals up to about 100 Hz. The gain distribution of the two paths is controlled

by two gain controls: an overall gain control common to both paths, and a crossover gain

control in the fast path only.

Feedback electronics

Both gain controls are implemented as the circuit shown in figure3.13with each having

an additional scale factor. These gain controls can be represented mathematically as

G(p) = K330+ 1000p

330+ 10000− 1000p, (3.15)

wherep is the value of the gain control knob ranging from 0 to 10, andK is a scale factor;

K = 0.025105 for the overall gain control andK = 27.7728679 for the crossover gain

control. The response of the two gain knobs is shown in figure3.14. Using equation

3.6 A model of the Michelson locking servo 98

3.15 in the calibration scheme allows us to simply input the actual values on the gain

knobs. This is much more convenient for keeping the calibration scheme in line with the

real detector parameters.

−+

10 kΩ 330Ω

330Ω

Figure 3.13:The simple gain control cir-

cuit used for the two gain controls of the

Michelson servo. The individual scale

factors of the two gain controls are not

included. This gain control gives an ap-

proximately logarithmic gain factor over

most of its range.

870 1 2 3 4 5 6 9 10

870 1 2 3 4 5 6 9 10

Overall gain

Gain Knob Value

10 0

10−1

10−2

10−3

10−4

10 3

10 2

10 1

10 0

Crossover gain

Gai

nG

ain

Figure 3.14:The gain curves of the two

Michelson gain knobs, including the indi-

vidual scale factors of each.

The rest of the servo loop electronics were modelled by fitting a pole-zero model of the

circuits to measured transfer functions. A model for both sets of electronics (fast and

slow path) was developed. Figures3.15and3.16show the measured transfer functions

of the slow and fast electronics respectively. The pole-zero model determined by the

fitting procedure is also plotted for comparison. The slightly poorer fit for the fast path

electronics could be due to a poorer measurement of the electronic transfer function. Due

to time constraints before the start of the S1 run, no further investigation or measurement

was possible; after the run, the electronics were changed to accommodate changes in the

pendulums that suspend the two test masses. In any case, we will see later in this section

that the contribution of the fast path electronics to the final calibrated signal is negligible

at the frequencies where the fit is poor. The determined parameters of the fitting process

are given in table3.4.

3.6 A model of the Michelson locking servo 99

Magnitude data

Phase data

Magnitude fit

Phase fit

−90

180

135

90

45

0

−45

−135

−180100100.1

50

30

20

10

0

Mag

nit

ude

(dB

)60

Frequency (Hz)

Phas

e(

)

40

1 1000

Figure 3.15:The measured transfer func-

tion of the slow path electronics. A pole-

zero fit to the data is also shown. The fit

is clearly excellent showing that the pole-

zero model adequately represents the sys-

tem.

Magnitude data

Phase data

Magnitude fit

Phase fit

−90

180

135

90

45

0

−45

−135

−18010001001010.1

30

50

25

35

45

55

Frequency (Hz)

Phas

e(

)

Mag

nit

ude

(dB

)

40

Figure 3.16:The measured transfer func-

tion of the fast path electronics. A pole-

zero fit to the data is also shown. Al-

though there is some deviation of the

model from the measured data at high fre-

quencies, the low frequency fit is good.

Slow path Fast path

Pole (Hz) Q Zero (Hz) Q Pole (Hz) Q Zero (Hz) Q

1 2.798 0.632 34.5 2.87

2 22.65 5.80 111.9 1.17

3 0.404 5.87 0.0 0.5

4 128.4 49.19 0.469 5.52

5 304.7 4.6 596.1 22.8 305.2 3.1 304 60

6 586.5 2.3 303.9 80 137.2 2

7 1193 1.5

Table 3.4:A list of poles and zeros for the feedback path electronics. The

pole and zeros are determined from a fit to the measured transfer function of

the electronics. The fit was performed using LISO[Heinzel99].

3.6 A model of the Michelson locking servo 100

Complete feedback paths

A model of the electro-static drive response was described in section3.3.1. For the inter-

mediate mass drives, we can adopt the same policy: for modelling, shift the fundamental

pendulum modes to 5 Hz and apply a gain factor so that the response well above 5 Hz

matches the ‘real’ pendulum response with a 0.8 Hz resonance.

The model of the intermediate mass drives is similar to that of the ESDs. For the IMDs,

the frequency response is 1/ f 4 for frequencies above the pendulum resonance. This

response is achieved in the model by using two complex poles at 5 Hz, again with the

gain factor applied to match 0.8 Hz poles above 10 Hz. The DC calibration factor of the

IMDs is then applied. The DC calibration of the IMDs will be discussed in section3.8.3.

Using the model of the electronic transfer functions and of the actuators, we can compute

the overall transfer functions of the two feedback paths from Volts to displacement.

Figure3.17 shows bode plots of the two feedback paths for an overall gain setting of

9.65 (×0.3685) and a crossover gain setting of 5 (×27.7)—the nominal settings around

S1.

SIMULINK model

In order to facilitate rapid prototyping and consistency checking for the many elements

of the Michelson servo model, a [SIMULINK ] model was developed containing all the

main features of the Michelson servo. In the model, the pendulum resonances were set

to 0.8 Hz. All displacement factors are applied in micrometres and all voltages are given

in Volts. Using the model, all transfer functions of the system can be simulated.

At each gain stage of the model, normally distributed white noise is added to allow the

overall noise level of the detector to be changed in a realistic way. The effect of grav-

itational waves is simulated by adding signals in after the feedback actuators (labelled

external displacementin figure3.12).

3.6 A model of the Michelson locking servo 101

‘Slow Path’ [m/V]

‘Fast Path’ [m/V]

10 410 310 210 110 010−1

Frequency (Hz)

−100

−150

−200

−250

−300

−350

−400

−450

−500

180

0

−180

−360

−540

−720

Mag

nit

ude

(dB

)P

has

e(

)

Figure 3.17:Bode plots of the Michelson servo feedback paths from input to

the first (overall) gain stage to displacement produced by each actuator. The

transfer functions are generated from the SIMULINK model of the MI loop.

3.6 A model of the Michelson locking servo 102

3.6.3 Closed-loop transfer function

The closed-loop transfer function of the Michelson servo is what we must correct for in

the calibration process. The open-loop gain of the Michelson servo can be expressed as

an S-domain transfer function by

GOL(s) = H1(s)Hrel(s)[H2(s)H3(s) + H4(s)H5(s)]. (3.16)

From this we can write down the closed-loop transfer function,C(s), as

C(s) =1

1 + GOL(s). (3.17)

The frequency response of the transfer function,T(s), from differential mirror displace-

ment to the detector (error-point) output, in Volts, can be calculated fors = i ω, to be

T( j ω) = H1( j ω)C( j ω). (3.18)

The system is then described by a list of poles and zeros and five gain parameters. The

poles and zeros are simply the poles of the pendulum responses plus the lists given in

table3.4. Table3.5shows the five gain parameters complete with typical values.

Parameter Nominal value Description

H1 15 [V/nm] The optical gain.

H2 9.5 The overall gain.

|H3|DC 200 [µm/V] The DC calibration of the IMD.

H4 5.0 The crossover gain.

|H5|DC 0.18 [µm/V] The DC calibration of the ESD.

Hrel 0.01 The relative gain.

Table 3.5:The gain parameters that make up the variable part of the Michel-

son servo. Typical values for the six parameters are given.

Figure3.18shows the closed-loop transfer function for various optical gain values. The

optical gains values are given in [V/nm]. We can see that the loop gain of the servo is

negligible above about 300 Hz for this (sensible) range of optical gains.

3.7 Real-time calibration 103

The closed-loop transfer function was measured for a particular state of the detector

and equation3.17was fitted by allowing the gain parameters shown in table3.5 to vary

around their nominal values. The result of the fit is shown in figure3.19. We can see

that around 100 Hz the model does not perfectly match the data. This is not understood

but may be related to the deviation of the fast path electronics from the model around

this frequency or it could be related to the quality of the measured data or the presence

of noise. The important thing here is that the model is a good approximation to reality.

10 310 2

Frequency (Hz)

10 0

−50

0

50

100

150

Mag

nit

ude

Phas

e(

)

H1 = 35

H1 = 15H1 = 10

H1 = 20H1 = 25H1 = 30

Figure 3.18: The modelled Michelson

servo closed-loop transfer function for

various optical gain settings. The gain

settings are given in[V/nm] in the key.

The nominal value of the optical gain dur-

ing S1 was around 15V/nm.

Model fit

Measured data

10 2

Frequency (Hz)

10 0

−200

−100

0

100

200

Phas

e(

)M

agnit

ude

Figure 3.19: The measured Michelson

servo closed-loop transfer function. A fit

was made to the data by allowing the gain

parameters of the Michelson loop to vary

around their nominal values. The fit is

also shown.

3.7 Real-time calibration

Having modelled the Michelson closed-loop transfer function as described above, we

can use this model to derive a calibration function that takes the recorded detector error-

point as an input and outputs a calibrated strain signal. When first implementing such

a system, it is important to gain confidence that the process does what is required as

well as what is expected. It is therefore very important that a continuous record of the

calibration process is kept, allowing the process to be reconstructed and validated after

the fact. This can be achieved by archiving various by-products of the process as separate

data channels. These channels can then be included in the main detector data stream and

3.7 Real-time calibration 104

archived in the normal way.

To achieve a continuous and real-time calibration system, we must perform all the cal-

culations necessary to calculate the calibrated strain channel in faster than real-time,

allowing some time to record the strain channel and the other by-products to disk. The

natural unit of time in the process is the second since the GEO raw data files contain one

second of data. In other words, we must complete the calculation and subsequent storage

of the calibrated strain channel—as well as the storage of the additional by-products—in

less than one second. To achieve this, the calibration process was implemented using

IIR filters.

This section details the derivation of the calibration function and looks at the complexi-

ties involved in implementing the system.

3.7.1 Deriving the calibration equation

f ′1

f4

frel

f2 f3

f5

‘fast path correction’

d(t)

v(t)

Relative gain

‘slow path correction’

optical gain correction

Figure 3.20: A system diagram of the time-domain calibration function.

Each blue box represents the application of the filter contained in the box to

the input. The input to the system is the de-whitened version of the recorded

error-point; the output is the derived differential mirror displacement.

We wish to find a function,F , that takes in the de-whitened detector error-point time-

series,v(t), and outputs a calibrated displacement time-series,d(t). Hence, the calibra-

tion equation can be written as

d(t) = Fv(t). (3.19)

3.7 Real-time calibration 105

The calibration function,F , can be written as the application of a time-domain filter,f ,

to the error-point signal,i.e.,

d(t) = f ⊗ v(t), (3.20)

where⊗ denotes filter application. We saw in section3.6.3that the error-point signal,

v(t), can be derived from any differential arm-length changes,d(t), the optical gain,H1,

and the closed-loop transfer function,C(s). If we conceptually create a time-domain

filter, c, that has the same transfer function asC(s), then we can write the error-point

signal as

v(t) = H1(c ⊗ d(t)). (3.21)

From equations3.20and3.21we can immediately see that the calibration filter we seek

must have a frequency response equal to the inverse of the transfer function given by

equation3.18. This can be written in the S-domain as

C′(s) =1 + H1Hrel[H2H3 + H4H5]

H1(3.22)

=1

H1+ Hrel[H2H3 + H4H5]. (3.23)

So, in the frequency domain at least, we can compute the mirror displacement,D(ω),

from the recorded error-point spectrum,V(ω), by

D(ω) = C′( j ω)V(ω) (3.24)

=V(ω)

H1( j ω)+ Hrel( j ω)H2( j ω)H3( j ω)V(ω)

+Hrel( j ω)H4( j ω)H5( j ω)V(ω). (3.25)

This equation can be implemented in the time domain by replacing each S-domain filter,

H∗, by an equivalent time-domain filter,f∗, and by replacing each multiplication by the

application of a filter. In this way, a discrete time-domain calibration function can be

written as

d(n) = f ⊗ v(n) (3.26)

= v(n) ⊗ f ′

1 + frel ⊗ f2 ⊗ f3 ⊗ v(n)

+ frel ⊗ f4 ⊗ f5 ⊗ v(n), (3.27)

wheren represents individual samples and the primed filter,f ′

1, represents a time-domain

filter with a frequency response equal to the inverse frequency response ofH1—in the

3.7 Real-time calibration 106

case of the power-recycled interferometer calibration,f ′

1 is just the reciprocal of the

optical gain. A system diagram for this calibration function is shown in figure3.20.

fast path correction

sum

optical correction

slow path correction

10 2 10 3

Frequency (Hz)

10−4

10−6

10−8

10−10

200

150

100

50

0

−50

−100

−150

−200

Phas

e(

)M

agnit

ude

(µm

/V)

Figure 3.21:Transfer functions of the three terms of the calibration function.

The sum of the three terms (the overall transfer function) is also shown.

Looking at each term in equation3.26, we can see that the majority of the high-frequency

content of the calculatedd(n) comes from the first term,v(n) ⊗ f ′

1, (since the filters

f3 and f5 are strong low-pass filters). The low frequency content is determined by

correcting for the loop-gain of the system. In equation3.26we can see that the last two

terms generate signals that are the same as the feedback paths of the Michelson servo.

Adding these to the first term ‘corrects’ for the suppression of the loop-gain.

Each time-domain filter,f , can be implemented as an IIR filter. The design of each

IIR filter has already been discussed in the previous sections of this chapter. Again,

using a SIMULINK model of the calibration equation, we can examine the individual

3.7 Real-time calibration 107

terms graphically. Figure3.21shows the transfer function from the input (voltage) to

the output of the three terms (displacement). The sum of the three terms is also shown.

Comparing the sum with closed-loop transfer function shown in figure3.19, we can see

that they are just the reciprocal of each other, as expected.

Using a short section of the S1 data, spectra were made of the time-series that correspond

to the three individual terms of calibration function. Figure3.22shows these spectra.

The ‘sum’ spectrum represents the differential arm length change. This can be converted

to strain directly using equation3.1.

sum

fast path correction

slow path correction

optical correction

10 2 10 3

Frequency (Hz)

10−6

10−7

10−9

10−10

10−11

10−12

Am

plitu

de

(µm

/√H

z)

10−8

Figure 3.22:Measurements of the individual terms of the calibration func-

tion. The data are taken from a short section of the S1 data set. The detector

was locked at the time.

3.7.2 Up-sampling the optical and relative gains

The first term of the calibration function (equation3.26), shows the de-whitened error-

point signal,v(t), filtered with the time-domain filter having the same transfer function

as the inverse ofH1. This time-domain filter,f ′

1, is just a time varying gain and must

have a value at every sample instant of the error-point signal if we wish to apply it.

However, the transfer functionH1 represents the optical gain of the detector which, as

3.7 Real-time calibration 108

was stated earlier, is flat in frequency and is only estimated/updated once per second.

We then have two options as to what value to set each sample of the filterf ′

1: we can set

each sample equal to the reciprocal of the estimated optical gain for that second of data,

or we can somehow interpolate between successive 1 Hz optical gain estimates to give

smoothly changing values at each sample instant of the filter.

It is clear that if we make all samples off ′

1 the same value, then at a second boundary,

when the optical gain value changes, there will be a step discontinuity in the output of the

filter application, and hence in the derived strain channel. To avoid this, we must take the

second option of interpolating the optical (and relative) gain so that it varies smoothly

from second to second but with a sample rate equal to that of the recorded error-point

(16384 Hz). The inverse of the interpolated optical gain then becomes the filterf ′

1.

The first experiments were done using a spline interpolation over 20 optical gain esti-

mates to get values for the optical gain at each sample instant of the error-point. The 20

estimates were taken such that the middle estimate pertained to the second of data to be

calibrated. To calibrate the following second, the ‘spline window’ is moved along one

second. This means that the resulting calibrated data is smooth across second bound-

aries. This method is explained graphically in figure3.23.

There we see the up-sampling of the optical gain for a particular data set from the S1 data.

Each second of the process is colour coded. The first calibration second shown is for

t = −1 s; this is the red data in the plot. Twenty estimates of the optical gain surrounding

this time are up-sampled to 16384 Hz using a spline interpolation (as described above);

these interpolated values are shown as the thin red line that is terminated at both ends

by a red circle. The thick red line that spans one second and is labelled ‘segmentn’, is

the data used to create thef ′

1 filter for this second. Successive segments are shown for

calibrating successive seconds of data.

The relative gain filter (frel), is calculated in the exactly the same way.

The method is fast and effective but more tests need to be carried out to check the effect

of this interpolation method on the frequency content of the resulting optical gain mea-

surements. In principle, the correct way to do this is with a symmetric interpolating FIR

3.7 Real-time calibration 109

13.420

13.415

13.410

13.405

13.400

13.395

13.390

13.385

13.380

13.375

13.370−10 −5 0 5 10 15

Time (s)

segment n + 2

segment n + 1

segment n

Opti

calG

ain

(V/n

m)

Figure 3.23:A diagram showing how the optical gain is interpolated for

successive estimates. Each successive second of data that is to be calibrated

uses the data from a particular colour. For example, to calibrate second n,

the red data is calculated from 20 estimates of the optical gain and segment

n is used to do the calibration. The short horizontal lines associated with

each colour/segment represent the discrete estimates of the optical gain for

those seconds.

filter, but due to time constraints it was not possible to implement this for the S1 data.

Figure3.24shows a section of the optical and relative gains with and without interpola-

tion. Figure3.25shows the section of the calibrated strain channel calculated with the

gains of figure3.24. In the case where the interpolation is not performed, each sample

of the f ′

1 filter is set to the same value. These two methods are referred to as ‘discrete’

updating and ‘continuous’ updating of the optical gain.

3.7.3 Issues of signal dynamic range and numerical precision

One important consideration when storing signals digitally is how the dynamic range of

the signal compares to the storage capabilities of the chosen numerical format, and what

is the resulting precision with which the numbers are stored?

Data is stored numerically as a certain number of bits. The number of bits determines the

dynamic range and precision of the numbers that can be stored. For example, in standard

3.7 Real-time calibration 110

continuous gain updating

discrete gain updating

12.05

12.00

11.95

11.90

11.85

11.75

11.70

11.80

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8

Time (s)

0.0078

0.00776

Rel

ativ

eG

ain

(V/V

)O

pti

calG

ain

(V/n

m)

Figure 3.24: The optical and relative

gains determined with and without inter-

polation. The result is a discrete or con-

tinuous updating of the gains in the cali-

bration scheme.

continuous gain updating

discrete gain updating

5.014.99 5

Time (s)

3

2

1

0

−1

−2

−3

Str

ain

(10−

16/√

Hz)

Figure 3.25:A comparison of the effect of

updating the optical gain in a discrete or

continuous way.

GNU C programming language5, double precision floating point numbers are stored in

64 bits. The storage is split so that 52 bits are used for the mantissa, 11 bits store the

exponent, and the remaining bit gives the sign. A 52-bit mantissa has a precision of 1

part in 252≈ 4.5× 1015. This means that storing data in this format can introduce noise

at the level of about 1 part in 1015. Because of the way floating point notation works,

the dynamic range of the signal affects the precision with which numbers are stored and

hence the amount of noise introduced.

A simple experiment can be done to demonstrate these issues. We start by generating two

identical vectors of white noise. If we then form the difference between these two vectors

we expect to, and do, get a vector of zeros. Now, instead of calculating the difference, we

add a large amplitude sinusoid to one of the vectors and Fourier transform each vector

to get a spectrum of each. Performing the Fourier transform does two things: firstly

it allows the noise signal to be examined separately from the sinusoid, and secondly it

performs a large number of numerical operations on the data which may highlight any

artifacts that could appear due to the storage format. If we now form the difference

between these two spectra we can see what effect the sinusoid has on the noise in bins

corresponding to different frequencies. If the data were perfectly represented in the

5The entire calibration scheme is written in standard GNU C.

3.7 Real-time calibration 111

computer we would expect the difference of the two spectra to yield a non-zero value in

the bin corresponding to the frequency of the original sinusoidal signal, and zero in all

other bins. In practise this is not the case. Tests were done with sinusoids of different

amplitudes and the residuals inspected to get a qualitative measure of the effect of storing

large dynamic range signals.

To investigate the behaviour, the percentage power in the residuals, compared to the

power in the original noise vector, was stored for different amplitudes of sinusoid. Figure

3.26 shows the results. As we can see, as we increase the amplitude of the sinusoid,

the percentage error increases. What this means is, if we had a sinusoid signal in a

background of noise such that the dynamic range of the total signal is 1013 say, then

errors would appear in the data at the level of about 0.1% of the noise power. If the

amplitude of the sinusoid was around 1015 higher than the noise, we see contamination

at levels above 10%. This situation could arise in gravitational wave detectors if we have

some high amplitude low-frequency sinusoidal signal imposed on a noise spectrum that

is many orders of magnitude smaller at high frequency.

1012 1013 1014 1015 101610−3

10−2

10−1

10 0

10 1

10 2

10 3

10 4

Peak Sinusoid Amplitude

Per

cent

age

erro

r

Figure 3.26:A plot of the error introduced when storing signals with large

dynamic range as double precision floating point numbers.

In order to avoid such errors, it is desirable to restrict the dynamic range of numerical

signals at all times to about 7 orders of magnitude. Then, even if we were to integrate

data for a year or more (in the case of pulsar searches), we would not risk any serious

3.7 Real-time calibration 112

contamination of the data.

Due to seismic noise, the noise spectra of all interferometric gravitational wave detectors

increases dramatically towards lower frequency. In the case of GEO, one simple way to

decrease the dynamic range of the calibrated strain signal is to cut off all frequencies

below the detection band. This is done in the calibration scheme by applying a high-

pass filter with corner frequency of 10 Hz to the recorded error-point, thus ensuring that

the dynamic range of the signal is minimised before the signal is de-whitened and before

any of the calibration filters are applied.

3.7.4 Suppressing the calibration lines

An additional reduction in dynamic range can be achieved by suppressing the calibration

lines in the recorded error-point, again, prior to application of the de-whitening and

calibration filters.

This is done by simulating the calibration peaks as they would appear in the recorded

high-power error-point and then subtracting them from the recorded high-power error-

point. For frequencies above the unity gain point of the Michelson length control servo,

the simulation of the high-power error-point is a linear process. Since we have a record-

ing of the injected calibration signal, we can pass that signal through a series of filters

that model the detector. These steps are shown schematically in figure3.27.

high-power

error-point

simulatedrecordedcalibration

signal

ESD response Whitening filter

×Gopt

Optical gain

Figure 3.27:A schematic of the process that simulates the recorded high-

power error-point signal at the frequencies of the calibration lines.

First of all we must convert the recorded calibration signal into a displacement by fil-

tering it with the filter that simulates the response of the electro-static drive (labelledf5

in figure3.20). We then convert this displacement into a voltage by multiplying by the

3.7 Real-time calibration 113

estimated optical gain for the particular second we are dealing with. Having done that,

we simulate the recording process by applying an IIR version of the high-power photodi-

ode whitening filter. The result is a time-domain signal whose frequency content closely

matches that of the recorded high-power error-point at the frequencies of the calibration

peaks, and is essentially zero at all other frequencies. Figure3.28shows snap-shot am-

plitude spectral densities of the simulated and recorded high-power error-point signals.

recorded error-point

simulated error-point

10 2 10 3

10−2

10−4

10−6

10−8

10−10

Am

plitu

de

(V/√

Hz)

Frequency (Hz)

Figure 3.28: Snap-shot amplitude spectral densities of the simulated and

recorded high-power error-point signals.

In principle, if the process simulating the high-power error-point was perfect, we could

now suppress the calibration lines in the recorded high-power error-point signal by

adding the two time-series together. This would reduce the presence of the calibration

lines to a level consistent with the signal-to-noise ratio of each line. Visual inspection of

the calibration peaks in figure3.28show that the practical situation is not quite as good

as this ‘perfect’ case. In particular, the inability of IIR filters to match the 1/ f 2 ESD

response at higher frequencies (> 1 kHz), means that the calibration peaks in the sim-

ulated error-point have increasingly wrong amplitudes towards higher frequency. Since

these peaks do not dominate the dynamic range of the signal, the effect is not important.

Another point to note is that the fundamental calibration peak (244 Hz) appears with

slightly higher amplitude in the recorded error-point than in the simulated error-point.

The reason for this can be deduced from figure3.19; here we see that there is a small

3.7 Real-time calibration 114

amount of gain due to the length control servo which is not modelled in the simulation

of the error-point.

Figure3.28also shows that the biggest possible contaminant introduced in adding the

simulated error-point to the recorded error-point will appear at the level of 1 part in 1000

(the minimum distance between the two noise floors).

10 310 2

Frequency (Hz)

10−15

10−16

10−17

10−18

10−19

10−20

Am

plitu

de

(rm

s/√

Hz)

h(f ) with no cal line suppression

h(f ) with suppressed cal lines

Figure 3.29:Snap-shot amplitude spectral densities of the calibrated strain

signal with and without suppression of the calibration lines.

Figure3.29shows snap-shot amplitude spectral densities of the calibrated strain channel

with and without suppression of the calibration lines. We can see clearly that we reduce

the dynamic range of the signal by about one order of magnitude by suppressing the

calibration lines in this way. Furthermore, with this defined detection band we could not

really do any better since the noise floor around 50 Hz dominates the dynamic range.

We can examine the effect of suppressing the calibration lines on the calibrated strain

signal. Figure3.30 shows the percentage difference of the two traces of figure3.29.

The positions of the calibration lines are indicated by the red vertical lines. We can see

that the process of suppressing the calibration lines introduces contamination at a level

significantly below 1%.

3.7 Real-time calibration 115

10 2 10 3

Frequency (Hz)

0.3

0.25

0.2

0.15

0.1

0.05

0

Per

cent

age

diff

eren

ce

Figure 3.30:The contamination introduced as a result of the process that

suppresses the calibration lines. The contamination is expressed as a per-

centage of the strain channel with calibration line suppression. The (red)

vertical lines are the calibration line frequencies.

