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Modelling of Rainfall Drop Size Distributions for Application in Fade Prediction on Earth-Space Paths

Doctor of Philosophy Thesis

K'ufre-Mfon Ebong Ekerete

May 2017


“pauca sed matura”Johann Carl Frederich Gauss (30 April 1777 – 23 February 1855)

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DeclarationThis thesis has not been previously accepted in substance for any degree and is not

being concurrently submitted in candidature for any degree.

Signed: …………………………………… (Candidate)

Date: ………………………………………

STATEMENT

This thesis is being submitted in partial fulfilment of the requirements for the degree of

Doctorate of Philosophy (PhD).

This thesis is the result of my own independent investigations, except where otherwise

stated, other sources are acknowledged by explicit references. A bibliography is

appended.

I hereby give consent for my thesis, if accepted, to be available for photocopying and

for inter-library loan, and for the title and summary to be made available to outside

organisations.

Signed: …………………………………… (Candidate)

Date: ………………………………………

K’ufre-Mfon Ebong Ekerete | Page ii


DedicationThis thesis is dedicated to all my Ks.

Kokoma, Karen, Kevin and Karla.

I do look forward to reading each of yours,

and to the memory of the following:

Margaret O. Ekerete, PhD (1964 – 2000)

Ruth Jeanette Hardage (1931 – 2016).

This work is now dedicated to the memory of my mother, Inyang E. Ekerete, (11th

November 1937 to 24th May, 2017) who passed on the day my PhD was confirmed.

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AcknowledgementsProjects are built on already available body of knowledge; hence I cannot lay claim that this

work was all mine effort. I stood on the shoulders of giants so that I could see farther.

One of the giants that let me stand on his shoulder was my indefatigable Director of Studies,

Prof. Ifiok E. Otung. My thanks go to my co-supervisors Dr. Francis Hunt and Mrs. Judith

Jeffery, who bore the brunt of my questions.

My sincere gratitude goes to the Chilbolton Observatory and its staff member for providing the

data used in this thesis and some commendable support. This thesis was funded by the Niger

Delta Development Commission (NDDC) Overseas Scholarship, my eternal thanks to the

Commission and its staff members for extending me the scholarship to carry out this research.

I can’t forget to thank my family and friends; my parents and siblings, Anietie, Joe, Seifon,

Nkopuduk and Nseno, my aunty Mrs. Margaret Ekerete, my cousins Ken and Russell, my

other mum Mrs. R. Jeanette Hardage and my mother-in-law who didn’t live to see me

complete this journey. Special mention of my old roommate and friend at a previous

University, Mr. Eyo Ansa who showed me much friendship and the much-needed

encouragement when things looked the other way.

I also thank my teachers, past and present for their patience in parting with their hard-earned

knowledge. I also thank the staff of the University of South Wales, without them there

wouldn’t be the University. I thank my colleagues and co-travellers Dr. Leshan Uggalla,

Abdulkareem Karasuwa, and Burak Unal.

I wouldn’t dream of going back home if I don’t give a big hug to my dear wife, Kokoma who

became daddy and mummy to my children during my long absences. My children too, Karen,

Kevin and Karla who were the inspiration for this journey.

Please bear with me if I have not mentioned you, as there were too many people who offered

help one way or another. To all, those I’ve remembered and those I’ve forgotten, a very big

THANK YOU from my heart of hearts.

Table of Contents

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Declaration................................................................................................................................... ii

Dedication........................................................................................................................iii

Acknowledgements...........................................................................................................iv

Table of Figures................................................................................................................ix

List of Tables....................................................................................................................xi

Abbreviations.........................................................................................................................xii

Table of Symbols..................................................................................................................xiii

Abstract..................................................................................................................................xv

Chapter 1 : INTRODUCTION............................................................................................1

1.1 Introduction..................................................................................................................2

1.2 Motivation for the research..........................................................................................3

1.3 Research Aims and Objectives.....................................................................................5

1.4 The Chilbolton Observatory, instruments and the datasets..........................................6

1.5 Thesis contents overview...........................................................................................10

1.6 Novelty in the Research and Contributions to Knowledge........................................14

1.7 Publications from the research...................................................................................15

Chapter 2 : RADIO WAVE PROPAGATION....................................................................17

2.1 Introduction................................................................................................................18

2.2 The Earth’s Atmosphere.............................................................................................19

2.2.1 The Layers of the Earth’s Atmosphere..................................................................19

2.3 Propagation Impairments...........................................................................................21

2.3.1 Signal losses in free space......................................................................................21

2.3.2 Dispersion of electromagnetic waves.....................................................................22

2.3.2.1 Absorption.........................................................................................................22

2.3.2.2 Diffraction.........................................................................................................22

2.3.2.3 Reflection..........................................................................................................22

2.3.2.4 Refraction..........................................................................................................23

2.3.2.5 Scattering...........................................................................................................23

2.3.3.6 Multipath...........................................................................................................23

2.3.2.6 Scintillation.......................................................................................................23

2.4 Principles of radiometry.............................................................................................24

2.5 Attenuation.................................................................................................................27

2.5.1 The ITU Model Relationship between Rain Rate and Attenuation.......................27

2.6 Scattering by spherical raindrops...............................................................................28

2.7 Extinction (or attenuation) cross-section....................................................................30

2.8 Specific Attenuation...................................................................................................30K’ufre-Mfon Ebong Ekerete | Page v


2.9 Effective Rainy Slant Path Length.............................................................................32

2.10 Fade Dynamics...........................................................................................................33

2.11 Fade Mitigation Techniques.......................................................................................35

2.12 Conclusion..................................................................................................................35

Chapter 3 : RAINFALL DROP SIZE DISTRIBUTIONS AND DSD MODELS...................37

3.1 Introduction................................................................................................................38

3.2 Rain Dynamics...........................................................................................................38

3.2.1 Rain formation........................................................................................................38

3.2.2 Rainfall types..........................................................................................................39

3.2.3 Raindrop sizes and shapes......................................................................................40

3.2.4 Fall velocity............................................................................................................42

3.2.5 Rainfall rates..........................................................................................................44

3.3 Rainfall Drop Size Distributions and DSD Modelling...............................................46

3.3.1 Disdrometer Bin Calibration..................................................................................47

3.3.2 Possible Errors........................................................................................................47

3.4 DSD modelling...........................................................................................................48

3.4.1 Negative Exponential Distribution.........................................................................49

3.4.2 The Lognormal Distribution...................................................................................49

3.4.3 The Gamma Distribution........................................................................................50

3.4.3.1 Method of Moments...........................................................................................52

3.4.3.2 Maximum Likelihood Estimates........................................................................53

3.5 Goodness-of-Fit Tests................................................................................................53

3.6 Limitations and Gaps Identified.................................................................................55

3.7 The Gaussian Mixture Model (GMM).......................................................................55

3.8 DSD and Multimodality.............................................................................................59

3.9 Conclusion..................................................................................................................61

Chapter 4 : RAINFALL DSD MODELLING.....................................................................63

4.1 Introduction................................................................................................................64

4.2 Data............................................................................................................................64

4.3 Fitting Methodology...................................................................................................65

4.4 Procedure....................................................................................................................65

4.5 Results and Analyses..................................................................................................68

4.6 Possible Interpretations..............................................................................................74

4.7 Conclusions................................................................................................................76

Chapter 5 : MULTIMODALITY IN RAINFALL DSDs AND THE GAUSSIAN MIXTURE

MODEL..........................................................................................................................78

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5.1 Introduction................................................................................................................79

5.2 Data and Procedure used in this Study.......................................................................80

5.2.1 Data Collection.......................................................................................................80

5.2.2 Procedure................................................................................................................81

5.3 Results and Interpretations.........................................................................................82

5.4 Conclusions................................................................................................................93

Chapter 6 : SPECIFIC AND TOTAL RAIN ATTENUATION............................................95

6.1 Introduction................................................................................................................96

6.2 Data Collection and Processing..................................................................................97

6.3 Procedures..................................................................................................................97

6.4 Results and Interpretations.........................................................................................98

6.5 Rain Attenuation......................................................................................................109

6.5.1 Data and procedure...............................................................................................109

6.5.1.1 Data collection..................................................................................................109

6.5.1.2 Procedure..........................................................................................................109

6.6 Results and possible interpretations.........................................................................115

6.7 Rain Aloft.................................................................................................................121

6.8 Conclusions..............................................................................................................123

Chapter 7 : EFFECTIVE RAINY SLANT PATH LENGTH MODELLING AND

APPLICATION TO RAIN FADE PREDICTION............................................................125

7.1 Introduction..............................................................................................................126

7.2 Effective rainy slant path length...............................................................................126

7.3 Results and Interpretations.......................................................................................127

7.4 Application to Rain Fade Prediction........................................................................133

7.5 Conclusions..............................................................................................................134

Chapter 8 : CONCLUSIONS..........................................................................................136

8.1 Introduction..............................................................................................................137

8.2 Thesis Summary.......................................................................................................137

8.3 Work mapped against the initial objectives.............................................................140

8.4 Limitations...............................................................................................................142

8.5 Suggestions for further work....................................................................................142

References............................................................................................................................146

Appendix I............................................................................................................................153

Appendix II..........................................................................................................................154

Appendix III.........................................................................................................................155

Appendix IV.........................................................................................................................156

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Appendix V..........................................................................................................................156

Appendix VII........................................................................................................................158

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Table of Figures

Fig. 6.1: Specific attenuation over a 24-hour period (4-Jul-2012).............................................98

Fig. 6.2: DSD and specific attenuation, with meteorological readings for 04-Jul-2012@1629 99

Fig. 6.3: Average contribution of bins to the specific attenuation...........................................101

Fig. 6.4: Beacon signal data.....................................................................................................110

Fig. 6.5: Beacon signal data and rain rates...............................................................................112

Fig. 6.6: Gaseous attenuation, zero dB templates and attenuation for both beacon signal and radiometer-obtained data for 5th September 2011............................................................117

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Fig. 1.1: Chilbolton Observatory, Southern England...................................................................7

Fig. 1.2: Impact Disdrometer .....................................................................................................8

Fig. 1.3: Chilbolton Observatory’s Radar, radiometer, rain gauge and disdrometer...................9

Fig. 1.4: Chapter Layout and Flow............................................................................................14

Fig. 2.1: Generation and propagation of electromagnetic waves ..............................................18

Fig. 2.2: Layers of the Earth’s atmosphere ..............................................................................20

Fig. 2.3: Atmospheric gas absorption and frequencies .............................................................26

Fig. 2.4: The geometry of scattering .........................................................................................29

Fig. 2.5: Effective rainy path length scenarios for stratiform and convective rain types...........33

Fig. 2.6: Characteristics of fade dynamics ................................................................................34

Fig. 3.1: The rain cycle...............................................................................................................39

Fig. 3.2: A 2D Video Disdrometer.............................................................................................41

Fig. 3.3: Aerodynamic Forces on Raindrops..............................................................................42

Fig. 3.4: Comparing the terminal velocity equations.................................................................43

Fig. 3.5: A Gaussian (Normal) Distribution...............................................................................56

Fig. 3.6: Failure of the unimodal distributions...........................................................................57

Fig. 3.7: A Gaussian Mixture Model..........................................................................................58

Fig. 4.1: Total rain drops and meteorological data and a 1-minute drop size distribution........69

Fig. 4.2: Seasonal % variation of DSD fits (MFR/ETH_ALL/ETH_TRUNC)...........................71

Fig. 4.3: Rain rate % variation of DSD fits (MFR/ETH_ALL/ETH_TRUNC)...........................72

Fig. 4.4: Wind speed % variation of DSD fits (MFR/ETH_ALL/ETH_TRUNC)......................73

Fig. 4.5: Negative exponential distribution results when the 0.1 mm/h threshold is lifted........75

Fig. 5.1: An underlying multimodal distribution......................................................................80

Fig. 5.2: GMM with three clusters for 26 July 2003 at 16:46....................................................83

Fig. 5.3: Multimodal DSD from a 2D Video Disdrometer at Graz, Austria..............................85

Fig. 5.4: Average number of modes for different rain rates and wind speeds...........................87

Fig. 5.5: GMM fits for the new model based on the predicted number of modes.....................88


Fig. 6.7: Cumulative Distribution for attenuations from both methods...................................118

Fig. 6.8: Rain rates and rain attenuation for a 24-hour period.................................................122

Fig. 7.1: Effective rainy path length over a 24-hour period (4-Jul-2012)................................127

Fig. 7.2: Effective slant path length from GMM (Predicted Nm)’s specific attenuation..........128

Fig. 7.3: Effective slant path length variation with rain rates..................................................129

Fig. 7.4: Effective slant path length and wind speeds..............................................................130

Fig. 7.5: Measured and model-obtained attenuations over a 1-hour period (17-Sep-2011)....132

Fig. 7.6: Predicted and measured attenuation for a 24-hour period.........................................134

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List of TablesTable 1.1: Hotbird 13A Satellite Beacon Parameters..................................................................9

Table 1.2: Datasets (all captured at Chilbolton Observatory) ..................................................10

Table 3.1: Rain rates and the different classifications...............................................................45

Table 3.2: UK Met Office’s Colour Coding for different rain rates..........................................45

Table 4.1: Percentage differences between the rain gauge and disdrometer rain rates..............68

Table 4.2: Chi Square Test: 29-Oct-2003/19:23 @ 95% Confidence Interval..........................70

Table 4.3: Meteorological sensor readings for the above 1-minute time slice..........................70

Table 4.4: The fit of the DSDs with varying temperature..........................................................74

Table 4.5: The Chi-square fit of the DSDs for each distribution...............................................74

Table 5.1: Estimated Parameters for the GMM.........................................................................83

Table 5.2: Chi-square Goodness-of-fit test for the GMM..........................................................84

Table 5.3: Summary of results for Chi-square Goodness-of-fit test for different clusters.........84

Table 5.4: Average number of modes at different rain rates and wind speeds..........................86

Table 5.5: Average i, i and wi for GMM fits for different rain rates......................................90

Table 5.6: Model equations for i, i and wi at various number of modes................................91

Table 5.7: Seasonal unimodal and multimodal distributions for observed and predicted Nm....91

Table 5.8: Unimodal and multimodal distributions for both observed and predicted Nm..........92

Table 5.9: Wind speeds unimodal and multimodal distributions for Nm...................................92

Table 5.10: Unimodal and multimodal distributions for both observed and predicted Nm........93

Table 6.1: Summary of results for the specific attenuation......................................................100

Table 6.2: RMSPE variation of specific attenuation from the ITU’s aRb specific attenuation102

Table 6.3: Specific attenuation across seasons.........................................................................105

Table 6.4: Specific attenuation across different rain regimes..................................................106

Table 6.5: Specific attenuation across wind classes.................................................................107

Table 6.6: Specific attenuation across temperature..................................................................108

Table 6.7: Radiometer specifications.......................................................................................109

Table 6.8: Ninth-order polynomial fit......................................................................................116

Table 6.9: Seasonal variation for attenuation...........................................................................119

Table 6.10: Variation for attenuation with rain rates...............................................................119

Table 6.11: Variation for attenuation with wind speeds..........................................................120

Table 6.12: Variation for attenuation with temperature...........................................................121

Table 7.1: Effective slant length model equations from different models...............................129

Table 7.2: Root mean square percentage errors for different models......................................132

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Abbreviations

BADC British Atmospheric Data Centre

CAMRa Chilbolton Advanced Meteorological Radar

dB Decibel

DLPC Downlink Power Control

DSD Rainfall Drop Size Distribution (or (Rain)drop Size Distribution)

EIRP Equivalent Isotropically Radiated Power

EM Expectation-Maximisation Algorithm

ETH Eidgenössische Technische Hochschule

FMTs Fade Mitigation Techniques

FSPL Free Space Path Loss

GMM Gaussian Mixture Model

Hz Hertz

IWV Integrated Water Vapour

JWD Joss-Waldvogel Impact Disdrometer

K-S Kolmogorov-Smirnov test

LWP Liquid Water Path

MLE Maximum Likelihood Estimates

MoM Method of Moments

NERC Natural Environment Research Council

NetCDF Network Common Data Form

Pdf Probability Density Function

QoS Quality of Service

RADAR RAdio Detection And Ranging

RAL Rutherford Appleton Laboratory

SSE Sum of Squared Errors

STFC Science and Technologies Facilities Council

ULPC Uplink Power Control

WVP Water Vapour Path

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Table of Symbols

Symbol MeaningA(t) Attenuation time series

Ameasured Measured attenuationAmodel Attenuation for the model

aRb ITU specific attenuation power law (or γ(R))

BdBm Received beacon signal level (dBm)BDC Received beacon signal level (DC)

c Velocity of light in a vacuumdDi Bin widthsDi Drop diameters

Dm Volume weighted mean diameterDmax Maximum drop diameter

E Electric fieldETH_ALL ETH bin boundaries (all bins)

ETH_TRUNC ETH bin boundaries (truncated)f Frequency

g(x) Fit polynomialH Magnetic fieldK Attenuation coefficient

Kcls Clear sky temperatureLeff Effective rainy slant path length (or distance)

Leff_model Rainy slant path length for the modelm Refractive index of water

MFR Manufacturer’s bin boundariesmn(t) nth moment of the DSD

N0 Negative exponential interceptni(t) Total drop countsNm Predicted number of modes

N(D) or N(D,t) or Nm(Di,t)

Spatial drop densities

NT Scaling parameter of a DSDp(D) Probability density function (pdf)

pi Fit polynomial coefficientsQt Attenuation cross section

R(t) or Rm(t) Rain time seriesS Resultant propagation wave energy

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Symbol MeaningT0 Cosmic background temperature

Teff Effective medium temperatureTsky Sky temperature

vi Terminal fall velocity of raindropsW Wind speedwi Gaussian weightsγ Specific attenuation

γ(Di) Bin Di specific attenuationγITU ITU’s aRb specific attenuation

γmodel Specific attenuation for the modelΔt Disdrometer integration timeε Spectral emissivity (or radiance)

θi Electromagnetic wave angle of incidenceΛ Slope parameter in the DSDλ Wavelength

Λ = αRβ Negative exponential coefficient function of rain rateμg Geometric meanμi Gaussian mean for ith clusterσg Geometric variance

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AbstractFuture earth observation systems that continuously capture sub-metre resolution Earth imagery data spanning a wide swath must have the capability to transfer large volumes of data from the low earth orbiting satellite to a ground terminal during the short window of visibility between the two. This will require operation at Ka-band where there is sufficient allocated radio bandwidth (e.g. 25.5 GHz – 27 GHz) to support the required multi-gigabit per second downlink bit rate capacity. However, at Ka-band the downlink becomes highly susceptible to disruption by rain during each satellite pass when path elevation angles will range anywhere between 90 degrees and 20 degrees or less. The resulting rain attenuation will sometimes exceed what can be feasibly combated through link dimensioning to include a sufficient fixed fade margin. Adaptive fade mitigation will, therefore, be essential to ensure a reliable and robust downlink. The required knowledge of the instantaneous rain attenuation on the downlink could be gained by using ground-based measurement of rain rate and rainfall drop size distribution (DSD) at the earth station location.