3.7.5 Sign convention of GEO

If multiple detector analysis is to be performed, then a convention for the sign of the

strain channel must be followed. This means that if one detector records a signal with

increasing strain, then all other detectors should agree. In the case of GEO, the sign of

the derived strain channel was determined prior to the S1 run and is valid for that data

taking period. Changes to electronics and software can easily change the sign so it must

be recalculated prior to each data taking session.

A definition was agreed on within the GEO collaboration stating that an increase in the

length of the North arm of the detector would be considered as a positive increase in

strain. In other words,

↑ h(t) =↑ 1LNorth. (3.28)

By tracing the phase of the calibration signals through both the Michelson control sub-

system and the data acquisition system, we deduced that there is a 180 phase offset

between the applied calibration signal and the derived calibrated strain channel. Since

3.8 Application to the S1 science run 116

the application of the calibration signal has the same effect as a gravitational wave signal

with the same waveform, then there is also a 180 phase offset between any gravitational

wave signal and the calibrated strain channel. So by the definition made above, the cali-

brated strain channel produced by the calibration scheme represents−h(t). The reason

for this sign change is due to the non-standardisation of the sign convention at the time of

the S1 run. The LIGO detectors chose a convention that has the opposite sign to the GEO

convention, hence a sign flip was introduced into the GEO data to make multi-detector

analysis more simple. For future runs, a more globally accepted convention should be in

place.

3.8 Application to the S1 science run

The eighteen day science run (S1) proved an excellent opportunity to test the calibration

scheme over long time periods. The calibration scheme ran for the entire time during the

science run, producing, in near real-time, a calibrated strain channel that could be viewed

on-line with the standard data viewing tools. This section looks at some of the results that

come from studying the calibration scheme and its associated data products. A validation

of the calibration scheme is also presented, as are the systematic and random errors

present in the system at the time of the run. The salient points of the characterisation

work presented in this section are brought together in [PRcalS1].

3.8.1 Stability of the input calibration lines

Before we can study the behaviour of the detector from the point of the view of the cal-

ibration procedure we must be sure that we understand the source of any other possible

effects. One possible effect that could affect the result of the calibration procedure comes

from variations in the recording of signals due to, for example, temperature related gain

changes in the pre-amplifier stage of the ADC boards of the DAQ system (see section

2.2.1). This would lead us to believe that we induced a displacement different from what

was actually induced—the result being an error in the estimation of the optical gain.

3.8 Application to the S1 science run 117

However, this is only the case if the channel that records the detector error-point would

be affected differently by any temperature changes. Since we can not measure this effect

at the same time as recording a signal, we must look for another method to determine if

a problem exists, and if it does, to determine if it is significant.

Another possible way to introduce a mis-match between the recorded injection signal

and the signal actually injected, is at the point in the electronics where the signal is

prepared for recording; this is after the point where the signal is split to be both recorded

and injected into the ESDs.

Figure3.31shows the time evolution of the 732, 1220, and 1708 Hz calibration peaks

as measured in the recording of the injected calibration signal. The peaks are measured

every minute.

0.216

0.215

0.235

0.2346

0.2342

0.1958

0.1956

0.1954

0.195210 2 3 4 5 6 7 8 9 10

Time from the start S1 (days)

Am

plitu

de

(V)

732Hz

1220Hz

1708Hz

Figure 3.31: The time evolution over

10 days of the three injected calibration

peaks used to determine the optical gain

during S1. The data are sampled every

minute.

33

32

31

30

Tem

per

ature

(C

)

92.45

92.40

92.35

92.30

Am

plitu

de

(mV

)

5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 7

Time from start S1 (days)

Figure 3.32: A comparison of the time

evolution of the injected 732 Hz calibra-

tion with the temperature in the electri-

cal rack housing the electronics that pro-

duces the calibration signal.

Figure3.32shows 2 days of data for the 732 Hz calibration peak. Here we can see three

dominant fluctuations: one a daily change due to external temperatures; a faster change

(around 2 minutes) due to the switching of the cooling fans mounted in the top of the

electrical rack that houses the electronics producing the calibration signal; and a secular

drift associated with the (very slow) drift of the calibration line through the spectral bins

of the analysis (see the discussion on page65, section2.4.5in the context of DAQ timing

3.8 Application to the S1 science run 118

accuracy tests).

In the worst case, if all of the observed fluctuation were to arise after the point where the

signal splits to go to the actuator and the DAQ system, then we would mis-calculate the

optical gain because of this. The question is, ‘How much?’

Let us consider the larger, daily, fluctuation. We can calculate the optical gain for the

two extremes of the daily variation. We can consider this for only one calibration line

at first to get an idea of the size of the effect. For the 732 Hz line, the daily amplitude

variation is from around 0.2157 V to 0.2156 V. In calculating the optical gain, we need

to use the values after correcting for the recording process. From table3.3 we see the

appropriate correction factor is 2.17 yielding values for the change of the 732 Hz peak

from 0.4680 V to 0.4678 V. Using a typical value for the size of this peak as it appears in

the high-power error-point signal, and for the response of the electro-static drive actuator

at 732 Hz (typical values taken from table3.1), and assuming that the value of the peak

in the error-point doesn’t vary, we can compute an upper and lower limit of the optical

gain for this variation of input calibration signal as

GoptL =1.23× 10−3 [V]

0.4680 [V] × 0.02× 10−11 [m/V](3.29)

= 1.3169× 1010 [V/m] (3.30)

GoptU =1.23× 10−3 [V]

0.4678 [V] × 0.02× 10−11 [m/V](3.31)

= 1.3175× 1010 [V/m]. (3.32)

This corresponds to a relative change in the optical gain estimation of around 0.05%—

clearly a negligable effect.

3.8.2 Results and characterisation of the calibration process

Some of the intermediate data products of the calibration process, for example, the op-

tical gain, provide a useful way of probing the behaviour of the detector. First let us

look at the time evolution of the two measured parameters of the calibration process: the

optical and relative gains. Figure3.33shows time-series of the gains sampled every 10

minutes for the entire duration of the run.

3.8 Application to the S1 science run 119

0 2 4 6 8 10 12 14 16 18

18

16

14

12

0.02

0.015

0.01

0.005

0

Time (days)

Rel

ativ

eG

ain

(V/V

)O

pti

calG

ain

(V/n

m)

10

Figure 3.33:Time evolution of the estimated optical and relative gains for

the entire S1 data run. The data are sampled every 10 minutes.

Optical gain behaviour

For the optical gain (the upper trace of figure3.33), we can immediately see that there

are a number of fluctuations that occur on different time-scales and with different am-

plitudes. In addition we see a general decrease in the optical gain of about 20% over

the run. Another striking feature is the daily variation of around 7% that can be seen.

The reason for this daily variation becomes more clear when one investigates the tem-

perature fluctuations in the interferometer. Figure3.34shows a plot of a shorter section

of the optical gain (around 6 days) with the temperature as measured on the laser bench

overlaid. We see a clear correlation between optical gain and temperature. This could

arise from power fluctuations in the laser system that are related to temperature.

The easiest way to conceive of changing the optical gain is to change the light levels in

the optical cavity. The most obvious way that light levels in the cavity can change is

by either fluctuations in the input light power, or by small mis-alignments of the cavity.

Figure 3.35 shows amplitude spectra of the optical gain and the Michelson differen-

tial feedback signal. Two prominent features are visible in the differential feedback: a

3.8 Application to the S1 science run 120

11

10

12

13

14

15

Opti

calga

in(V

/nm

)

26

25.8

25.6

25.4

25.2

25

Las

erB

ench

Tem

per

ature

(C

)

0 1 2 3 54 6

Time (days)

Figure 3.34: A comparison of the esti-

mated optical gain over a period of 6

days with the temperature measured on

the laser bench.

Estimated optical gain

Michelson differential feedback

(lunar tides)12 H 25 min

0

Am

plitu

de

(dB

arb.

unit

s)

200

100

Am

plitu

de

(dB

arb.

unit

s)

−1001 10 100

Frequency (µHz)

24 H

Figure 3.35:Low frequency spectra of the

optical gain and the Michelson differen-

tial feedback signal.

peak corresponding to 24 hours, and a peak corresponding to the lunar tides (12 hours

25 minutes). We can see that the optical gain spectrum shows no prominent peaks at

these frequencies. This means that the optical power in the cavity is not modulated by

these length change fluctuations. So, either the residual coupling of the longitudinal

motion into cavity misalignment is too low to be seen in this spectrum, or the cavity

auto-alignment system sufficiently compensates for any coupling that does exist.

If we look at the optical gain on time-scales of about 1 hour, we see a fluctuation in the

optical gain with a period around twenty minutes. Figure3.36shows the optical gain

over a period of 200 minutes around the start of the S1 run. Figure3.37shows a low-

frequency power spectral density of the optical gain. Both figures show clearly a signal

around 19.5 minutes. This signal is believed to originate from the switching of the air-

conditioning unit in the central station. Temperature changes due to this switching could

couple into power fluctuations in the laser or to alignment changes in the interferometer.

The width of the peak in the spectrum may suggest that the switching is not precisely

19.5 minutes but drifts around this nominal value. This air-conditioning system has now

been replaced by a more stable system.

3.8 Application to the S1 science run 121

20m

14.2

14.1

14.0

13.9

13.8

13.7

13.5

13.6

13.4

13.3200 220 240 260 280 300 320 340 360 380 400

Opti

calG

ain

(V/n

m)

Time (minutes)

Figure 3.36: A closer view of the esti-

mated optical gain showing a signal with

a period of around 20 minutes believed

to come from the switching of the central

station air-conditioning unit.

ndata:nfft:navs:fs:enbw: 0.00007

1/6052

36018720

1 10

10 7

10 6

10 5

Frequency (mHz)

19.5 minutesOpti

calG

ain

(V2/m

2/H

z)

Figure 3.37:A low-frequency power spec-

tral density of the optical gain showing a

peak around 19.5 minutes. The peak is al-

most certainly the result of the switching

of the air-conditioning unit in the central

station.

Relative gain behaviour

The relative gain (the lower trace of figure3.33) shows rather large changes of up to

100% at various times during the S1 run. Towards the end of the run the behaviour

becomes particularly erratic. Sometimes these changes occur over very short time-scales

but there are also much slower changes occurring over periods of a few days.

The reasons for the sharp changes are as yet unknown but one possible explanation could

be due to shifts of the beam position on the output bench. Saturation of various beam

position sensors can cause the position of the output beam to drift slightly. This in turn

could mean that the beam could be clipped at some point in the optical path after the

point where the beam is split to go to the high-power photodiode. The result being that

the power level on the quadrant photodiode would change whereas the level on the high-

power diode would remain constant. Since the relative gain is essentially the ratio of

these two power levels, sudden changes in the relative gain would not be unexpected.

3.8 Application to the S1 science run 122

1708 Hz

1220 Hz

732 Hz

244 Hzndata:nfft:navs:fs:enbw:

1638400010010

163840.015

10−15

10−16

10−17

10−19

−2 −1.5 −1 −0.5 0.50 1 1.5 2

Frequency (Hz)

Str

ain

( rms/√

Hz)

1.1

0.55

−0.

55

−1.

110−18

Figure 3.38: High resolution spectra of

the calibration lines in the calibrated

strain signal.

with line suppression

no line suppressionndata:nfft:navs:fs:enbw:

1638400010010

163840.015

734733.5733732.5732731.5731730.573010−19

10−18

10−17

10−16

10−15

Str

ain

( rms/√

Hz)

Frequency (Hz)

Figure 3.39:The 732 Hz calibration line

as it appears in the calibrated strain

channel with and without line suppression

switched on.

Study of the calibration lines

Studies of the calibration lines as they appear in the detector output can give information

about any low frequency modulation present in the detector. For example, if there is any

residual pendulum motion that couples into misalignments of the optical cavity, then we

would see amplitude modulation of the calibration lines due to the modulation of the

optical gain. If we look at a high resolution amplitude spectrum of the calibrated strain

signal, then we clearly see amplitude sidebands on the calibration lines (see figure3.38).

These sidebands show that the calibration lines are being modulated at 0.55 Hz. The

question that naturally arises is, is it a real effect or an artifact of the calibration process.

For example it could come from the process of suppressing the calibration lines. Figure

3.39shows the 732 Hz calibration line as it appears in the calibrated strain channel, with

and with line suppression. We can clearly see that the line suppression does not cause

the sidebands. If we look at the calibration line in the recorded high-power error-point

signal (figure3.40), we see that the sidebands persist so they must be a real effect of

the detector. Measurements of the alignment motion of individual mirrors shows that the

beamsplitter has a rotational resonance around 0.6 Hz. This could explain the modulation

we see if the rotational fluctuations produce slight changes in the optical gain.

What this modulation also shows is that we are not updating the optical gain often

enough. If we were, we would see the modulation in the recorded high-power error-

3.8 Application to the S1 science run 123

point but not in the calibrated strain channel.

ndata:nfft:navs:fs:enbw:

1638400010010

163840.015

734733.5733732.5732731.5731730.5730

Am

plitu

de

( Vrm

s/√

Hz)

Frequency (Hz)

10−1

10−2

10−3

10−4

10−5

Figure 3.40:A high resolution spectrum

of the 732 Hz calibration line as it ap-

pears in the recorded high-power error-

point signal.

3.8.3 Validating the calibration scheme

An important aspect of the calibration work is to validate each point that makes up the

whole scheme. Figure3.41 shows a schematic of the steps necessary to validate the

calibration process. The figure also provides a way to see where the possible errors in

the system lie.

Calibration of the electro-static and intermediate mass actuators

The calibration scheme as outlined here relies on the ability to induce a known mirror

displacement. To do this, we need to know (accurately) the response of the actuator used

to induce the motion, in this case, the electro-static drives. Calibration of the electro-

static drives was done using two methods: via the intermediate mass calibration, and via

the master laser piezo calibration.

The validation process starts with the calibration of the intermediate mass drive. This

was done using a fringe counting technique. With the Michelson unlocked, the inter-

3.8 Application to the S1 science run 124

DC calibration

of ESD

DC calibration

of IMD

Frequency

response

of ESDof the applied

calibration signal

Accurate recording

Actuator Response

Determination of

induced mirror

displacment

Response of the

optical cavity

Design of stable

IIR filters

Equivalence of

time and frequency

domain methods

Model of the

Michelson

control servo

Valid Calibration

Calibration of

laser piezo

Figure 3.41:A schematic of the chain of dependence that leads to a valid

calibration.

mediate mass was driven longitudinally with a low frequency signal (around 100 mHz,

well below the pendulum’s main longitudinal resonance) and the Michelson fringes were

observed on the quadrant photodiode. By counting the number of fringes between suc-

cessive turning points of the pendulum, we determined the number of fringes we get per

Volt applied to the IMD. Since one fringe corresponds to 0.532µm displacement of the

mirror (λ/2), we get a calibration of the IMD to be around 100µm/V per drive. The

measurement was repeated many times and the error on the calibration of each of the

two drives was determined to be around 1.5%.

The calibration of the electro-static drives can be done using the calibration of the IMD.

Since GEO has two electro-static drives, it is possible to lock the interferometer using

only one of the drives, leaving the other test mass free except for the action applied by

the intermediate mass drive. If we then drive the ‘free’ ESD, again at 100 mHz, the

Michelson servo will try to suppress that motion via both intermediate mass actuators.

By observing the size of the correction signal in the intermediate mass feedback signal,

we can determine how much motion we introduced at the test mass (because the loop

gain of the Michelson servo is high at this frequency). This in turn gives us a calibration

factor for one of the electro-static drives. For the East electro-static drive, the calibration

3.8 Application to the S1 science run 125

factor was determined to be 100 nm/V, and for the North, 98 nm/V.

732 → 1220

732 → 1708

1220 → 1708

6000

4000

2000

0

6000

6000

4000

4000

2000

2000

0

043210−1−2−3−4

Residuals (%)

Cou

nt

Figure 3.42: The deviations of the ESD

from the1/ f 2 model.

10 310 2

Frequency (Hz)

1.04

1.02

1

0.98

3

2

1

0

−1

Am

plitu

de

Rat

ioP

has

eD

iffer

ence

()

Figure 3.43: A comparison between the

time- and frequency-domain models of the

Michelson control servo.

We can also check the calibration of the ESDs against the (known) calibration factor of

the master laser piezo. By driving one ESD, an asymmetric length change is introduced

in to the Michelson cavity which results in a length change in the power-recycling (PR)

cavity. Variations in the length of the PR cavity result in changes in the resonant fre-

quency of the cavity. In order to keep the carrier laser light resonant in the PR cavity,

the frequency of the master laser is changed. By observing the induced signal in the

feedback to the master laser, a calibration factor can be determined for the ESD. The

calibration of the master laser is done by driving the piezo at a known frequency to pro-

duce frequency modulation side-bands on the carrier light. Analysis of these sidebands

and knowledge of the applied voltage can lead to a calibration of the master laser piezo.

In addition, the piezo can be calibrated against the (known) free-spectral-range of the

first mode-cleaner. The two methods for calibrating the master laser piezo agreed to

within about 3%. The ESDs were calibrated in this way and the values for each drive

were within 15% of the values obtained by the IMD/fringe counting method.

Frequency response of the ESDs

The calibration scheme uses the model that well above the pendulum resonance, the

frequency response of the electro-static actuators has a 1/ f 2 frequency response. This

3.8 Application to the S1 science run 126

needed to be checked since at the time of the S1 run, the main test masses (which are

acted upon by the ESDs) were suspended in steel wire slings. Driving these test masses

by a varying electro-static field undoubtedly results in forces being applied to the steel

wires as well. Depending on the size of the interaction, the nominal 1/ f 2 response could

be modified. An experiment was done to check the validity of the 1/ f 2 model and to

confirm that the interaction between the drive and the steel suspension wires was suitably

small.

By measuring the values of the 732, 1220, and 1708 Hz peaks in the recorded high-

power error-point spectrum at 1 Hz for around 24 hours, the response of the ESD was

checked. By assuming the 1/ f 2 model, it should be possible to use any one of the peak

measurements,e.g., at 732 Hz, to predict the amplitude of the other two peaks for every

sample of the measurements.

Let us consider predicting the 1220 Hz peak,m2, from the measured values of the 732 Hz

peak,m1. The first thing to do is to normalise the measured peaks by the amplitude of

the peak applied to the ESD. The normalised peak values are then given by

mi =mi

ci, (3.33)

whereci is the amplitude of the input peak. The predicted value,P12, of the 1220 Hz

peak is then given by

P12 = m1f 21

f 22

, (3.34)

where f1 = 732 Hz, andf2 = 1220 Hz. We can now form the residual,R12, between the

amplitude of the normalised measured peaks at 1220 Hz and the predicted amplitudes

of the 1220 Hz peak. Such that, expressed as a percentage of the mean value of the

measurements of the 1220 Hz peak, we get

R12 = 100×m2 − p12

〈m2〉. (3.35)

Doing this for the three possible cases yields the histograms shown in figure3.42. We

see that in all three cases that the mean of the measured responses doesn’t deviate sig-

nificantly from zero (the nominal value). The scatter we see in the predictions is entirely

consistent with the noise in the measurement of the calibration peaks (around 1%). So,

as far as it is possible to tell with this experiment, the ESD response is correctly modelled

as a 1/ f 2 response above the pendulum resonance.

3.8 Application to the S1 science run 127

Frequency response of the optical cavity

15.8

15.6

15.4

15.2

15.0

14.8

14.6

Gopt + σ

Gopt − σ

±1%

0.0105

0.01

0 0.02 0.12 0.14 0.16 0.180.10.080.060.04

Time (hours)

Grel − σ

Grel + σ

±1%

Rel

ativ

eG

ain

(V/V

)O

pti

calG

ain

(V/n

m)

Figure 3.44:A short section of the estimates of the optical and relative gain

showing±σ levels. A±1% level is also shown for reference.

Throughout the entire calibration discussion, the frequency response of the optical cavity

from mirror displacement to demodulated detector output has been assumed to be flat.

This response is composed of a true optical part and the response of the photodetector

and RF-demodulation system. The main optical response is that of the interferometer

arm which has a flat frequency response below 6 kHz (the first feature being a null at

the inverse light travel time—around 125 kHz). The photodiode and RF system have a

bandwidth around 100 kHz and were measured to have a flat response below 6 kHz. The

only contribution to the calibration accuracy is the error associated with measuring the

optical gain. This was typically less than 1% for the S1 run. Figure3.44shows a short

section of the optical gain plotted as±σ limits (plus and minus one standard deviation).

The relative gain is also shown.

Equivalence of time- and frequency-domain models

The low frequency calibration depends on the validity of the time-domain model of the

Michelson servo that was developed to simulate the S-domain model. For a particular

set of the gains, the closed-loop transfer function of the Michelson servo was calculated

3.8 Application to the S1 science run 128

using a purely frequency domain model and compared to the response generated by the

IIR filter time-domain model. The comparison is shown in figure3.43. The differences

at low frequency can be attributed to the inability of IIR (Z-domain) filters to perfectly

match the response of equivalent S-domain filters. These differences represent a system-

atic error in the calibration.

3.8.4 Calibration accuracy

All of the systematic and random errors discussed throughout this chapter can be brought

together to give an impression of the overall calibration accuracy achieved for the S1

data set. Table3.6shows the various errors that contribute to the overall accuracy of the

calibration process. The table approximately follows the route through figure3.41.

Source Systematics (%) Random errors (%)

IMD calibration 3 (lf)

ESD calibration 5 (lf, hf)

ESD response 1 (lf,hf)

Optical gain 1 (lf, hf)

Relative gain 1 (lf)

Time/Freq. equivalence 4 (lf)

Signal recording/whitening 3.5 (lf, hf)

Table 3.6: A table summarising the observed systematic and random er-

rors in the time-domain calibration procedure. For each error, a relevant

frequency range is given: ‘lf ’ (low-frequency) means< 200Hz; ‘hf ’ (high-

frequency) means> 200Hz.

By crudely adding the errors from the table we can estimate that low frequency calibra-

tion (<300 Hz), has a systematic error of around 8% and a random error around 9%. For

the high frequency calibration, the systematic errors add up to about 4% and the random

errors are around 5%.

3.8 Application to the S1 science run 129

3.8.5 Calibration bandwidth

Another useful measure of the success of the calibration process is to look at the residual

fluctuations of the calibration peaks in the calibrated strain signal. For a perfect calibra-

tion process these fluctuations should be the same as the fluctuations of the input calibra-

tion peaks—essentially zero. Measurements were made of the 1220 Hz peak amplitude

in amplitude spectra of the recorded high-power photodiode signal and the calibrated

strain channel every second for the entire S1 run. The peak amplitudes are recorded and

then amplitude spectral densities of these peak values are computed. Figure3.45shows

the resulting spectra. We can see that the fluctuations present in the pre-calibrated spec-

trum are significantly reduced by the calibration process. The spectra are scaled so that

the fluctuations at the Nyquist frequency (0.5 Hz) are the same amplitude.

8.7× 10−6

ndata: 1555200nfft: 172800navs: 9fs: 1enbw:

h(t) at 1220 Hz

high-power error point at 1220 Hz

ndata: 1555200nfft:navs:fs: 1enbw:

14400108

1.04× 10−4

10−210−310−410−5

10 5

10 4

10 3

10 2

10 1

10 0

Am

plitu

de

(arb

.unit

s)

10 0

10 1

10 2

10 3

Frequency (Hz)

10−110−210−3

Figure 3.45:A comparison of the amplitude spectral density of

the 1220 Hz calibration peak amplitude measured in both the

recorded high-power photodiode error-point signal and the cal-

ibrated strain signal. Both spectra are made from the same data

set; different lengths of ffts are performed to show the perfor-

mance at higher and lower frequencies.

3.9 A summary of the calibration procedure 130

3.9 A summary of the calibration procedure

The method developed to calibrate the power-recycled GEO can be summarised with the

following recipe:

Step 1: Read one second of the three input data streams: the injected calibration signal,

the low- and high-power photodiode detector output signals.

Step 2: Pre-process these signals to remove any effects of the recording process (details

in section3.5).

Step 3: Transform each signal to the Fourier domain and form the transfer function

from the input signal to the high-power detector output at calibration line fre-

quencies. This gives us a measurement of the overall optical gain,Gopt. Do

the same thing for the two output signals to get an estimate of the relative gain,

Hrel (details on page88and96).

Step 4: Up-sample these 1 Hz optical gain estimates to the input data rate (16384 Hz)

(details in section3.7.2).

Step 5: Divide the pre-processed detector output signal by the up-sampled optical gain

data to give a data stream that represents differential arm-length changes well

above the unity gain point of the Michelson servo.

Step 6: Generate the two feedback signals (fast and slow) by filtering the pre-processed

detector output signal through model IIR filters of the Michelson servo elec-

tronics after multiplying by the up-sampled relative gain estimates.

Step 7: Add these two feedback signals to the data stream calculated in step 5 to correct

for the effect of the Michelson servo to give a signal that represents differential

arm-length changes at all frequencies where the detector model is valid.

Step 8: Divide output of step 7 by 1200 to convert the signal to a strain signal,h(t).

The various input/output data streams and parameters are summarised in table3.7.

3.9 A summary of the calibration procedure 131

Input data Description Reference

EP-P HP The error-point signal demodulated

in-phase at the high-power photodi-

ode.

section3.3.2

EP-P The error-point signal demodulated

in phase at the quadrant photodiode.

section3.3.2

CAL The calibration signal applied

differentially to the electro-static

drives.

section3.3.2

Fixed input parametersDescription Reference

H2 The overall electronic gain. section3.6.3

|H3|DC The DC calibration of the IMD. section3.8.3

H4 The crossover gain. section3.6.3

|H5|DC The DC calibration of the ESD. section3.8.3

Hrel The relative gain calculated as the

ratio between the two photodiode

outputs at the calibration line fre-

quencies.

section3.6.1

Variable parameters Description Reference

H1 The optical gain calculated at the

calibration line frequencies from ra-

tio of the high-power photodiode

output and the input calibration sig-

nal.

section3.6.3

Output data Description Reference

h(t) The strain signal. section3.7.1

Table 3.7:A summary table of the input/output data streams and parameters.