This thesis fits the standard lognormal and gamma distributions to one-minute slices of rainfall DSDs captured between 2003 and 2013 at the Chilbolton Observatory in southern England. Unlike previous studies, goodness-of-fit of the models is tested by calculating chi-square goodness-of-fit for distributions fitted. Results show that failure to fit is greater than would normally be expected. This failure to fit is explored and broken down and examined against seasonal variations, different rain rates, atmospheric temperature and wind speed. This thesis investigates the occurrence of multimodality in the rainfall data observed by various researchers, as a possible explanation for the failure to fit. It proposes a Gaussian Mixture Model in this case, predicting the number of modes from rain rates and wind speeds.

Specific attenuation at corresponding instants is computed using the attenuation cross-section of rain drops and their size distributions at ground level. Both disdrometer-measured and standard-modelled DSDs are employed and the results are validated by comparison to the ITU specific attenuation for the beacon frequency. Results show that specific attenuation tends to increase with the drop sizes, and the smaller drops contribute little to the overall attenuation experienced by signals.

Effective rainy slant path length is defined as the ratio of the total instantaneous slant path rain attenuation and the specific attenuation. Assuming a latitude-dependent fixed rain height, the variability of instantaneous effective rainy slant path length with rain rate is modelled. From this length and the specific rain attenuation, the instantaneous total rain attenuation can be estimated. This in principle enables telecommunications service providers to provide better quality of service at reduced cost.

Keywords: Theoretical Modelling, Space and Satellite Communications, Estimation and Forecasting, Probability Distributions, Rainfall Drop Size Distribution (DSD), Gaussian Mixture Model (GMM), Attenuation, Rainfall, Moments, Maximum Likelihood Estimates, Goodness-of-Fit, Effective Rainy Slant Path Length, Meteorological Factors, Microwave Radio Propagation, Lognormal Distribution, Gamma Distribution.

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Chapter 1 : INTRODUCTION

K’ufre-Mfon Ebong Ekerete: PhD ThesisPage 1


CHAPTER 1

INTRODUCTION

1.1 IntroductionModern life entails the generation and dissemination of vast quantities of data – data

relating to banking, weather, news, education, health, entertainment and much more.

This data comes to us in several formats – text, images, voice, video, and multimedia –

and may be in different locations. It has to be transmitted from one place to the other

for it to be useful, enhancing the modern person’s well-being, productivity, education,

and leisure. In that light, the faster and more accurately the information is sent and

delivered, the more the modern person feel at ease with himself.

There are several methods for the transmission of digital information including wired

(involving laid down wires), electromagnetic waves (involving radio and free space

optical waves), and fibre optics (involving the sending of light in an optical medium).

This thesis is concerned with the sending of information using microwave radio,

particularly through artificial satellites that reside in a particular region above the

earth’s surface.

Satellite types include the geostationary, otherwise known as GEO satellites, positioned

35,785 km above the equator and moving at the same rate as the rotation of planet

earth, the highly eccentric elliptic orbit (HEO) satellites inclined at an angle of 63.4° to

the equatorial plane with varying apogees and perigees, the medium earth orbit (MEO)

satellite that rotates about the earth at an altitude of between 5,000 km to 20,000 km,

and the low earth orbiting (LEO) satellite, that is in circular orbit at altitudes between

500 km and 1,500 km above the earth [1].



With satellites, information can be relayed to great distances at the speed of light.

Satellites provide a faster way to transmit data across vast, sometimes unfriendly (e.g.

war zones) or uneconomical (e.g. hilly, rocky or swampy) terrains. These great

distances covered by signals carrying information impacts on the speed as well as the

quality of the finally beamed-back signal. Amongst the problems encountered will be

propagation delays, high bit error rates, distortion due to the presence of other materials

in the atmosphere, fading signals, asymmetry of the channels, and others. Despite these

problems, these satellites still do the job of carrying electronic signals from point to

point.

In the current data-hungry world, there is an enormous demand for telecommunication

services (cable television, Internet services, etc.), and there is an increasing need to

utilise more transmission bandwidth. Telecommunication service providers increasingly

use microwave transmission frequencies above 10 GHz since lower frequency bands

are now congested [2] and the higher frequency provides a higher data rate and larger

transmission bandwidth, as well as reduced interference potential and smaller

equipment sizes [3]. The transmission of radio wave signals through the atmosphere at

frequencies above 10 GHz is however much more susceptible to attenuation due to

precipitation; the signals when beamed from the transmitter to the receiver during

satellite transmissions must pass through cloud formations. These cloud formations are

often water-laden due to precipitation, thereby leading to degradation in the desired

quality of service (QoS) and link availability (or attenuation).

1.2 Motivation for the researchIn a bid to design reliable systems for the provision of better QoS to the consumer,

there is the need to properly estimate the attenuation of signals due to rainfall, as over-

estimating is wasteful of resources, and the consumer unnecessarily pays more, while

under-estimating may lead to system outages. In order for system engineers to design

reliable systems, there is the need to reliably predict how precipitation, and rain

particularly, attenuates transmitted signals. The design of more reliable systems with

shorter or no outages presents the users with more availability, and the



telecommunications service providers with a larger market, both leading to reduced cost

in services and equipment.

Modelling the rainfall drop size distribution (DSD) is a key ingredient in the

characterisation of the rainfall drop spectra for the prediction of rain attenuation. This

allows telecommunications service providers to design mitigation techniques to counter

attenuation due to these rain events.

Any rain event is composed of several raindrops distributed over space and time. A

simple sampling of any patch of these drops falling will reveal that they are not all the

same diameters; some are finer than others and fall at different speeds. It is the

distribution of these different sizes of raindrops that is the basis of the study and

characterisation of the DSD. At any given moment, how many drops of what diameters

are in a given volume interests the researcher as raindrops of different sizes absorb and

scatter radio wave energy differently. Over time, one can characterise the composition

and say that it follows a particular pattern. The density of drops predicted as a function

of the diameter in a sample represents the drop size distribution [4].

Typically, the DSD is modelled as a function, N, of drop diameter D, and time t. The

function gives the density of drops per unit volume of atmosphere, per unit diameter of

the drop, and per unit time. In this formulation, when comparing this model to reality, it

is assumed that N(D,t) gives the expected density and that the actual density varies by a

small random amount from this expected value.

Identifying the most appropriate form of N(D,t) is central to this study. Accurate

identification will allow accurate predictions, particularly of attenuation. It is

recognised that the form may depend on the locality of the rain, e.g. tropical or

temperate, and the rainfall rate, e.g. heavy or light.

Also of interest is the change of N(D,t) with time as this would allow for better

modelling of various rain dynamics such as fade margin (the available allowance of

received power before outage), fade slope (the rate of change of attenuation with time,

in dB/s) and fade duration (the time interval between consecutive crossings on a



defined attenuation threshold). Fade slope and fade duration, as used here, are metrics

used to style the dynamic behaviour of attenuation experienced by radio links [5].

This thesis continues a personal research and professional practice in the field of

Information and Communications Technology, particularly in the field of Satellite

Communications. This thesis is aimed at providing a better understanding of the

attenuation of satellite signals, and contributing to the understanding of attenuation of

signals due to rainfall, with the hope of the provision of better telecommunication

services by the design of more efficient link budgets. The thesis aim of a better

prediction model of rainfall rate in order to achieve a better rain attenuation prediction

is achieved.

1.3 Research Aims and ObjectivesIn order to arrive at the attenuation caused by rain, there is the need to measure the

received signal by setting up equipment to monitor the satellite’s beacon strength. A

satellite beacon is a low-power signal that is sent out by a satellite for research or for

tracking and control purposes. By monitoring the signal strength in various weather

conditions, it is possible to make more accurate signal strength models.

The research aim was:

to predict attenuation better than existing methods, in order to enable

telecommunications service providers to provide better quality of service at

reduced cost

This is broken down into the following subsidiary research objectives:

i. to develop a good model of the DSD based on measurements in southern

England.

This is built upon analyses of disdrometer measurements at Chilbolton; and

investigation of the variation of model parameters with time and meteorological

parameters such as wind speed and direction; investigation of possible seasonal

and climatic trends.

ii. to express the DSD model parameters as functions of rain rate.



This will allow anyone to know the distribution of raindrops from the rain rate in

their locality.

iii. to compute the specific rain attenuation using the DSD model developed.

This will be compared with the specific attenuations computed using other DSD

models such as the lognormal and gamma distributions. This will involve the

determination of the attenuation cross section of raindrops at the 19.7 GHz of the

Eutelsat Hotbird 6 satellite beacon and ground-measured temperature for the

numerical integration involving the drop densities and the attenuation cross-

section.

iv. to estimate the effective rainy slant path length, as a function of rain rate.

This is achieved by pre-processing the Hotbird 6 beacon data to extract time

series of rain attenuation along the path, averaged over one-minute intervals in

each rain event. A radiometer-based algorithm is used to establish a clear-air

reference level for satellite beacon attenuation measurements. Using the estimate

of the specific attenuation ,and known total attenuation, the estimated effective

rainy slant path length can be calculated. This can then be found as a function of

rain rate.

v. to combine the estimated specific rain attenuation and effective rainy slant

path length to estimate the instantaneous total rain attenuation as a function

of rain rate.

Detailed work plan, and the final outcomes are given in Section 9.3.

1.4 The Chilbolton Observatory, instruments and the datasetsThe Chilbolton Observatory (51.14° N, 1.44° W, 84 m above sea level), located in

Chilbolton, Hampshire, southern England houses some of the world’s most advanced

meteorological measurement facilities, including the world’s largest fully steerable

meteorological radar, the Chilbolton Advanced Meteorological Radar (CAMRa). There



are other sophisticated instruments. These collect data for research in atmospheric

sciences, radiocommunications, astronomy and space science and technology. Included

in the collection of equipment is the RD-69 Joss-Waldvogel impact disdrometer (JWD)

which measures the DSD, the Radiometrics MP1516A Water Vapour Profiler and

several rain gauges, including the RAL Rapid Response Drop Counting rain gauges.

The Chilbolton Observatory (Fig. 1.1), a part of the Science and Technologies Facilities

Council (STFC) is run by the Rutherford Appleton Laboratory (RAL Space) and

receives funding from, among others, the Natural Environment Research Council

(NERC).

The disdrometer in Chilbolton is the RD-69 Joss-Waldvogel impact disdrometer1

attached to an ADA90 analyser. The disdrometer has an exposed 50 cm2 surface area

Styrofoam® cone that senses falling raindrops. The disdrometer works by converting

impacts from raindrops to electrical pulses, where the amplitude of the pulses is used to

estimate the diameter of the rain drops. The converted diameters are sorted into a 127

grouping called bins (or channels) ranging from 0.3 mm to 5.2 mm, with the lower

boundary of the bins distributed exponentially. The number of drops is tallied every 10

seconds, with an accuracy of ±5% [6].

1 Unless otherwise stated, in the rest of this thesis every reference to “the disdrometer” will refer to the RD-69 Joss-Waldvogel Impact Disdrometer


Fig. 1.1: Chilbolton Observatory, Southern England


The vibration of the Styrofoam® surface takes time as it resets itself. This may result in

“dead-time”, where the instrument misses its recording. The problems of “dead-time”,

re-splashed drops and the recalibration of the bin sizes are discussed further in the

work. An illustration of the disdrometer is shown in Fig. 1.2.

The MP1516A Water Vapour Profiler is a passive, self-calibrating microwave

radiometer that measures the sky brightness temperature radiation within a conical

volume of the sky by scanning the sky at 21 different frequencies ranging from 22 GHz

to 30 GHz (which traverses the water vapour absorption line), with an adjustable time

resolution of between 30 seconds and 120 seconds. The radiometer has a 300 MHz

bandwidth for each channel integrated over a band spanning 10 MHz to 160 MHz

above and below the centre frequency, with a 3 dB beamwidth that varies from 6.3 to

4.9 across the full frequency range and a brightness temperature accuracy of between

0.5 K and 1 K per channel [7].

Though the primary data captured by the profiler is the sky brightness temperature,

secondary data like the integrated water vapour along the path, the integrated liquid


Fig. 1.2: Impact Disdrometer [6]


water along the path, and the water vapour profile along the path can also be deduced

from the measurements.

The RAL Rapid Response Drop Counting Rain Gauge (RAL RRDC Gauge) deployed

at Chilbolton measures rain using the volumetric drop technique, and has a 150 cm2

collection area, a 5 – 200 mm/hr range and a 0.004 mm/drop calibration with a ±10%

rain rate error and samples at 10 seconds.

The Observatory also hosts met sensors, taking measurements every 10 seconds for air

temperature, relative humidity, atmospheric pressure, wind speed and direction, rainfall

and solar radiation. Some of the equipment at the Observatory is shown in Fig. 1.3.

The Hotbird 13A satellite beacon signals are recorded at the Chilbolton Observatory at

19.7 GHz, with an elevation angle of 29.9 and an azimuth of 13 E and recordings are

made every second. Measurements along the path to the Hotbird satellite using the

Radiometrics microwave radiometer started on 18th July 2011 and ended on 18th April

2016 (although satellite transmissions at 19.7 GHz ended some time before this).

The parameters for the Hotbird satellite beacon signals captured at Chilbolton are given

in Table 1.1 below.


Fig. 1.3: Chilbolton Observatory’s Radar, radiometer, rain gauge and disdrometer


PARAMETER VALUELatitude 51.14 NLongitude 358.74 EFrequency 19.7 GHzElevation angle 29.9Tilt angle 13Height above sea level 84 mRAIN HEIGHT [8] 2.4251 km

The Chilbolton data are published and made available to subscribers. The datasets are

hosted by The British Atmospheric Data Centre (BADC), the NERC-designated Data

Centre for the Atmospheric Sciences (badc.nerc.ac.uk). The datasets, instruments and

quantities shown in Table 1.2 were utilised in this thesis:

Dataset Instrument Data Period Format Notescfarr-disdrometer_chilbolton Disdrometer Drop counts 2003-

2013 NetCDF ±5% accuracy

cfarr-radiometer-radiometrics_chilbolton Radiometer

Sky tempGround tempRel. Humidity

2010-2013

NetCDFMS Excel

0.5 K – 1 K accuracy

cfarr-raingauge_chilbolton Rain gauge Rain rates 2003-

2013 NetCDF ±10% accuracy

cfarr-met-sensors_chilbolton Met sensors

Ground tempRel. HumidityPressureWind speedWind direction

2003-2013 NetCDF

chhbNovella tracking receiver

DC beacon signals

2010-2013 Text files

The available data period did not always coincide. For example, data from the

Radiometer was only available from April 2010.


Table 1.1: Hotbird 13A Satellite Beacon Parameters

Table 1.2: Datasets

(all captured at

Chilbolton

Observatory) [9]


1.5 Thesis contents overviewThis report describes a detailed study of the rainfall drop size distributions based on

data collected from the disdrometer, rain gauges and radiometer located at the

Chilbolton Observatory from 2003 to 2013. Table 1.2 also shows the period for each of

the datasets utilised in the work.