3.10 Future requirements and improvements 132

3.10 Future requirements and improvements

The calibration procedure developed to calibrate the power-recycled Michelson config-

uration of GEO was successfully employed to produce a calibrated strain signal for the

S1 science run. The accuracy of the calibration was evaluated to be of the order 5% for

frequencies above 300 Hz, with most of this uncertainty arising from the calibration of

the electrostatic drive actuators used for injecting the calibration lines. The calibration

of the electrostatic drive is one of the most difficult aspects of the calibration to improve

upon. The development and use of a photon-pressure actuator may provide a way to re-

duce the uncertainty in the electrostatic drive calibration, and may even be used to inject

the calibration lines directly. The electrostatic drive calibration still enters at low fre-

quencies where the loop-gain of the Michelson length-control servo is significant. The

use of a photon-pressure actuator would also allow the calibration of the electro-static

drive to be tracked in time (so far, the calibration has been assumed constant).

This work has laid the foundations for calibrating the final configuration of GEO—the

dual-recycled configuration. The tests and validation procedures used, together with

all the tools for generating and testing IIR filters can be carried forward to the dual-

recycled calibration scheme. Some areas of improvement are clear for the future calibra-

tion scheme. In particular, the up-sampling of the optical gain estimates should be done

using a more robust method, for example, band-limited interpolation. A condensed form

of the principles employed in the power-recycled GEO calibration scheme is presented

in [PRcal].

Chapter 4

Calibration of the dual-recycled

GEO 600 : principles

The final configuration of GEO 600 uses signal-recycling to achieve the design sen-

sitivity. An interferometer that uses both power-recycling and signal-recycling is of-

ten referred to as a dual-recycled Michelson interferometer (DRMI). The principles of

signal-recycling are covered in detail in [Heinzel99] and [Grote03] and are not repeated

here. A brief discussion of the implementation of signal-recycling is given in chapter

1.1.2.

The calibration of the dual-recycled GEO is significantly more complicated than the

calibration of the power-recycled GEO discussed in chapter3. The optical layout of

the dual-recycled interferometer is the same as the power-recycled layout with the ad-

dition of the Signal-Recycling (SR) mirror placed at the output port of the detector.

The addition of this mirror creates an optical cavity between the SR mirror and the

power-recycled Michelson interferometer. This cavity resonantly enhances the signal

sidebands, and hence any gravitational wave signals that are present. The presence of

the signal-recycling cavity means that the signal transfer from differential mirror dis-

placement to detector output signal is no longer flat (as in the PR case). This means that,

in order to recover an optimally calibrated strain signal, we need to determine many more

parameters that describe the optical response. The response of the cavity (and thus, the

4.1 Improving techniques from the power-recycled GEO calibration scheme 134

parameters) depends on the microscopic position, or tuning, of the signal-recycling (SR)

mirror and on the mirror reflectivity.

Fluctuations in the optical response can arise, for example, from alignment drifts due to

tidal, seismic, and thermal effects, as well as from fluctuations in the input laser power.

In order to account for such drifts in the calibration process, the various parameters that

describe the detector response need to be estimated as often as possible. As in the case

of calibrating the power-recycled GEO, the calibration of the dual-recycled GEO is best

approached in the time-domain so that these fluctuations in the optical response of the

detector can be smoothly corrected for. This chapter presents a time-domain calibration

scheme for recovering a strain signal from the detector output(s). The following two

chapters show application of the scheme to simulated data and data from an extended

data-taking period. A summary of the designed calibration procedure is given at the

end of this chapter (section4.3) while a more general summary of the the three DR

calibration chapters is given at the end of chapter6.

4.1 Improving techniques from the power-recycled GEO

calibration scheme

The calibration scheme applied to the power-recycled GEO was discussed in chapter3.

The principles of that calibration scheme can be built upon in developing a scheme for

calibrating the dual-recycled interferometer. This section discusses the various improve-

ments that were made based on the experience gained in PR calibration scheme. The

extra complexities present in calibrating a dual-recycled interferometer are discussed in

the following sections along with a possible calibration scheme.

4.1.1 Improvements carried forward

The time-domain calibration scheme for the power-recycled GEO used a spline interpo-

lation method to smoothly correct the detector output for changes in the overall optical

4.1 Improving techniques from the power-recycled GEO calibration scheme 135

gain of the instrument. This technique was a quick solution that was implemented in or-

der to conform to the time constraints of the experiment. The process of up-sampling the

1 Hz optical gain estimates to 16384 Hz is, in principle, best done using a band-limited

interpolation method. This method takes into account the implicit assumption that the

optical gain does not vary on time scales shorter than two seconds. (This assumption

arises because we sample the optical gain at only 1 Hz.) This method was explored and

implemented for the calibration scheme presented in this chapter. The method is used

for up-sampling both the overall optical gain estimates and the error-point signal prior

to some parts of the signal processing pipeline (see section5.2.2).

For the power-recycled calibration, the estimated optical gain samples were smoothed

using a Hanning window of length 60 seconds. This is replaced in the DR scheme with a

2 pole IIR low-pass filter with corner frequency at 0.2 Hz. Here we assume that, although

fluctuations in the alignment of the detector result in variations of the overall optical gain

on time-scales down to a few seconds, the estimations of such variations would be noisy

and difficult to correct for using this scheme. By applying the low-pass filter, we reduce

the effect of the measurement noise which would make the recovered strain signal more

noisy on time-scales of a few-seconds. This effect was discussed at the end of chapter3

(page129) and is explored for the case of the dual-recycling calibration towards the end

of chapter6.

4.1.2 Modelling the optical response of the DRMI

In the power-recycled setup of GEO, the optical gain was assumed to be flat across the

detection band of the instrument (see section3.1, page78). The introduction of a signal-

recycling mirror, which converts GEO into a dual-recycled interferometer, means that

this is no longer the case and so, as discussed above, we need to determine, and cor-

rect for, this non-flat (resonant) response. An additional complication arises because the

optimal demodulation phase for the output photodiode is no longer independent of fre-

quency. This means that signals are spread between the two demodulation quadratures

of the output photodiode. By selecting a particular demodulation phase, we maximise

signals of a particular frequency in one quadrature and minimise them in the other; sig-

4.1 Improving techniques from the power-recycled GEO calibration scheme 136

Q

P

200

100

101

100

10−1

10−2

Frequency (Hz)

0

−100

−200101 102 103 104

Mag

nit

ude

(arb

.unit

s)P

has

e(

)

Figure 4.1: The in-phase and out-of-phase responses of the signal-

recycling cavity to differential armlength changes. The response is

shown for a de-tuning of around 1200 Hz with a SR mirror reflectivity

of 99%. The demodulation phase is set to give optimal signal size in

Pat 0 Hz.

nals at other frequencies appear with different ratios in the two quadratures. If we were

to use the output photodiode as the loop sensor for the Michelson length-control servo,

then we would typically choose a demodulation phase that maximises one quadrature at

DC. This quadrature is known asin-phaseor Pquadrature; the other quadrature is termed

out-of-phaseor Q . These terms are also used for all other demodulation phases with the

general definition that the signal content inP is maximum at a chosen frequency and is

minimum inQ . Throughout this text, bold notation is used to differentiate the quadra-

turesPand Q from p and q used traditionally to refer to poles and zeros in transfer

function equations.

The response of this cavity to differential armlength changes can be modelled using the

interferometer simulation software, [FINESSE]. Figure4.1 shows the response of the

signal-recycling cavity with an SR mirror of 99% reflectivity and microscopic tuning of

around 1200 Hz. Both demodulated quadratures are shown. The demodulation phase

is chosen in this simulation to give a minimum response of theQ quadrature at 0 Hz.

With the phase set like this, we can see that the demodulation at all other frequencies is

4.1 Improving techniques from the power-recycled GEO calibration scheme 137

P

Q

0 20 40 60 80 100 120 140 160 180

1.5

1

0.5

0

6

5

4

3

2

1Mag

nit

ude

at12

00H

zM

agnit

ude

at1

Hz

Demodulation phase ()

Figure 4.2:Michelson output photodiode demodulation phase curves

for two signal frequencies. Using such curves, the demodulation phase

can be chosen such thatP is maximum at the chosen signal frequency.

non-optimal and signal power is spread between the two demodulation quadratures. To

recover the full signal with optimal signal-to-noise ratio, we may need to consider both of

the demodulation quadratures (see later). We can also see that the optical gain parameter,

Gopt, introduced in section3.4, is no longer independent of frequency in the detection

band. So we introduce two new parameters,HP and HQ , which represent the transfer

functions from differential mirror motion to the two demodulated output signals; both of

these may need to be determined in order to be able to perform an optimal calibration.

Using the same simulation software, we can see how the demodulation phase can be

chosen to maximiseP (and minimiseQ ) at particular frequencies. Figure4.2shows two

examples: one to optimise signal sizes at 1 Hz, the other to optimise signals at 1200 Hz.

Figure4.1shows the optical response of the detector to differential mirror displacements

for the optimal demodulation around DC (1 Hz). Figure4.3 shows the same detector

setup demodulated to optimisePat 1200 Hz.

Having the signal spread between two quadratures means that we must somehow com-

bine these signals, either before or after calibration, in order to recover all of the available

signal information. In chapter6, the scheme presented here is applied to data recorded

4.1 Improving techniques from the power-recycled GEO calibration scheme 138

Q

P

101

100

10−1

200

0

−100

−200103 104101

100

102

Frequency (Hz)

Mag

nit

ude

(arb

.unit

s)P

has

e(

)

Figure 4.3: The in-phase and out-of-phase responses of the signal-

recycling cavity to differential arm-length changes. The response is

shown for a de-tuning of around 1200 Hz with a mirror reflectivity of

99%. The demodulation phase is set to give optimal signal size inPat

1200 Hz.

during an extended science run (S3). For the first part of the science run, the demodu-

lation phase was optimised for DC; for the remainder, the demodulation phase was set

to optimise the signals inParound 1 kHz (the most sensitive frequency range for GEO

at that time). OptimisingParound the peak sensitivity meant that we could essentially

neglect the signal content of the other quadrature,Q . In general this is not optimal, but

for these first experiments it was desirable to do a simple calibration so as to reduce the

uncertainty in the resulting calibration. This is discussed more in chapter6.

Parameterising the optical response

If we are to correct for the optical response of the signal-recycling cavity as measured in

both quadratures, then we need to be able to model the two responses. For the purpose

of modelling the system, it is convenient to consider the two demodulation quadratures

as separate transfer functions from differential mirror displacement to detector output

signal.

4.1 Improving techniques from the power-recycled GEO calibration scheme 139

P simulated

P fit

Q simulated

Q fit

10 310 2

10 2

10 0

10−2

10 1−200

−100

0

100

200

Frequency (Hz)

Am

plitu

de

(arb

.unit

s)P

has

e(

)

Figure 4.4:A pole/zero fit to a 600 Hz de-tuned, 1% SR response with de-

modulation phase set to give optimal signals inPat DC. Higher order modes

are activated in this Finesse model. These are the cause of the features at

higher frequencies that are not included in the pole/zero model.

Simulations of the response of the dual-recycled interferometer to differential armlength

changes were generated using a Finesse model of the detector. The response of the two

quadratures can be modelled (ignoring higher-order optical modes) with one complex

pole pair and one real zero together with a suitable gain factor. The complex pole pair

has a characteristic frequency that depends on the microscopic detuning of the SR mir-

ror, and aQ factor related to the bandwidth of the cavity which is determined by the

reflectivity of the SR mirror. The poleQ is also weakly related to the microscopic tun-

ing of the SR mirror (see optical response measurements in section6.1.1, page177). So

a suitable model of the two optical responses is,

HP (s) = GP(s − zP )

(s − pP )(s − p∗

P ), (4.1)

HQ (s) = GQ(s − zQ )

(s − pQ )(s − p∗

Q ), (4.2)

where ‘*’ denotes complex conjugation,zP is the zero in theP response,zQ is the zero

in theQ response, and so on. These equations can be parameterised in a more convenient

way using a frequency/Q description of the poles and zeros. The frequency response of

a complex pole pair,p and p∗, with characteristic frequency,f0, quality factor,Q, and

4.1 Improving techniques from the power-recycled GEO calibration scheme 140

with unity gain at DC, can be expressed as

H(s) =G

(s − p)(s − p∗), (4.3)

where

G = 4π2 f 20 , (4.4)

<(p) =−2π f0

2Q, (4.5)

=(p) =2π f02Q

√4Q2 − 1. (4.6)

The response of a real zero of characteristic frequency,f0, again with unity gain at DC,

is given by

H(s) =s − 2π f0

2π f0. (4.7)

So, characterising the complex pole pair by a frequency,p f , and a quality factor,pQ,

and the real zero by a frequency,z f , means that we have 8 parameters to determine in

order to characterise the response of the two demodulated optical signals to differential

mirror displacement; these areGP , P z f , P p f , P pQ, GQ , Q z f, Q p f

, Q pQ.

Figure4.4 shows the simulatedPandQ responses of a particular dual-recycling setup

with a pole/zero fit overlaid. The fit was made in MATLAB using the output data from

a Finesse simulation and the model described above. The case shown is for a SR mirror

with 99% reflectivity de-tuned to around 600 Hz with a demodulation phase optimised

for DC.

4.1.3 Obtaining the optimal signal-to-noise ratio of the calibration

There are two main issues regarding the recovery of the strain signal with optimal signal-

to-noise ratio. One of the points was discussed in detail in section3.2.1. There we saw

that, for a low-noise detector, it may be necessary to recover the gravitational wave signal

from both the feedback and the error-point signals of the Michelson control servo in

order to maintain an optimal signal-to-noise ratio (SNR). For the dual-recycled GEO, this

may well be the case, especially as the detector comes closer to its design sensitivity. We

are then left with the challenge of combining the information from the three calibrated

4.1 Improving techniques from the power-recycled GEO calibration scheme 141

signals (from the two feedback signals and the error-point signal) if we want to produce

a single, optimal, strain signal.

An additional consideration arises because of the frequency dependent optimal demod-

ulation phase. At a particular frequency, a gravitational wave signal will appear in both

quadratures. In the absence of noise, the full underlying gravitational wave signal can be

recovered by calibrating either of these quadrature signals back to a displacement signal.

However, in the presence of noise, the situation is not as clear; we can still calibrate both

demodulated output signals back to displacement but, depending on the frequency of

interest, the optimal signal-to-noise ratio may only be gained by a suitable combination

of the two recovered displacement signals. Depending on where noise is added in the

system, the SNR of a particular signal can be maintained or degraded by the presence of

the noise. For example, the noise added due to the thermal noise present in the mirrors

would be subject to the same signal transfer as any gravitational wave signal and so the

difference between the SNRs of both quadratures would be independent of frequency.

If however, noise is added at the photon detection stage,i.e., shot noise, then the result-

ing difference between the SNRs of the two quadratures will be frequency dependent.

For example, looking at the detector responses shown in figure4.1, we see that low

frequency gravitational wave signals will appear with a much higher amplitude in the

Pquadrature than in theQ quadrature. Although the recovered displacement signals will

contain the same gravitational wave signal level, the noise in the one recovered from the

Q quadrature would be significantly higher at low frequencies than in the one recovered

from thePquadrature. So for low frequencies we would want to strongly weight the

Pdisplacement signal and weakly weight theQ signal. In general, the recoverable SNR

will be frequency dependent and the optimal SNR displacement signal can be achieved

by summing the four (if we include the 2 strain signals recovered from the feedback

signals) recovered displacement signals with suitable, frequency dependent, weighting

functions.

The final goal of calibrating the dual-recycled GEO is to recover four calibrated dis-

placement signals, and then (if necessary) combine them to get one displacement signal

that has the optimal SNR at all frequencies in the detection band. This can be expressed

4.2 Time-domain calibration of DR GEO 142

mathematically as

doptimal = WP dP + WQ dQ + Wsfbdsfb + Wffbdffb, (4.8)

where the frequency dependent weighting factors,W∗, must be determined from consid-

erations of the signal-to-noise ratio and transfer functions of each signal. The subscripts

‘P ’ and ‘Q ’ refer to the respective quadratures, and the subscripts ‘sfb’ and ‘ffb’ refer

to the slow feedback signal and the fast feedback signal respectively. In the early experi-

ments presented in this thesis, these considerations are set aside and all effort is devoted

to calibrating only thePquadrature as accurately as possible so as to thoroughly test the

signal-processing pipeline.

4.2 A time-domain calibration scheme for calibrating the

dual-recycled GEO 600

Before attempting to calibrate the detector, it is essential to have a good model of the

relevant subsystems that directly affect the transfer of strain signals to the detector out-

put(s). Such a model is shown in figure4.51. Four relevant output signals are highlighted,

as well as the input for the calibration signal(s); the diagram also shows the split feed-

back path of GEO. The model is slightly different to that used in the power-recycling

calibration scheme (see figure3.12, page95): the part of the model that imitates the

servo electronics has been split into three parts (previously two) to better mimic the real

electronics.

A convenient way to approach the problem of calibrating the dual-recycled interferome-

ter is to consider it as four separate processes. Figure4.6shows how the problem can be

split into four separate tasks that, together, lead to a calibrated strain signal. Each of the

main tasks will be discussed in detail in the following sections.

1This is a simplified version of the setup of the sensing and demodulation scheme used at GEO during

S3; see chapter6 for further details.

4.2 Time-domain calibration of DR GEO 143

fastfeedbackoutput

slowfeedbackoutput

‘slow path’

Detector

error-point

output in-phase (P)

Detector

error-point

output out-of-phase (Q)

[V]

[m]

[m]

[V]HIMD

HESD

IMD

ESD

calibrationinput

displacementexternal

signalse.g., h(t) induceddisplacement

[V]

ESD electronics

[V]

‘fast path’

Common electronics

[V][V] Hcom

IMD electronics

[V][V] Hslow

Hfast

[m] [V]HP

[m] [V]HQ

Figure 4.5:A model of the length control servo of the dual-recycled GEO.

Included are four relevant output signals and the input point for calibra-

tion signals. The optical response of the detector is shown as two separate

transfer functions for the two demodulations,PandQ .

system

identification

optical response

correction

h(t)

calculation

correction

MI servo

inputs outputs

Figure 4.6:A schematic of the tasks involved in the calibration of the dual-

recycled GEO.

4.2 Time-domain calibration of DR GEO 144

4.2.1 System identification process

The aim of this process is to determine the parameter set that best describes the detector

at any particular instant in time. In the power-recycled calibration scheme this task

(though not named as such) only determined the optical gain of the detector (see section

3.4, page88). In the case of the dual-recycled IFO, the minimum parameter set that must

be determined consists of the 4 (or 8 if theQ quadrature is included) optical parameters

described in4.1.2. In principle, this parameter set can be extended to include the other

gains in the system: the two electronic gain settings (see page97) and the two actuator

calibrations (see page123).

If we are to correct for any fluctuations in the underlying parameters of the system, then

we have to estimate the parameters on similar time-scales as the fluctuations occur. The

current scheme, as was the case for the calibration of the power-recycled GEO, assumes

we can inject calibration lines that will have a detectable SNR of around 100 in the high-

power error-point using one second of data. (The amplitude with which we can inject the

calibration lines is limited by the dynamic range of the feedback actuator.) For S3, (see

chapter6), this was easy to achieve and it was possible to make low-noise parameter

estimations at 1 Hz. The remainder of this text assumes measurements of calibration

lines and subsequent parameter estimations occur at 1 Hz.

The approach chosen for estimating the parameter set is to sample the underlying transfer

function from differential mirror motion to demodulated detector output (Pand/orQ )

at a number of different frequencies and then to fit a model of the transfer function

to these data points using a least-squares optimisation routine. For the remainder of

this discussion we will consider only the parameters forP ; the determination of the

Q parameters can be done in the same way.

Continuous transfer function estimation

Estimation of the optical transfer function is done by measuring the magnitude and phase

of the calibration lines in both the injected calibration signal and the recorded demodu-

4.2 Time-domain calibration of DR GEO 145

lated error-point signal (P ). From these measurements we get a set of complex numbers,

EC f from the calibration signal, andEPf from thePerror-point signal (where the subscript

f denotes the frequency of the calibration line being measured).

From figure4.5we can see that the transfer function,TCP, from calibration input toPcan

be written as

TCP(s) = HESD(s)HP (s)CLTF(s), (4.9)

whereCLTF(s) is the closed-loop transfer function of the servo.

The closed-loop transfer function can be written as

CLTF(s) =1

1 + OLG(s), (4.10)

where the open-loop gain,OLG(s), is given by

OLG(s) = HP (s)Hcom(s)[Hslow(s)HIMD (s) + Hfast(s)HESD(s)]. (4.11)

From the measurements of the calibration lines we can then form the transfer function

Tmeasuredf =

EPf

EC f EESDf, (4.12)

where the values EESDf are taken from a model of the electrostatic drive actuator (see

section3.3.1). This transfer function is the transfer function from differential arm-length

change toP .

From equation4.9 we can see that this measured transfer function is equivalent to the

transfer functionTmodel = HP (s)CLTF(s) for the correct choice of parameters. Using

an optimisation routine, we can determine the best parameter set that matches the model

transfer function,Tmodel to the measured transfer function,Tmeasured.

If we continuously inject a sufficient number of calibration lines (more than the num-

ber of parameters we try to recover), then we can estimate the underlying parameters of

the detector once per second. To do this, we must have a frequency-domain model of

the Michelson servo loop, including a full model of the electronics. We have already

discussed a possible model of the optical response of the detector in section4.1.2; the

models for the two actuators are essentially those used in the case of calibrating the

4.2 Time-domain calibration of DR GEO 146

power-recycled IFO (with appropriate calibration factors). As the detector is still in

the commissioning phase (during which time the Michelson locking-servo electronics

change regularly), models of the electronics need to be generated for the particular data

segment being calibrated. Details of this process are given in chapter6. An implemen-

tation of this optimisation system is presented in chapter5.

4.2.2 Correcting for the optical response

The optical response of the detector can be considered conceptually as two separate

responses. Looking at equation4.1, we see that it can be split into a varying gain factor

that is flat in frequency, and a frequency dependent response that has fixed overall gain.

The parameters that describe both parts are obtained in the system identification routine.

To correct the detector output in the time-domain, we need to divide the output signal

by the flat gain factor and then filter the result through the inverse of the frequency

dependent part. When the error-point signal is ‘corrected’ in this way, it represents (at

least at frequencies where there is no loop gain) the differential displacement of the end

mirrors of the interferometer. Stating this mathematically, we can write the correction

for the flat time-varying gain as

Popt[n] =P[n]

GP [n], (4.13)

wheren is the sample being processed, P[n] are the recorded samples of the high-power

error-point signal,P , sampled at 16384 Hz, andGP [n] are the samples of the overall

optical gain factor (which we so far have at only 1 Hz). From this we can recover a

representation of the differential mirror displacement by

dxopt[n] = Fopt ⊗ Popt[n], (4.14)

where⊗ denotes filter application, andFopt is a time-domain filter that has a frequency

response equal to the inverse of the estimated optical response of the detector.

We can see that two problems immediately arise. Firstly, we need a sample ofGP at

each sample instant of the error-point if we are to avoid discontinuities in the recovered

strain signal (see the discussion for the calibration of the power-recycled GEO, section

4.2 Time-domain calibration of DR GEO 147

3.7.2, page107). The error-point signal is sampled at 16384 Hz whereas the estimates of

GP are only obtained every second. We must therefore up-sample the estimates ofGP

to 16384 Hz. The details of a possible method for doing this are presented in chapter5.

The other problem concerns the inverse of the optical response. The inverse of the optical

response will have two zeros and one pole2. While such a filter would be unstable if

implemented as an analogue filter, it can be stable as an IIR digital filter. A IIR time-

domain filter with the response of the inverse of the optical response is, however, difficult

to design; the response of the filter typically deviates from the desired response towards

the Nyquist rate of the filter. One method of dealing with this is to design the filter for a

much higher sample rate of data; the response of this filter can be controlled by placing

additional poles at high frequency (much higher than the band we are interested in): this

stops the filter blowing up at high frequencies. This method is not so useful when the data

are sampled at a rate of 16384 Hz and our band of interest is from 50 to 6000 Hz so the

data needs to be up-sampled. After filtering, the corrected signal can be downsampled

to 16384 Hz with minimal introduction of errors. The design and details of this filtering

are presented in chapter5 where the method is demonstrated on simulated data.

4.2.3 Correcting for the length-control loop

The principle of loop gain correction was introduced in the calibration of the power-

recycled interferometer (see section3.7.1on page104) and is no different for the case

of calibrating the dual-recycled interferometer. Another possible method can also be

explored that, in principle, could provide a better representation of the loop correction

signals. If we look at a detailed system diagram for the calibration process (figure4.7),

we can see that there are two possibilities for recovering the signals labelledsfbandffb.

One method uses the recorded error-point signal filtered through a model of the feedback

electronics. While this provides a completely noise-free representation of the feedback

signals, it also relies on a accurate model of the (complicated) feedback electronics. The

other method is to record the feedback signals that are applied to the two actuators. This

2Inverting a filter response can be considered as converting all poles to zeros, and all zeros to poles

(with a suitable change of gain).

4.3 Summary of the procedure 148

method, in principle, provides the best representation of the feedback signals, especially

as it also includes any gain drifts of the electronics. However, if the electronics add any

noise to the error-point signal, then these two recorded feedback signals would add noise

to the recovered strain signal. These issues are explored in the context of the calibration

scheme applied to the S3 science run data (see page197, section6.2.3).

4.2.4 Calculating the strain signal

From the optical correction and loop correction tasks, we recover three data streams that

each contain some information about the differential displacement of the end mirrors

of the interferometer:dxopt, dxsfb, anddxffb. These three signals, when appropriately

combined, give a calibrated strain signal. This was presented in detail in the calibration

scheme for the power-recycled GEO (see section3.7.1on page104) and is stated here

just for completeness. The recovered strain signal is then given by

h[n] = (dxopt[n] + dxsfb[n] + dxffb[n])/1200. (4.15)

4.3 Summary of the procedure

The suggested calibration procedure can be summarised using the following recipe. Only

thePquadrature is considered but the recipe can be applied similarly to theQ quadrature.

Step 1: Transform one second of the input calibration signal and the detector output

signal into the Fourier domain and form the complex transfer function at the

calibration line frequencies.