Analyses of the DSD are done by attempting to fit standard statistical distributions to

the disdrometer data. This thesis goes beyond the fitting of these statistical distributions

by attempting to determine the goodness-of-fit of each of these distributions. The work

goes ahead to study the presence of multimodality encountered in the data, as well as

proposing a novel method of modelling these multimodal data the current unimodal

statistical distributions fail to cater for. The work determines both the specific

attenuation and the total rain attenuation modelled with each of these distributions, as

well as compare them with the attenuation obtained using the ITU aRb model. It

concludes by modelling the effective rainy slant path length, investigating how this

varies with rain rates, wind speeds and ground-measured atmospheric temperature.

And the remaining chapters are arranged thus:

Chapter 2 discusses radio wave propagation, the earth’s atmosphere and the different

signal dispersion mechanisms and details of how signals are scattered are discussed. It

looks at how rainfall affects propagation. It also introduces radiometry as well as

discusses specific attenuation – how the ITU recommendation obtains specific

attenuation for a given rain rate, as well the derivation of specific attenuation from the

distributions by integrating the DSD and the extinction (or attenuation) cross-section of

the drops over the entire bin. The Chapter also introduces and defines the effective

rainy slant path length. It concludes with a discussion on fade dynamics and fade

mitigation techniques.

Chapter 3 explores the physics of rainfall; the formation, composition and distribution

of rainfall, including the shape and sizes of raindrops. It looks at measuring equipment

and measurement techniques and examines the problems with the measurement

techniques. Terminal drop velocities are discussed. The Chapter also looks at rainfall



rates and discusses the statistical modelling of rainfall drop size distributions obtained

from drop counts measured with the disdrometer, with several unimodal models that are

used to characterise the rainfall drop size distributions looked at. It also looks at

possible seasonal and climatic variations in DSD measurement. The Chapter also

discusses and presents results for the goodness-of-fit of the rainfall data using the

standard unimodal statistical distributions. The Gaussian Mixture Model (GMM) is

introduced as a solution for the modelling of multimodal data. The underlying GMM’s

Expectation-Maximisation (EM) algorithm is discussed.

Chapter 4 looks at the rainfall drop size distributions of the data gathered from the

disdrometer at Chilbolton, comparing the disdrometer-obtained rain rates with that of a

co-located rain gauge. The Chapter also looks at the results of using three different bin

delineations on the same dataset, viz the manufacturer’s recommended bin boundaries

with all the bins considered (MFR), the ETH-recomputed bin boundaries with all the

bins considered (ETH_ALL), and the ETH-computed bin boundaries with smaller-sized

bins eliminated (ETH_TRUNC). This was necessary as it was discovered that the

disdrometer’s voltage calibration introduced some errors in the bin boundaries, and also

that the instrument’s sensitivity to small-sized drops may be unreliable.

Chapter 5 introduces the presence of multimodality encountered in rainfall data. It goes

on to show how prevalent it is as well as exploring how these vary with meteorological

parameters. The Gaussian Mixture Model is proffered as a solution in the modelling of

the multimodal data encountered in the rainfall DSD data, and results of applying the

GMM in the data are presented. It proposes a new equation in the prediction of the

number of modes as well as equations for the modelling of relationships between the

rain rates and other usable parameters (means, standard deviations and weights).

Chapter 6 computes the specific attenuation based on the ITU P R 838 model and

compares the specific attenuation obtained from the GMM as well as the other

statistical models aforementioned. The method of Mätzler is used in the determination

of the extinction cross-section from the Mie coefficients. The contribution of individual

drops to the overall specific attenuation is explored. The Chapter concludes with the



comparative analyses of the specific rain attenuation variation with other quantities;

seasons, rain rates, wind speeds, and temperature. This Chapter also looks at total rain

attenuation. The different compositions of the rain attenuation are discussed, and sky

temperature measurements from the radiometer are utilised in the derivation of the

attenuation. The measured beacon signals are also converted from raw DC voltage

values to dB values. A new method is proposed for the determination of the clear sky

zero dB reference level, here used in the calculation of the rain attenuation. The

Chapter also does a comparison of the attenuation estimated from the different

statistical models (including the GMM), and comparing it with that estimated from the

ITU model. It concludes by looking at the variation of attenuation with meteorological

parameters.

Chapter 7 computes the effective rainy slant path length, Leff, a ratio of the rain

attenuation and the specific attenuation. It investigates the variation of the Leff with

meteorological parameters. It proposes new power law equations for the slant path

length and rain rates relationship. This Chapter also looks at the derivation of

attenuation for the different models considered, as well as comparing the obtained

attenuation from measurements. The Chapter discusses the application of the results to

fade predictions, where prediction here is defined as the ability to use the instantaneous

rain rate measurements to determine the instantaneous attenuation. The Chapter also

looks at various fade mitigation techniques, and determines the performance of the

proposed model vis-à-vis the probability of encountering rain aloft while the signal

travels the slant path. It discusses these results, and proposes the application of these in

a recommended fade mitigation technique (FMT).

Chapter 8 summarises the research and proffers recommendations and suggestions for

the extension of the work.

The flow of the Chapters is in Fig. 1.4 given below:



1.6 Novelty in the Research and Contributions to Knowledge Testing the goodness-of-fit of DSD models formally against the empirical DSD

using statistical methods. (As opposed to informally assessing the fits visually or

via the accuracy of the rainfall or other predictions obtained from the DSD as done

by previous researchers);

Firmly establishing the presence of multimodality in rainfall data. Even though this

has been suggested by other researchers, the presence is still disputed or dismissed

as an artefact of disdrometer calibration. This thesis confirms the presence of

multimodality in the rainfall data, both using an impact disdrometer with the so-

called ETH-revised calibration, and using a more sophisticated 2D Video

Disdrometer in a different location with a smaller sample;


Fig. 1.4: Chapter Layout and Flow

Chapter 1: Introduction

Chapter 2: Radio Wave Propagation

Chapter 3: Rainfall DSDs and DSD Models

Chapter 4:Rainfall DSD Modelling

Chapter 6:Specific and Total Rain Attenuation

Chapter 5:Multimodality in Rainfall DSDs and the GMM

Chapter 7:Effective Rainy Slant Path Length Modelling

and Application to Fade Prediction

Chapter 8:Conclusions and Suggestions for

Further Work


Proposing the Gaussian Mixture Model (GMM) as a model for multimodality when

encountered in rainfall data. No way to model multimodal rain data has been

proposed before in the literature. This thesis, as a follow up of the above point tests

how well the data fits the model using the chi square test;

Novel method to establish a zero dB reference template for extraction of rain

attenuation from satellite beacon measurements. The method proposed here builds

a zero dB reference template from corresponding ten-minute segments of

neighbouring clear days, instead of entire 24-hour clear days, as the nearest 24-hour

clear day may be too far away to be useful given the diurnal movement of satellites;

1.7 Publications from the researchThis work so far has been showcased and published in various fora and journals. The

following is a list of publications generated by this research:

Journal Papers

1. Ekerete, K. E., Obi, C. C., Hunt, F. H., Jeffery, J. L., Otung, I. E., (In Preparation), “Slant Path Attenuation Prediction from Point Rainfall Measurement in Southern England”, IEEE Transactions on Antennae and Propagation.

2. Ekerete, K. E., Hunt, F. H., Jeffery, J. L., Otung, I. E., “Variation of multimodality in rainfall drop size distribution with wind speeds and rain rates”, IET Journal of Engineering. 2016. doi:10.1049/joe.2016.0013

3. Ekerete, K. E., Hunt, F. H., Jeffery, J. L., Otung, I. E., 2015, “Modelling of rainfall drop size distribution in southern England using a Gaussian Mixture Model”, Journal of Radio Science. 2015. doi: 10.1002/2015RS005674.

Book Chapters from Conferences:

1. K’ufre-Mfon Ekerete, Francis Hunt, Judith Jeffery, Ifiok Otung, 2016, “Specific Rain Attenuation derived from a Gaussian Mixture Model for Rainfall Drop Size Distribution”, 8th International Conference on Wireless and Satellite Systems (WiSATS 2016) Cardiff, United Kingdom, 19th – 21st September 2016.

2. K’ufre-Mfon Ekerete, Francis Hunt, Judith Jeffery, Ifiok Otung, 2015, “Multimodality in the rainfall drop size distributions in southern England”, 7th International Conference on Wireless and Satellite Systems (WiSATS 2015) Bradford, United Kingdom, 9th – 10th July 2015.



Conference Papers:

1. K’ufre-Mfon Ekerete, Francis Hunt, Judith Jeffery, Ifiok Otung, (Submitted for Peer Review), “Fade Prediction Modelling using a Gaussian Mixture Model”, 9th International Conference on Wireless and Satellite Systems (WiSATS 2017), Oxford, United Kingdom, 24th – 25th July 2017.

2. K’ufre-Mfon Ekerete, Francis Hunt, Judith Jeffery, Ifiok Otung, 2016, “Modelling of Effective Rainy Slant Path Length in Southern England for Instantaneous Rain Attenuation Computation on Satellite Links”, 21st Ka Conference, Cleveland, USA, October 2016.

3. K’ufre-Mfon Ekerete, Francis Hunt, Judith Agnew, Ifiok Otung. 2014, “Experimental study and modelling of rain drop size distribution in southern England”, IET Colloquium on Antennas, Wireless and Electromagnetics, United Kingdom, May 27th, 2014.

Poster Presentations:

1. Judith Jeffery, K’ufre-Mfon E. Ekerete, Ifiok Otung and Francis Hunt, 2016, “The measurement and modelling of rainfall drop size distributions at the Chilbolton Facility for Atmospheric and Radio Research”, Royal Meteorological Society/National Centre for Atmospheric Science (RMetS/NCAS) Conference 2016, University of Manchester, Oxford Road, Manchester, Greater Manchester, M13 9PL, United Kingdom, 6 – 8 July 2016.

2. Ekerete K.E, Hunt F.H, Agnew J.L. and Otung I.E. 2014, “Modelling of rainfall drop size distribution in southern England using a Gaussian Mixture Model”, IET Satcom Security and Efficiency Conference, IET Birmingham, Austin Court, United Kingdom, 26th November 2014.



Chapter 2 : RADIO WAVE PROPAGATION



Chapter 2

RADIO WAVE PROPAGATION

2.1 IntroductionModern digital messages are sent wirelessly. How are these messages generated and

sent? Electromagnetic waves are a combination of oscillating electrical and magnetic

fields. The electrical and magnetic fields are perpendicular to each other and to the

direction of the waves. They do not require a physical medium in order to be

propagated, they can travel through free space, solid medium as well as across vacuum.

Heinrich Hertz and James Clerk Maxwell’s earlier works demonstrated these radio

waves travel at the same velocity as that of light, c [10].

Fig. 2.1 above shows the electromagnetic waves travelling to the right of the page with

the electric field, usually represented as E, vertically polarised, while the magnetic

field, usually represented as H, is horizontally polarised. The frequency of the wave is

described as the rate of vibration of the wave per second, measured in Hertz (Hz), while

the amplitude is the height of the crest of the wave from its position of equilibrium and

the wavelength is the distance between two crests (or troughs). The higher the

frequency (or the shorter the wavelength), the higher the energy and the resultant

energy, is given as:


Fig. 2.5: Generation and propagation of electromagnetic waves [11]


S=E× H (2.1)

The atmospheric content and conditions affect the propagation of radio waves as it

traverses through space. The transmission of radio signals will be affected by the

frequency, ambient temperature, atmospheric pressure and the atmosphere’s water

contents.

2.2 The Earth’s AtmospherePlanet Earth, man’s current home is a near-spherical geoid enveloped in layers of gases.

The commonest of the gases are nitrogen (at 78%) and oxygen (21%) [12]. Other gases

like argon, carbon dioxide, neon, helium and methane exist in varying quantities. The

atmosphere gets thinner and colder and the pressure becomes lower as one recedes from

the terra firma. The Earth’s atmosphere is divided from the ground up into the

troposphere, stratosphere, mesosphere, thermosphere and exosphere.

2.2.1 The Layers of the Earth’s AtmosphereThe troposphere, the most dense of the layers, starts at 10 km and goes as much as 20

km above the Earth’s surface (depending on the geographical location and season) [13]

and cloud formation, dust particles, and water vapour, hence the weather, occur here.

The top of the troposphere is the tropopause and forms the boundary between the

troposphere and the stratosphere, and the thickness of the tropopause may depend on

the latitude, season and diurnal period. The tropopause is higher at the equator that at

the poles.

The stratosphere, immediately above the troposphere, occurs from the tropopause to

about 50 km above the Earth’s surface, and this hosts the ozone layer, shielding the

Earth from harmful ultraviolet rays from the sun [14]. The temperature variation is the

reverse of the troposphere as it rises with altitude. This layer is quite stable, containing

very little cloud formation and water vapour enabling commercial airlines to fly at this

level. The top of the stratosphere is called the stratopause, merging with the next layer,

the mesosphere.



The mesosphere, starting at about 50 km above the Earth’s surface has the same

temperature variation as the troposphere, it gets colder with higher altitude, with the

mesopause, the top layer getting as cold as -90° C [15]. It is in this layer that the

meteors from outer space burn up.

The thermosphere starts at about 85 km above the Earth’s surface [16]. Solar activities

are experienced in this layer, hence a steep rise in the temperature of the lower

thermosphere with altitude which however steadies in higher altitude. It is at this layer

that the International Space Station and space shuttles orbit the Earth because of the

very low air density. It is here that the auroral displays occur. The upper layer is called

the thermopause, which merges with the next layer, the exosphere. The exosphere,

considered outer space, has a very thin atmosphere. A graphic representation of the

Earth’s atmosphere is shown in Fig. 2.2 below.

This image is courtesy of Nick Strobel at www.astronomynotes.com

Fig. 2.6: Layers of the Earth’s atmosphere [17]

In sending messages via satellites, the signals must pass through these layers of the

Earth’s atmosphere. Signals are impaired by the constituents of the ionosphere. The


http://www.astronomynotes.com/


ionosphere is a parallel classification of the Earth’s atmosphere, beginning from about

75 km above the Earth’s surface to about 1,000 km [18]. The ionisation of this layer is

mainly due to the Sun’s radiation (ultraviolet rays, gamma rays and cosmic particles).

This region, an electrified region in the upper atmosphere, is divided into the D (60 km

above the Earth’s surface), E (90 km above the Earth’s surface) and F1 and F2 (from

150 km above the Earth’s surface) regions in the day time, and into the E and F regions

at night. During the night time, the D layer is however not observed, and the F1 and F2

layers merge into a single F layer, mainly due to the recombination of ions.

The ionosphere is made up of a large number of charged particles and free electrons.

The ionosphere’s free electrons affect radio wave signals propagated through space as

they have the capability of absorbing and reflecting radio waves in lower frequencies,

hence frequencies of signals must be increased to overcome these impairments, even

though there may still be varying degrees of signals degradation depending on the

geographic location and time of the day [19].

2.3 Propagation ImpairmentsRadio wave signals, when propagated along the Earth-space path experience

impairment due to a number of reasons; loss, interference, distortion, the presence of

materials in the path, absorption, fading, scattering, refraction, diffraction, multipath,

scintillation, etc.



2.3.1 Signal losses in free spaceIn a line-of-sight (LoS) transmission through free space, signals may be weakened even

where there are no obstructions, and this loss is usually expressed as a power ratio, or

decibels (dB), a logarithmic unit expressing the ratio of the values of two physical

quantities, with one of the quantities taken as the reference value. The weakening of

signals or the free space path loss (FSPL) is due to the spread of the signal’s energy

over the distance that it has to travel between the transmitter and the receiver. This can

be computed as [20]:

FSPL=( 4 πdλ )

2

=( 4 πdfc )

2

(2.2)

or expressed in dB as

FSPL (d B )=10 log10(( 4 πdfc )

2

)=20 log10 (d )+20 log10 ( f )−147 (2.3)

where d (in metres) is the path length, λ (in metres) is the wavelength of the signal f (in

Hz), the frequency, and c (in metres per second) the velocity of light in a vacuum.

2.3.2 Dispersion of electromagnetic wavesIn the course of the signal travelling between the transmitter to the receiver along the

Earth-space path, signals may encounter obstacles or traverse media that are not

transparent to these electromagnetic waves thereby dispersing it. The result is the

weakening (or attenuation) of the signal’s energy or power density. The causes of this

attenuation may include, but not limited to:

2.3.2.1 AbsorptionThis is where the signals are totally or partially absorbed by the medium that they are

passing through. Here, energy is converted by the medium and dissipated through heat.

The commonest culprits are oxygen and water vapour, especially when the signal’s

frequency is close to the vibrating frequencies of these molecules (water at 22 GHz, and

oxygen at 63 GHz).



2.3.2.2 DiffractionThis is when the signal encounters irregular obstacles, especially one with dimensions

bigger than its wavelength, or there is a restriction in the aperture of the transmission

path. This results in the production of a secondary wave bending it around the obstacle.