Step 2: Fit a parameterised model of the detector response function to the measured

transfer function using an optimisation routine. This procedure gives estimates

for the parameters:GP , P z f , P p f , andP pQ.

Step 3: Up-sample parameterGP from 1 Hz to 16384 Hz and divide each sample of the

detector output signal by the corresponding sample ofGP .

4.3 Summary of the procedure 149

Step 4: Up-sample the output of step 3 by a factor of 4.

Step 5: From parametersP z f , P p f , andP pQ, construct IIR filters for a sample rate

of 16384× 4 Hz that correspond to the inverse of the optical response of the

detector and filter the output of step 4.

Step 6: Down-sample the output of step 5 back to 16384 Hz.

Step 7: Filter the detector output signal through model IIR filters of the feedback elec-

tronics and actuators to give two correction signals that, when added to the

output of step 6, give the differential arm-length changes of the detector for all

frequencies where the detector model is valid. This signal can then be converted

to a strain signal.

Table4.1summarises the input data streams of the calibration scheme, together with the

fixed and variable parameters that are used calibrate the data. The full procedure is also

shown graphically in figure4.7.

4.3 Summary of the procedure 150

Input data Description Reference

Detector output The error-point signal demodulated

in-phase at the high-power photodi-

ode.

section4.2

Injected calibration signal The calibration signal applied

differentially to the electro-static

drives.

section4.2

Fixed input parameters Description Reference

Hcom A model of the electronics common

to both feedback paths including the

overall electronic gain.

section4.2

HIMD A model of the IMD including a DC

calibration factor.

section4.2

Hslow A model of the slow path electron-

ics.

section4.2

HESD A model of the ESDs including the

DC calibration factor.

section3.8.3

Variable parameters Description Reference

GP The overall optical gain. section4.1.2

P z f The frequency of the real zero in the

Poptical transfer function.

section4.1.2

P p f The frequency of the complex pole

pair in thePoptical transfer func-

tion.

section4.1.2

P pQ The quality factor of the complex

pole pair in thePoptical transfer

function.

section4.1.2

Output data Description Reference

h(t) The strain signal. section4.2.4

Table 4.1:A summary table of the input/output data streams and parameters

of the calibration scheme designed to calibrate the dual-recycled GEO 600.

4.3 Summary of the procedure 151

downsa

mpl

e÷

4er

rorp

oint

downsa

mpl

ing

ups

ampl

ing

optica

lga

in

ups

ampl

e×

1638

4

HIM

D

IIR

filter

HE

SD

IIR

filter

optica

lga

ines

tim

ate

mea

sure

peak

s&

ep[i]

optg

[i]

corr

ection

optica

lga

in

Hco

m

IIR

filter

Hsl

ow

IIR

filter

Hfa

st

IIR

filter

corr

ection

optica

lre

spon

se

IIR

filter

tost

rain

conve

rt÷

1200

ups

ampl

ing

erro

rpoi

nt

ups

ampl

e×

4

h(t

)

Res

pon

seco

rrec

tion

Sys

tem

iden

tific

atio

n

GP

Pp

f,P

pQ,P

z f

Ser

voco

rrec

tion

h(t

)co

nst

ruct

ion

LSC

MID

CA

L

LSC

MID

EP

-PH

P

LSC

MID

FP

-MC

EI-

MC

EI

LSC

MID

FP

-MC

E-M

CE

ffb

sfb

dx

sfb

dx

ffb

dx

opt

Figure 4.7:A simplified schematic of the data processing pipeline for cali-

brating the dual-recycled GEO. The two possible loop correction procedures

are shown, with ‘mechanical’ switches representing the conceptual switch

between the two methods. Input signals are labelled using their GEO DAQ

labels (with the G1: prefix dropped). These signals, and their recording, are

discussed in more detail in chapter6.

Chapter 5

Calibration of the dual-recycled

GEO 600 — Simulations and software

In order to test the various aspects of the signal processing pipeline designed to cali-

brate the dual-recycled GEO, it is desirable to test the process on data for which the

correct output of the processing is known. As we don’t know what strain the detector

experiences (that being the point of calibration), we must use simulations to validate the

pipeline. Using simulated data allows most of the pipeline to be tested and its accuracy

determined. In addition, the processing of simulated data allows any systematic errors

present in the calibration scheme to be quantified.

This chapter describes the development of a time-domain simulation code that produces

output signals comparable to those from GEO. The output of the simulation is stored in

frame files (as with the ‘real’ GEO data) such that the developed calibration software

can be applied directly to simulated and real data. The development of a time-domain

calibration code is discussed and used to calibrate the simulated data. The results of the

calibration are presented at the end of this chapter.

5.1 Simulation setup 153

5.1 Simulation setup

The time-domain simulation is an open-loop model of the detector as it is difficult to

imitate a feedback servo in the time-domain. This means that the simulation can test all

aspects of the calibration scheme described in chapter4, apart from the loop correction

part. The loop correction can be independently verified (see chapter6). Fluctuations

in the frequency dependent response of optical transfer function are also omitted; only

the overall gain of the optical response is varied, allowing the calibration process to be

tested in the presence of large overall gain fluctuations. However, the presence of noise

in the system means that optical variations appear to be there: the recovered parameters

(as we shall see) are noisy. As this noise cannot be distinguished from other fluctuations,

the determination of, and correction for, the optical response can be tested completely

(see experiment 2 on page5.3.2).

The open-loop simulation was developed in MATLAB and a schematic of the process is

shown in figure5.1. This is very similar to the MI servo loop model of figure4.5with the

obvious exception that the feedback paths are missing. The electrostatic drive actuator

is included so that the on-line transfer function measurements detailed in section4.2.1

are the same. The difference lies in the model that is fitted to the data: it only contains

the optical transfer function of thePquadrature.

HESD

ESD

[V][m]

calibrationinput

displacementexternal

signalse.g., h(t) induceddisplacement

[m] [V]G

DX

CA

L

N EP-P HP

OPTG

[V][V] FP error-point

output in-phase (P)

detector

Figure 5.1: A block diagram of the open-loop simulation implemented in

MATLAB to allow detailed testing of the calibration software. The output

signals of the simulation are shown in red with their prefixes G1:LSCMID

omitted for clarity. These signals are described in section5.1.1.

5.1 Simulation setup 154

The optical response of the detector,HP , is split into a flat (possibly varying) gain stage,

G, and a fixed gain frequency dependent part,FP . Both the ESD and the frequency

dependent part of the optical response are implemented as IIR filters. An additional time-

varying gain is included in the optical transfer function to imitate changes in the overall

gain of the system. Throughout the following experiments, the calibration procedure

concentrates on thePquadrature only.

5.1.1 Simulation outputs

Channel name Description Sample rate

G1:LSCMID EP-PHP The error signal demodulated in-

phase. (P )

16384 Hz

G1:LSCMID CAL The injected calibration signal. 16384 Hz

G1:LSCMID LS The lock status of the detector. 1 Hz

G1:LSCMID MAINT The maintenance status of the de-

tector.

1 Hz

G1:LSCMID OPTG The overall optical gain of the de-

tector.

16384 Hz

G1:LSCMID DX The injected displacement noise of

the detector, including the injected

calibration signal.

16384 Hz

G1:LSCMID N The injected displacement noise of

the detector, excluding the injected

calibration signal.

16384 Hz

Table 5.1:The various signals output from the open-loop simulation of GEO.

The table is split into two parts: the upper part lists the signals that can

be measured at the real detector; the lower part lists those signals which

are hidden (unmeasurable) in the real detector — they are recovered by the

calibration process.

The simulation outputs the signals listed in table5.1. Two of the detector status channels

are also simulated (lock status and maintenance channel). These allow the calibration

5.1 Simulation setup 155

code to be tested for different detector states (for example, across losses of lock). In

the experiments and tests discussed in this chapter, these two channels are fixed at their

nominal values: lock status high (1), maintenance channel low (0). They are included

for completeness in this list and are discussed in more detail in chapter6.

Typical snap-shot spectra of the main signals are shown in figure5.2. A time-series of

the overall optical gain is also shown.

104

100

10−4

10−8

103102

Am

plitu

de

(V/√

Hz)

10−10

10−11

10−12

10−13

10−14

102 103

Am

plitu

de

(m/√

Hz)

Frequency (Hz) Frequency (Hz)

G1:

LSC

MID

CA

L

100

10−2

10−4

10008006004002000

7

7.5

6.5

6

5.5

5

Time (s)102 103

Frequency (Hz)

Am

plitu

de

(V/√

Hz)

Am

plitu

de

(×10

9V

/m)

G1:

LSC

MID

EP

-PH

P

G1:

LSC

MID

OP

TG

G1:

LSC

MID

DX

Figure 5.2: Typical spectra and time-series of the main simulation output

signals. The calibration lines are marked with (blue) circles.

The frequency dependent part of the optical response is implemented using an IIR filter

designed with LISO to have a complex pole at 600 Hz with a Q of 2.77, and a real zero

at 730 Hz. This choice is somewhat arbitrary as the optimisation routine will recover any

parameter values (physically plausible or not) that make the model fit the data.

5.2 Calibration software 156

5.2 Calibration software

Development of the calibration code follows closely the calibration pipeline detailed in

chapter4. This section looks in more detail at the implementation of the signal process-

ing pipeline. The entire calibration pipeline is implemented as a C code.

5.2.1 System identification

The system identification routine designed to calibrate the open-loop simulated data uses

a simplified version of the model discussed in section4.2.1. From the output channels

G1:LSCMID CAL and G1:LSCMID EP-PHP, and using a model of the electrostatic

drive actuator, we can form the measured transfer function

Tmeasuredf =

EPf

EC f EE SDf, (5.1)

where the subscriptf represents the frequencies of the calibration lines. The optimi-

sation routine tries to fit the pole/zero model of the optical response to this measured

transfer function. From this we recover the parameter set:G; P p f ; P pQ; P z f .

The optimisation routine is implemented as C code and is based on the (sophisticated)

core fitting algorithm of LISO [Heinzel99]. The algorithm uses various minimisation

routines to determine the set of parameters that best describes the data. When the al-

gorithm reaches a set tolerance that represents no change in the new parameter guesses

compared to the previous guesses, it returns these parameter estimates, together with a

measure of success (calledχ2), which is the sum of the squares of the differences be-

tween the model (using the final parameter set) and the data. Thisχ2 value is a good

indication of whether or not the model is a suitable description of the data. In the case

where the measured data is non-stationary (for example, in the real the detector output

data), the fluctuations in theχ2 value also represent, to some extent, the quality of the

underlying data.

As the optimisation routine is intended to run once per second, it is limited in the number

of parameter guesses it makes. If it reaches the maximum number of iterations without

5.2 Calibration software 157

reaching the required parameter-change tolerance, then the routine returns an error. In

this way, the main calibration code knows that the optimisation did not converge to a

suitable solution in the time allowed; it can then decide what to do in calibrating the

current second of data. Having such a limitation means that the optimisation routine al-

ways returns either a solution or an error in much less than one second. This is important

because we want the final calibration code to run in real-time.

5.2.2 Optical correction

The code that corrects for the optical response of the detector is implemented exactly as

described in section4.2.2. This section looks in more detail at the implementation and

in particular at the performance of the up- and down-sampling filters.

Up-sampling the overall optical gain estimates

As discussed earlier, the 1 Hz estimates of the overall optical gain need to be up-sampled

to 16384 Hz so that the error-point signal can be smoothly corrected. This is done using

a series of FIR filters designed to perform band-limited interpolation. Each FIR filter

allows the data to be up-sampled by a factor 4. After 7 such up-samplings, the data

is at 16384 Hz. Up-sampling repeatedly by small factors allows the use of ‘shorter’

filters that have a better response and introduce fewer numerical errors. The higher

the up-sampling factor, the higher the number of filter coefficients required to perform

successful band-limited interpolation. After a number of experiments, it was found that

splitting the up-sampling into stages with factor 4 each stage, was the most robust and

computationally efficient way of getting from 1 Hz to 16384 Hz.

Figure5.3 shows a sample time-series of the recovered overall optical gain estimates,

together with the up-sampled version that is used to correct the error-point signal. The

spectral content of the up-sampled version is shown in figure5.4. The underlying overall

optical gain has three spectral components at frequencies 10 mHz, 1 mHz, and 0.1 mHz.

The first two of these modulations can be seen in the spectrum; all other spectral content

5.2 Calibration software 158

Upsampled to 16384 Hz

1 Hz optical gain estimates

7000

6700

6800

6900

660010 12 14 16 18 20

Am

plitu

de

(V/µ

m)

Time (s)

Figure 5.3: A sample time-series of the

overall optical gain estimates recovered

by the calibration software from the sim-

ulated data. The time-series up-sampled

to 16384 Hz is also shown.

105

104

103

102

101

100

10−3 10−2 10−1 100 101

Am

plitu

de

(V/µ

m/√

Hz)

Frequency (Hz)

G1:DER DATA OPT-UP

Figure 5.4:The spectral content of the up-

sampled optical gain estimates. The spec-

trum is made from 1000 s of data with no

averages; a Hanning window is applied.

The underlying optical gain fluctuations

were at 10,1,and 0.1 mHz.

introduced by the up-sampling process is 2-3 orders of magnitude below.

Inverse optical filters

If we wish to allow for the possibility that the optical response of the detector may vary

(for example, the pole frequency may drift in frequency), then we need a robust way

to calculate new filters as the need arises, without stopping the calibration process. In

principle we want the inverse optical filter to smoothly change its response to follow

any changes in the optical response of the detector. A good way to do this is to use the

bilinear transform to calculate the filter coefficients. The details of the bilinear transform

are presented in appendixA.1.1. As was previously mentioned, the inverse of the optical

response is difficult to implement with a sample rate of 16384 Hz. To improve the filter,

we can design it to work at a higher sample rate, and add extra poles at high frequency in

order to temper the response. The resulting transfer function of the inverse optical filter

can be written in the S-domain as

H ′

P (s) = G(s − pP )(s − p∗

P )

(s − zP )×

1

(s − phf)(s − p∗

hf), (5.2)

5.2 Calibration software 159

where the complex pole,phf, is added to control the filter response, and the gainG is

chosen so that the filter has unity gain at DC. The complex zero,pP is that pole deter-

mined by the system identification process from the forward optical transfer function;

likewise, the real pole,zP , is the real zero recovered from the forward optical transfer

function.

30

25

20

15

10

5

00 2 4 6 8 10 12 14 16 18 20

Sample rate (16384 Hz)

Per

cent

age

erro

r(%

)

Figure 5.5:The magnitude difference of an S-domain filter and an IIR fil-

ter designed with the bilinear transform. The difference is expressed as

a percentage error and is plotted again multiples of the base sample rate

(16384 Hz).

An additional consideration is that the bilinear transform method of designing filters

produces filters with responses that can be quite poor approximations of the desired

S-domain response as compared to other filter design methods. Typically, filters de-

signed using the bilinear transform method deviate from the desired response towards

the Nyquist rate of the filter. This deviation in the gravitational wave band, reduces as

the sample rate of data to be filtered increases (since the Nyquist rate of the filter in-

creases). This can be quantified, at least for an example filter. To do this, we take an

S-domain filter whose frequency response is defined by a complex pole at 800 Hz with

a quality factor 2. Now we construct, using the bilinear transform, a set of IIR filters

with approximately the same frequency response but designed to filter data of different

sample rates. We can then compare the magnitude response of each IIR filter with the

S-domain filter. If we form the ratio of the two filters and then sum over all frequencies

in the detection band, we have a measure of how well the IIR filter mimics the S-domain

5.2 Calibration software 160

Ideal filter

Bilinear IIR filter

10 410 310 210 110 0

20

−20

0

−40

−60

−80

0

−50

−100

−150

−200

−250

Frequency (Hz)

Phas

e(

)M

agnit

ude

(dB

)

Figure 5.6: A Bode plot of an ‘ideal’ S-

domain filter and an IIR filter designed

using the bilinear transform for a sample

rate of 16384 Hz.

10 310 2

1

0.9

0.8

0.7

0.6

0.5

50

40

30

20

10

0

Mag

nit

ude

rati

oP

has

ediff

eren

ce(

)

Frequency (Hz)

Figure 5.7: The magnitude ratio and

phase difference between the S-domain

and IIR filters shown in figure5.6.

filter at the frequencies of interest. Figure5.5shows a plot of the differences, expressed

as a percentage, as a function of multiples of the base sample rate (16384 Hz). An exam-

ple of the deviation of the two filters for a sample rate of 16384 Hz is shown in figures

5.6and5.7.

As we can see from figure5.5, a sample rate of between 10 and 20 times 16384 Hz

allows a filter to be used which results in only a small total percentage difference from

an optimal filter across the detection band. For this reason, and because we wish to place

the ‘tempering’ poles at high frequency, a sample rate of 16×16384 Hz was chosen to

do the inverse optical filtering. The additional complex pole was added at 100 kHz with

a Q of 3. Figure5.8shows an example of the inverse filter used to correct for the optical

response of the detector, complete with the response of a perfect filter. The differences

between this inverse optical filter and the perfect filter are shown in figure5.9.

Up-sampling the error-point signal

In order to apply the inverse optical correction filter described above, the detector error-

point (P ) needs to be up-sampled by a factor 16. This is done using the same method that

was applied in up-sampling the 1 Hz optical gain samples (see section5.2.2). The same

5.2 Calibration software 161

IIR filter response

Perfect filter response

102

101

100

150

100

50

0

−50

−100105104102 103

Mag

nit

ude

Phas

e(

)

Frequency (Hz)

Figure 5.8:An example of the inverse op-

tical IIR filter together with the response

of a perfect filter. The IIR filter is designed

to operate at 16×16384 Hz.

1.01

1.005

1

0.995

0.99

5

4

3

2

1

0

−1

102 103

Frequency (Hz)

Mag

nit

ude

rati

oP

has

edifer

ence

()

Figure 5.9:The phase and magnitude dif-

ferences between an inverse optical IIR

filter designed using the bilinear trans-

form and a perfect filter. The IIR filter is

designed to operate at 16×16384 Hz.

×4 filter is used in two stages to up-sample by the required factor 16. We can do the

same tests that were done for the optical gain samples to see how well the up-sampling

performs. Figure5.10shows snap-shot spectra of the error-point signal at 16384 Hz and

of the up-sampled version (at 16×16384 Hz). The spectra are made from 1 second of

data. The differences between the simulated error-point and the up-sampled version are

small across the detection band: less than 1% in magnitude and less that 1 degree in

phase (see figure5.11).

5.2.3 Calculation of the strain signal

Since the simulation is open-loop, correcting the error-point signal for the effect of the

optical response produces a signal that represents differential armlength changes at all

frequencies. So, for calibrating the simulated data, the strain signal is simply given by

h[n] = dxopt[n]/1200. (5.3)

5.2 Calibration software 162

G1:LSC MID EP-P-UP

G1:LSC MID EP-P HP100

10−1

10−2

10−3

10−4

10−5

10−6

102 103 104 105

Am

plitu

de

(V/√

Hz)

Frequency (Hz)

Figure 5.10: Snap-shot spectra of the

simulated error-point signal and the up-

sampled version. The spectra are made

from 1 second of data.

1.01

1.005

1

0.995

0.99

1

0.5

0

−0.5

−1103102

Phas

ediff

eren

ce(

)M

agnit

ude

rati

o

Frequency (Hz)

Figure 5.11: The magnitude and phase

differences between the two spectra

shown in figure5.10. The differences are

only shown across the detection band.

5.2.4 Output of software

To aid in debugging and to help with characterising the detector, the calibration soft-

ware was designed to output a number of diagnostic channels that are derived during the

calibration process. Table5.2shows a list of the channels that are output from the cali-

bration software and recorded in standard frame format. Those not previously introduced

are discussed in the remaining text of this subsection.

Peak measurement channels

The system identification process makes amplitude and phase measurements of the cal-

ibration peaks in both the injected calibration signal and the detector error-point signal.

These peak measurements are very useful for understanding the performance of the de-

tector over long time-scales. Currently only the magnitude of the peak measurements

are stored as a channel but this can easily be extended to include the phase as well. In

addition the peaks are also measured in the recovered strain signal. The measurements

are recorded at 1 Hz in double precision. The magnitude and phase is calculated from

a one second amplitude spectrum. The data are windowed prior to performing the FFT

using a [HFT95] window. The data are stored with channel names that are automatically

5.2 Calibration software 163

Signal name Description Data type

DER DATA H The recovered strain signal. double

DER DATA QUALITY The encoded quality channel. short

DER DATA OPT-DIRECT The direct measurement of the optical gain (see

text).

double

DER PARAM 0 The recovered overall optical gain. double

DER PARAM 1 The recovered pole frequency in Hz. double

DER PARAM 2 The recovered pole Q. double

DER PARAM 3 The recovered zero frequency in Hz. double

DER PARAM CHISQ The recoveredχ2 parameter. double

DER PEAK CAL-XXX The calibration peaks as measured in the de-

whitened calibration signal. The measured val-

ues are in Vrms. A channel is created for each

calibration peak frequency with the -XXX suf-

fix being replaced by the calibration line fre-

quency.

double

DER PEAK PEP-XXX Peak measurements for each calibration line in

the high-power diodeP .

double

DER PEAK QEP-XXX Peak measurements for each calibration line in

the high-power diodeQ .

double

DER PEAK H-XXX Peak measurements for each calibration line in

the recovered strain channel.

double

DER LOCK STATUS The binary representation of the analogue lock

status channel.

short

DER MAINT STATUS The binary representation of the analogue main-

tenance switch channel.

short

Table 5.2:The output channels of the calibration process. Each channel is

recorded with the standard GEO prefix ‘G1:’ (omitted here for clarity). All

data are recorded at 1 Hz with the exception of DATAH which is recorded

at 16384 Hz.

5.2 Calibration software 164

calculated from the details of the peak using the format

G1:DER_PEAK_***-nnn

where ‘***’ is replaced by the channel in which the peaks are measured and ‘nnn’ is re-

placed by the frequency at which the measurement is made. A typical channel name for a

peak measured at 600 Hz in the injected calibration signal is then G1:DERPEAK CAL-

600.

The χ2 channel

Theχ2 channel stores theχ2 values returned from the system identification optimisation

routine. The samples are stored in double precision at 1 Hz.

The quality channel

The quality channel aims to encode information about the quality of theh(t) data stream

with a one second resolution. At the most basic level, this channel includes information

about whether the detector is locked or not. Figure5.12shows the first 5 bits of the 16-bit

word in the way that it is used. In general, interpreting this channel requires looking at

the individual bits except in certain cases where the value of the channel can be directly

interpreted. In the case that a sample of the quality channel is precisely zero, then the

data quality can be considered as highest. If the sample is precisely 1, then the detector

is unlocked; precisely 2 and the maintenance switch is set; 3 means that the detector is

unlocked and the maintenance switch is on, and so on.

The lock status bit is derived from the detector lock status channel. The detector lock

status channel (G1:LSCMID LS) is an analogue signal produced by the microprocessor

that governs the locking system of the detector. This signal is around 0 V when the

detector is unlocked and is around 5 V when the detector is locked and detuned to the

target frequency response. During the encoding of this information into the lock status

5.2 Calibration software 165

0 1 2 3 4BIT

mai

nten

ance

stat

us

χ2

thre

shol

d1

χ2

thre

shol

d2

lock

stat

us

χ2

thre

shol

d3

Figure 5.12:A bitwise representation of the quality

channel. Currently, the quality channel is a 16-bit

integer but in principle can be extended to longer

data types.

bit, the logic is inverted such that the bit being set (equal to 1) corresponds to unlocked,

and the bit being unset (equal to 0) corresponds to the detuned locked state.

In general, operation of the detector is done from a remote building. If it is necessary

for an operator to enter the building during a science run, a maintenance switch is set.

The output of the switch is recorded in one of the data acquisition channels. The voltage

level as seen by the DAQ system is around 5.0 V if the switch is set to ‘maintenance

mode’, and around 0 V when the switch is not set. This information is encoded in the

quality channel such that bit 1 of the 16-bit word is set (equal to 1) if the detector is in

maintenance mode, and unset (equal to 0) if not in maintenance mode.

The χ2 parameter recovered from the system identification routine is compared to a

number of threshold values that are set in the configuration file of the calibration soft-

ware. If theχ2 crosses any one of these thresholds, the corresponding bit in the quality

word is set.

5.3 Simulation experiments 166

5.3 Simulation experiments

A number of different experiments were performed with the aim of validating the full

signal processing pipeline. Figure4.7shows an overview of the designed signal process-

ing pipeline. For the simulations, this pipeline is simplified as described below. Each

experiment is discussed below where the aim of each is given together with the results.

5.3.1 Experiment 1 – Timing aspects of the pipeline

The first step in validating the pipeline is to confirm that all of the timing aspects are

correct. The use of up- and down-sampling filters means that data streams are shifted in

time relative to each other (this requires buffering of the data). Since we wish to combine

various data streams and then end up with a strain data stream that is appropriately

phased with respect to the recorded detector error-point, we must use the data from the

correct places in the data buffers. For example, in up-sampling the error-point signal by

a factor of 16, we delay the signal by around 1.2 ms (315 samples at the high sample

rate).

To deal with these time-shifts, each signal is buffered long enough to allow the correct

segment of data to be accessed. This process is somewhat complicated and relies on

correct determination of the filter delays, as well as on correct implementation of the

buffering system.

To test the buffering of the error-point signal, all of the IIR filtering was replaced in

the pipeline by simple copying so that the strain signal that is produced by the process

should (if all of the pipeline timing is correct) be a copy of the input error-point signal

(taking into account any noise introduced in the up- and down-sampling processes). This

does not test the up-sampling and buffering of the overall optical gain estimates as this

part of the code is deactivated—this is tested in a later experiment. Figure5.13shows

the core elements of the calibration pipeline, highlighting which elements are bypassed

in this experiment.

5.3 Simulation experiments 167

opticaloverall

gainopticalinverse

filter

ep[n]opt[n] fs× 16

up-sample down-sample

bypassed bypassed

fs÷ 16 DATA HEP-P HP

Figure 5.13:A schematic of the core calibration pipeline for experiment 1.

Those elements which are bypassed in this experiment are indicated.

−1.5

−1

−0.5

0

0.5

1

1.5

1.011.0081.0061.0041.0021

Am

plitu

de

(V)

Time (s)

simulation inputsimulation output

Figure 5.14:A time-series showing the in-

put and output of the simulation for the

conditions described in experiment 1.

0

1× 10−3

−1× 10−3

1 1.002 1.004 1.006 1.008 1.01

Time (s)

Am

plitu

de

diff

eren

ce(V

)

Figure 5.15:The amplitude difference be-

tween the input and output time-series for

experiment 1.

Figure5.14shows a section of time-series of both the input data (error-point signal) and

the output data (nominally the strain signal). The differences between the two are shown

in figure5.15and are well bellow 1%. From this experiment, we see that the timing and

buffering aspects of the calibration pipeline are implemented correctly.

5.3.2 Experiment 2 – Performance of the optical filter

The second experiment extended the conditions of experiment 1 to include the fixed

optical response. Both the simulation and the calibration software include a fixed optical

response. This allows the inverse optical filtering to be tested. We compare now the

output of the calibration procedure to the underlying strain signal given bydx[n]/1200.

The core calibration pipeline is as in figure5.13with the exception that the inverse opti-

5.3 Simulation experiments 168

DER DATA H

LSC MID DX

10−14

10−13

10−12

10−11

102 103

Am

plitu

de

(m/√

Hz)

Frequency (Hz)

Figure 5.16:Snap-shot spectra of the un-

derlying differential displacement and the

output of the calibration process used in

experiment 3. Each spectrum is formed

from 1 second of data with a Hanning

window applied.

1.05

1

0.95

5

0

−5102 103

Phas

ediff

eren

ce(

)M

agnit

ude

rati

o

Frequency (Hz)

Figure 5.17: The magnitude and phase

differences between the two spectra of fig-

ure5.16.

cal filtering is no longer bypassed. Figure5.16shows snap-shot spectra of the underlying

differential displacement signal (LSCMID DX) and of the output of the calibration pro-

cess. The effect of the down-sampling filter around 8 kHz can clearly be seen. Figure

5.17 shows the magnitude and phase difference between the two spectra. The differ-

ences are small (well below 5%) in the detection band indicating that the inverse optical

filter designed using the bilinear transform is a good approximation to the inverse of

the underlying optical response. It should be noted, however, that this systematic error

includes the error in the IIR filter used to imitate the optical response of the detector.

Figure5.18shows the magnitude and phase differences between the IIR filter that gives

the simulated optical response of the detector, and a perfect S-domain filter with the

precise underlying pole/zero values. We can see immediately that the IIR filter does not

have a complex pole at precisely 600 Hz. We also see an unexpected notch-like struc-

ture around 2.2 kHz. The errors in figure5.16are therefore more properly interpreted

as an upper limit for the errors introduced by the inverse optical filtering since they are

dominated by the systematic errors of the optical response filter (from figure5.18).

This experiment was also repeated to test the possibility of updating the inverse optical

filter to account for fluctuations in the estimated parameters. In this experiment, the

5.3 Simulation experiments 169

102 103

1.1

1

0.9

10

5

0

−5

−10

Mag

nit

ude

rati

oP

has

ediff

eren

ce(

)

Frequency (Hz)

Figure 5.18:The magnitude and phase difference between the IIR optical

filter used in the simulations and an ideal optical response modelled using

poles and zeros.

underlying optical response was fixed so the recovered parameters vary only due to the

presence of noise. However, the process of updating the inverse optical filters can still

be tested since noise cannot be distinguished from fluctuations due to other sources. The

details of the filter updating and results of this experiment are discussed further in section

6.2.2on page195.

5.3.3 Experiment 3 – Overall optical gain correction

The third simulation experiment introduces variations in the overall optical gain. Vari-

ations at three frequencies were introduced: 10, 1, and 0.1 mHz. The calibration code

now includes the correction for the recovered overall optical gain.

Again we compare the underlying input differential displacement to the calibration out-

put signal. Figures5.19and5.20show the spectra and magnitude and phase differences

of the two signals. The results are similar (as expected) to experiment 3 but it is clear

that in correcting for the overall optical gain, a small amount of additional noise is added

in the process.

5.3 Simulation experiments 170

DER DATA H

LSC MID DX

10−11

10−12

10−13

10−14

102 103

Frequency (Hz)

Am

plitu

de

(m/√

Hz)

Figure 5.19:Snap-shot spectra of the un-

derlying differential displacement and the

output of the calibration process used in

experiment 3. Each spectra is formed

from 1 second of data with a Hanning

window applied.

0

−5

5

0.95

1

1.05

102 103

Phas

ediff

eren

ce(

)M

agnit

ude

rati

o

Frequency (Hz)

Figure 5.20: The magnitude and phase

differences between the two spectra of fig-

ure5.16.

As we have a simulation, we can also compare the recovered overall optical gain esti-

mates to the underlying optical gain (LSCMID OPTG) since we record it in the simula-

tion. Figure5.21shows a short time-series of the recovered overall optical gain estimates

together with the underlying values for the same time stretch. The agreement is good,

confirming again the timing aspects of the pipeline and the ability of the system identifi-

cation routine to correctly determine the state of the detector.

5.3.4 Experiment 4 – System identification performance with fixed

optical gain

This experiment aims to quantify the performance of the system identification routine in

the case where the overall optical gain is fixed. The amplitude of each injected calibra-

tion line is set to give an SNR of around 50 in the error-point signal. The simulation was

performed for a much longer time (3 hours of data) so that reasonable statistics could be

gathered on the parameter estimations.

In this experiment, the only thing affecting the parameter recovery process is the pres-

5.3 Simulation experiments 171

PARAM 0

LSC MID OPTG

6800

6600

6400

6200

6000

5800

5600

5400

52000 10 20 30 40 50 60 70 80 90

Ove

rall

opti

calga

in(V

/µm

)

Time (s)

Figure 5.21: The overall optical gain estimates plotted together with the

underlying overall optical gain.

ence of noise on the measured peaks. We would therefore expect the recovered param-

eters to be normally distributed about the underlying values. In the case of the overall

optical gain, the underlying value was set to 6000. The optical parameters,P z f , P p f ,

andP pQ, were set to have approximate underlying values of 600, 2.8, and 730 respec-

tively. However, the underlying values are not known exactly and we already know from

figure5.18that they are not precisely 600, 2.8, and 730.

Using 10000 seconds of data, the following results were generated. Figure5.22shows

the time-series of the four parameters and theχ2 channel. The time-series clearly show

that the recovered parameters have mean values close to the underlying values. We need

to look more carefully at the distributions to quantify how good the recovery process is.

Figure5.23shows amplitude spectral densities of the time-series shown in figure5.22.

The units on the y-axis are deliberately omitted since each spectrum has different units.

The plot still, however, gives useful information. In particular, one can see clearly the

effect of low-pass filtering the parameter estimations. For the overall optical gain,G,

we see the corner of the filter was at 200 mHz; for the other parameters the corner of the

filter was set to 10 mHz. The plot also shows the absence of any spectral features. This

is as expected where the only noise present in the system is white. It is also important to

note that theχ2 values are correctly distributed.

5.3 Simulation experiments 172

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

1000

500

0

720

700

680

2.9

2.7

2.8

605

600

595

6200

6000

5800

GP

olef

Pole

QZero

fχ

2

Hz

Hz

Time (s)

Figure 5.22: Time-series of the four re-

covered parameters and theχ2 channel

from experiment 4.

χ2

Zero f

Pole Q

Pole f

G

101

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10−8

1 10 100

Frequency (mHz)

Am

plt

ide

(1/√

Hz)

Figure 5.23:Amplitude spectral densities

of the time-series shown in figure5.22.

We can also form histograms of the time-series and calculate the mean and standard

deviation of the recovered parameter set. Figure5.24shows the distributions of the four

optical parameters; figure5.25shows the distribution of theχ2 values. We can see that

the recovered parameters are distributed tightly around values which are close to the

underlying values. Remember, we do not know precisely the underlying values of the

parameters since they are represented by an IIR filter that only approximates the desired

response. The differences we see between the intended underlying values, and the mean

recovered values can easily be attributed to the approximate response of the IIR filter.

In addition, the coherence between different parameters was explored. All possible pa-

rameter pairs showed a very low coherence except for the pole and zero frequencies:

these showed a coherence around 0.8. More investigation is required to understand this

coherence.

5.3.5 Experiment 5 – System identification performance with vary-

ing optical gain

This is a repeat of experiment 4 but with the addition of a varying overall optical gain (as

described in experiment 3). In addition, the SNR of the calibration peaks was increased

5.3 Simulation experiments 173

600

500

700

400

200

100

0599 600 601 602 603

G Pole f

µ = 5984σ = 36

600

400

300

200

100

05800 5900 6100 62006000

500

300

Zero f500

400

300

200

100

0695 700 705 710 715 720

600

500

400

300

200

100

02.792.78 2.8 2.81 2.82 2.83

Pole Q

Hz

Hz

V/µm

σ = 0.4µ = 601.6

σ = 4µ = 709µ = 2.8

count

sco

unt

s

σ = 0.006

Figure 5.24:Histograms of the recovered

parameters from experiment 4. The mean

and standard deviation of each distribu-

tion is also shown.

1600

1400

1200

1000

800

600

400

200

00 100 200 300 400 500 600 700 800 900

χ2

count

s

Figure 5.25:A histogram of the recovered

χ2 values from experiment 4.

by a factor 2 to give each peak an SNR around 100 in the error-point signal. This should

give information about the dependence of theχ2 and parameter distributions on the SNR

of the calibration peaks. A calibration peak was also added at 2 kHz to see if the resulting

value for the zero frequency is affected.

The same results are generated as for experiment 4. Figure5.26shows the time-series

of the recovered parameters. The effect of increasing the SNR of the injected calibration

peaks can clearly be seen in theχ2 values which have reduced from values around 100 to

values around 4. Figure5.27shows the amplitude spectral densities of the time-series.

Now we can see effect of the underlying overall optical modulation on the recovered

overall optical gain,G. More surprising is that theχ2 values also show peaks at the

modulation frequencies. This is as yet unexplained and needs more investigation to

discover the source of this coupling.

Figure5.28shows histograms of the recovered values. The first thing to note in these

plots is that the recovered optical gain is no longer normally distributed (since we in-

troduced modulation) so it is not so straightforward to interpret the mean and standard

deviation of this distribution. Looking at the other parameters, we see that the increase

in the SNR of the calibration lines by a factor 2 has led to a reduction in the spread of the

distributions by about a factor of 2. It is not clear if the additional pole at 2 kHz made

5.4 Summary 174

4000

6000

8000

G

600

602

604

Pole

fHz

2.78

2.8

2.82

Pole

Q

720

710

700

Zero

fHz

0

20

40

100000 1000 2000 3000 4000 5000 6000 7000 8000 9000

χ2

Time (s)

Figure 5.26: Time-series of the four re-

covered parameters and theχ2 channel

from experiment 5.

χ2

Zero f

Pole Q

Pole f

G

101

100

10−1

10−2

10−3

10−4

10−5

10−6

1 10

Frequency (mHz)

100

103

102

Am

plt

ide

(1/√

Hz)

Figure 5.27: Amplitude spectra of the

time-series shown in figure5.26.

any impact or not on the recovery process.

5.4 Summary

The use of the open-loop detector simulation allowed a calibration program to be almost

entirely written, debugged, and tested. In addition, it was possible to get some quanti-

tative measure of the systematic errors and noise introduced by the calibration pipeline.

From these simulations we might expect the process to give errors around the 1-5% level

when used on real data.

The system identification procedure performs well and consistently recovers parameter

values that are close to the underlying values. In all of the simulations, the system

identification routine was able to recover a parameter set for every second of data (more

than 20000 seconds). We also saw that theχ2 ‘measure of goodness’ returned by the

optimisation algorithm is strongly dependent on the SNR of the injected calibration lines

as they appear in the error-point signal. The dependence of the parameter spread on the

SNR of the injected lines is almost one to one, at least for SNR values greater than 50.

The simulation has not tested the part of the calibration procedure that deals with the

correction of the Michelson length control servo; this must be tested separately using a

5.4 Summary 175

1000

800

600

400

200

601.5601600.5 602 602.5 6030

Pole fG

σ = 0.2

Zero f600

500

400

300

200

100

0706 712 714708 710 716

Hz

σ = 1.3µ = 711

400

300

1000

200

700

600500

Pole Q

2.78 2.79 2.8

σ = 0.003

4000 5000 6000 7000 8000

400

300

200

100

0

count

s

σ = 737µ = 6018 µ = 602

µ = 2.8

count

s

HzV/µm

Figure 5.28:Histograms of the recovered

parameters from experiment 5. The mean

and standard deviation of each distribu-

tion is also shown.

0 5 10 15 20 25

1200

1000

800

600

400

200

0

count

s

χ2

Figure 5.29:A histogram of the recovered

χ2 values from experiment 5.

frequency domain model of the detector (see chapter6).

Further simulations could lead to a better understanding of how the recovered values,

and their associatedχ2 values, depend on the calibration lines used. In particular, very

little investigation has been done to determine the effect of the frequency of the injected

calibration lines on the recovered parameters. In any case, the use of such a simulation

is invaluable for further optimising and extending of the calibration software. Currently

the calibration software runs at around 2-4 times real-time, making these investigations

quite time consuming. As the calibration code is optimised, we can expect significant

improvements in performance, opening the door to more extensive and systematic tests

using simulated data.

Chapter 6

Calibration of the dual-recycled

GEO 600 — application to the S3

science run

On the 5th of November 2003, GEO started to participate in a coincident science run

with other detectors in the LSC (for example, LIGO and TAMA). As for S1 (see section

3, page77), the aim of the science run was to collect high quality data continuously for

an extended period of time. Unlike in S1, GEO did not plan maintenance periods for S3;

any maintenance time that was required (for repairs or realigning) was taken as detector

down time. The S3 run was planned to span around 2 months. As GEO had only a

very short time (about 2 weeks) to prepare for the run after getting the dual-recycled

locking scheme finalised and working robustly, the plan was to run for the first week of

S3 and then go offline to allow a period of further optimisation of the detector. When the

sensitivity improvement of the instrument was deemed sufficient, GEO rejoined the S3

run. GEO resumed taking science quality data on the 30th December 2003 and continued

to run for a further 2 weeks. The two parts of S3 attended by GEO will be referred to as

S3 I and S3 II; S3 will be used when both parts are relevant.

For both S3 I and S3 II the DAQ system of GEO ran with 100% duty cycle and with

no reported errors on the central station DCU. The duty cycle of the instrument was

6.1 Experimental setup and detector details 177

similarly impressive: for S3 I, GEO achieved a duty cycle of around 95% with the longest

lock being around 27 hours; for S3 II, the duty cycle was 97% with a longest lock of 95

hours.

As the detector configuration changed between S3 I and S3 II, the calibration scheme

had to be adjusted to suit both states. This mostly involved changing the frequency

domain model of the detector used in the system identification routine, as well as the IIR

filters that mimic the servo electronics. This chapter will concentrate mostly on S3 II and

present the details of the detector and the implemented calibration scheme. Results from

the calibration procedure, together with some validation procedures and results, will be

presented in section6.3.

6.1 Experimental setup and detector details

For S3, GEO used a second photodiode to detect the output light field of the interferom-

eter. As was the case in S1, the second photodiode was a high-power diode implemented

out-of-loop such that around 80% of the output light was seen by it. The quadrant cam-

era of the auto-alignment system was used in-loop to generate a suitable error signal to

control the differential length of the interferometer. Figure6.1shows a schematic of the

length-control servo and differential-displacement sensing scheme that was used during

S3.

The following subsections will look in detail at various aspects of this model, including

measurements and model fits where appropriate.

6.1.1 Optical response of the detector

So far we have relied on the simulations from Finesse to determine the optical response

of the dual-recycled GEO. However, with the detector locked in DR configuration, we

can indirectly measure this transfer function. By injecting signals into the electrostatic

drives at the calibration injection point, we can measure a transfer function between this

6.1 Experimental setup and detector details 178

fast

feed

bac

kou

tput

slow

feed

bac

kou

tput

Hig

h-p

ower

phot

odio

de

outp

ut

out-

of-p

has

e(Q

)

Hig

h-p

ower

phot

odio

de

outp

ut

in-p

has

e(P

)

Low

-pow

er

phot

odio

de

outp

ut

in-p

has

e(q

uad

)

‘slo

wpat

h’

[V]

[m]

[m]

[V]

HIM

D

HE

SD

IMD

ESD

calib

rati

onin

put

dis

pla

cem

ent

exte

rnal

sign

als

e.g.

,h(t

)in

duce

ddis

pla

cem

ent

[V]

ESD

elec

tron

ics

[V]

‘fas

tpat

h’

[V]

[m]

HP

[m]

[V]

HQ

Com

mon

elec

tron

ics

[V]

[V]

Hco

m

IMD

elec

tron

ics

[V]

[V]

Hsl

ow

Hfa

st

[V]

[m]

Hquad

Hig

h-po

wer

Det

ecto

r(o

ut-of

-pha

se)

Hig

h-po

wer

Det

ecto

r(i

n-p

hase

)

Low

-pow

erD

etec

tor

(in-p

hase

)

Figure 6.1:A schematic of the length control servo and differential displace-

ment sensing setup of GEO for S3. The recording points of some relevant

signals are also indicated.

6.1 Experimental setup and detector details 179

input and the output at the high-power diode demodulated in-phase. If we then correct

that measurement for the response of the ESDs, then we are left with the optical response

of the detector combined with the response of the Michelson length-control servo. Figure

6.2 shows measured transfer functions of the optical response, measured as described

above using a swept-sine method; the legend shows the target detuning frequency. We

can see that the target detuning and the peak response are slightly different. This is

due to the algorithm that controls the many parameters of the detuning process; as the

procedure was very new at the time of these measurements, the scaling between the target

frequency that is set on the controls and the resulting detuned frequency was not properly

determined. The measurements were made prior to S3 I. At lower frequencies (below

400 Hz or so), the influence of the Michelson length-control servo can be seen, especially

on the phase. For the S3 I run, the SR cavity was tuned to give a peak resonance around

1200 Hz. We can also see that not only the pole frequency depends on the microscopic

de-tuning of the SR mirror, but the pole Q is also affected (although to a lesser degree).

5500 Hz

3000 Hz

2000 Hz

1000 Hz103

1011

1010

200

100

0

−100

−200

Mag

nit

ude

(V/µ

m)

Phas

e(

)

Frequency (Hz)

Figure 6.2:Measured optical transfer functions of GEO for different detun-

ings of the signal-recycling mirror. The measurements are made including

the response of the electrostatic drives; this response is subtracted out in the

plots.

For S3 II, the demodulation phase of the high-power diode was optimised for a frequency

6.1 Experimental setup and detector details 180

around the SR resonance which was set be approximately 1 kHz (see below). Transfer

functions were measured using the above technique with white noise instead of a swept-

sine (for speed). A MATLAB code was developed to fit the pole/zero model of the optical

response to the measured transfer functions. The fit was only done using the magnitude

data. The main purpose of this is to get good initial values for the system identification

routine to start from and to confirm that the optical response model is appropriate. Figure

6.3shows a measured transfer function taken at the end of S3 II; a pole/zero fit is shown

on the same plot together with the recovered values. This fit confirms two things: firstly,

the pole/zero model of the optical response seems to be an accurate representation of

the real optical response; secondly, as the data is corrected for the response of the ESD

prior to fitting using a model for the ESDs, then this ESD model must also be a good

representation of the real actuator response.

G = 44203V/µm

Pole f = 1035 Hz

Pole Q = 2.39

Zero f = 823 Hz

Fit

Measured

105

0

−20

−40

−60

−80

−100

103

Frequency (Hz)

P fit

Phas

e(

)M

agnit

ude

(V/µ

m)

Figure 6.3:A transfer function of the optical response of the detector as seen

by the high-power diodeP . A pole/zero fit to the magnitude is also shown

together with the recovered parameters.

6.1.2 Length-control servo electronics

When the detector is down-tuned to, for example, 1 kHz, we require calibration lines at

lower frequencies, say down to 500 Hz, in order to properly resolve the optical transfer

6.1 Experimental setup and detector details 181

function. At these lower frequencies the effect of the Michelson servo must be taken

into account in the model. In order to do this, the servo electronics must be modelled. In

addition, if we are to correct for the loop-gain using time-domain filters, then we must

also have good models of the servo electronics in order to design appropriate IIR filters.

During the period between the two parts of GEO’s participation in S3, the servo electron-

ics were changed slightly so different models of the electronics were used for the two

calibrations. Only the S3 II models are discussed here since the procedure and results

are similar for S3 I.

The common electronics(Hcom)

Some of the servo electronics are common to both feedback paths. This stage includes

an overall gain setting and a small amount of filtering (a notch around 500 Hz and a low

frequency integrator). These were measured twice as part of the two (fast and slow)

feedback paths. During the fitting of the fast and slow electronics, they were forced to

be the same in the two cases.

Fast path electronics(Hfast)

The fast path electronics include another gain setting (the crossover gain setting intro-

duced in section3.6.2) together with many filters. For reasons as yet unknown, fitting

a pole/zero model to the measurement of the fast path electronics has proved difficult.

Since the loop-gain of the Michelson servo becomes negligible after 400 Hz or so, and

since we only wish to calibrate accurately down to 50 Hz, some compromise was made

in the fitting to give the best fit possible around 100 Hz.

Figure 6.4 shows the measured response of the fast path electronics together with a

pole/zero fit. The fitting for all the electronics was done using LISO. The magnitude

ratio and phase differences between the fit and the measurement are shown in figure6.5.

6.1 Experimental setup and detector details 182

Model fit

Measurement

102

Frequency (Hz)

100

150

50

0−50−100−150−200

102

101

100

Phas

e(

)M

agnit

ude

(V/V

)

Figure 6.4:A pole/zero fit to the measured

transfer function of the fast path feedback

electronics used during S3 II.

102

−10

−5

5

0

10

Phas

ediff

eren

ces

()

Frequency (Hz)

1.1

1

0.9

Mag

nit

ude

rati

o

Figure 6.5:Magnitude and phase differ-

ences between the measured and mod-

elled fast path electronics for S3 II.

As was already stated, the model of the fast path electronics is important for two areas of

the calibration procedure. Firstly it is used in the system identification procedure where

the loop-gain of the Michelson servo is modelled in order that low frequency calibration

lines are correctly interpreted. The other use of the electronic model is in designing the

IIR filters that are used to do the loop-gain correction. Since the fast feedback signals

are filtered through the 1/ f 2 response of the electrostatic drive actuators, the effective

strain produced by the feedback becomes negligible towards higher frequencies; it is

most important that the model is accurate at lower frequencies (50-200 Hz). From figure

6.5we can see that the model is accurate to within 10% in magnitude over the important

frequency band.

Slow path electronics(Hslow)

The slow path electronics model is made up only of filters; no additional gain controls are

implemented. The influence of the slow path on the loop-gain above 50 Hz is very small

(since it is filtered through a 1/ f 2 response of two pendulum stages), and so this feedback

path could in principle be excluded for this configuration of the servo. It is included,

however, for completeness and to allow for the fact that the low frequency feedback may

be increased in future detector configurations. Figure6.6 shows the measured transfer

function for the slow path electronics together with a pole/zero fit. The magnitude ratio

6.1 Experimental setup and detector details 183

Model fit

Measurement

200

100

0

−100

−200

Phas

e(

)

101

100

Mag

nit

ude

(V/V

)

102

Frequency (Hz)

Figure 6.6:A pole/zero fit to the measured

transfer function of the fast path feedback

electronics used during S3 II.

20

10

0

−10

−20

1.5

1

0.5

102

Frequency (Hz)

Phas

ediff

eren

ces

()

Mag

nit

ude

rati

o

Figure 6.7:Magnitude and phase differ-

ences between the measured and mod-

elled fast path electronics for S3 II.

and phase differences between the fit and the measurement are shown in figure6.7.

6.1.3 Closed-loop transfer function of the Michelson servo

Having developed models of the Michelson length-control servo electronics, we can

generate a model of the closed-loop transfer function (CLTF) of the servo by including

the models of the two types of feedback actuators from section3.6.2together with the

model of the optical transfer function. (See equations4.10and4.11for the formulation

of the open-loop and closed-loop transfer functions.)

The measurements of the Michelson servo electronics shown above are made at con-

venient points in the electronics; in doing this, some gain stages are omitted from the

measurements. In order to determine these gain stages (which include switchable inte-

grators used only in lock), the model of the closed-loop transfer function can be fitted

to a measurement of the closed-loop transfer function. The closed-loop transfer func-

tion was measured as close as possible in time to a measurement of the optical transfer

function so that the optical response model is appropriate for the fit (we already know

that the overall optical gain fluctuates in time). We need to keep the optical response

fixed because a change in overall optical gain is indistinguishable from an electronic

gain change in the closed-loop.

6.1 Experimental setup and detector details 184

102

Frequency (Hz)

150

100

50

0

−50

Phas

e(

)

100

Mag

nit

ude

Model fit

Measurement

Figure 6.8:A measurement of the closed-

loop transfer function of the Michelson

servo used during S3 II. A model fit is also

shown.

102−10

−5

0

5

10

0.9

1

1.1

Mag

nit

ude

rati

oP

has

ediff

eren

ces

()

Frequency (Hz)

Figure 6.9:The magnitude and phase dif-

ferences between the measured and mod-

elled closed-loop transfer function of the

Michelson servo for S3 II.

Figure6.8shows a measurement of the closed-loop transfer function. The model of the

CLTF was fit to the data. Three overall gain factors were allowed to vary: one gain in

the common part of the servo model, one in the fast path, and one overall gain to account

for the fact that the closed-loop transfer function measurement was measured across a

flat-gain stage (with gain around 3) in the servo. Figure6.9 shows the magnitude and

phase differences between this model fit and the measured data. The model is clearly an

accurate representation of the real transfer function at the time of the measurement.