This is more noticeable when the obstacle is sharper (like a knife-edge) than in rounded

obstacles and lower frequencies tend to diffract more than higher frequencies.

2.3.2.3 ReflectionJust like mirrors reflect light, signals can be reflected by various media. In reflection,

the angle of incidence equals the angle of reflection. Unless there is a total reflection,

there may be signal loss due to some signals passing through or being absorbed by the

reflecting medium. This will likely occur when the signal encounters a large object with

dimensions larger than its wavelength.

2.3.2.4 RefractionRefraction occurs when a signal passes from one medium to another. The angle of

refraction depends on the refractive index of the media the signal traverses. Refraction

will only occur if the angle of incidence is not 90°. The relationship between the angles

of incidence and refraction is described by Snell’s law, which is given as [21]:

sin Asin B

=v A

v B=

λA

λB=

nB

n A(2.4)

where A and B are the angles of incidence and refraction respectively, and vA, vB, λA, λB

and nA, nB are the velocities, wavelengths and refractive indices of the incident and

refracted signals respectively.

2.3.2.5 ScatteringWhen signals encounter obstacles or pass through different media the energy of the

electromagnetic waves may be dispersed in different directions, especially when they

encounter inhomogeneous materials in the transmission path with wavelengths smaller



than the objects. The signals’ energy arriving at the receiving antenna may be greatly

weakened as some may have been dispersed in different directions.

2.3.3.6 MultipathSignals may take more than one path to arrive at its destination. In the course of

traversing different paths, the signals may experience different path lengths and media

and may experience signal losses due to some other attenuation factors like absorption,

diffraction, reflection, and refraction.

2.3.2.6 ScintillationMainly observed in the F layer of the ionosphere, especially above the equatorial region

and in high altitudes, radio waves may undergo rapid fluctuations in its amplitude

and/or signal phase and direction of propagation. This is mainly caused by the passing

of signals through a region of irregularities in electron density. Signals may be scattered

due to these fluctuations causing shifts in the signal’s phase and amplitude. This is

noticeable in satellite communications as the signals would of necessity have to pass

through the ionosphere. The effect of scintillation peaks during periods of high sunspot

activity [19].

Radio waves may be also affected by other phenomena; frequency dispersion (when a

frequency and phase of transmission changes, e.g. due to a disruptive medium),

depolarisation (when the polarisation of a transmission may change, e.g. due to

precipitation), noise (unwanted signals interfering with the propagated signals), and

more.

Troposheric scintillation is the distortion to signals caused mainly by turbulent

irregularities in temperature, humidity and pressure occurs in the ionosphere, especially

for signals with frequencies above 4 GHz.

2.4 Principles of radiometryRadiometry is the science of measuring the quantity of electromagnetic radiation

received by a body at all wavelengths based on the principle that all material media



absorb and emit electromagnetic energy. A blackbody is a body whose surface absorbs

all radiation incident upon it. In this light, the brightness temperature, sometimes called

the black-body equivalent radiometric temperature is the temperature at which a black

body in thermal equilibrium emits the same energy as the body being measured, and it

is used in this case to give an insight into the amount and temperature of water vapour

laden in the atmosphere. The radiometer uses microwaves to determine the temperature

and humidity of the atmosphere in clear and cloudy conditions. Microwaves are used in

radiometry since they can go through non-precipitating clouds with small drop sizes

compared to the wavelength. Dust and solar radiation do not seriously affect the

measurements, but the instruments are sensitive to liquid water [22].

An ideal black body is one that absorbs all energy incident upon it, while real materials

or “grey bodies” reflect a portion of the incident energy. A black body’s brightness

temperature is always lower than the physical temperature since the brightness

temperature is the maximum possible power radiated by the body in question. The

physical temperature is the temperature at equilibrium [23].

For real materials, the spectral emissivity (or radiance), ε, the ratio of the brightness

temperature and the thermodynamic temperature is always less than unity, and equals

unity in the ideal black body.

Radiometric techniques are applicable to the estimation of liquid water path (LWP),

clear air attenuation and rain attenuation along the Earth-space slant paths, based on the

notion that the absorption rate of water vapour (the main constituent of rain) and the

brightness temperature can be used to retrieved the LWP, thereby predicting rain

events. Fig. 2.3 shows the attenuation rate against different frequencies. The 22 GHz

frequency is most sensitive to water vapour while the oxygen complex peaks at 60

GHz. This thesis experiment’s frequency is 19.7 GHz, and therefore special note is

taken of the closeness of this to the high attenuation of the water vapour peak

attenuation.



Fig. 2.7: Atmospheric gas absorption and frequencies [24]

Radiometric attenuation is estimated using [25-27]:

A (d B )=10 logT eff −T0

T eff −T sky(2.5)

where

T eff=Kcls

1−10−0.1∗A g(2.6)

Teff is the effective medium temperature, T0 is the cosmic background temperature, Tsky

is the measured sky temperature (sample measurement range for July 2012 is shown in


Frequency (GHz)

Spe

cific

Atte

nuat

ion

(dB

m)


Appendix VI), Kcls is the clear sky temperature (excluding the cosmic background

temperature) and Ag is the clear sky (gaseous) attenuation.

The total attenuation is composed of the sum of attenuations due to oxygen, water

vapour and liquid water. Hence, the water vapour and liquid water can be estimated

from the water vapour measurements of a dual channel radiometer [27].

2.5 AttenuationRain attenuation is the greatest factor that is taken into consideration in the design of

high-availability satellite communication links, especially for frequencies above 10

GHz.

2.5.1 The ITU Model Relationship between Rain Rate and AttenuationAs seen earlier, rain attenuation is the reduction in the transmitted signal amplitude, and

this degrades the reliability and performance of the communication links. It is the most

significant propagation impairment on satellite links operating at frequencies above 10

GHz [28, 29]. The attenuating effects of the troposphere are affected by the

characteristics of the rain systems, which include the size and distribution and

movements of rain cells and density. An approximation of the specific rain attenuation,

γ, (or attenuation per unit distance) is calculated as a function of the instantaneous

rainfall rate, R, and is given as [30], and may be obtained by the power law relationship

[31]

γ (R)=aRb dB/km (2.7)

where R is the rainfall rate in mm/h and γ(R) and the coefficients a and b are parameters

dependent on the signal frequency and polarisation respectively, given by ITU-R

recommendations [30]. The values of a and b in equation (2.7) are determined for 1

GHz ≤ f ≤ 1000 GHz as:

log10 a=∑j=1

4

a jexp {−( log10 f −b j

c j )2}+mk log10 f +ca (2.8)



b=∑j=1

5

a j exp {−( log10 f −b j

c j )2}+mα log10 f +cb (2.9)

where f is the frequency in GHz, a and b are either horizontal or vertical constants of

polarisation. The coefficients mk, mα, aj, bj, cj, ca and cb are given in ITU-R 838-11 [30].

The rainfall rate is one of the fundamental input parameters in deriving the ITU-R rain

attenuation model. The specific attenuation depends on the rain rate and a and b, some

frequency dependent parameters [31-33]. The ITU model is best an estimate, and even

though universally accepted may still be improved upon. Also, this may be applicable

in to a limited frequency range.

There are other attenuation models (e.g. the Olsen et al. Specific Rain Attenuation

Model [34], the Crane Models [35-37], the Lin model [38]). The analysis of these

models is however beyond the scope of this thesis. The ITU model was chosen for its

universality, even with the known limitation that it is best an estimate of rain

attenuation based on long-term calculated probabilities.

2.6 Scattering by spherical raindropsScattering is generally the redirection of signals away from the intended propagation

path. Refraction, reflection and diffraction are examples of scattering, as each of these

tends to redirect the signal in one way or another away from its intended path. For

simplicity, the assumption is made of the spherical shape of the raindrops. Chapter 3

discusses the physics of rain, its formation, shapes and sizes, and one observes that the

shape of rain droplet is more oblate than spherical.



Fig. 2.8: The geometry of scattering [39]

Scattering results in the weakening of propagated signals hence is of importance during

propagation. In the interaction of an incident radio wave signal with a rain-filled

medium, the signals may be scattered backwards (back scattering) in the direction of

the incident wave, forward scattered (in the intended direction), side scattered (at an

angle to the incident wave), and some may be absorbed by the medium and converted

to heat to be dissipated to the environment. An example of this is shown in Fig. 2.4,

where θi is the angle of incidence.

Scattering may be classified as elastic or coherent (where there is little or no change in

the wavelength of the incident radio wave signal), inelastic or Raman lidar (where the

wavelength of the scattered light is changed by the medium). Scattering may be

affected by the signal’s frequency, the size of the scattering particle (or wave number,

the ratio of the circumference to the wavelength), and the complex refractive index of

the medium.

The wave number determines the dominant type of scattering. It will be considered a

Rayleigh scattering if the particle radius is much less than the wave length, a Mie



scattering if the particle radius is the same as the wavelength, and a Fraunhoffer

diffraction if the radius is much higher than wavelength [40]. Rayleigh scattering

involves particles with diameters much less than the wavelength of the incident signal,

and the result of it is the blue sky, whereas the presence of larger particulate matter in

the air presents Mie scattering.

2.7 Extinction (or attenuation) cross-sectionExtinction is the sum of the processes of scattering and absorption. The extinction (or

attenuation) cross-section, Qt, is dependent on the wavelength, λ, the complex refractive

index of water, m ≡ p + iq and the drop diameter, D. Qt = Qt(λ, m, D) [41].

Mie’s theory assumes spherical raindrops. This gives the complex forward scattering

amplitude for the spherical raindrop function as [42]:

S (0 )=12∑n=1

∞

(2n+1)(an¿+bn)¿ (2.10)

where an and bn are the Mie scattering coefficients, complex functions of λ, m and D.

The extinction cross-section is thus given as:

Qt=4 πk 2 R e {S (0 ) }= λ2

πℜ {S (0 ) }= λ2

2 π ∑n=1

∞

(2n+1 ) ℜ(an+bn) (2.11)

The attenuation coefficient is determined by integration over all the drop sizes since the

drop sizes are not all equal, the attenuation coefficient is given as:

k=∫0

∞

Qt(¿ λ ,m, D) N ( D ) dD¿ (2.12)

2.8 Specific AttenuationSpecific attenuation (measured in dB/km) is the attenuation per unit distance and may

depend on a number of factors affecting the propagation of the electromagnetic wave



signals. The specific attenuation chiefly depends on rain rate and will also depend on

the size distribution, the temperature, shape, size and orientation of the raindrops as

well as the frequency, polarisation and the propagation direction of the incident wave.

Signals degrade more with higher frequencies as the wavelength to raindrop diameter

ratio reduces.

Integration of the specific attenuation along the path gives the total attenuation [34].

The specific attenuation can be estimated by using Mie’s scattering theory, assuming

the raindrops are spherical [43].

If Pr is the received power on a uniformly distributed spherical water drop of radius r

with transmitted power, Pt, over length L, then the attenuation, A (in dB) will be given

as:

A=10⋅ log10 ( Pt / Pr )=10⋅ lo g10exp ( kL )=4.343 ⋅ kL (2.13)

where the attenuation coefficient, k, is

k=N (D)⋅Qt (2.14)

N(D) is the drop density, and Qt is the extinction cross-section of a drop of diameter D

[31].

The attenuation (or extinction) cross-section Qt, as shown before, is a function of the

wavelength λ, the complex refractive index m ≡ p + iq, and the diameter D of the rain

drop, hence Qt = Qt(λ, m, D) [41].

Since the drop sizes vary, the coefficient of attenuation is obtained by integrating over

the drop size spectrum, and the attenuation coefficient is given as in equation (2.12),

and hence the specific attenuation of a radio wave coursing through rain is then given

as [44]:



γ=4.343 ×103∫0

∞

Qt ( λ , m, D ) N ( D )dD dB /km (2.15)

where Qt is the attenuation cross-section of a single rain drop as a function of the drop

diameter D, with wavelength λ, and the refractive index of water, m, depending on both

frequency and temperature and N(D) is the rainfall drop size distribution.

The dependence of rain attenuation on drop size, DSD, rain rate, and attenuation cross-

section is clearly shown in the equation (2.15). The first three parameters (λ, m and D)

are characteristics that depend on the rain structure only. The frequency and

temperature dependence of rain attenuation is obtained through the attenuation cross-

section only. All the parameters show temporal and spatial variations that are not

deterministic or predicted directly, hence most researchers investigating rain

attenuation depend on statistical analyses to calculate the quantitative impact of rain on

communications systems.

2.9 Effective Rainy Slant Path LengthThe effective rainy slant path length, Leff, is defined as the ratio between instantaneous

total rain attenuation A(t) at time instant t and the corresponding specific attenuation

γ(t). Challenges to overcome in the determination of Leff along the slant path include the

correct determination of the rain height as well as the horizontal extent of the rain cell,

as this varies with the rain type and rain rate. It is generally assumed that the higher the

rain rate, the shorter its horizontal extent since stratiform rain type is typically lighter

and more widespread than the convective rain type as shown in Fig. 2.5 below.



Because of this variation of the rain type and intensity, the effective rainy slant path

length will vary, and rain rate will generally not be uniform throughout a slant path. It

may well be that the signal may encounter rain in its path, even when the ground

equipment may report no such rain. This is investigated later in this thesis.

It is worthy of note that the standard ITU solution for long-term statistical distribution

gives a good prediction of rain attenuation determined with measurements taken over a

period. This thesis investigates the instantaneous total attenuation instead of the long

term so that telecommunication service providers may be able to estimate link

conditions to adjust power or other communication resource either proactively or

reactively based on prevailing conditions.

2.10 Fade DynamicsRain fade (or attenuation) refers to the disruption of signals due to rain events. This is

experienced when the signal power level goes below a certain level marked as the

availability threshold. This may be caused by inadequate link budgeting to provide for

the time when there is a higher attenuation than clear air.


Fig. 2.9: Effective rainy path length scenarios for stratiform and convective rain types


The understanding of this fade dynamics (fade duration - the temporal length of the

fade, fade slope - a rate of change of the rain fade, fade margin – the allowance between

the received power and the sensitivity of the antenna before outage) is an important

factor as it is useful in the “design [of] a control loop that can follow signal variations,

or to allow a better short-term prediction of the propagation conditions” [45]. The fade

slope distribution may depend on the type of climate, influenced by the DSD and the

wind velocity [46].

The systems designer must make allowance for these crossings between the marked

threshold of availability so as to ensure the continuous provision of services to the

consumer. He has to design in fade mitigation procedures to cater for these times. This,

therefore, means that a thorough understanding of the fade dynamics is necessary.

Fig. 2.6 above shows a brief schematic of signal levels, indicating fade over time.

Signals fade when the threshold is surpassed, and the fade duration is the length of time

it takes for the signal to again cross the threshold and restore communication, whereas

the interfade interval is the duration between consecutive fades, whilst the fade slope

refers to the rate of fade of the signal over time. Several crossings of the threshold may

occur in the same precipitation event before a sustained availability of communication

leading to another precipitation event. The duration between these precipitation events

is the inter-event interval. It is of importance to study this fade information as this


Fig. 2.10: Characteristics of fade dynamics [45]


enables for the provision of the best mitigation schemes to counter the disruption in the

communication.

2.11 Fade Mitigation TechniquesTo allow for a satisfactory QoS, a proper fade margin (the allowance made in power

between the received power and outage) needs to be designed to ensure the continuous

provision of services. Fade Mitigation Techniques (FMTs) are usually deployed to

ensure this continuity. FMTs are mainly classed as [2]:

i) Equivalent Isotropically Radiated Power (EIRP) control techniques;

ii) Adaptive Transmission Techniques (where the signals are modified before transmission), and;

iii) Diversity Protection Schemes (including time, site, frequency and orbital diversity – where the parameters are stretched to counter failures in the signal transmission).

The preferred FMT, the EIRP control techniques involve the control of power (Uplink

Power Control (ULPC) and Downlink Power Control (DLPC)) to overcome the fade

experienced in a link. The ULPC’s power is adjusted at the earth station’s transmitter as

against the DLPC’s adjustment at the satellite. These adjustments are done based on the

available link information, enabling continuous availability.

EIRP control may be implemented as open loop (where the power is adjusted based on

a feedback from the receiving antenna) or the closed loop (where the power is adjusted

based on the information available from the link channel). The open loop is

disadvantaged by the round-trip time feedback, which may be enormous and

impracticable in some instances. The closed loop, however does the adjustment more

dynamically based on fade measurements of the channel.

2.12 ConclusionThis Chapter has looked at radio wave signals, how they are generated and propagated

through the Earth-space path. It discusses the Earth’s atmosphere and the various

propagation impairments through it and space. The Chapter discusses the subject of

radiometry, its use in the measurement of sky brightness temperature. Its help in the



determination of rain-free days as it is susceptible to liquid water is discussed in

Chapter 6 that discusses rain attenuation.

The relationship between attenuation and rainfall rain rates is explored, with the ITU

specific attenuation model introduced. The Chapter also looks at the scattering of

signals by spherical raindrops and discusses the extinction cross-section and how

specific attenuation scales with raindrop sizes. The extinction (or attenuation) cross-

section is computed. Finally, the effective rainy slant path length is discussed, and its

variation with rain intensity is introduced. The Chapter concludes with a brief

discussion of fade dynamics.

This Chapter addresses the background to the subject of the thesis, Chapter 7 draws on

the principles of radiometry and the equations discussed here. Chapters 6 and 7 draw

from the specific and total attenuation discussion, especially the computation of the

specific attenuation using the extinction cross-section in Chapter 6. Chapter 8’s

effective rainy slant path length, Leff, uses the discussion in this Chapter. The Chapter

concludes with discussions on fade dynamics and fade mitigation techniques.



Chapter 3 : RAINFALL DROP SIZE DISTRIBUTIONS

AND DSD MODELS



Chapter 3

RAINFALL DROP SIZE DISTRIBUTIONS AND

DSD MODELS

3.1 IntroductionTo understand how signals fade in the Earth-space path when they encounter

precipitation, it is fundamental that one understands what constitutes precipitation.

Precipitation in the Earth-space path takes various forms; liquid rain, clouds, snow,

graupel, sleet, hail, drizzle, etc. Rain, the main precipitation type, is the one universally

seen all over the planet, and its geographic distribution, type and causes are location-

dependent.

3.2 Rain Dynamics3.2.1 Rain formationRain is formed when water on the earth’s surface is heated by sunlight, the warm air

rises and as it ascends, the temperature drops down to a certain level, termed the dew

point and the water vapour condenses into liquid cloud droplets by attaching to dust

particles and other materials suspended in the atmosphere. As the particles are carried

upward, they collide into each other and merge to form larger droplets. Sunlight is

scattered from the resulting droplets and that is what is seen as clouds [47].

Typically, above the melting layer level, ice crystals are formed in the below-freezing

air. As the ice crystals droplets and particles continue to collide with each other, they

grow in size and weight, they are pulled down by gravity, they melt as they descend

below the melting layer and hit the ground in the liquid state, in a form that is known as

rain. A schematic of the rain cycle is shown in Fig. 3.1.



3.2.2 Rainfall typesThere are three main rainfall types: convective, stratiform and orographic. (Another

sub-classification of rain may be relief, convectional and frontal rain types). Orographic

rain is mainly seen in topographies where there is an abrupt change in altitude, e.g.

mountains or cliffs. In this type of rainfall, the side of the changed topography lifts the

rain-bearing breeze, and the lifted moist breeze is frozen when it ascends to the melting

layer and under the right conditions falls back to the ground as rain.

Convective rainfall is generated when warm air rises from the ground due to the sun’s

heat. As this rises, the warm air cools and form clouds, and at condensation or dew

point, forms clouds. This result in a heavy rainstorm and is usually accompanied with

thunder and lightning. This kind of rain, mostly common in warmer climates, is more

intense, less widespread and does not last very long.

The stratiform rain, less intense and more widespread than the convective rainfall

comes about when there is a forced upward movement of air. As the unsaturated air

rises, there is an increase in the relative humidity. On saturation, clouds form as the air

continues its rise and eventually results in precipitation. The stratiform rain tends to last

longer than the convective rain.


Fig. 3.11: The rain cycle (adapted from [47])


3.2.3 Raindrop sizes and shapesThe earliest record for the measurement of rainfall drop sizes is Lowe in 1892 [48]. He

observed splashed rainfall drop patterns on slates. Wiesner [49] exposed absorbent filter

papers treated with water-soluble dye to rain, where the raindrops marked on the paper

were measured and the results analysed. Bentley [50] exposed a layer of smoothened,

uncompacted flour to rainfall. The resulting flour pellets were dried and measured, with

each pellet representing a raindrop. Laws and Parsons [51] modified the method of

Bentley by separating the pellets into groups of several sizes using sieves. Eigel and

Moore [52] measured raindrop diameters using an oil immersion method.

Joss and Waldvogel [53] developed the impact disdrometer that collects raindrops on an

exposed surface and the sound resulting from the impact of the raindrops on the surface

is thus translated into the size of the raindrops. Jones [54] developed a method to

photograph raindrops using two short-exposure 35 mm cameras positioned at different

angles.

The 1D and 2D Video Disdrometers (described in [55]) provide more accurate

measurements for both small (D < 1.5 mm) and large (D > 5 mm) drops sizes, D being

the drop diameter. The 2D video disdrometer (Fig. 3.2) uses two cameras placed at right

angles to measure raindrops with diameters from 0.17 mm in a 100 × 100 mm2

sampling area. The 2D Video Disdrometer has an adjustable integration time of from 15

seconds to 12 hours.


Fig. 3.12: A 2D Video Disdrometer [56]


Raindrops typically range in diameter from 0.1 mm to 7 mm [57], and the drops grow

bigger due to collision with other drops. Drops larger than 5 mm tend to be unstable

and break up due to forces acting on them as they fall.

Raindrops of sizes up to 9.7 mm have been reported [58, 59]. Rainfall sizes are found to

vary with rain intensity. Large raindrops with diameters more than 5 mm are likely to

occur in tropical, sub-tropical and high-altitude continental locations, especially during

spring.

Traditionally, raindrops are assumed to be shaped like teardrops. However, they are

near spherical when their cross-sectional diameter is less than 2 mm, and larger drops

assume an oblate shape, flattened at the bottom because of forces acting underneath it

as shown in Fig. 3.3. As the raindrop size increases due to coalescence, the forces

tending to divide the drop also increase, tending to split it when the diameter gets above

5 mm.



3.2.4 Fall velocitySmall raindrop sizes fall to the ground at between 3 m/s and 8 m/s depending on wind

speeds, and these speeds vary with the size of the raindrops as well as the carrying

wind. Gunn and Kinzer [61] computed the terminal velocities of different rain

diameters at sea level conditions at 20 °C and 1013 millibars (see Appendix I).

Beard [62] expressed relationships for the terminal velocity of different sizes of rainfall

drops, and for the range of drop radii 1.2 mm ≤ r ≤ 4 mm, and suggested that the

terminal drop velocity will be:

vi=k √r (3.1)


Fig. 3.13: Aerodynamic Forces on Raindrops [60]


where the constant k is dependent on the drag coefficient, gravity, and the density and

dynamic viscosity of air.

However, from the measurement data gathered by Gunn and Kinzer [61], the terminal

fall velocity of the raindrops, vi, (a function of the raindrop’s diameter, Di (in mm)), is

used by some researchers as [63-65]

vi=9.65−10.3 exp (−0.6 Di ) (3.2)

It is to be noted that different researchers [66-68] have used the equation:

vi=3.78 D i0.67 (3.3)

Both equations (3.2) and (3.3) are from the measurements of Gunn and Kinzer [61].

This thesis, however, uses equation (3.2), (which has an asymptotic limit of 9.65 m/s

for terminal velocities) as Fig. 3.4 suggests that equation (3.3) fails for larger drop

sizes.



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Drop Diameter (mm)

0

2

4

6

8

10

12D

rop

Velo

city

(m/s

)

9.65 - 10.3 * exp(-0.6 * D)3.78 * D0.67

This study, using equation

(3.2) as the reference, found that there is an 8.46% difference between both equations

for the drop size range (0.3 mm to 4.79 mm) measured in the rainfall DSD dataset used

in this study. The difference is biggest with small drops with diameters below 0.6 mm

and with large drops with diameters above 4.1 mm. Even though the analysed data fall

within the range of 0.6 mm and 4.0 mm, the decision to use equation (3.2) is based on

the reason of the asymptote of the equation, as physically, the velocity of a body does

not increase ad infinitum with increasing size.

3.2.5 Rainfall ratesRainfall rate, or intensity, measures the quantity of rain that falls in a place on the

surface of the Earth per unit area per unit time. It is typically measured in millimetres

per hour (mm/h). This is taken to mean the depth of water that would accumulate in the

considered flat, horizontal area if all the liquid rain were captured (assuming no run-

off) in the given unit of time. A 1 mm/h of rainfall would be equivalent to a litre of

rainwater falling in an area of 1 square metre within one hour [69].


Fig. 3.14: Comparing the terminal velocity equations


Rain intensity can be measured using a number of instruments; the several different

types of rain gauges, where the captured rain is measured and the rain rate calculated,

the disdrometer (sometimes called the distrometer) where the rain hits an exposed

surface and the rain impact sets off electrical impulses which are converted to raindrop

sizes, and the rain rate can be computed from the number and sizes of raindrops hitting

the exposed surface; the radar, where electromagnetic waves are bounced off

precipitation in the atmosphere, and the returning waves are used to determine the

quantity and sizes of the water particles thereby giving an estimate of the rain rate [70].

This thesis uses the RD-69 impact disdrometer with a 50 mm2 exposed area and a 10

second integration time, the rain rate is expressed as [70]:

Rm( t)=∑ NT ∙ {4 π3 ( D

2 )3}/( A ∙S ∙ Δt

3600) (3.4)

where NT is the drop count, D (in mm) is the drop diameter, A (in mm2) is the area of

the exposed disdrometer cone, S the number of time-sliced samples to merge to give the

one-minute time interval considered (here taken as 6: 6 ×10 seconds = 1 minute), and 𝛥t is the integration time of the disdrometer.

The UK Met Office rain accumulation rate classification is shown in Table 3.1 below.

For this thesis, rain intensities are classed using the UK Met Office’s classification [71]

of rainfall types. This is chosen as the boundaries gave fewer rain classifications, and

suits the purpose as the work set a threshold of rain as 0.1 mm/h.

Rain Class Rain Rate (mm/h)Slight 0 < R ≤ 2Moderate 2 ≤ R ≤ 10Heavy 10 ≤ R ≤ 50Violent R > 50


Table 3.3: Rain rates and the different classifications [71]


The UK Met Office specifies colour codes for use in rain forecast maps is shown in

Table 3.2 below.

Table 3.4: UK Met Office’s Colour Coding for different rain rates [72]

Rain Rates (mm/h) Colour0.01 ≤ R ≤ 0.5 Blue0.5 ≤ R ≤ 1.0 Azure1.0 ≤ R ≤ 2.0 Olive2.0 ≤ R ≤ 4.0 Yellow4.0 ≤ R ≤ 8.0 Orange

8.0 ≤ R ≤ 16.0 Red16.0 ≤ R ≤ 32.0 Fuchsia

R > 32.0 Aqua

The measurement of rainfall intensity comes in useful in several applications. The

climate can be modelled and the weather predicted if one has accurate measurements of

rain intensities in different geographical locations. The water cycle can be well

understood. Runoffs and evaporation can equally be modelled from accurate

measurements of rainfall intensities. [ref?]

In Section 3.2.2, different classifications of rain (orographic, convective and stratiform)

were given. In this thesis, the terms stratiform and convective rain regimes are not used.

This thesis refers to rain regimes as light, moderate, heavy and very heavy. This is due

to the complex nature of classifying stratiform or convective rain. A lot of

meteorological factors ˗ not just the rain rates ˗ interplay to really determine if a rain

type is stratiform or convective.

3.3 Rainfall Drop Size Distributions and DSD ModellingAs shown in previous Sections, rainfall causes dispersion in electromagnetic signals as

it passes through the Earth-space path [36, 73], thereby causing a weakening in the

received signal, otherwise termed attenuation. A portion of the signal is absorbed by the

raindrops and converted to heat, and part may be scattered in different directions,

causing attenuation [73, 74]. The rate of absorption and dispersion of these radio wave

signals is dependent on the size of the raindrops, with bigger drops causing more

attenuation [75]. Attenuation is especially noticeable in the higher frequencies (above



10 GHz), and may affect even lower frequencies (7 GHz) in the tropics due to the

presence of larger diameter drops [76-78].

As mentioned earlier, in the study of the rainfall drop size distribution, different

instruments are used in the measurement of the falling raindrops; rain gauges, impact

disdrometers, video disdrometers, and radar. This thesis looks at the measurements

made by a Joss-Waldvogel RD-69 impact disdrometer located at Chilbolton

Observatory, in southern England.

The disdrometer, as discussed earlier, is an electronic instrument that measures raindrop

sizes continuously and automatically. In the impact disdrometer, raindrops fall on an

exposed Styrofoam® cone. The vertical momentum of the drops that impact the cone

are converted into electrical pulses. These generate voltages which are measured and

the sizes of the raindrop are determined from the amplitude of pulses [6].

The knowledge of raindrop sizes and their distribution becomes important as this

distribution determines the rain intensity, thereby leading to the modelling and

computation of attenuation due to rain [79]. As stated earlier, modelling the rainfall

drop size distribution (DSD) is important for understanding, predicting and mitigating

rain-induced attenuation in satellite signals in the millimetre band along the Earth-space

path.

Other factors may affect the DSD, though marginally. Meteorological factors (wind

speed, temperature, pressure, wind direction, rain intensity) all play their roles in the

shaping of the drop size distributions.

3.3.1 Disdrometer Bin CalibrationIt has been shown that the manufacturer calibration of the RD-69 JWD disdrometer bin

sizes can create peaks in the number density of drops observed by the disdrometer.

Therefore, for this thesis, two possible calibrations of the bin sizes are used: the original

calibration provided by the manufacturer (MFR) and the ETH calibration of the RD-69

JWD disdrometer bin sizes (ETH_ALL (using ALL the bins) and ETH_TRUNC



(eliminating the lower-sized bins)) [80]. The ETH-calculated bin boundaries are shown

in Appendix II.

3.3.2 Possible ErrorsThere are several possible sources of error in the disdrometer measurement [81]. The

effect of the wind on an impact disdrometer is an issue, and this may affect both

volume size and number concentration [81, 82]. The horizontal component of the wind

is not a problem as it does not directly affect fall velocity, but the vertical turbulence

also increases with wind speed. The up- and down-draughts caused by turbulence will

affect the actual fall speed of the particle, so the assumption that it is falling at terminal

velocity is invalid. To some extent, the velocity differences will work to both increase

and decrease the measured momentum and hence the assigned size bin of the drops, but

some errors are likely to remain. The effect will be worse for small drops which already

have a small terminal velocity.

This thesis considered the effect of large and small drops falling simultaneously on the

disdrometer and the breaking of large drops into smaller drops by re-splashing after the

first impact had been recorded, thereby increasing the count of smaller drops. Recall

that the disdrometer converts the momentum of the drops to a pre-determined voltage

rating to determine the diameter of the drops. The smaller drop may be masked by the

larger drop, depending on the time delay. The effect is likely to be negligible except at

very high rain rates. The smaller secondary drops are also unlikely to be included in

this analysis as they are not falling at terminal velocity and as such would be assigned

to a smaller bin than their true size.

Another issue is “dead-time” of the disdrometer due to the ringing of the Styrofoam®

cup. This is the time it takes for the disdrometer to reset itself after the impact of a drop.

This causes the number of small drops detected by the disdrometer to be depressed.

This effect will be more significant at higher rain rates. It would tend to produce a

nearly constant error across the DSD.



Sheppard and Joe [81] provide a formula for estimating and correcting for this effect.

However, for the data used in this thesis, fitting gamma or lognormal distributions

suggested that there were usually too many drops in the lower bins to fit the model.

Therefore, rather than apply a dead-time correction, this thesis simply considers the

case that the counts in the lower 15 bins are unreliable and examines the fit to the data

using the ETH calibration but with these bins ignored. The unreliability of the lower-

sized bin is suggested by the works of Thurai, et al. [83], Kumar, et al. [84].

3.4 DSD modellingVarious standard classical statistical distributions have been proposed in literature as

models for the positive continuously DSD. Several researchers have tried to fit

statistical distributions to captured rainfall data. The DSD (measured in mm-1 m-3) is

defined as the number concentration (m-3) of raindrops per unit volume per unit

diameter, centred on D (in mm). Here N(D) is the number of such drops per unit

volume with diameters in the infinitesimal range (D – dD/2, D+dD/2) with size dD

centred on D.

3.4.1 Negative Exponential Distribution

Marshall and Palmer (1948) [85] proposed the relationship

N ( D )=N0 exp (−ΛD ) ,0<D≤ Dmax (3.5)

with Dmax as the maximum drop diameter. They note that for their data, the coefficient

Λ (in mm-1) is a function of the rainfall rate R (mm/h), according to the relationship

Λ=α Rβ (3.6)

In equation (3.5), N0 = 8000 mm-1 m-3, while in equation (3.6), α = 4.1 and β = -0.21.

However, it should be noted that at small diameters (D < 1.5 mm), this relationship fails

[85]. Their method involved measuring the size of rainfall drops that impacted on

stained dye paper that was briefly exposed to rain. There is the likelihood that small

drop stains could have been covered by larger drop diameter stains, thereby making the

data for the small diameter rainfall drops unreliable.



This should however be considered against a different work [86] which says that these

smaller diameter rain drops contribute to the rain attenuation in millimetre and sub-

millimetre radio wave transmissions. Marshall and Palmer’s data was collected over the

summer of 1946 from Canada [87].