6.1.4 Demodulation phase for high-power diode

Since the setup of the sensing system of the detector during S3 uses two output photo-

diodes (one looking at an attenuated version of the output beam in-loop for locking, and

one out-of-loop for sensing a high-power, unattenuated, version of the output beam), we

can independently set the demodulation phase of the two mixers used in the demodula-

tion. This means that, while the in-loop diode demodulation phase is optimised to give

maximum signal power in one quadrature at low (DC) frequencies, the high-power diode

can be demodulated to optimise one of the quadratures (sayP ) at any chosen frequency.

Some investigations were done for S3 I to see the effect of the demodulation phase.

6.1 Experimental setup and detector details 185

By injecting a calibration line into the differential feedback, we can measure the effect

of the demodulation phase at that frequency. Two sets of measurements were made: one

set with a calibration peak at 10 Hz (an effective DC optimisation)1, and one set with

a calibration line around 1200 Hz. By varying the demodulation phase and measuring

the calibration peak amplitude in bothPandQ we can investigate the relationship be-

tween demodulation phase andP /Q amplitude at the calibration line frequency. We can

compare the results to the simulation shown in figure4.2. To test the validity that a cal-

ibration peak at 10 Hz provides a good DC optimisation, some measurements were also

made using a 5 Hz calibration peak.

0 1 2 3 4 5 6 7 8 9 10

Demodulation Phase (arb. units)

0

20

40

60

80

100

120

Pea

kA

mplitu

de

(mV

rms)

P@10 Hz

Q@10 Hz

P@5 Hz

Q@5 Hz√

P2 + Q2@10 Hz√

P2 + Q2@5 Hz

Figure 6.10: A plot of the demodulation phase for the high-power diode

against the amplitude of a 10 Hz calibration line as measured inPandQ .

Figure 6.10 shows the amplitude of a calibration line as measured in the high-power

diode,PandQ , for different demodulation phases. The demodulation phase settings

are shown as the value on the potentiometer control, not in degrees. Also shown are

measurements for a 5 Hz calibration line. We can see that the optimal demodulation

phase (that maximisesP ) for 5 Hz and 10 Hz is the same within the resolution of the

experiment. This tells us that the optimal demodulation for DC should be very close to

the value determined at 5 and 10 Hz. The plot also shows the total power(√

P2+ Q 2),

for each measurement point. This is useful to check the stability of the locked detector

throughout the duration of the measurements. Figure6.11shows the same experiment

1Using a calibration peak at 10 Hz is only for convenience and speed of measurement since using a

very low frequency (less than 1 Hz say) requires many seconds of data per data point.

6.1 Experimental setup and detector details 186

for a calibration peak around the peak sensitivity of the instrument for S3 I (1200 Hz).

The demodulation phase for the high-power diode was determined by choosing the value

that gives the minimum signal size inQ at DC on these plots: namely, 4.15 for S3 I, and

5.6 for S3 II.

4 4.5 5 5.5 6 6.5 7

Demodulation Phase (arb. units)

1200

1100

1000

900

800

700

600

500

400

Pea

kA

mplitu

de

(µV

rms)

P@1200 Hz

Q@1200 Hz√

P2 + Q2@1200 Hz

Figure 6.11: A plot of the demodulation phase for the high-power diode

against the amplitude of a 1200 Hz calibration line as measured inPandQ .

The general shape and behaviour of the measured curves agrees well with the expecta-

tions from the Finesse model, though the curves would need to be calibrated to perform

a more detailed comparison.

The choice of demodulation phase needs to be made based on many factors. So far,

there has been only preliminary investigations into the noise in the two quadratures,

and how the noise is affected by different choices of demodulation phase. In principle,

many more experiments are needed to understand the signal and noise content of the two

demodulation quadratures and how to best recover an optimal strain signal.

For S3 II, the demodulation phase was so as to maximiseParound the peak sensitivity

(1000 Hz). This was done to gain the maximum SNR for thePquadrature since the

sensitivity of the instrument had increased by around a factor of 10 from S3 I, making the

data more scientifically important at these frequencies. In S3 I the choice of optimising

at DC was more as an experiment so as to have a clear understanding of the transfer

function ofPandQ as given by the Finesse models. In addition, at that time it was not

clear how much the optical transfer functions would differ between the high- and low-

6.1 Experimental setup and detector details 187

power diodes if one was optimised at DC and the other around 1 kHz. This could be

important since the high-power diode response is used as the in-loop optical response in

the system identification model. As it turns out, the response of the low-power diode in-

phase demodulation is similar to that of the high-power diode (apart from a scale factor)

but further tests are required to determine the effect on the loop-gain and therefore on

the fitted parameters returned by the system identification routine.

6.1.5 Injected calibration lines

As we have seen in the experiments described in section5, the calibration of the dual-

recycled interferometer requires the ability to inject calibration lines at different fre-

quencies and with different amplitudes in order to ‘map’ the response of the detector

well enough to allow a good parameter fit to be made. Since the required frequencies

of the calibration lines depend on what frequency the peak SR response is tuned to, we

wish to be able to adjust the frequencies in a simple way. For these reasons, a signal

generator was designed and built which outputs a signal that is the sum of multiple, in-

dividually defined, sine waves. The hardware design of the signal generator is discussed

below, followed by a brief look at the controlling software.

Hardware and software design

The hardware design uses solid-state memory to store the desired waveform. The wave-

form, which is the sum of the required sine waves, is clocked out through a DAC to

provide an analogue signal suitable for driving the actuator used to inject the calibration

signal. The signal waveform is generated on a controlling computer and then down-

loaded to the memory on the generator. At any time, the waveform can be regenerated

on the computer with different parameters and then downloaded to the generator, replac-

ing the existing waveform. In this way, the injected calibration lines can be adjusted in

amplitude, frequency, and phase. Figure6.12shows a schematic overview of the signal

generator.

6.1 Experimental setup and detector details 188

GPS

unit

output

signal

Processor

DA

C

Calibration Signal Generator

data control

timing routine

4 MHz

1 PPS16 kHz

RAM

PC

output electronics

Figure 6.12:A schematic overview of the signal generator designed to in-

ject calibration lines with user-chosen frequencies, amplitudes, and phases.

The device takes two timing inputs (1 PPS and 4 MHz) from the DAQ GPS

card, and one data connection to a controlling PC. The output signal is an

analogue representation of the digital signal.

To ensure the stability of the injected calibration lines, the whole unit receives its timing

signals from a GPS unit. The processor, RAM, and ultimately the DAC chip, are all

clocked by signals from, or derived from, the 222 Hz (∼ 4 MHz) frequency standard

of the external GPS unit (in this case, the DAQ GPS card). A 16384 Hz timing signal

is derived from the 4 MHz GPS signal to clock the DAC such that each output sample

is coincident with the sample instances of the DAQ system. The 1 PPS signal from the

GPS clock provides the reference to which the 16384 Hz signal is phased, giving the first

output sample just after the second boundary. The DAC used is an AD1862 [ANALOG]

20-bit converter which claims ultra-low noise performance. The generator is designed to

use non-volatile [SRAM] (part no. DS1245AB) so that in the event of a power failure,

the generator returns to its previous operational state. The SRAM can store 128k 8-bit

words.

Since the calibration lines are always integer Hz values (so that all the power is in one

bin in any subsequent analysis), only one second of the waveform needs be downloaded

to the generator at one time.

6.1 Experimental setup and detector details 189

The injector is controlled by a LabVIEW virtual instrument that allows the frequency,

amplitude, and phase of each line to be chosen individually; an overall gain factor is also

included to allow the amplitude, and therefore the SNR, of all lines to be easily adjusted.

Injected signals for S3 I and S3 II

For both S3 I and S3 II calibration lines were injected into the electrostatic drive feedback

path to give a known differential displacement. The signals were pre-conditioned in an

external adding/whitening module designed to allow other signals to be injected at the

same time. The adding/whitening module also has a monitor output that was connected

to the DAQ system to provide a record of any injected signals.

freq. (Hz) Amp. (digital) Amp. P HP (Vrms) Amp. CAL (Vrms)

609 0.06 1.7e-03 0.115

817 0.1 2.0e-03 0.19

1010 0.15 2.9e-03 0.287

1213 0.22 3.8e-03 0.42

1309 0.27 3.6e-03 0.52

1411 0.3 2.7e-03 0.57

2132 0.8 8.5e-04 1.5

Table 6.1: The injected calibration lines for the S3 I science

run. Typical values of the amplitudes of the calibration lines

are shown as measured by the DAQ system in both the injected

calibration signal and the high-power error-pointPquadrature

signal.

Table6.1 shows (for S3 I) the frequency of the injected calibration lines together with

their amplitudes (as measured by the DAQs) at the adding/whitening module monitor

output and in the high-power error-pointP2. Table6.2shows the same data for the S3 II

configuration with additional (digital) phase information; for S3 I all injected lines had

2The amplitudes measured in the high-power error-pointPare only example measurements since these

vary in time as the optical gain of the detector varies.

6.1 Experimental setup and detector details 190

digital phase of 0.

freq. (Hz) Amp. (digital) Phase (cycles) Amp. P HP (Vrms) Amp. CAL (Vrms)

417 0.1 0 0.031 0.050

609 0.06 1000 0.012 0.030

787 0.06 2000 0.011 0.031

1011 0.1 3000 0.017 0.053

1213 0.12 4000 0.011 0.066

1309 0.15 5000 0.010 0.084

1717 0.8 6000 0.007 0.187

2032 0.9 700 0.010 0.555

2569 0.3 500 0.007 0.786

Table 6.2: The injected calibration lines for the S3 II science

run. Typical values of the amplitudes of the calibration lines

are shown as measured by the DAQ system in both the injected

calibration signal and the high-power error-pointPquadrature

signal.

Snap-shot spectra of the recorded calibration signal and high-power error-pointPare

shown in figures6.14and6.13for S3 I and figures6.16and6.16for S3 II. In figure6.13

we can clearly see the optical response of the detector in the calibration lines since all the

lines were approximately uniform in displacement. After S3 I, the injected calibration

lines were changed to better map the new detuning point (1000 Hz); in S3 II the mag-

nitudes of the lines were changed so that the lines had approximately uniform SNR in

the high-power error-pointP . Due to the flexibility of the signal generator, this process

can be done on-line while looking at the error-point spectra. The phase of the individual

calibration lines was set to 0 for all lines in S3 I; in S3 II this was changed so as to try and

minimise the presence of additional noise in the injected signal caused by beats between

the individual lines.

6.1 Experimental setup and detector details 191

navs:16384nfft:

ndata: 163840

163841.50

fs:enbw:

10

10−1

10−2

10−3

10−4

10−5

10−6

102 103

G1:LSC MID EP-P HPA

mplitu

de

(Vrm

s)

Frequency (Hz)

Figure 6.13: A snap-shot spectrum of

the high-power error-point signal during

S3 I. The injected calibration lines are

highlighted with (blue) circles.

navs:16384nfft:

ndata: 163840

163841.50

fs:enbw:

10

103102

Am

plitu

de

(Vrm

s)

Frequency (Hz)

101

100

10−1

10−2

10−3

10−4

10−5

10−6

G1:LSC MID CAL

Figure 6.14:A snap-shot spectrum of the

injected calibration signal during S3 I.

navs:16384nfft:

ndata: 163840

163841.50

fs:enbw:

10

G1:LSC MID EP-P HP

103102

Frequency (Hz)

10−1

10−2

10−3

10−4

10−5

10−6

Am

plitu

de

(Vrm

s)

Figure 6.15: A snap-shot spectrum of

the high-power error-point signal during

S3 II. The injected calibration lines are

highlighted with (blue) circles.

navs:16384nfft:

ndata: 163840

163841.50

fs:enbw:

10

G1:LSC MID CAL100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

Am

plitu

de

(Vrm

s)

102 103

Frequency (Hz)

Figure 6.16:A snap-shot spectrum of the

injected calibration signal during S3 II.

6.1 Experimental setup and detector details 192

Signal name Description Whitening process

LSC MID CAL The injected calibration signal. ×2 from differential

sending.

LSC MID EP-PHP The in-phase demodulation of the

high-power photodiode. Demodu-

lation phase was set to maximise

the signal content of this quadrature

at 1300 Hz.

Analogue filter with 2

zeros at 0.1 Hz, 2 poles

at 100 Hz with gain 100

at high frequency.

LSC MID EP-QHP The out-of-phase demodulation of

the high-power photodiode.

Analogue filter with 2

zeros at 0.1 Hz, 2 poles

at 100 Hz with gain 100

at high frequency.

LSC MID LS The lock status indicator. The chan-

nel is around 0 V when the detec-

tor is unlocked and rises to around

5 V when the detector is locked and

fully detuned to the target operating

point.

-10 dB from attenuation

patch panel.

PEM CB MAINT The maintenance channel switch.

This is 0 V when the switch is off

- i.e., no one is in the building.

When someone enters the building,

this switch is switched and the level

goes to -1.2 V.

None.

Table 6.3:The recorded signals relevant for the calibration process for the

S3 data run. The standard GEO channel-name prefix ‘G1:’ is omitted from

the table.

6.2 Calibration software 193

6.1.6 Recording of relevant signals

The recording of the relevant signals plays an important role in the calibration process.

Each signal needs to be recorded with the highest fidelity possible if we are to avoid

losing any gravitational wave information. Typically, this means that a certain amount

of analogue signal conditioning occurs before the signals are digitised, usually in the

form of signal whitening. The details of how each signal is recorded must be included

in the calibration process so that accurate parameter estimation and strain calculation is

possible.

The signals relevant to the calibration process for S3 I are summarised in table6.3. For

S3 II, the high-power error-point signals (both quadratures) were further attenuated by

10 dB.

In the calibration process we need to be able to correct for the whitening process as

accurately as possible. The whitening filter forPandQ was modelled and inverted to

allow an IIR filter to be designed. This ‘de-whitening’ filter was designed in LISO. The

response of the IIR filter is shown in figure6.17 together with a pole/zero model; the

phase and magnitude differences between the two are also shown.

6.2 Calibration software

The calibration software has already been described in detail in chapters4 and5. In

this section only those details that are specific to the S3 science runs are presented. This

includes details of the filters used in the pipeline as well as details of how the filters were

applied. Both parts of the science run (S3 I and S3 II) require their own version of the

calibration software since various aspects of the detector and signal recording changed

between the two runs.

6.2 Calibration software 194

IIR filter

Analogue filter

0

−40

−80

−120

−160

102 103

Frequency (Hz)

−12

101

−8

−4

0

Phas

ediff

.(

)P

has

e(

)

101 102 103

Frequency (Hz)

100

10−2

0.95

1

1.05

Mag

nit

ude

Rat

ioM

agnit

ude

Figure 6.17:A comparison between the IIR dewhitening filter

used for the high-power error-pointPand an analogue model.

6.2.1 System identification

Since we can only measure the optical parameters of the detector once per second with

the scheme described so far, we will have a certain amount of measurement noise on the

recovered estimates. For these first experiments, it was decided that it would be better to

estimate the optical parameters more accurately by smoothing the recovered estimates.

Again we implicitly assume by doing this that the variations in the parameters are slower

that the time-scale of the smoothing process. In the case of the overall optical gain, we

have some evidence that fluctuations do occur on fast time-scales (1 Hz or more from the

pendulum resonances). If we smooth these estimates on time-scales of, say, 10 seconds,

then we reduce our ability to correct for fluctuations at 1 Hz. Other measurement tech-

niques are possible that will allow faster estimation of the optical parameters by making

faster measurements of the calibration lines. For example, since we know precisely the

frequency of the calibration lines we can perform a complex heterodyne of the error-

point signal at the calibration line frequency and, after some filtering, get measurements

of the lines with higher sample-rates than with using a simple FFT method. Initially,

however, the more simple FFT method was used and the estimates were smoothed to fo-

6.2 Calibration software 195

cus the calibration process on the larger, slow drifts in the optical parameters caused by

thermal and tidal effects. For the other optical parameters (pole frequency, pole Q, and

zero frequency), we also assume that there are no significant fluctuations on time-scales

less than a few minutes so these parameters were smoothed even more.

All of the smoothing of the optical parameters was done using IIR filters. The filter for

the overall optical gain estimates was designed to have 2 poles at 0.2 Hz whereas the

filters for the other three optical parameters were designed to have 2 poles at 0.01 Hz.

The effect of these filters can clearly be seen on the spectral content of the recovered

parameter estimates (see section6.3.2).

6.2.2 Optical correction

The optical gain correction filter used during the S3 runs was implemented as two IIR

filters: one filter, f 11p, composed of a complex zero representing the inverse of the

complex pole recovered from the system identification routine and a (tempering) com-

plex pole placed at 100 kHz with a Q of 3; the other filter,f 12p, is a simple real pole

representing the inverse of the real zero recovered from the system identification rou-

tine. This is a useful way to split the problem since it is relatively straightforward to

analytically calculate the coefficients for these two filters using the bilinear transform

(see appendixA.1.1).

From the experiments carried out in calibrating the power-recycled GEO configuration,

we already saw fluctuations in the overall optical gain of the instrument and so we in-

clude the estimation and correction of these fluctuations in the calibration scheme for the

dual-recycled configuration. Since the model of the optical response of the dual-recycled

IFO has three extra parameters, we must also assume that these will vary in time. With

this in mind the system identification process has been designed from the beginning to

include the estimation of these extra parameters. This process has been discussed at

various places in the last three chapters. What has not been discussed so far is the cor-

rection for time variations in the frequency dependent part of the optical response. In

the simulations presented in chapter5, the optical response was simulated using a fixed

6.2 Calibration software 196

IIR filter. Consequently, the correction of this response was done using a fixed filter

based on the known underlying parameter values. In the case of the real detector, the

parameters of underlying optical response will vary in time and so we need to include

these variations in the inverse optical filter. One way to do this is to adjust the coeffi-

cients of the inverse optical filter as new estimates of the parameters are measured. If we

carry forward the history from the last application of the filter, we can smoothly change

the filtering response. In this way, the filter coefficients become functions of time. At

each time step, new coefficients are calculated by performing a bilinear transform on the

recovered optical parameter estimates for that second.

Experiment 2 discussed on page167was redone using this filter updating method. The

results were indistinguishable from those obtained from the fixed filter experiment. As

we shall see in section6.3.2, the recovered parameter estimates for the frequency part

of the optical response vary only very slightly between successive updates. This results

in only slight changes in the filter coefficients at each time step. Figure6.18shows the

time evolution over one hour during S3 II of the variable coefficients of the complex-

pole/complex-zero filter (f 11p) 3. We can see that the changes in the coefficients are

small, much less than 1%. In figure6.19 we see the time-evolution of the real pole

coefficients (f 12p) over the same hour. Again the changes in the coefficients are much

less than 1% between successive updates.

Apart from correcting for fluctuations in the parameters of the optical response, updat-

ing the filters in this way allows the calibration procedure to follow, on-line, the target

detuning of the SR response. Once GEO is a fully commissioned gravitational wave

observatory, one of the main advantages of having a signal-recycled IFO is the ability

to change the detector’s most sensitive frequency band depending on what astrophysical

source may be under investigation. In addition, if a variable-reflectivity SR mirror is

installed, the bandwidth of the detector’s most sensitive band can be adjusted. For ex-

ample, if a particular pulsar signal were to be investigated, then the microscopic position

of the SR mirror could be adjusted to target the pulsar frequency and then the reflectiv-

ity of the SR mirror could be increased to enhance the signals in a narrow band around

the pulsar frequency. These long term considerations, together with the desire to have

3The ‘b’ coefficients are constant in time since they represent the fixed high-frequency complex pole.

6.2 Calibration software 197

10.90.80.70.60.50.40.30.20.10

25

20

15

10

5

0

−5

−10

−15

−20

−25

Coe

ffici

ent

Val

ue

Time (Hours)

a[0], µ = 3350

a[1], µ = −6665

a[2], µ = 3317

Figure 6.18: Time evolution of the in-

verse optical filter, f11p, over one hour

of S3 II. The mean of each time-series is

subtracted and indicated in the legend.

10.90.80.70.60.50.40.30.20 0.1

4

3

2

1

0

−1

−2

−3

−4

Coe

ffici

ent

Val

ue

(×10−

4)

Time (Hours)

a[2], µ = −0.98

a[0], µ = 0.01

a[1], µ = 0.01

Figure 6.19: Time evolution of the in-

verse optical filter, f12p, over one hour

of S3 II. The mean of each time-series is

subtracted and indicated in the legend.

on-line adaptable calibration, drives the development of such a filter update scheme as

described above.

6.2.3 Length-control servo correction

As we saw in section4.2.3there are two possible methods for correcting the loop-gain

of the Michelson servo. For S3 (I and II), it was only possible to implement the digital

correction method,i.e., to use the simulated correction signals derived by filtering the

error-point signal through filters describing the Michelson servo electronics. This is be-

cause in S3 I, the feedback signals were recorded with a low sample-rate (1024 Hz); for

S3 II, the sample-rate was increased to allow investigation of this method but it turned

out that the measured feedback signals were considerably more noisy than those gener-

ated by the filtering method. Figure6.20shows snap-shot spectra of the recorded fast

feedback signal and the simulated fast feedback signal (filtered error-point). Noise added

in the feedback electronics is filtered by the response of the feedback actuators mean-

ing that the noise does not appear at the error-point. If we filter the error-point through

model filters of the electronics we end up with ‘clean’ feedback signals that can be used

to correct the Michelson loop-gain while, at the same time, avoiding any contamination

of the resulting strain signal. In addition to adding noise, we also add the calibration

6.2 Calibration software 198

Simulated fast feedback

Measured fast feedback

102 103

100

10−2

10−4

10−6

10−8

Am

plitu

de

(V/√

Hz)

Frequency (Hz)

Figure 6.20:Spectral density plots of the simulated and measured fast feed-

back signals. The spectra show that the feedback electronics introduce noise,

making the recorded signals unsuitable to perform the loop-gain correction

in the calibration routine.

lines before the point where the feedback signals are picked off for recording. If we

are to use these recorded feedback signals, they should be extracted from the electronics

before the point where calibration lines are added.

The filters used for simulating the feedback correction signals are designed using the

poles and zeros of the servo electronics models from section6.1.2. Using LISO, these

poles and zeros were split into shorter stable sections and IIR filters were designed.

The differences between the IIR filter responses and response of the model is an impor-

tant consideration in determining the systematic errors present in the calibration system.

Only the filters for S3 II are discussed here; the filters for S3 I are similar in accuracy.

Figure6.21shows the differences between the response of the IIR filters that mimic the

fast path electronics and the model of the fast path electronics. (The slow path is not in-

cluded since its influence on the calibration process is minimal for the current Michelson

servo configuration.) We can see that converting the model of the fast path electronics

into IIR filters introduces negligible error over the frequency band of consequence.

6.3 Results and validation 199

IIR filter

Model

−200

−100

0

100

0.9

1

1.1

−0.4

−0.2

0

Frequency (Hz)Frequency (Hz)

102 102

Mag

nit

ude

Mag

nit

ude

rati

o

Phas

ediff

.(

)P

has

e(

)Figure 6.21:The magnitude and phase differences between the

model of the fast path electronics and the designed IIR filters.

6.3 Results and validation

One of the most important aspects of the calibration work is the characterisation and

quantifying of the various errors in the process. These errors come in various places;

some of them systematic (for example, in the fitting of a model to measurements of the

feedback electronics), some of them random (as in the case of measuring the optical

parameters in the presence of noise). This section presents some of many possible tests

and experiments that can be done to determine the calibration accuracy, together with

the results of the calibration of the S3 I and S3 II data sets.

6.3.1 Frequency-domain calibration comparison

One useful test of the calibration process is to test it against a (standard) frequency-

domain calibration. This allows, among other things, the loop-gain correction routine to

be tested (this was not tested in the simulation experiments of chapter5). Calibrating in

the frequency domain is somewhat simpler that calibrating in the time-domain since it

bypasses the need for designing stable IIR filters. Since most of the work necessary in

6.3 Results and validation 200

performing a frequency-domain calibration is done in order to develop the time-domain

calibration pipeline, it is a relatively easy process to complete the frequency domain

calibration method. The frequency-domain model is essentially that calculated in the

fitting of the optical response (as in section6.1.1) and the closed-loop transfer function

fit shown in section6.1.3.

The comparison of the two methods takes place in the frequency domain. We take the

recorded error-point and make an FFT of the time-series which we then calibrate to

a complex strain using a frequency domain model of the transfer function fromP to

differential displacement. This transfer function can be written as

TPdx =1

HP CLTF, (6.1)

whereCLTF is the closed-loop transfer function of the servo given by equation4.10.

If we take the parameters of the optical response estimated by the system identification

routine, then we have all the parameters necessary to generate a frequency-domain rep-

resentation of the transfer function given by equation6.1. We get the complex strain

quantity by

h = TPdx × s, (6.2)

wheres( f ) is the complex vector resulting from the FFT of the error-point data. We can

then take an FFT of theh(t) signal produced by the time-domain calibration process and

compare it to the result of equation6.2.

Figures6.22and6.23show the magnitude and phase differences between the frequency-

and time-domain methods for S3 II data in two bands of width 100 Hz, one centred on

100 Hz, the other on 1230 Hz. (Two magnified bands are shown to allow the detail to be

visible but there is nothing special about the chosen bands; the remainder of the band is

comparable.) We can see that the time-domain calibration process agrees very well with

the frequency domain calibration. We also see that the time-domain calibration process

introduces noise at the level of a few %. This method provides a good way of optimising

the time-domain calibration procedure. Changes in up- and down-sampling filters can

affect the noise we see in this comparison, while the design of the various IIR filters can

influence any systematic effects we see.

6.3 Results and validation 201

80 90 100 110 120 80 90 100 110 120

Frequency (Hz) Frequency (Hz)

20

10

0

−10

−20

1.2

1.0

0.8

10−15

10−14

10−13 200

100

0

−100

−200Mag

.(a

rb./√

Hz)

Mag

.R

atio

Phas

e(

)P

has

eD

iff.

()

Figure 6.22:A comparison of the time- and frequency-domain calibration

methods around 100 Hz. The magnitude is shown in arbitrary units because

the data was not scaled after the FFT was performed. As this was done the

same for both time-series, the effect is removed from the comparison.