3.4.2 The Lognormal DistributionThe lognormal distribution, a well-known positive unimodal statistical distribution, first

proposed by Feingold and Levin (1986) [88] takes the general form

N ( D )=NT

√2 π σ g(D−θ)exp{−(log ( D−θ )−μg )2

2 σ g2 }

(3.7)

for a random variable X if log (X – θ) is normally distributed with μg and σg as the

geometric mean and variance respectively for some θ [89]. In the use of the lognormal

distribution to characterise the rainfall DSD, θ is set equal to zero, and the distribution

is multiplied by a scaling parameter, as it is a probability distribution.

Kolmogorov [90] first suggested that breaking droplets are lognormally distributed.

Other later researchers also modelled rainfall data using the lognormal distribution [88,

91-93]. The two estimated parameters (μ and σ ) were later found inadequate to describe

the volumetric distribution of raindrops, as it does not provide for a comparison of

DSDs as well as the recognition of raindrop shapes in different rain rates [67].

3.4.3 The Gamma DistributionLater studies showed a gamma distribution yielded better rainfall predictions than the

lognormal, especially when combined with radar data [94]. Ulbrich and Atlas (1984)

proposed the gamma distribution for the DSD, given as:

N ( D )=NT D μ exp (− Λ D );0 ≤ D≤ Dmax (3.8)



with Λ as the slope parameter, µ is the shape parameter and NT the scaling parameter,

and these enable the characterisation of various rainfall scenarios. Note that the gamma

distribution is a generalisation of the exponential distribution, and for µ=0 the gamma

distribution collapses to an exponential distribution.

Ulbrich and Atlas do not actually claim the DSD is a gamma distribution, but rather

assume that a gamma distribution yields more accurate rainfall predictions. They accept

that other distributions might serve equally well.

The normalised form for the gamma distribution for any rain rate R is given when NT is

substituted in equation (3.8) with [95]

NT=N0Γ (μ+1)

Λμ+1

(3.9)

Later researchers [66, 67, 96, 97] prefer to use the normalised form of the gamma

distribution, as the parameters are more meaningful, where the D (in equation (3.8)) is

replaced with D/Dm, with Dm being the volume weighted mean diameter, or mean

volume diameter. This now gives the number of rainfall drops as

N ( D )=N w f (μ)( DDm )

μ

exp [−( 4+μ ) DDm ] (3.10)

where N(D) (in m-3 mm-1) is the number of drops per unit volume per unit size interval,

D (in mm) is the sphere equivalent diameter, Nw (in m-3 mm-1), µ and Dm (in mm) are the

scaled intercept, shape and mass-weighted mean diameter parameters respectively.

f(μ¿, a function of the shape parameter, μ is defined as

f ( μ )=6(4+μ)μ+4

44 Γ ( μ+4 )(3.11)



where Γ is the complete gamma function. This is useful when comparing the shapes of

DSDs that have different liquid water content and/or the same diameter [67].

The lognormal and gamma distributions are dependent on two parameters. These are

the parameters to be estimated to achieve the models of the distributions. The

lognormal’s parameters are the geometric mean and the geometric variance, whereas

the gamma’s parameters requires the use of the method of moments to estimate the

parameters or an iterative process to estimate the most likely value of the parameters.

The distribution of the raindrops at the disdrometer surface is not the distribution per

unit volume because the big drops fall faster. Another point to note is that the

disdrometer measures within a certain diameter range, meaning that there is a

truncation in the captured data [98], and this should be considered when estimating the

parameters.

Another distribution proposed in the modelling of rainfall drop size distribution is the

Weibull, first proposed by Sekine and Lind (1982) [99], where N(D) is defined as

N ( D )=N0ησ ( D

σ )η−1

exp {−( Dσ )

η} (3.12)

with D (in mm) being the drop diameters, σ, η are functions of the rain rate. With η=1

in equation (3.12), the Weibull becomes the exponential distribution set out earlier in

equation (3.3).

For all the distributions considered, the assumption is made that the DSDs are spatially

homogeneous and stationary over short periods, e.g. one (1) minute [98], and these

have implications when carrying out the estimation, chiefly being that the resolution of

the results may be unreliable. Based on these studies and others, it has been suggested

that different models may be needed for different rain types [44, 100, 101] and regions

[94, 102, 103].



3.4.3.1 Method of MomentsParameters for a normalised gamma distribution fit were calculated using the method of

moments (one of the common methods for the estimation of parameters of a

population) used by Islam, et al. (2012) [67] and Montopoli, et al. (2008) [66], where

the parameters Nw (the generalised intercept parameter), Dm (the mass weighted mean

diameter) and μ are computed using the moments of the DSD as follows:

The method of moments defines the nth moment of the DSD at time instant t as

mn ( t )=∫0

∞

D n∙ N (D , t ) dD=∑i=1

n

Din . N m ( D i , t ). Δ D i (3.13)

for n channels (or bins) of the collecting instrument, with the right-hand side

summation being a near approximate of the left-hand side integral.

If N(D,t; Dm,Nw, μ) is a gamma distribution, then the mean diameter Dm is defined as

Dm=m4

m3(3.14)

and Nw may be computed from

Nw=44 ∙ m3

5

6 ∙ m44 (3.15)

and μ is equally expressed in terms of the moments [68].

If N(D,t) is a lognormal distribution, then its parameters can be estimated, using

standard methods of estimating parameters of a normal distribution, but applied to a

sample of log(D) values.



3.4.3.2 Maximum Likelihood EstimatesIn the fitting of the rainfall DSD data to the model, the Maximum Likelihood Estimates

(MLE) method is used to estimate the parameters of a model. It finds the likelihood of

the observed set of data, as a function of the parameters for the probability distribution,

it then finds the model parameter values that maximises this likelihood. In other words,

it tries to get the values of the parameters that would be most likely to have produced

the data under consideration. MLE finds the parameters that maximise this likelihood.

The advantage here is that MLE is unbiased (if its mean equals the estimated

parameter) for large samples, consistent (if the estimator converges as the sample size

grows) and efficient (if its mean square error is minimum). MLE yields more accurate

estimates of DSD parameters than the moment method as the moment method is biased

[98].

Note that the disdrometer drop sizes are sorted into discrete bins. This should also be

considered when estimating parameters [104]. Equally, observe that there is truncation

in the data. There is an upper and lower threshold drop diameters captured by the

disdrometer, even if there are raindrops with diameters outside these thresholds. This

should be considered when doing the MLE estimation [98].

3.5 Goodness-of-Fit TestsWith the modelling of the rainfall DSD data to fit standard statistical distributions, there

is still the need to determine the goodness-of-fit of the data to the distribution. While

many researchers use the accuracy of the rainfall or other predictions drawn from the

DSD, as proxy measures for goodness-of-fit of the model [44, 66, 85, 94, 95, 105-107],

this thesis uses the Pearson chi-square goodness-of-fit test to determine how well the

data fits the assumed distributions. Pearson’s chi square was chosen for the goodness-

of-fit to test if the models fit the data with acceptable confidence.

Several researchers use some variant of the sum of squared errors (SSE) [66, 85, 88].

SSE is not necessarily a good measure. The standard deviation of a drop count is

proportional to the square root of the expected drop count but the SSE treats all

deviations equally. Note that the SSE is used to compare distributions rather than test



the fit in these papers. Owolawi (2011) [93] did test the fit (using the Kolmogorov-

Smirnov (K-S) test due to small sample size).

The standard goodness-of-fit test determines how well the data and the proposed model

agree. The Kolmogorov-Smirnov test is used to determine whether two samples are

taken from the same distribution while the Pearson’s chi-squared test treats the

equivalent for frequencies, or works with binned data. Since this thesis deals with a

very large sample, hence frequencies, it was ideal to use the Pearson’s chi-squared test,

where the observed and the expected counts are determined and compared.

The K-S test works better for small samples than chi-square test. Advice on the use of

the chi-squared test suggests merging frequency bins where counts are less than 5

before computing the chi-square statistic [108]. This is stricter than the minimum of 5

counts in 80% of the bins rule considered by statisticians. The computed statistic is

compared with the confidence threshold (here taken at 95% confidence) with n – k - 1

degrees of freedom (where k is the number of parameters to be estimated). The

hypothesis is either rejected or not rejected depending on the comparison of the

computed chi-squared statistic and the confidence threshold.

The Pearson chi-squared test can be used to test the fit of a distribution, by using the

distribution to calculate the probability of the value falling into a particular class, the

distribution of the counts in the classes then being multinomially distributed. This thesis

uses the standard chi-square goodness-of-fit test [108] to evaluate the fit of the model to

the DSD. To do this, the DSD model was converted back to give the expected drop

counts. The bins were merged to ensure that all the expected counts were more than

five, and the chi-square statistic was computed as

χ2=∑i=1

n (O¿¿ i−Ei)2

Ei¿ (3.16)

with Oi and Ei as the observed data and the model’s expected values respectively in the

ith class, with n classes, and the results were compared with the 95% confidence

threshold with n − 3 degrees of freedom, with n being the number of bins. Since the



Chi-square estimates the mean and the variance, and since convention says the degree

of freedom must be less by one for estimated parameters, the degrees of freedom here

will be estimated as n – 3 because of the two estimated parameters.

3.6 Limitations and Gaps Identified

As stated earlier, many researchers use proposed mathematical models for the DSD.

The normalised gamma, (with D/Dm in place of D) is used for more meaningful

parameters [97, 107], as well as allowing for easy comparison of the DSDs.

While literature seems to favour lognormal and gamma distributions since the 1980s,

these classical probability distributions as presented have not been tested for goodness-

of-fit against the data. The data captured using disdrometers need to support calculated

DSDs for impact disdrometer and drops binned and truncated.

There is equally little study done in the delineation of DSDs between the seasons,

atmospheric temperatures and wind classes. The question needs to be asked as to how

well these data really fit the assumed distributions, and how they vary with seasons,

temperature and wind speeds. There is the need to do the actual testing of the fit of the

distribution to the data, rather than assume the fit by proxy.

The processed rainfall data exhibits multimode (more than one mode). The statistical

distributions discussed earlier seem ill-equipped to handle multimodal data. This thesis

suggests the use of the Gaussian Mixture Model (GMM) in the handling of such data

when encountered.

3.7 The Gaussian Mixture Model (GMM)Mathematical modelling involves the attempt to fit data into a theory model. If one

were to look at the physical height of several plants in a nursery, it is assumed that not

all will be the same height and that a small sample (say a section in a shed) may

represent the general approximation of the plants’ heights. The numbers of short and

tall plants are likely to be less than the number of plants with average height.



6 7 8 9 10 11 12 13 14Plant height (cm)

0

5

10

15

20

25

The distribution of these heights may well be represented by a bell-curved shape with

tails on either side, (Fig. 3.5) called the Normal or Gaussian distribution, or the plants’

heights can be said to be distributed Normally, with the mean μ, taken to be the centre

(or peak) of the distribution, and σ, the spread or variance of the data [109, 110].

In some instances, like in the rainfall drop size distribution here, it was realised that

there may exist more than one peak (Fig. 3.6), unlike the plant example demonstrated

above. Where there is more than one peak (or mode, in statistical language), the

Gaussian and other standard unimodal (one mode) distribution fail to model it

accurately. It now calls for a multimodal (multiple mode) distributions to attempt to

model the data.


6 7 8 9 10 11 12 13 14

Plant Height (cm)

25

20

15

10

5

Cou

n

tμ

Fig. 3.15: A Gaussian (Normal) Distribution


This leads to the introduction of the Gaussian Mixture Model. The GMM, in its

simplest term, is the sum of individual Gaussians with different means, variances and

weights giving one “umbrella” distribution. This is used in this thesis to model

multimodal data encountered in the rainfall drop size distribution data measured.

In the rainfall DSD data that visually exhibits multi-peaks, there is a need to group

these into clusters with discernible means and variances and for each cluster estimate

the mean, μi and variance, σi and the weight parameters. Fig. 3.7 shows three Gaussians

(in red, blue and green), and the mixture of the Gaussians (in dotted black) that clearly

models the data shown in Fig. 3.6 above better than the unimodal distributions as it

describes the data better. This therefore leads to a better prediction of the rainfall DSD.


Dro

ps p

er m

3 per

mm

-1

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8Drop diameter (mm)

0

1000

2000

3000

Dro

ps p

er m

3 per

mm

Time 16:46 Rainrate 4.1 mm/h Lognormal=REJECT Gamma=REJECT MLE=REJECT

Drop DensitiesGamma (Moment)Gamma (MLE)Lognormal

Fig. 3.16: Failure of the unimodal distributions


-1.5 -1 -0.5 0 0.5 1Log drop diameter (mm)

0

10

20

30

Dro

p c

ou

nts

A Gaussian Mixure Model with 3 centres: 26-Jul-2003@16:46

Drop counts

1= 0.06, 1=0.14, w1=0.33

2= -0.53, 2=0.12, w2=0.45

3= -0.85, 3=0.18, w3=0.23

Sum of Gaussians

0.4 0.6 0.8 1 1.2 1.4 1.6Drop diameter (mm)

0

1000

2000

3000

4000

Dro

p d

en

sitie

s

A Gaussian Mixure Model with 3 centres

Drop densitiesSum of GaussiansTo determine the clustering of the data into the different Gaussians, the first step is to

seed k random means, μi,,| i ∊ {1, 2, …, k} as the initial cluster centres, with the

covariance set for each cluster as the covariance of the entire dataset and an equal prior

probability for each cluster, the prior probability being the weight or fraction of the data

points in each cluster. Each data point is assigned to a cluster, with its Euclidean

distance measured to the mean, μi.

In the GMM, each cluster (or individual distribution) in the population will have its

own parameters, and these may be determined using the iterative Expectation-

Maximisation (EM) algorithm [111]. The parameters to be estimated for each cluster

are the mean, μ and variance, σ and the weight.

The GMM uses the EM algorithm to determine the membership of each data point to a

cluster. For the “Expectation” step, the probability that each data point is a member of

each cluster is determined based on the last estimated means and covariances, and in

the “Maximisation” step, the cluster means and covariances are determined based on

the probabilities earlier determined from the Expectation step and assigns an adjusted

weight to each individual cluster. The GMM seeks to combine several individual

Gaussian distributions into one distribution [111].


Fig. 3.17: A Gaussian Mixture Model


In a GMM of k clusters, 3 × k − 1 parameters need to be estimated since the weights

must be normalised to 1. For 3 clusters, the GMM requires 8 parameters to be estimated

(3 μs, 3 σs for each of the clusters, and weights w and wi for the population).

The probability density function (pdf) for a GMM is of the form [112]

p(D)=∑i=1

k

wi ∙1D

exp [−( ln (D)−μi )2

2σ i2 ] (3.17)

where μi and σi are the mean and standard deviation of the ith mode respectively and the

weightsw i have the property ∑i=1

k

wi=1. In this thesis, since the measured data, the

rainfall drop size diameter is always positive2, the Gaussians is fitted in the log domain.

It needs to be pointed out that there may be different kinds of mixture models. It is

possible for the constituents of the mixture models to be different classes of

distributions; a Gaussian mixed with a Pareto, say. The complexity of the computations

in this kind of mixture model is however beyond the scope of this thesis. The

application of the GMM and the results are further discussed in Chapter 5.

3.8 DSD and MultimodalityMultimodality has been observed in the DSD by previous researchers [113-115].

Sauvageot and Koffi (2000) [115] attribute the presence of multimodality in DSDs to

the overlapping of different rain shafts resulting from cloud volumes at different

heights, and they also show that the number of peaks, Nm, of a DSD, depends on the

rain rate variations, and not on the mean rain rate, but this was for rain with D i > 2 mm,

where large Nm are inversely related to values of the slope parameter, λ and with large

values of the intercept, N0 of the exponential distribution.

2 The term “Gaussian Mixture Model” as used here may not be exactly correct. The Gaussian (or Normal) distribution allows for negative values, whereas drop diameters cannot be negative. This term is used only to describe the model, a mixture of Gaussians, as the correct description of lognormal mixture model may be strange in literature. So, throughout the thesis, GMM means Gaussian Mixture Model fitted in the log domain.



Steiner and Waldvogel (1987) [114] also studied multimodal behaviours in DSDs and

reported that these modes existed for different drop size diameters in convective rain

regimes. Radhakrishna and Rao’s [113] study indicated that the appearance of

multimodal distributions in the DSDs are dependent on the rain height, and varies with

different rain systems, with multimodal distributions frequently encountered in

convective rain systems, where multimodality is observed to occur due to a transition

from convective to stratiform rain. They classified the rain systems as convection,

stratiform, and transition. This thesis, as discussed in Section 3.2.5, however, classes

rain regimes as light, moderate, heavy and very heavy. Jones (1992) [116] reports of

secondary peaks at some rain rates in samples examined, though he says this was not

uniformly present in all samples.

McFarquhar (2010) [117] says that multimodal peaks are observed in computed

models, but not systematically in physical data. Åsen and Gibbins (2002) [118] argues

against the existence of multimodal peaks in observed rainfall, and say that the

presence of multimodal peaks may be due to an error in the calibration of the bins.

McFarquhar and List (1993) [80] recomputed the bin boundaries based on the

recalibration of the RD-69 JWD disdrometer done at the Laboratory of Atmospheric

Physics at the Eidgenössische Technische Hochschule (ETH) in Zurich, Switzerland.

However, Chapter 3 showed that even with the ETH calibration, there was still the

presence of multimodality, and not just a consequence of the instrumental artefacts as it

compared the Distromet® (manufacturers of the disdrometer) calibration with the

ETH’s recalibration. This Chapter, however, relies on the ETH recalibration of the bins

boundaries.

Steiner and Waldvogel [114] define a frequency as a mode if “... the concentration of

raindrops per unit volume and unit diameter interval of a given interval was

significantly larger than the concentrations of the two neighbouring diameter intervals”.

Sauvageot and Koffi [115] and Radhakrishna and Rao [113] similarly treat N(Di) as a

mode if N(Di-1) < N(Di) > N(Di+1), where N(Di) is the density of drops with diameter Di,

and Di are the diameters measured by the disdrometer.



This definition, while valid overlooks the issue of sampling from a distribution. A

random sample drawn from a distribution may be multimodal even though the

underlying distribution is unimodal. This is just a consequence of the “noisy” nature of

the sample. This effect can be reduced by merging neighbouring bins.

This thesis, however, employs a different method to determine the number of modes in

a multimodal distribution by identifying each individual mode from the troughs

surrounding it. N(Di) is defined as a trough, when

N(Di-1) > N(Di) < N(Di+1) < N(Di+2) (3.18)

This ensures a steady rise to determine the beginning of the next cluster. This was

found to be more dependable by visual inspection when compared with the method

used by other researchers. For example, if this method is used to identify peaks instead

of troughs as the determinants of modes, there is the chance of counting false peaks.

This study has attempted to fit the Gaussian Mixture Model (GMM) to the data. The

GMM, as seen earlier in Section 3.7, seeks to combine several individual Gaussian

distributions into one distribution.

3.9 ConclusionThis Chapter looked at the physics of rainfall; analysing the formation and dynamics of

rainfall as a prelude to understanding the underlying concept of the rainfall drop size

distributions. Rainfall types, shapes and sizes and fall velocities were mentioned and

the developing history of the measurement of rainfall drop sizes and their distributions

were briefly discussed. The Chapter looked at rain classifications and rain rate

calculations. The Chapter introduced the different unimodal DSD models considered in

this thesis, viz. the negative exponential, the lognormal and the gamma distributions.

The derivation of rainfall rates was discussed, including the thesis’s definition and the

reason for the choosing of the rain classifications.



The Chapter looked at the recalibration of the bins based on the ETH study, different

from the bin boundaries provided by the manufacturers of the disdrometer. It discussed

possible errors in the measurement of raindrop counts using the RD-69 JWD impact

disdrometer. In the measurement of the DSDs, multiple modes were observed, and

since standard unimodal statistical models do not make room for these multiple modes,

this thesis has suggested the Gaussian Mixture Model as a solution, with a hope of

providing better estimates for the DSDs. Modes are redefined, with a trough taken as

the end of a mode, a departure from the traditional definition of a peak determining a

mode. The Chapter concludes with a discussion of multimodality and the Gaussian

Mixture Model, which forms a great part of this thesis in later Chapters.



Chapter 4 : RAINFALL DSD MODELLING



Chapter 4

RAINFALL DSD MODELLING

4.1 IntroductionThis Chapter continues with the modelling of the rainfall DSD with real data. It looks at

the procedure and the fitting of the model to the data using the three datasets identified

earlier, viz the dataset using manufacturer’s bin boundaries for all bins, the dataset using

the recalibrated bin boundaries suggested by McFarquhar and List for all bins, and

finally the dataset using the suggested alternative bin boundaries but with the smaller-

sized bins eliminated.

4.2 DataExtensive weather data has been collected at the observatory in Chilbolton in south-east

England. The work in this Chapter assesses the goodness-of-fit of two of the standard

proposed statistical models, the lognormal and gamma distributions (using the method

of moments and the maximum likelihood estimates), to the disdrometer measurements

collected at Chilbolton Observatory between 2003 and 2009. It does a goodness-of-fit

test using Pearson’s chi-squared test. The success of the model fit is also examined with

respect to values of other meteorological factors such as wind speed and temperature.

As stated earlier, this thesis utilised data captured by the RD-69 Joss-Waldvogel Impact

Disdrometer connected to an ADA90 analyser at the Chilbolton Observatory in

southern England (51.1445 N, 1.4386 W) between 2003 and 2009 (with gaps in

between – as the data does not fully cover the entire stated period)3. The disdrometer

has a surface area of 50 cm2 and measures raindrops with diameters from 0.3 mm to 5.0

mm in 127 gradations otherwise termed bins and records the count of the different

raindrop diameters at 10-second intervals. The 127 size classes are distributed more or

3 There was no data for January to March, 2003. There was also no data for July 2005 to May 2006 as between those dates the equipment was returned to the manufacturer for repairs [67]. In the other months, there were a total of 73 days that data was not captured.



less exponentially over the range of drop diameters and the accuracy rate of the

readings is ±5% of measured drop diameter [6].

To increase confidence in the reliability of the disdrometer data, the disdrometer-

obtained rain rate was compared with that of a rain gauge in the same vicinity. The

wind measurements were made using a cup and vane anemometer at a height of 10 m

above ground level, 3 m above a cabin roof. They are located at a distance of

approximately 50 m from the disdrometer.

4.3 Fitting MethodologyThe drop counts at the disdrometer surface were converted into spatial drop densities,

Nm(Di,t), i.e. the density of drops (or drop concentration) per m3 per mm at time instant

t, by dividing the bin drop count by the sensor area, sample time, estimated drop speed,

and bin width [66] using the relationship [66, 67]

Nm ( Di ,t )=ni(t)

A .dt . vi . d Di(4.1)

where at a discrete time instant t, Di is the central drop diameter of the ith channel, ni(t)

being the total drop count in that channel, A is the exposed area of the disdrometer’s

sensor, dt the time interval, dDi is the width of the bin, and vi the terminal fall velocities

of the raindrops (a function of the raindrop’s diameter), given in equation (3.2).

4.4 ProcedureThis study aggregated six 10 second readings into one-minute samples, this was done to

achieve a smaller sample size as well as strike a balance as a lower time period may

yield insufficient data for sampling and a longer period may yield erroneous data due to

changes in rainfall dynamics. This is the approach taken by previous researchers [66,

67]. This implicitly assumes that the underlying distribution is approximately stationary

over a one-minute time scale.



Furthermore, three adjoining bins were merged to smooth the data. Merging too many

bins will result in the loss of the data’s resolution. It should be noted that Radhakrishna

and Rao [113] worked with a 5-minute distribution interval, rather than the 3-bin merge

within one-minute distribution intervals used here.

Given the unreliability of small diameter drops measurements [83, 112], even with

more sophisticated instruments than the impact disdrometer [83], all bins with drop

diameters less than 0.6 mm were eliminated from the data under consideration. Because

of this unreliability, three versions of the data set are considered viz. (i) using the

default manufacturer calibration (MFR), (ii) using the ETH calibration (ETH_ALL), and

(iii) using the ETH calibration but ignoring the counts in the lower 15 bins

(ETH_TRUNC).

The 127th bin measures drops with diameters above the capability of the instrument,

rather than within a small interval so was eliminated and not used in this study. The

samples were chosen with the condition that the rain rate should be more than 0.1 mm/h

[66], with the rain rate, Rm (mm/h) in 10 second integration time, at time instant t

computed as in equation (3.4). Any interval that had rain rates readings below that

threshold was considered not to have rain. The ETH bin boundaries were used

throughout in this thesis.

The average drop sizes were computed from averaging drop volumes across the bins

using the relationship:

Di=3√ (binBottoms+binSizes )4−binBott oms4

4 ∙binSizes(4.2)

with Di being the drop diameters (in mm). The drop velocities were as given in

equation (3.2) and the bin volumes (in mm3) given as vbin=43

∙ π ∙( Di

2 )3

, with the rain

rates given as: Ri=3600 ∙ vbin ∙ allDropCounts

Δt ∙ sensorArea, as in equation (3.4).



For each minute, σi was expressed as

σ i=√∑ ( probDensity ∙ Δ Di ∙ log ( Ddiam)2−μ2 ) (4.3)

and the nth moment of the drop densities, mn (n = 2,3,4 and 6) given as in equation

(3.13). The weighted mean diameter, Dm is as given in equation (3.14) and Nw is given

in equation (3.15).

Defining η=m4

2

m2 ∙m6, then the mean of the gamma distribution is given as:

μγ=7−11 ∙η−√ (7−11 ∙ η )2−4 ∙(η−1)∙(30 ∙ η−12)

2 ∙(η−1)

(4.4)

For the maximum likelihood estimated (MLE) gamma fit, the parameters were

computed by minimising the negative log likelihood, using the MATLAB® function

fminunc for finding the local minima of functions with unconstrained parameters

using the following as initial seeds

init1=√μγ+1 and init2=√ Dm

4+μγ(4.5)

With the gamma (MLE) values given as:

γmle=S ∙D i

(k−1) ∙ exp (−Di

t)

tk ∙[ Γ ( Ut

, k )−Γ ( Lt

, k )](4.6)

where S is the scaled trapezoidal integration of the drop densities, Nm with respect to the

drop diameters, Di, and k and t are the first and second parameters of the incomplete



gamma function, Γ, with L and U defined for n bins as L=D1−Δ D1

2 , and

U=Dn−Δ Dn

2.

Parameters for the lognormal fit were computed by taking the mean and standard

deviation of the log of the N(D) values, as shown in equation (3.7), and the gamma

(using the method of moments) is fitted as described in Section 3.4.3, equations (3.8) to

(3.11). The density function in the log domain was converted to a density function of

the drop diameters, and the expected drop counts were recovered from the drop

densities by multiplying the densities by the product of the integrated area, the sensor

area of the disdrometer, the integration time, the drops speeds and the bin sizes.

And finally, the goodness-of-fit was done using Pearson’s chi-squared test as described

in Section 3.3.4 to determine how well the data matches the models.

4.5 Results and Analyses To increase confidence in the reliability of the disdrometer data, the disdrometer-

obtained rain rate was compared with that of a rain gauge in the same vicinity, and it

was discovered that the difference in readings from the disdrometer was about 15%

higher on the average and this is consistent with other works [67, 119]. This can be

explained by the fact that the disdrometer, discounting for environmental noise, debris,

insects and re-splashed raindrops record more rain than the standard rain gauge. The

resulting differences may be a consequence of environmental factors, variations in

rainfalls as well as the precision of the instruments [119].

The differences between the overall means of the disdrometer rain rates and that of a

rain gauge (with the rain gauge readings used as the base reference) in the same vicinity

are shown in Table 4.1 below for different rain regimes.


Table 4.5: Percentage differences between the rain gauge and disdrometer rain rates


Rain TypeLight

(< 2 mm/h)Moderate

(2 - 10 mm/h)Heavy

(10 - 50 mm/h)Very Heavy

(> 50 mm/h) ALL% Difference 17% 6% 6% 7% 15%

Using the data and procedure described in Sections 3.6 to 3.8, distributions were fitted

to a total of 92,100 one-minute samples and tested using the Pearson chi-squared test

for all the distributions with n-3 degrees of freedom. However, the test was not done for

n ≤ 3, n being the number of bins.

Fig. 4.1 shows a typical one-minute sampled data (29th October 2003 at 19:23) fit of the

three methods for the two distributions (lognormal, gamma using method of moments

and gamma using maximum likelihood estimates). The upper figure shows the total rain

drops (in drops per minutes) over a 24-hour period (29th October 2003) as well as other

meteorological data over the same period (scaled rain rate-obtained from the

disdrometer (in mm/h), equivalently scaled rain rate from a nearby rain gauge (in

mm/h), scaled wind speed (in m/s) and scaled temperature (in °C)). The vertical red line

indicates the time of the 1-minute time slice shown in the lower figure. The lower

figure shows the distribution of the raindrop sizes, the drop densities (in mm -1 mm-3)

plotted against the drop diameters (in mm).



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (Hours)

0

200

400

600

800

1000

1200

29-Oct-2003

Total Rain Drops/minDisd RainRate (mm/h)(x100)RG RainRate (mm/h)(x100)Wind Speed (m/s)(x100)Temp (°C)(x100)

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2Drop diameter (mm)

0

100

200

300

400

500

600

Dro

ps p

er m

3 per

mm

Time=19:23; Rainrate=2.4 mm/h: Lognormal=Don't Reject; Gamma=Don't Reject; MLE=Don't Reject

Drop DensitiesGamma (Moment)Gamma (MLE)Lognormal

The three curves are the statistical models used to fit the observed 1-minute data. The

models considered here are the lognormal (shown in red, with diamond markers), the

gamma using the method of moments (shown in blue with circle markers), and the

gamma using the maximum likelihood estimates (shown in green with square markers).

All three distributions for this one-minute sample time slice were not rejected by the

chi-squared test with the chi-squared statistic, the threshold and the degrees of freedom

for each model at 95% confidence interval shown in Table 4.2 below.


Fig. 4.18: Total rain drops and meteorological data and a 1-minute drop size

distribution


Table 4.6: Chi Square Test: 29-Oct-2003/19:23 @ 95% Confidence Interval

Model Chi Square Threshold ResultLognormal 20.58 49.80 Don’t Reject4

Gamma (MoM) 29.57 53.38 Don’t RejectGamma (MLE) 28.99 50.99 Don’t Reject

The difference in the degrees of freedom is accounted for by the different number of

bins considered, as the bins with very few counts were merged with others, reducing

the total number of bins considered, hence a lower number of the degrees of freedom.

Corresponding meteorological data for the above one-minute sample is shown in Table

4.3 below.

Table 4.7: Meteorological sensor readings for the above 1-minute time slice

Meteorological Factor ValueWind Speed: 3.47 m/s

Wind Classification: Gentle BreezeWind Direction: 310.5°

Temperature: 279.1 K (5.95 °C)Disdrometer Rain Rate: 2.39 mm/h

Rain Type: ModerateSeason: Autumn

The percentage of fits for each model that were not rejected by the chi-square test is

shown in Appendix III (and Fig. 4.2), in the row entitled “overall”. (The column

labelled GMM is discussed in later Section). It is noticeable that the gamma MLE fit

improves with the use of the ETH calibration and the removal of the lower size bins but

perhaps surprising that standard models fit at best only 56% of the time. (To check this

was not simply an implementation error of the chi-squared test, pseudo samples of rain

drops were generated with parameters matching those calculated, and these were tested

and were not rejected around 90% of the time). Possible reasons for this are discussed 4 In Statistics, the common convention is the use of “Don’t Reject” or “Reject” in the testing for a hypothesis. The meaning of this is that although the statistic in question has satisfied the condition of the hypothesis, it still does not mean that it is true because the test only gives which is most likely. Hence, instead of using the term “Accept”, Statisticians use the “Don’t Reject” term to reflect this.



later in Section 4.5, but this result is consistent with Montopoli, et al., [66] which found

that MLE gamma fits appeared to fit the peak of the observed data distribution well, but

not the tail; and with Radhakrishna and Rao, [113] which observed bimodal DSDs that

were certainly not well fitted by standard distributions.

It is interesting to see how the percentage of distribution fits varies with parameters

such as the season of the year, the rain rate and the wind speed. Fig. 4.2 (Appendix III)

shows the percentage of non-rejects for spring (March-May), summer (June-August),

autumn (September-November), and winter (December to February). It is noticeable

that all the standard models are worse at modelling the summer rainfall. The

percentages are given for the three datasets earlier described: MFR - using the default

manufacturer calibration, ETH_ALL - using the ETH calibration, and ETH_TRUNC -

using the ETH calibration but ignoring the counts in the lower 15 bins.

Lognormal

Gamma (MoM)

Gamma (MLE)

GMM

010203040506070

Spring

Summer

Autumn

Winter

Overall

The variation of the percentage of distribution fits with rain rate is shown in Fig. 4.3

(Appendix IV). If the rainfall is classified as light (less than 2 mm/h), moderate (2


%

Fig. 4.19: Seasonal % variation of DSD fits (MFR/ETH_ALL/ETH_TRUNC)


mm/h to 10 mm/h), heavy (10 mm/h to 50 mm/h), and very heavy (more than 50 mm/h)

then it is noticeable that the standard models are worse at modelling the higher rain

rates. However, it should also be noted that lower rain rates correspond to smaller

samples, so provide less data with which to reject an ill-fitting model. It should also be

noted however that the percentages at high rain rates may be unreliable since there were

few samples in these categories.

LognormalGamma (MoM)

Gamma (MLE)GMM

0

20

40

60

Light

Moderate

Heavy

V. Heavy

Light

Moderate

Heavy

V. Heavy

Fig. 4.20: Rain rate % variation of DSD fits (MFR/ETH_ALL/ETH_TRUNC)

The variation of the percentage of distribution fits with wind speed is shown in Fig. 4.4

(Appendix V). Unlike with the rain rate, the fit of standard models seems to improve

with increased wind speed. It should again be noted however that the percentages at

high wind speeds may be unreliable since there were few samples in these categories.