6.3.2 Recovered parameters

The S3 runs provided an excellent opportunity to test the calibration procedure over long

data segments. Typical lock lengths during S3 I were around 24 hours; in S3 II, locks

of much longer (up to 95 hours) where recorded. In both runs, a preliminary version

of the calibration code was run on-line giving (preliminary) on-line information about

the sensitivity of the detector. After the runs, the final details of the calibration process

were confirmed and adjusted where necessary before the two calibrated data sets were

regenerated. Results from the calibration of both data sets are presented in the following

sections.

S3 I

The recovered optical parameters from S3 I are shown plotted every 30 seconds for the

entire run in figure6.24. Included on the plot are the lock status information (bit 0 of the

quality channel) and the recoveredχ2 values. We can clearly see slow drifts in all of the

optical parameters, especially at the beginning and end of the week. An optical transfer

6.3 Results and validation 202

1210 1220 1230 1240 1250 1210 1220 1230 1240 1250

10−17

10−18

10−16

10−15

Frequency (Hz) Frequency (Hz)

200

100

0

−100

−200

20

10

0

−10

−20

1.2

1.0

0.8

Mag

.R

atio

Mag

.(a

rb./√

Hz)

Phas

e(

)P

has

eD

iff.

()

Figure 6.23:A comparison of the time- and frequency-domain calibration

methods around 1230 Hz. The magnitude is shown in arbitrary units because

the data was not scaled after the FFT was performed. As this was done the

same for both time-series, the effect is removed from the comparison.

function measured post S3 I gave optical parameters of 6.5 V/nm, 1238 Hz, 2.7, 1580 Hz

for the overall optical gain, pole frequency, pole Q, and zero frequency respectively; The

mean values of the four parameters were 6.033 V/nm, 1220 Hz, 2.7, and 1370 Hz. We see

that the pole frequency and Q are estimated very well by the system identification rou-

tine whereas the zero frequency is systematically offset from the measured value. This

could be attributed to an inaccurate model of the electrostatic drive response. Some mea-

surements of the ESD response have shown that the magnitude response deviates from

a 1/ f 2 response at frequencies above 2 kHz. Since we have calibration lines at these

frequencies, we could easily be biased in the model. More investigations and on-line

calibration runs are required to investigate the differences between optical parameters

recovered from measured transfer functions and optical parameters recovered from the

system identification routine.

S3 II

Figure6.25shows the recovered parameters plotted every 30 seconds for the duration of

the S3 II run. Once again, the plot includes the lock status of the detector and the recov-

eredχ2 values from the system identification routine. There are a number of differences

6.3 Results and validation 203

120

140

020

4060

8010

016

0

Tim

e(H

ours

)

500 01

0.5 0

2000

1500

1000 5002

2.53

1200

10008 6 4

Ove

rall

optica

lga

in

Pol

efreq

uen

cy

Pol

eQ

Zer

ofreq

uen

cy

Loc

kSta

tus

χ2

HzHzV/nm

Figure 6.24:The recovered optical parameters from the S3 I science run. The

data are plotted every 30 seconds for the duration of the run. The bottom two

traces show the lock status of the detector (1 is locked) and the recoveredχ2

values respectively.

6.3 Results and validation 204

immediately apparent between the S3 II parameters and the S3 I parameters. The most

obvious difference is in the values of the recovered parameters. These changes are ex-

pected since the detector was detuned to a different operating point: 1000 Hz instead

of 1200 Hz. In addition, we see that the optical gain is about an order of magnitude

higher than in S3 I. This change is almost entirely due to the sensing setup of the detec-

tor; changes in the photodiode amplification circuit led to an increase in the optical gain

since we measure the optical response of the detector from differential mirror motion to

Volts after the demodulation of the detected optical signal.

Another noticeable point is that around hour 110, the values of the optical parameters all

make a step change. This was due to the detector being detuned to a different operating

point. The control software that sets the operating parameters of the dual-recycled lock

reset to its default settings after a loss of lock; at this time the default setting was to de-

tune to 1200 Hz instead of 1000 Hz. The period where the detector was detuned like this

is highlighted by the vertical (red) dotted lines. This turned out to be a good test of the

system identification routine; as we can see, it successfully recovered the parameters that

describe the different detuning frequency, allowing the data to be successfully calibrated.

Theχ2 values recovered from the system identification routine are somewhat larger than

those from S3 I. This is partly attributed to the placing of the calibration lines and partly

to the fact that the model may not properly describe the detector when the high-power

photodiode is demodulated to give optimal signals around 1 kHz while the demodulation

of the in-loop diode is optimised to give maximum signals at DC. The model used in the

fitting routine implicitly assumes that the optical transfer functions of the detector as

seen by the high- and low-power diodes demodulated in-phase are the same except for

an overall gain factor. In S3 I this was a very good assumption since the high-power

diode was demodulated to give optimal signals at DC; in S3 II, this was no longer the

case. We can still threshold theχ2 values for the quality channel but in this case we have

to start from a higher overall value.

The recovered parameter values from S3 II are shown as histograms in figure6.26. The

mean and standard deviation of the distributions are shown on each subplot. Data from

the 1200 Hz de-tuned state are excluded from the distributions. We can see that the

6.3 Results and validation 205

010

050

150

250

200

300

Tim

e(H

ours

)

0

1000

20000

0.51

500

1000

1500

2000

Hz

3

2.5 2

1200

Hz

60 50 40

V/nm

Ove

rall

optica

lga

in

Pol

efreq

uen

cy

Pol

eQ

Zer

ofreq

uen

cy

Loc

kSta

tus

1000

χ2

Figure 6.25:The recovered optical parameters of the S3 II science run plot-

ted every 30 seconds for the duration of the run. The detector lock status and

the recoveredχ2 values from the system identification routine are shown on

the bottom two traces. The red, vertical, dotted lines indicate the period of

time when the detector was detuned to a different operating point: 1200 Hz

instead of 1000 Hz.

6.3 Results and validation 206

1020 1025 1030 1035

Hz

µ = 1028.8 Hz

σ = 1.6 Hz

40 45 50 55 60

V/nm

G

σ = 2 V/nmµ = 47.8 V/nm

Pole Q

σ = 0.02

µ = 2.51

2.4 2.5 2.6 900 920 940 960

Hz

Zero f

Pole f

µ = 934 Hz

σ = 6.6 Hz

Figure 6.26:Histograms of the four optical parameters recov-

ered from the calibration of the S3 II data set. The mean and

standard deviation of each distribution are shown on the plots.

three ‘frequency’ parameters (pole frequency, pole Q, zero frequency) are distributed

approximately normally with standard deviations of around 1% or less. The overall

optical gain shows a distribution that looks like multiple normal distributions. This is

expected since we know that the overall optical gain of the detector varies at certain

frequencies due to known physical effects and, as we can see in figure6.25, reduces

with time over the run. For example, the daily temperature fluctuations cause the optical

gain to fluctuate around a certain mean value, whereas, faster fluctuations due to the

micro-seismic motion of the mirrors produce a distribution of the the optical gain around

a different mean value.

6.3.3 Calibration suppression performance

Another way to study the performance of the calibration procedure is to consider the

effect the calibration process has on the induced differential displacement signals,i.e.,

the calibration lines. For a perfect calibration, the amplitude and phase of the calibra-

tion lines as they appear in the calibrated strain signal should be constant to a degree

consistent with the stability of the injected calibration lines. The calibration code has to

6.3 Results and validation 207

609 Hz peak in G1:DER DATA H

609 Hz peak in G1:LSC MID EP-P HP

−5× 10−18

0

5× 10−181× 10−4

0

−1× 10−4

0 10 20 30 40 50 60 8070 90

Am

plitu

de

(Vrm

s)

Am

plitu

de

(Vrm

s)

Time (Hours)

Figure 6.27: The time evolution of the

peak amplitude of the 609 Hz calibration

line as it appears in the high-power error-

point Psignal and the calibrated strain

channel.

10−19

10−20

10−18

10−17

1 10 100 1000

Am

plitu

de

(Vrm

s)

Frequency (mHz)

DATA H@609 Hz

EP-P HP@609 Hz (scaled)

Figure 6.28:The spectral content of mea-

surements of the peak amplitude of the

609 Hz calibration line as measured in the

high-power error-pointPand the calibra-

tion strain signal.

measure the peak values of the injected calibration lines in both the injected signal and

in the high-power error-point in order to determine the optical parameters of the system.

This is done every second and the values are recorded as 1 Hz time-series in the output

frame files (see table5.2). In addition, the calibration code measures the calibration

peaks in the calibrated strain spectrum and records those values in the same way.

Using these calibration line measurements we can look at the time-evolution of individ-

ual lines in the calibrated and un-calibrated data. In order to compare the spectral content

of the time-evolution of the two signals at particular calibration lines, it is necessary to

scale one of them. In the following results, the spectra of the line amplitudes measured in

the high-power error-point are scaled such that the fluctuations around 0.5 Hz are equal

to those in the spectra of the strain line amplitudes.

Figure6.27 shows the time evolution of the peak amplitude of the 609 Hz calibration

line as measured in the high-power error-pointPsignal and the calibrated strain signal.

The peak measurements are made every second by the calibration software. The data

shown are for a 94 hour lock of S3 II. We can immediately see that the low frequency

fluctuations present in the high-power error-point data are not visible in the calibrated

data. If we take this data and form an amplitude spectrum, we can see on what time-

scales we correct for fluctuations in the optical gain. Figure6.28shows amplitude spectra

6.3 Results and validation 208

0 10 20 30 40 50 60 8070 90

Time (Hours)

00

1

1011 Hz peak in G1:LSC MID EP-P HP

1011 Hz peak in G1:DER DATA H

-1

-2

Am

plitu

de

(×10−

4V

rms)

2

-2

-4

Am

plitu

de

(×10−

18V

rms)

Figure 6.29: The time evolution of the

peak amplitude of the 1011 Hz calibration

line as it appears in the high-power error-

point Psignal and the calibrated strain

channel.

10−19

10−20

10−18

10−17

Am

plitu

de

(Vrm

s)

1 10 100 1000

Frequency (mHz)

EP-P HP@1011 Hz (scaled)

DATA H@1011 Hz

Figure 6.30:The spectral content of mea-

surements of the peak amplitude of the

1011 Hz calibration line as measured in

the high-power error-pointPand the cal-

ibration strain signal.

of the two time-series shown in figure6.27. The spectrum of the high-power error-point

measurements is scaled such that the fluctuations are the same as the fluctuations in the

calibrated data at 0.5 Hz. The underlying idea here is that the calibration procedure does

very little to correct for fluctuations on time-scales shorter than 1 second since we only

make estimates of the optical response every second and then smooth these estimates

further using a low-pass filter. The figure shows clearly that as we go to longer time-

scales, the calibration process does better at removing optical gain fluctuations. Figures

6.29 and6.30 show the results for the 1011 Hz calibration line where we see similar

performance of the calibration routine.

6.3.4 Calibration accuracy

As was the case in the calibration of the power-recycled interferometer, the calibration

accuracy is frequency dependent and has to be considered carefully in different fre-

quency bands. For the power-recycling calibration, the errors in the calibration process

(both systematic and random), were presented for frequencies below the unity gain point

of the Michelson servo (< 200 Hz) and for higher frequencies where the loop gain of the

Michelson servo was negligible (> 200 Hz). The extra complexity of the dual-recycling

6.3 Results and validation 209

calibration scheme requires that we consider the errors in the calibration process in more

frequency bands. For example, the measurement of the optical response of the detector

can introduce a systematic error which will be more significant at intermediate frequen-

cies (around the SR cavity resonance).

In the case of the power-recycling calibration, one of the main sources of uncertainty was

the calibration of the electrostatic drive actuators used to inject the calibration lines. This

source of uncertainty remains for the calibration of the dual-recycled interferometer.

A useful way to consider the additional errors present in the dual-recycling calibration

process is to look back through the last three chapters and to consider at various stages of

the calibration pipeline, what errors (random and/or systematic) are introduced, and in

what frequency bands. The following discussion follows through the development of the

calibration pipeline and looks at any errors identified in its application to simulated data

and science run data (S3). Table6.4 summarises the main errors that were highlighted

through chapters5 and6; discussion of each table entry follows. In some cases, possible

improvements to reduce the errors are presented.

Up-sampling of overall optical gain

The up-sampling of the optical gain samples from 1 Hz to 16384 Hz introduces a small

amount of noise due to the finite suppression of frequencies higher than 0.5 Hz by the

FIR low-pass filter that is applied in the band-limited interpolation process. If we look

at the noise spectral density shown in figure5.4, we can see that this noise is at the level

of about 1 part in 200. This noise is introduced broad-band (over the whole detection

band) since it is applied to all samples in the time-series of the high-power error-point

Pduring the correction for the overall optical gain.

The FIR filter used in the band-limited interpolation process can be improved to give

better performance. Filters with different numbers of coefficients and designed using

different algorithms can be tried and tested to attempt to lower this source of noise.

6.3 Results and validation 210

Source Freq. band (Hz) Type Value Reference

Electrostatic drive calibration 50-6000 random 5% figure3.8.4

Up-sampling of overall opti-

cal gain

50-6000 random < 1% figure5.4

Inverse optical response filter

and the use of bilinear trans-

form

1000-6000 systematic < 1% figure5.9

Overall optical gain correc-

tion

50-6000 random 1-5% figure5.20

Uncertainty in optical param-

eters (SNR∼50, sim)

50-6000 random 1% figure5.24

(SNR∼100, sim) 50-6000 random < 1% figure5.28

(SNR∼100, S3 II) 50-6000 random 1% figure6.26

Fast path electronics (mea-

surement to model)

50 systematic 3% figure6.5

(model to IIR) 50 systematic < 1% figure6.21

MI Closed-loop transfer func-

tion (measurement to model)

50-6000 systematic 3% figure6.9

Pdewhitening filter 2000 systematic 2% figure6.17

Frequency-domain to Time-

domain

50 systematic /

random

5 ± 5% figure6.22

1200 systematic /

random

0 ± 5% figure6.23

Table 6.4: A summary of the systematic and random errors highlighted

through the development and testing of the dual-recycling calibration pro-

cess.

6.3 Results and validation 211

Inverse optical response filter and the use of the bilinear transform

The use of a time-domain filter to correct for the optical response of the detector requires

the addition of one or more poles at high frequency in order to make a stable filter.

In doing this, the response of the filter must deviate from the response of a ‘perfect’

analogue filter above some frequency. When using filters designed for 16× 16384 Hz,

this deviation is less than 1% over the whole of the detection band. The up-sampling of

the high-power error-point signal by a factor 16 (in order that this filter can be applied)

introduces noise at a level below 1% in the detection band (see figure5.11).

Overall optical gain correction

Estimating the effect of correcting for the overall optical gain is done by considering

the results of experiment 3 of the simulated calibration runs (see page169). This ex-

periment shows that the correction for the optical gain introduces noise at the level of

1-5% at different frequencies across the detection band. The results shown in figure5.20

include the underlying differences between the model of the optical response and the

implementation of the optical transfer function as an IIR filter.

Uncertainty in optical parameters

The effect on the calibration accuracy of incorrectly estimating the optical parameters

depends on which parameter is being considered. For example, incorrectly estimating

the overall optical gain,G, affects the calibration across the full detection band, whereas,

an error in estimating the frequency of the real zero will affect the calibration accuracy

most at high frequencies. Figures6.31, 6.32, and6.33show the magnitude differences

between an underlying optical response (using values similar to S3 I) and one which

deviates by a fixed percentage error in one of the three parameters: pole frequency, pole

Q, and zero frequency. Curves are shown for 1%, 5%, and 10% errors.

One way to assess the uncertainty in each parameter is to look at the distribution of the

recovered parameter values about their mean. Attributing the spread in the distributions

6.3 Results and validation 212

102 103

1.3

1.2

1.1

1

0.9

0.8

Mag

nit

ude

rati

o

Frequency (Hz)

+10%

−10%

−5%

−1%

+1%

+5%

Figure 6.31: The systematic magnitude

error introduced by incorrectly estimating

the pole frequency of the optical response.

102 103

1.1

1.06

1.02

1

1.04

1.08

0.98

0.96

0.94

0.92

+10%

+5%

+1%

−1%

−5%

−10%

Mag

nit

ude

rati

o

Frequency (Hz)

Figure 6.32: The systematic magnitude

error introduced by incorrectly estimating

the pole Q of the optical response.

to measurement noise makes the assumption that the underlying parameters do not vary

in time. In the case of experiment 4 on page170, this is the case and we can then see how

the spread in the parameter distributions can be interpreted as a frequency dependent

systematic error. The errors quoted in table6.4 represent upper limits to the error at

any particular frequency introduced due to an incorrect estimation of any one optical

parameter. The values shown are for two simulation experiments (with calibration line

SNRs of 50 and 100) and for the S3 II results.

Another way that the estimation of the optical parameters can introduce error into the

calibration process is if the recovered values are far from the underlying values. Since we

don’t know the underlying values, we can only gain evidence of this effect indirectly, for

example, by considering theχ2 values returned by the system identification algorithm or

by comparing the recovered values to those got from fitting the model to a measurement

of the optical response (as in figure6.3). If the model used in the system identification

routine is not a true representation of the detector, then we could recover biased optical

parameters, possibly resulting in highχ2 values.

Fast path electronics

The modelling of the fast path electronics affects two areas of the calibration routine:

the system identification routine, and the loop-gain correction. The effect on the system

6.3 Results and validation 213

102 103

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

Frequency (Hz)M

agnit

ude

rati

o

+10%

+5%

+1%

−1%

−5%

−10%

Figure 6.33: The systematic magnitude

error introduced by incorrectly estimating

the zero frequency of the optical response.

identification routine will be to bias the recovered optical parameters if we have calibra-

tion lines at frequencies where there is loop-gain. This in turn will contribute to higher

values of theχ2 number returned by the optimisation routine.

The correction for the loop-gain of the Michelson servo requires that the model of the fast

path electronics be as accurate as possible. We have two possible sources of systematic

error here: modelling the measured fast path electronics with a pole/zero model and

then converting that model into suitable IIR filters. Table6.4 shows one entry for each

of these cases. Each error is evaluated at 50 Hz where the loop-gain has a significant

effect in the detection band. The model of the measured fast path electronics introduces

a systematic error that is of the order 3% around 50 Hz; in converting this model to IIR

filters, we introduce only negligible additional error.

Michelson servo closed-loop transfer function

The closed-loop transfer function of the Michelson servo is one of the effects we try to

correct for in the calibration process. This transfer function is dominated by the com-

bination of the transfer functions of the fast path electronics and the electrostatic drive

actuator, together with the overall optical gain. At low frequencies where the Michel-

son servo has significant loop gain, the 1/ f 2 model of the electrostatic drive response

is accurate and therefore not considered as part of this error analysis. The errors in

6.4 Summary 214

the closed-loop transfer function are then dominated by any errors in the fast path elec-

tronics. From figure6.9 we can see a systematic error of around 3% at 50 Hz; this is

consistent with the error in the model of the fast path electronics at this frequency. Im-

provements in the modelling of the fast path electronics should lead to a better fit for the

MI closed-loop transfer function.

Frequency-domain to time-domain

All of the errors discussed so far have been evaluated in the frequency domain as differ-

ences between underlying models/parameters and implemented filters or measurements.

Since the calibration process is implemented as a time-domain pipeline, we have to con-

sider any errors that may enter due to the application of the time-domain filters, since

it is possible that the analytically-evaluated response of an IIR filter can differ from the

response of the filter when it is applied to data. The comparison of the time-domain cal-

ibration to a frequency-domain calibration was discussed in section6.3.1. We can see in

that comparison that there is a systematic difference around 50 Hz that could arise from

the performance of the IIR filters; around 1200 Hz we see no systematic differences be-

tween the two methods. It is therefore plausible that the observed systematic error at low

frequency arises in correcting for the loop-gain of the Michelson servo. At both high and

low frequencies, we see random errors at the level of about 5%; these errors are present

across the whole of the detection band and most probably arise in the correction for the

overall optical gain.

6.4 Summary

Using the methods, tools, and experience gained in calibrating the power-recycled con-

figuration of GEO, a scheme was implemented to perform a time-domain calibration of

the dual-recycled configuration of GEO. These initial experiments focused on the sim-

plest case of calibrating this more complex interferometer configuration (using only one

demodulated quadrature of the output photodiode to recover a strain signal). A simple

6.4 Summary 215

time-domain simulation was developed to allow most of the calibration pipeline to be

rigorously tested and debugged. The developed calibration scheme was applied to an

extended science run (S3) with great success. The accuracy achieved in the calibration

process was determined to be between 5 and 10% for the whole frequency band, with an

additional 5% systematic offset towards lower frequencies (50-200 Hz).

The calibration scheme presented over the last three chapters can be improved in a num-

ber of areas. In particular, so far the signal information present in theQ demodulation

quadrature has been neglected. The next stage in the development is to include this

information either post- or pre-calibration so that we obtain an optimal signal-to-noise

ratio in the calibrated strain signal. This requires an extension to the system identifica-

tion routine so that measurements of the calibration lines inQ are included and used to

recover the optical parameters forQ . The appropriate inverse optical filters forQ must

then be developed and applied using the same techniques are forP .

One of the aims of the detector development is to improve the sensitivity of the instru-

ment at frequencies from 50 to 1 kHz. This necessarily requires that the noise perfor-

mance of the Michelson length-control servo electronics is greatly improved. When this

is done, it may be possible to include the recorded feedback signals in the calibration

process, rather than simulating them with time-domain filters of the servo electronics

(see discussion in section6.2.3). Further experiments and testing of this method are

required. Another aim of the detector development is to improve the autoalignment sys-

tem so that fluctuations of the optical gain on time-scales around 1 second and faster are

reduced to a level that means that the calibration process does not have to correct for

them.

One of the main sources of noise in the calibration process is the up-sampling of the

optical gain estimates from 1 Hz to 16384 Hz. This would be greatly reduced if we only

had to up-sample from, say, 16 Hz. As the sensitivity of the detector improves, so does

the SNR of the calibration lines in the error-point spectrum such that we may be able

measure the calibration line amplitudes much more often. Another way to obtain more

frequent estimates of the optical parameters is to use a complex heterodyne method

to measure the complex amplitude of the calibration lines; this method requires some

6.4 Summary 216

investigation. Other sources of systematic and random error highlighted in this text need

to be focused on so as to reduce their affect on the overall calibration accuracy.

Currently the output of the calibration process is standard GEO style frame files with

channels as defined in table5.2. The format of these frame files and what information

is stored in them needs to be formalised within the framework of the LSC data analysis

requirements. For example, a suitable history of the data processing of each second of

data must be stored. This might include such things as the coefficients of the fixed IIR

and FIR filters. It is also important to implement a version control system in the event

that the calibration procedure undergoes revisions and new data sets are produced. This

is particularly important as the data from the current gravitational wave detectors starts

to lead to analysis results of scientific importance.

Appendix A

Digital filters

An overview of designing and using IIR filters.

A.1 IIR filters

Recursive, or IIR, filters are described by a difference equation:

y[n] = a0x[n] +a1x[n−1]+a2x[n−2]+ . . .+b1y[n−1]+b2y[n−2]+ . . . , (A.1)

wherex[] and y[] are the input and output digital time-series respectively. The coeffi-

cients,ai andbi , are called the recursion coefficients or filter taps.

The Z-transform is defined as

X(z) =

∞∑n=−∞

x(n)z−n. (A.2)

Applying the Z-transform to the difference equation shown above yields the transfer

function of the IIR filter. This can be written as

H(z) =(z − z1)(z − z2)(z − z3) . . .

(z − p1)(z − p2)(z − p3) . . ., (A.3)

wherezi and pi are the zeros and poles respectively as defined in the Z-plane. By ex-

panding the above equation and collecting powers ofz, the transfer function can always

A.1 IIR filters 218

be re-written in the form

H(z) =a0 + a1z−1

+ a2z−2 . . .

1 − b1z−1 − b2z−2 . . .(A.4)

where the coefficientsai andbi are the recursion coefficients of the difference equation.

The frequency response of an IIR filter can be evaluated as

H(i ω) =a0 + a1e(iω/fs) + . . . + ane(i(n−1)ω/fs)

1 + b1e(iω/fs) + . . . + ame(i(m−1)ω/fs), (A.5)

where, for a stable filter,m ≥ n.

A.1.1 The Bilinear transform

The bilinear transform provides a way to approximately map the S-domain of analogue

filters to the Z-domain of digital filters. Full details of the derivation of the bilinear

transform and the definition of the S- and Z-domains are not provided here but are well

represented in [DSPGuide], and [Antoniou].

To convert an S-domain filter with transfer function,H(s), to a Z-domain filter with

transfer functionH(z), such thatH(s) ≈ H(z), apply the transform

s = 2 fsz − 1

z + 1, (A.6)

where fs is the sampling frequency of the data to be filtered by the resulting digital filter.

An example will better explain this.

Recursion coefficients for a real pole

The response of a single real pole with characteristic frequencyω0, can be written in the

S-domain as

H(s) =ω0

s + ω0. (A.7)

A.1 IIR filters 219

Using the bilinear transform, we can write the approximation to the transfer function in

the Z-domain as

H(z) =ω0

2 fsz−1z+1 + ω0

(A.8)

=ω0z + ω0

2 fsz − 2 fs + ω0z + ω0(A.9)

=ω0 + ω0z−1

(ω0 + 2 fs) + (ω0 − 2 fs)z−1(A.10)

=

ω0ω0+2 fs

+ω0

ω0+2 fsz−1

1 +ω0−2 fsω0+2 fs

z−1. (A.11)

From this, we can immediately write down the recursion coefficients for an IIR filter as

a0 =ω0

ω0 + 2 fs, (A.12)

a1 = a0, (A.13)

b0 = 1, (A.14)

b1 =ω0 − 2 fsω0 + 2 fs

. (A.15)

The following results are derived in much the same way but the results are just stated.

Recursion coefficients for a real zero

A real zero defined by a characteristic frequency,ω0, has recursion coefficients,

a0 =2 fs + ω0

ω0, (A.16)

a1 =ω0 − 2 fs

ω0, (A.17)

b0 = 1, (A.18)

b1 = 1. (A.19)

Recursion coefficients for a complex pole

A complex pole defined by a characteristic frequency,ω0, and quality factor,Q, has

recursion coefficients,

A.1 IIR filters 220

a0 = 1/k, (A.20)

a1 = −2/k, (A.21)

a2 = −1/k, (A.22)

b0 = 1, (A.23)

b1 =2ω2

0 − 8 f 2s

kω20

, (A.24)

b2 =Qω2

0 + 4Q f 2s − 2ω0 fs

kQω20

, (A.25)

where the factork is given by

k =Qω2

0 + 4Q f 2s + 2ω0 fs

Qω20

. (A.26)

Recursion coefficients for a complex zero

A complex zero defined by a characteristic frequency,ω0, and quality factor,Q, has

recursion coefficients,

a0 =−Qω2

0/2 − 2Q f 2s − ω0 fs

Qω20

, (A.27)

a1 =4 f 2

s − ω20

ω20

, (A.28)

a2 =−Qω2

0/2 − 2Q f 2s + ω0 fs

Qω20

, (A.29)

b0 = 1, (A.30)

b1 = −2, (A.31)

b2 = 1. (A.32)

A.1.2 Changes of Gain

Changing the gain of an existing IIR filter is simply a case of scaling all of thea coeffi-

cients by an appropriate factor. To decide what that factor is, one needs to know the gain

A.1 IIR filters 221

of the current filter,G0, and then the new gain,G, can be applied by

ai =G

G0ai . (A.33)

The gain of an IIR filter at zero frequency can be determined by using the difference

equation on one input sample of unit amplitude. If we force the output to beG0—the

gain we seek—then we get the equation

G0 =

n−1∑i =0

ai

1 −

m−1∑j =1

b j

. (A.34)

The gain at high frequency (the Nyquist frequency) can be shown (in [DSPGuide]) to be

G0 =a0 − a1 + a2 − a3 + . . .

1 − (−b1 + b2 − b3 + . . .). (A.35)

A.1.3 Cascading systems

If we have two IIR filters defined byH1(z) and H2(z), then the combined response,

H(z), is given by

H(z) = H1(z)H2(z), (A.36)

=a0 + a1z−1

+ . . . + an−1z−(n−1)

1 − b1z−1 − . . . − bn−1z−(n−1)

×A0 + A1z−1

+ . . . + Am−1z−(m−1)

1 − B1z−1 − . . . − Bm−1z−(m−1), (A.37)

wherea1...n−1 andb1...n−1are the recursion coefficients of the filter described byH1(z),

andA1...m−1 andB1...m−1 are the coefficients of the filter described byH2(z). Although

in principle the number ofa andb (or A andB) need not be the same (although we need

moreb coefficients thana for a stable filter), the number of coefficients can always be

made equal by setting the extraa coefficients required to zero. This makes the notation

and calculation of cascading systems much simpler. By expanding the terms and collect-

ing power ofz, the coefficients,αi andβi , of the combined filter,H(z), can be computed

using the following algorithm:

A.1 IIR filters 222

---------------------------------------------------

% Zero all the output coefficients

% Calculate the output ‘a’ coefficients as follows:

for i from 0 to n-1

for j from 0 to m-1

alpha[i+j] = alpha[i+j] + a[i]A[j]

end

end

% Calculate the ‘b’ coeffiecients as follows:

for i from 0 to n-1

for j from 0 to m-1

if i>0 AND j>0

beta[i+j] = beta[i+j] - 1.0*b[i]*B[j]

else

beta[i+j] = beta[i+j] + b[i]*B[j]

end

end

end

% Number of output coefficients

N = m+n-1

---------------------------------------------------

A.2 Measured transfer functions of the DAQ anti-alias filters 223

As an example, combining two systems, both with 3 coefficients (m = n = 3), givesα

andβ coefficients

α0 = a0A0, (A.38)

α1 = a0A1 + a1A0, (A.39)

α2 = a0A2 + a1A1 + a2A0, (A.40)

α3 = a1A2 + a2A1, (A.41)

α4 = a2A2, (A.42)

β0 = 1, (A.43)

β1 = b1 + B1, (A.44)

β2 = B2 − b1B1 + b2, (A.45)

β3 = −b1B2 − b2B1, (A.46)

β4 = −b2B2. (A.47)

A.2 Measured transfer functions of the DAQ anti-alias

filters

A.2 Measured transfer functions of the DAQ anti-alias filters 224

101 102 103

10−4

10−2

100

Frequency (Hz)

Mag

nitu

de (d

B)

Amplitude − Ch1 (16384 Hz) / Ch3 (8192 Hz)

101 102 103

−150

−100

−50

0

50

100

150

Frequency (Hz)

Pha

se (d

eg)

Phase

Figure A.1:A measured transfer function of the IIR anti-alias filter that is applied

in the data archiving software. In this case, the filter is used when decimating data

from 16384 Hz to 8192 Hz. Here the corner frequency can be calculated from equa-

tion 2.3to be 2457.6 Hz. The model response of the filter is overlaid for comparison.

101 102 10310−6

10−4

10−2

100

Frequency (Hz)

Mag

nitu

de (d

B)

Amplitude − Ch1 (16384 Hz) / Ch3 (2048 Hz)

101 102 103

−150

−100

−50

0

50

100

150

Frequency (Hz)

Pha

se (d

eg)

Phase

Figure A.2:A measured transfer function of the IIR anti-alias filter that is applied

in the data archiving software. In this case, the filter is used when decimating data

from 16384 Hz to 2048 Hz. Here the corner frequency can be calculated from equa-

tion 2.3to be 614.4 Hz. The model response of the filter is overlaid for comparison.

A.2 Measured transfer functions of the DAQ anti-alias filters 225

101 102 10310−6

10−4

10−2

100

Frequency (Hz)

Mag

nitu

de (d

B)

Amplitude − Ch1 (16384 Hz) / Ch3 (1024 Hz)

101 102 103

−150

−100

−50

0

50

100

150

Frequency (Hz)

Pha

se (d

eg)

Phase

Figure A.3:A measured transfer function of the IIR anti-alias filter that is applied

in the data archiving software. In this case, the filter is used when decimating data

from 16384 Hz to 1024 Hz. Here the corner frequency can be calculated from equa-

tion 2.3to be 307.2 Hz. The model response of the filter is overlaid for comparison.

101 102 10310−6

10−4

10−2

100

Frequency (Hz)

Mag

nitu

de (d

B)

Amplitude − Ch1 (16384 Hz) / Ch3 (512 Hz)

101 102 103

−150

−100

−50

0

50

100

150

Frequency (Hz)

Pha

se (d

eg)

Phase

Figure A.4:A measured transfer function of the IIR anti-alias filter that is applied

in the data archiving software. In this case, the filter is used when decimating data

from 16384 Hz to 512 Hz. Here the corner frequency can be calculated from equa-

tion 2.3to be 153.6 Hz. The model response of the filter is overlaid for comparison.

A.3 A design for IIR Chebychev/Butterworth filters 226

A.3 A design for IIR Chebychev/Butterworth filters

The following filter design is based on one presented in [DSPGuide]. It is implemented

in the DAQ code to provide software anti-alias capabilities. Here is the algorithm as it

is in the DAQ source code filerecursion coeffs.c . The pseudo code is given in

verbatim text while explanations appear in normal mathematical text.

Given the following input parameters:

fc: The corner frequency of the filter.

type: The type of filter to design: 0=high pass, 1=low pass. A low pass is

always designed and then converted, if necessary, to a high pass at the

end. For this reason, the code talks almost entirely about poles and not

zeros.

Rp: The amount of ripple allowed in the pass/stop bands. Set this to zero to

design a Butterworth filter.

Ntaps: The number of taps (pole or zeros) in the filter.

Calculate a Chebychev/Butterworth filter as follows:

• Define some variables and setup an identity filter.

/* define some local variables */

double rp, ip /* Real and imaginary coords */

/* of pole on unit circle. */

double ep, vx, kx, k /* Parameters used in */

/* circle->ellipse mapping. */

double T, D, M /* Parameters used in bilinear */

/* transform. */

double W /* The output cutoff frequency. */

double a[], b[] /* The final filter coefficients. */

A.3 A design for IIR Chebychev/Butterworth filters 227

double A[3], B[3], X[3], Y[3] /* Coefficients produced */

/* in each stage of the */

/* calculation. */

double ta[], tb[] /* Temporary storage of */

/* the filter coefficients. */

double sa, sb, gain /* Parameters for gain. */

/* Initialise the coefficients to the identity filter. */

for i=0 to N_taps

a[i] = 0.0

b[i] = 0.0

ta[i] = 0.0

tb[i] = 0.0

end

a[2] = 1.0

b[2] = 1.0

• Next we calculate some useful values.

ep =

√100

100− Rp×

(100

100− Rp− 1

)

vx =1

Ntapslog

1

ep+

√1

ep2+ 1

kx =1

Ntapslog

1

ep+

√1

ep2− 1

kx =expkx + exp−kx

2.0T = 2.0 × tan 0.5

W = 2.0π fc

And the code for this:

/* Calculate some useful parameters. */

A.3 A design for IIR Chebychev/Butterworth filters 228

ep = sqrt((100.0/(100.0 - R_p)) * (100.0/(100.0 - ripple_percent)) - 1.0)

vx = (1.0 / num_poles) * log((1.0 / ep) + sqrt(1.0 / (ep * ep) + 1.0))

kx = (1.0 / num_poles) * log((1.0 / ep) + sqrt(1.0 / (ep * ep) - 1.0))

kx = (exp(kx) + exp(-kx)) / 2.0

T = 2.0 * tan(0.5)

W = 2.0 * PI * f_c

Each pair of poles,p, must be processed separately and then cascaded together at the

end. The following description produces the coefficients for one set of poles and then

describes the cascading procedure.

First we calculate the position of the pole-pair on the unit circle:

rp = − cosπ

2Ntaps+

π(p − 1)

Ntaps,

i p = sinπ

2Ntaps+

π(p − 1)

Ntaps,

and then adjust these if we want to calculate a Chebychev filter by:

rp = rp ×(expvx − exp−vx)

2kx,

i p = i p ×(expvx + exp−vx)

2kx,

Next we apply a bilinear transform to each pole-pair to get:

M = rp2+ i p2,

D = 4 − 4rpT + MT2,

A.3 A design for IIR Chebychev/Butterworth filters 229

A set of coefficients for a filter with unity cutoff frequency are then given by:

X[0] =T2

D,

X[1] = 2X[0],

X[2] = X[0],

Y[1] =8 − 2MT2

D,

Y[2] =−4 − 4rpT − MT2

D.

Next we must convert these coefficients to ones with the desired cutoff frequency. This

process is different for a high pass or a low pass filter. Here is the high pass calculation

first:

k =− cos W

2.0 + 0.5

cos W2.0 − 0.5

,

D = 1.0 + Y[1]k − Y[2]k2,

A[0] = (X[0] − X[1]k + X[2]k2)/D,

A[1] = −(−2.0X[0]k + X[1] + X[1]k2− 2.0X[2]k)/D,

A[2] = (X[0]k2− X[1]k + X[2])/D,

B[1] = −(2.0k + Y[1] + Y[1]k2− 2.0Y[2]k)/D,

B[2] = (−k2− Y[1]k + Y[2])/D.

A.3 A design for IIR Chebychev/Butterworth filters 230

And for the low pass case:

k =sin 0.5 − W/2.0

sinW/2.0 + 0.5,

D = 1.0 + Y[1]k − Y[2]k2,

A[0] = (X[0] − X[1]k + X[2]k2)/D,

A[1] = (−2.0X[0]k + X[1] + X[1]k2− 2.0X[2]k)/D,

A[2] = (X[0]k2− X[1]k + X[2])/D,

B[1] = (2.0k + Y[1] + Y[1]k2− 2.0Y[2]k)/D,

B[2] = (−1.0k2− Y[1]k + Y[2])/D.

We must now cascade the coefficients for this pole-pair onto the others calculated so far.

Here is the code for this loop including the steps needed to complete the process.

/* loop through each pole-pair */

for pole = 1 to N_taps

/* calculate this pole’s position on the unit circle */

rp = -1.0 * cos (M_PIl / (2.0 * num_poles)

+(1.0 * pole - 1.0) * M_PIl / (1.0 * num_poles))

ip = sin (1.0 * M_PIl / (2.0 * num_poles)

+(1.0 * pole - 1.0) * M_PIl / (1.0 * num_poles))

/* is this a Chebychev or just a Butterworth? */

if R_p > 0.0

rp = rp * (exp (vx) - exp (-vx)) / (2.0 * kx)

ip = ip * (exp (vx) + exp (-vx)) / (2.0 * kx)

end

/* now apply bilinear transform */

M = rp * rp + ip * ip

D = 4.0 - 4.0 * rp * T + M * T * T

X[0] = T * T / D

X[1] = 2 * X[0]

A.3 A design for IIR Chebychev/Butterworth filters 231

X[2] = X[0]

Y[1] = (8.0 - 2.0 * M * T * T) / D

Y[2] = (-4.0 - 4.0 * rp * T - M * T * T) / D

/* now change these unity cutoff coeffs to the */

/* desired frequency cutoff */

if type == 1 /* high-pass */

k = -1.0 * cos (W / 2.0 + 0.5) / cos (W / 2.0 - 0.5)

D = 1.0 + Y[1] * k - Y[2] * k * k

A[0] = (X[0] - X[1]*k + X[2]*k*k) / D

A[1] = -1.0 * (-2.0*X[0]*k + X[1] + X[1]*k*k - 2.0*X[2]*k) / D

A[2] = (X[0] * k * k - X[1] * k + X[2]) / D

B[1] = -1.0 * (2.0*k + Y[1] + Y[1]*k*k - 2.0*Y[2]*k) / D

B[2] = (-1.0*k*k - Y[1]*k + Y[2]) / D

else /* low-pass */

k = sin (0.5 - W / 2.0) / sin (W / 2.0 + 0.5)

D = 1.0 + Y[1] * k - Y[2] * k * k

A[0] = (X[0]-X[1]*k + X[2]*k*k) / D

A[1] = (-2.0*X[0]*k + X[1] + X[1]*k*k - 2.0*X[2]*k) / D

A[2] = (X[0]*k*k - X[1]*k + X[2]) / D

B[1] = (2.0*k + Y[1] + Y[1]*k*k - 2.0*Y[2]*k) / D

B[2] = (-1.0*k*k - Y[1]*k + Y[2]) / D

end

/* now add these coeffs to the cascade */

for i = 2 to N_taps

a[i] = A[0] * a[i] + A[1] * a[i - 1] + A[2] * a[i - 2]

b[i] = b[i] - B[1] * b[i - 1] - B[2] * b[i - 2]

end

end

/* finish up the coeffs */

A.3 A design for IIR Chebychev/Butterworth filters 232

b[2] = 0.0

for i=0 to N_taps

a[i] = a[i + 2]

b[i] = -1.0 * b[i + 2]

end

/* normalise the gain to unity in the passband */

sa = sb = 0.0

if type == 1 /* high-pass */

for i = 0 to N_taps

sa = sa + a[i] * -1ˆi

sb = sb + b[i] * -1ˆi

end

else /* low-pass */

for i = 0 to N_taps

sa = sa + a[i]

sb = sb + b[i]

end

end

gain = sa / (1.0 - sb)

for i = 0 to N_taps

a[i] = a[i] / gain

end

Appendix B

Code listings

The following code listings are presented in a form of pseudo C code. Reference is made in each case to

the actual source file as it exists in the software package(s).

TableB.1 summarises the notation used in the code listings.

Operator Description

. Denotes accessing a field within a data structure

% Denotes modulo arithmetic.

/* */ Encloses comments.

& Gives the address of a variable.

|| Denotes logical OR.

&& Denotes logical AND.

!= Denotes ‘not equal to’.

Table B.1: Table of symbols and operators used in the pseudo code listings.

B.1 DCU code

These are the code listings for the DCU code. See section2.3.3for details of operation and design.

B.1 DCU code 234

B.1.1 Bounded buffer

The bounded buffer data structure is namedbbuffer within the DCU code. The code can be found in

bounded buffer.c .

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Get the current head position %

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function get_head(bbuffer b)

/* Get the current head position. */

head = b.head

/* Advance the head position ready for */

/* the next request. */

buffer.head = (buffer.head + 1) % b.maxLength

/* Return the value of the head to */

/* calling function. */

return head

end

The function to retrieve the current tail pointer of a bounded buffer is essentially the same

as that that retrieves the head. Increasing the length of the bounded buffer (the difference

between the head and the tail) is the responsibility of the reading/writing function. The

following code extract shows a read operation. Processes that read from the buffer need

to take care that the length of the buffer is greater than or equal to 1.

/* Wait for the buffer to contain data. */

while(bb.length < 1)

taskDelay(1)

end

/* get the current tail pointer */

tail = get_tail(bb)

/* Do something with the data */

/* that the tail points too. */

B.1 DCU code 235

print(tail.data[0])

/* Reduce the length of the buffer since */

/* we are done with this segment. */

/* The length of the buffer is protected */

/* from being changed by other processes */

/* during this operation. */

semTake(bb.bb_lock, WAIT_FOREVER) /* lock this buffer */

bb.length = bb.length - 1 /* change the length */

semGive(bb.bb_lock) /* release the buffer */

B.1.2 VMIC interrupt handler

The code proper appears in source filevmic.c of the DCU code.

function vmic_interrupt()

/* should we collect VMIC data? */

if(collect_vmic_data_flag)

/* is this the first sample of this second? */

if(vmic_samples_collected == 0)

/* get the current time from the gps card */

gps_get_time(&secs, &nsecs)

/* get the head of the bounder buffer */

vc_interrupt = get_head(bbs)

/* is the buffer full? */

if(bbs.length == bbs.maxLength)

/* increase the lost seconds counter */

vmic_lost_secs = vmic_lost_secs + 1

get_tail(bbs)

B.1 DCU code 236

bbs.length = bbs.length -1

else

/* always reset this flag if we can store data */

vmic_lost_secs = 0

end

/* time stamp this data packet */

vc_interrupt.gps_time = secs

vc_interrupt.gps_nsecs = nsecs

/* Check to see if the clock signals to */

/* the VMIC are still synchronised. We */

/* expect the first sample to be collected */

/* between t=1.5ms and t=2ms during normal */

/* operation (determined empirically). */

if( (nsecs < 1500000) || (nsecs > 2000000) )

message = ("### %s ### vmic_interrupt: OUT OF SYNC ERROR...")

queue_msg(bbm, message);

end

end /* end of first sample query */

/* Now collect this scan of data */

for chan=0 to VMIC_NUM_CHANS

/* get a pointer to the correct place */

/* in the VMIC bounded buffer. */

dataS = vc_interrupt.data[VMIC_WORD_SIZE *

(vmic_samples_collected + chan*VMIC_SAMPLE_RATE)])

/* Get the data from the VMIC board. */

*dataS = vmic_board->chan[chan]

end

B.1 DCU code 237

/* increase the sample counter */

vmic_samples_collected = vmic_samples_collected + 1

/* have we collected one second of data? */

if(vmic_samples_collected == VMIC_SAMPLE_RATE)

/* reset counters ready for next interrupt */

vmic_samples_collected = 0

/* increase the length of the bounded buffer. */

bbs.length = bbs.length + 1

end

end

end

B.1.3 Bounded buffer resynchronisation

/* bb = the ICS buffer */

/* bbs = the VMIC buffer */

/* dc = head of ICS buffer */

/* vdc = head of VMIC buffer */

/* If the two buffer heads point to */

/* data from different times. */

if(dc.gps_time != vdc.gps_time)

/* send a log message to the message queue */

message = ("### %s ### send_data: buffers are out of sync!...")

queue_msg(bbm, message)

/* Collect some data for resynchronisation process */

while(bb.length < 5 && bbs.length<5)

B.1 DCU code 238

taskDelay(10)

end

/* lock the two buffers */

semTake(bb.bb_lock, WAIT_FOREVER)

semTake(bbs.bb_lock, WAIT_FOREVER)

/* There are two possibilities: the ICS buffer is */

/* behind the VMIC buffer or vice versa. */

/* Drop data from the ICS buffer until it */

/* has caught up with the VMIC buffer. */

while( dc.gps_time < vdc.gps_time && bb.length > 1 )

gps_get_time(&secs, &nsecs)

/* send a message to the message queue */

message = ("### %s ### send_data: resynchronising buffers!...")

queue_msg(bbm, message);

/* drop one packet of data */

bb.length = bb.length - 1

dc = get_tail(bb)

end

/* Drop data from the VMIC buffer until it */

/* has caught up with the ICS buffer. */

while( dc.gps_time > vdc.gps_time && bbs.length > 1 )

/* again send a message */

gps_get_time(&secs, &nsecs)

message = ("### %s ### send_data: resynchronising buffers!...")

queue_msg(bbm, message);

/* Drop one data packet from the VMIC buffer. */

/* Because interrupt handlers in the VxWorks OS */

/* are not allowed to take semaphore flags, we */

/* must consider the number of samples collected */

/* by the vmic interrupt handler to ensure the */

/* buffer length variable is not being accessed */

B.1 DCU code 239

/* by the interrupt handler. */

if(vmic_samples_collected <10 || vmic_samples_collected > 500)

taskDelay(1)

/* now it’s safe to change the buffer length */

bbs.length = bbs.length - 1

vdc = get_tail(bbs)

end

/* release the buffers */

semGive(bb->bb_lock)

semGive(bbs->bb_lock)

/* send message to the message queue */

message = ("### %s ### send_data: buffers are synchronised!...")

queue_msg(bbm, message)

end

B.1.4 DAQ checking software

-----------------

How to use daqchk

-----------------

M Hewitson 20-01-03

$Id: README,v 1.1 2003/01/20 08:38:19 hewitson Exp $

This program performs continuous checks on the DAQ system. It uses the

notion of two particular objects: a DCU and a channel. Each of these

objects can have certain checks performed on them.

For DCU objects, the current time of the DCU is checked to be within 20

B.1 DCU code 240

secs of the current system time. This requires the system clock

(on morgan) to be synchronised to UTC. If no DCUs are in the list, no

other checks will be performed. If the DCUs are not in error, the

appropriate remote file (DCU_OK_FILE) is updated containing the default

error message, "DCU: ERROR". If a DCU is not on time, then the message,

"dcu_name: DCU is X seconds behind current time.",

is written into the appropriate remote file. A default error is written

so that if the program stops, an error will be reported by the monitoring

software.

For channel objects, the RMS value over a specified number of samples is

calculated. For each channel, the user specifies an upper and lower limit

for the RMS along with the number of samples to calculate over. The

calculation starts from the specified sample of the second and the

numbers of samples is coerced to be within 0->Fs. The DCU that contains

the channel is also specified in the configuration file so that the

appropriate remote file can be written to.

If the calculated RMS is within the user specified range, the appropriate

remote file is updated containing the defualt error message as above. If

the channel is out of range, the message,

"channel_name: channel is out of range.(RMS = X, l1 = Y, l2 = Z)",

is written into the appropriate remote file.

The internal diagnostic configuration structure has the following fields:

int nDCUs; // the number of dcus to check

int nChannels; // the number of channels to check

char dcus[][]; // store the dcu names

int dcuerror[]; // error state for each dcu

long int ctimes[]; // store the current times

char channels[][]; // store the channel names

char chan_dcus[][]; // store the dcu of a channel

int chanerror[]; // error state of channel

float ll[]; // store lower limits

B.1 DCU code 241

float ul[]; // store upper limits

int mi[]; // the measurement interval

int si[]; // start sample of RMS

The program can cope with upto 10 DCUs and 50 channels.

A typical config file looks like this:

-----------------------------------------------------------

# config file for daqchk

#

# M Hewitson 20-01-03

# we check 2 DCUs

DCU alchemist

DCU thot

# check one channel on each DCU

# calculate RMS over 100 samples starting at sample 10. Write

# the results in the remote file named MID_EP_OK

CHAN G1:LSC_MID_EP MID_EP_OK -0.1 0.5 10 100

# calculate RMS over all samples starting at sample 0

CHAN G1:Test TEST_OK -0.5 0.5 0 16384

-----------------------------------------------------------

Glossary of terms

1 PPS: A one Pulse Per Second signal that defines GPS second boundaries.

ADC: Analogue to Digital Converter.

AEI: Albert Einstein Institute.

CLTF: Closed-Loop Transfer Function.

DAC: Digital-to-Analogue Converter.

DAQ(s): Data AcQuisition (system).

DCU: Data collecting Unit.

DR: Dual-Recycling.

DRMI: Dual-Recycled Michelson Interferometer.

enbw: Equivalent Noise Bandwidth (used in figure legends).

EOM: Electro-Optic Modulator.

ESD: Electro-Static Drive.

ffb: Fast feedback.

FFT: Fast Fourier Transform.

FIFO: First In, First Out.

FIR: Finite Impulse Response filter.

fs: Sampling frequency.

243

GPS: Global Positioning System.

GR: General Relativity.

GW: Gravitational Wave.

ICS: Integrated Circuits and Systems.

IFO: Interferometer.

IIR: Infinite Impulse Response.

IMD: Intermediate-Mass Drive.

I/O: Input/Output.

LSB: Least Significant Bit.

MECL: Motorola Emitter Coupled Logic.

MI: Michelson Interferometer.

ndata: The number of data samples (used in figure legends).

nfft: The number of samples used for each FFT of an (averaged) amplitude or power

spectrum or spectral density (used in figure legends).

navs: The number of averages in an amplitude or power spectrum or spectral density

(used in figure legends).

OLG: Open Loop Gain.

PLC: Programmable Logic Chip.

PLL: Phase Locked Loop.

PR: Power-Recycling.

RAM: Random Access Memory.

RAID: ???

RF: Radio Frequency.

244

S1: The 18 day LIGO science run in August 2002.

S3: The LIGO science run from November 2003 to January 2004.

sfb: Slow feedback.

SMS: Short messaging system.

SNR: Signal-to-Noise Ratio.

SR: Signal-Recycling.

SRAM: Static RAM.

TTL: Transistor-Transistor Logic.

VME: VERSAmodule Eurocard.

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