The wind classification is adapted from the definition of the UK Met. Office [120].


%


Fig. 4.21: Wind speed % variation of DSD fits (MFR/ETH_ALL/ETH_TRUNC)

When the fitted data was looked at based on the atmospheric temperature of the

locality, it was discovered that the temperature range of between 280 K to 285 K had

the best fits (40% overall, 46% for the lognormal, 45% for the gamma using the method

of moments, and 43% for the gamma using the method of maximum likelihood

estimation.) The least fit was with the temperature range of between 270 K and 275 K,

ranging between 3% and 4%. This temperature range falls in the freezing zone (Table

4.4). The minimum temperature recorded within the period was 270.68 K, while the

maximum was 299.55 K.



Table 4.8: The fit of the DSDs with varying temperature

Temp (°C) Temp (K) Total Lognormal Gamma(MoM)

Gamma(MLE)

-3.15 to 1.85 270 to 275 3% 3% 4% 3%1.85 to 6.85 275 to 280 19% 22% 21% 25%

6.85 to 11.85 280 to 285 40% 46% 45% 43%11.85 to 16.85 285 to 290 34% 26% 27% 27%16.85 to 26.85 290 to 300 4% 3% 3% 3%

With further analyses of the data was, it was discovered that a large portion of the data

(~50%) did not fit any of the distributions (Table 4.5).

Table 4.9: The Chi-square fit of the DSDs for each distribution

Test %ageLognormal only 4Gamma (Method of Moments (MoM) only) 4Gamma (Maximum Likelihood Estimates (MLE) only) 14Lognormal and Gamma (MoM) 6Lognormal and Gamma (MLE) 8None 50

4.6 Possible Interpretations

Neither the gamma nor the lognormal distributions fit the actual DSDs particularly well

(one would expect around 95% non-reject rates). The gamma MLE fits better than the

others. The fits seem worse at higher rain rates but better with high wind speeds. A

possible explanation may be that the high winds break the drops into smaller sizes,

hence the preponderance of small drops. Best fits were in spring, whereas the worst

were in summer.

Possible explanations are:

1. The lognormal and gamma distributions, even with the best parameters, do not fit the DSDs (although they may yield reasonable rainfall predictions).

2. The disdrometer is not giving a correct picture of the DSD.

The disdrometers underreported small- to medium-sized drops, which most likely

caused the underestimation of rain totals, especially at D ≤ 0.5 mm [121]. Overall, it



was found that the disdrometer measurements largely agreed (with 15% deviation) with

the result of a co-located rain gauge, giving the confidence that the results from the

instrument are good enough for the proposed search for the distribution of the statistical

distribution that will better fit the data.

Results above show that the 10-second rainfall DSD samples can hardly be said to be

distributed exponentially. However, if all the data is aggregated over a longer period, as

opposed to the 1-minute intervals worked with so far (2003 to 2009 in the Fig. 4.5

below) and the 0.1 mm/h threshold is lifted and using the MFR dataset, the fit to a

negative exponential looks reasonable. This can be explained by the sum of all the

raindrops that do not constitute rain events, defined here by the threshold of 0.1 mm/h.

The interest of this thesis, however, is in the rain event that may cause disruption in

signals, hence the non-consideration of the lower (below 0.1 mm/h) rain rates.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

0

101

102

103

104

105

106

Chilbolton Observatory Disdrometer DSD: Total Rain Rate across all channels 01-Apr-2003 to 31-Dec-2009/10 secs

Drop diameter (mm)

Dro

p co

unts

(mm

-1 m-3 )

Drop CountNegative Exponential

While the above figure shows a near fit of the negative exponential to the data, there is

the need to point out that this fit is at best an approximation, as shown by the “bumps”

at the lower-sized rain diameters and in the middle. The abscissa in the figure above


Dro

p C

ount

s (m

m-1 m

-3)

Drop Counts

Negative Exponentials

Drop Diameters (mm)

[Type a quote from the document or the summary of an

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

106

105

104

103

102

101

100

Fig. 4.22: Negative exponential distribution results when the 0.1 mm/h threshold is

lifted


represents the different bins (127 of them) spaced exponentially (hence the dense

lower-sized diameters), while the ordinate is the number of drops, on a logarithmic

scale. The bumps here suggest the disdrometer problems highlighted by many authors

[83, 112, 121, 122].

In the event where about half of the data (Table 4.5) could not be fit into any of the

standard unimodal distribution, this implies that these classical distributions may be

inadequate to model the data, as there are underlying multimodalities present in the

data. This thesis suggests the use of the Gaussian Mixture Model (GMM) (discussed in

Section 3.7) as a probable solution in the modelling of the rainfall DSD data.

4.7 ConclusionsIt discussed the modelling of the rainfall drop size distribution, with different models

considered. The DSDs were computed and compared to empirical models. The

lognormal and gamma distributions (using both the method of moments and the

maximum likelihood estimates) were considered.

The Chapter went beyond the modelling of the collected data by attempting to measure

how well the data fitted the model using the Chi-square goodness-of-fit test for the data

captured by the disdrometer at Chilbolton Observatory between 2003 and 2009. It also

discussed the unreliability of the measurement of small drop diameters by the

disdrometer, and the truncation of the bins, eliminating the lower 16 bins.

The method of maximum likelihood for the gamma distribution estimates gave the best

fits for the data, with 56% (ETH_TRUNC) fits overall and when the data was delineated

into several categories (seasons, rain types, wind classes, atmospheric temperature),

results show that none of the considered statistical distributions gave a convincing fit at

95% confidence interval, hence the difficulty to draw the conclusion that the data were

either lognormal or gamma distributed.



The fundamental result, however, is that a large proportion of the data samples (~50%)

did not fit any of the standard statistical distributions, leading to the question of a

search for what distribution would best describe the captured data.

The Chapter proposed the use of the Gaussian Mixture Model as a solution to the

modelling of the large portion of data which exhibited multimodality and could not be

modelled by the unimodal distributions.



Chapter 5 : MULTIMODALITY IN RAINFALL DSDs

AND THE GAUSSIAN MIXTURE MODEL



Chapter 5

MULTIMODALITY IN RAINFALL DSDs AND THE

GAUSSIAN MIXTURE MODEL

5.1 IntroductionIn Chapter 3, it was seen that traditionally the DSD is said to have an exponential [51],

lognormal [88], or gamma [94] distribution. Recent work suggests multimodal

distributions may be needed to accurately model the rainfall DSD. This Chapter

examines the prevalence of multiple modes, and how it varies with weather parameters

such as wind speed and rain rate, and presents a model for the prediction of the number

of modes based on the rain rate and the wind speed.

A closer study reveals that the DSDs may not always fall into the traditionally defined

categories, and may show complex formations [113]. Some may be bimodal (two

peaks) or trimodal (three peaks), and in general, may form multi-peak rainfall drop size

distributions. This has been attributed to the transition between convective and

stratiform rain types or may be due to errors in measurement techniques [104]. The

captured rainfall data show evidence of multimodality.

Chapter 3 proposed using a Gaussian Mixture Model (GMM) with multiple modes as a

possible multimodal model. However, the number of modes is not set and this Chapter

investigates how the number of modes varies with wind speed and rain rate.

Fig. 3.6 demonstrated the difficulty of fitting multi-modal data to a classical unimodal

distribution, hence the need for the GMM with each cluster in the population having its

own parameters, determined using the iterative Expectation-Maximisation algorithm.



In Chapter 4, Table 4.5 showed that a significant number of samples did not fit any of

the distributions used to model the rainfall data. The question as to what distribution

will fit these samples arises. A thorough investigation of why the lognormal and

gamma distributions do not fit the data is needed.

It is worthy to note the difference between a multimodal sample and an underlying

multimodal distribution. Fig. 5.1 below demonstrates the difficulty of differentiating

from some underlying multimodal samples. The data is fitted with three Gaussians (in

green, blue and red), with different means, variances and weights, whereas the sum of

the Gaussians (in cyan) may be considered a Gaussian by visual inspection.

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2Log drop diameter

0

5

10

15

20

25

N(D

)


5.2 Data and Procedure used in this Study5.2.1 Data CollectionTo study the prevalence of multimodality in the data, this thesis again utilised data

captured by the RD-69 Joss-Waldgovel Impact Disdrometer located at the Chilbolton

Observatory in southern England (51.1 N, 1.4 W) between April 2003 and December

2009. The volume drops distribution was estimated as in equation (4.1) with the

terminal drop velocity given as in equation (3.2). The largest recorded drop diameter

for the period under consideration was 4.79 mm, while the mean temperature was 10.57

°C (proving it was not frozen). However, minimum and maximum temperatures

recorded for the period were -0.52 °C and 37.75 °C respectively.


Fig. 5.23: An underlying multimodal distribution


5.2.2 ProcedureBuilding on the procedure of Section 4.3 and 4.4, aggregating six 10 second samples

into a one minute sample to achieve a smaller sample size and merging three adjoining

bins to smooth the data, with all bins with drop diameters less than 0.6 mm eliminated

using the ETH bin boundary calibrations. The relationship in equation (3.18) was used

to determine the number of observed modes.

For each of the one-minute samples, the drop concentrations were computed using

equation (4.1) with the terminal drop speed as in equation (3.2). Based on the definition

of a mode given in equation (3.18), the number of modes in each one-minute

distribution was determined, with the maximum number of modes capped at four.

The study determined the average number of modes amongst the different wind type

and rain rates. The rain types, as before, were classed as light (with rain rates from the

threshold 0.1 mm/h to 2.0 mm/h), moderate (2 mm/h to 10 mm/h) and the merge of

heavy and very heavy rain due to very little data in the latter (rain rates above 10

mm/h). The wind speeds were classified from calm (less than 1 m/s) to severe gale (21–

24 m/s). The average of the number of modes in each delineated group was determined.

A correlation of the average rain rate and the average of the number of modes in each

rain type for all wind speeds were computed, with a multiple linear regression used to

determine the relationship between the number of modes and the rain rates and wind

speeds. A multiple linear regression was then fitted between the variables, where each

range of rainfall rates was represented by its average value.

For each one minute DSD, the number of modes determined was taken as the observed

number of modes, whereas the number of modes given by equation (5.1) was taken as

the predicted number of modes. Each of the one-minute DSDs was fitted with a

Gaussian Mixture Model with the number of clusters determined by the predicted and

observed number of modes.



This thesis utilised MATLAB®’s gmdistribution construct, which fits a given data

to a specified number of clusters. Fig. 5.2 shows the fit of the one-minute sampled data

in Fig. 3.6. Since the MATLAB® tools do not work with binned data; the procedure

used was as follows:

1. A pseudo-sample of size 1,000 in the log domain was generated based on the

distribution of drop densities measured.

2. The EM algorithm, described in Section 3.7, was employed to estimate the

parameters. Parameters determined were the mean, variance and the weight for

each cluster. The GMM distribution was computed in the log domain of the drop

diameters for each of the clusters, and the resulting distributions were added to

arrive at the overall model.

3. The density function in the log domain was converted to a density function of the

drop diameters.

4. The expected drop counts were recovered from the drop densities by multiplying

the densities by the product of the integrated area, the sensor area of the

disdrometer, the integration time, the drops speeds and the bin sizes.

The GMM probability density function (pdf) is given by equation (3.17), with the

condition that the sum of all weights must be unity, and the GMM modelled in the log

domain. For each rain regime (i.e. light, moderate and heavy/very heavy) the average

values of i, i and wi were computed.

5.3 Results and InterpretationsThe resulting overall sum of Gaussians model (or the GMM) with 3 clusters for the 1-

minute sample of 26 July 2003 at 16:46 is shown in Fig. 5.3 below. The red blue and

green curves are the individual clusters with their mean, μ and variance, σ shown in

Table 5.1, while the black dotted curve represents the overall summation of the three

individual curves.

For that one minute sample, the individual estimated μ, σ and weights for the sample

data were:



Table 5.10: Estimated Parameters for the GMM

Cluster μ σ WeightCluster 1 0.06 0.14 0.33Cluster 2 -0.53 0.12 0.45Cluster 3 0.85 0.18 0.23

The graph of the distribution represented by the above Table 5.1 is presented below.

The three Gaussian clusters are in red, green and blue, and the GMM (the mixture of

the three Gaussians) is shown by the black dashed lines. The means and variances are

also shown in the graph. This models the multimodal data better than the previous

unimodal distributions.

-1.5 -1 -0.5 0 0.5 1Log drop diameter (mm)

0

10

20

30

Dro

p co

unts

A Gaussian Mixure Model with 3 centres: 26-Jul-2003@16:46

Drop counts

1= 0.06, 1=0.14, w1=0.33

2= -0.53, 2=0.12, w2=0.45

3= -0.85, 3=0.18, w3=0.23

Sum of Gaussians

0.4 0.6 0.8 1 1.2 1.4 1.6Drop diameter (mm)

0

1000

2000

3000

4000

Dro

p de

nsiti

es


Drop densitiesSum of Gaussians

Fig. 5.24: GMM with three clusters for 26 July 2003 at 16:46



Do observe that since the GMM is a combination of Gaussians, meaning the underlying

distributions are still Gaussian distributions, it was necessary to fit the GMM in the log

domain, hence the negatives in the log of the drop diameters. However, to give it a

physical meaning, the drop diameters were recovered from the log domain, since drop

dimensions are not negative.

The chi-square goodness-of-fit test at 95% confidence interval for the above fit of the

overall curve to the data gave the data in Table 5.2 shown below

Table 5.11: Chi-square Goodness-of-fit test for the GMM

Parameter ValueChi-square statistic 61.79Threshold 62.83Degrees of Freedom 46Result Don’t Reject

The entire “orphaned” data (i.e. the 50% data in Table 4.5 that did not fit any of the

distributions) were fitted using the GMM. The results were as follows:

Table 5.12: Summary of results for Chi-square Goodness-of-fit test for different clusters

No. ofClusters

Parameters Estimated

%ageNon Reject

2 5 36%3 8 44%4 11 42%

The results show that the best fit was achieved with 3 clusters with 8 estimated

parameters (44%) (Table 5.3). Further work will be needed to determine the optimum

number of clusters for the distribution. Whilst more parameters would give a better fit,

the threshold for the chi-squared will however drop. Few clusters as possible is desired

as there is a possibility of assigning a cluster to each drop diameter, where each drop

diameter would be taken as a peak, albeit a false peak.

To establish that the reported multimodality in DSD is not merely an artefact of the

impact disdrometer employed at Chilbolton, England, this study also examined 1,587

one-minute DSDs recorded at Graz, Austria (47.1 N, 15.6 E, 353 m above sea level)



using a 2D video disdrometer over the period 22nd to 25th May 2015. This dataset was

treated the same way as the Chilbolton dataset. Even when five contiguous bins were

merged, the result showed that 440 (i.e. more than a quarter) of these DSDs were

multimodal. A sample DSD from this dataset with a GMM fit is shown in Fig. 5.3

below.

0.5 1 1.5 2 2.5 3 3.5 40

50

100

150

200

250

300

350

400

450

A Gaussian Mixure Model with 3 centre(s): 22-May-2015 @ 22:21, Rain rate=1.7 mm/h

Drop diameter (mm)

Dro

p de

nsiti

es (m

-3 mm-1 )

Drop densitiesGaussian 1Gaussian 2Gaussian 3Gaussian Mixture

Based on the data and methodology described in Section 4.3 and 4.4, the average

number of modes were determined for the rain classifications based on rain rates, R,

and wind speeds, W, as described earlier, and the results are as shown in Table 5.4. Fig.

5.4 shows a graphic of the same information.


Dro

p de

nsiti

es (m

-3 m

m-1)

450

400

350

300

250

200

150

100

50

0

Drop diameter (mm) 0.5 1 1.5 2 2.5 3 3.5 4

Fig. 5.25: Multimodal DSD from a 2D Video Disdrometer at Graz, Austria


Table 5.13: Average number of modes at different rain rates and wind speeds

Wind types Wind speeds Light rain Moderate Heavy/

V Heavy All RainRates

R < 2 mm/h 2 – 10 mm/h R ≥ 10 mm/hCalm W < 1 m/s 1.424 1.932 2.226 1.521

Light air 1 ≤ W < 2 1.461 1.997 2.256 1.554Light breeze 2 ≤ W < 3 1.438 1.889 2.457 1.522

Gentle breeze 3 ≤ W < 5 1.431 1.792 2.344 1.498Moderate breeze 5 ≤ W < 8 1.426 1.789 2.166 1.497

Fresh breeze 8 ≤ W < 11 1.543 1.986 2.369 1.636Strong breeze 11 ≤ W < 14 1.741 2.184 2.739 1.839

Near gale 14 ≤ W < 17 1.940 2.485 3.091 2.084Gale 17 ≤ W < 21 2.109 2.553 3.500 2.283

Severe gale 21 ≤ W < 24 1.667 2.333 4.000 2.286All Wind Speeds 1.469 1.882 2.327 1.549

Results show that the average number of modes for the entire data is 1.549 and that the

number of modes tends to increase with increasing rain rates and wind speeds.

It is noteworthy that the result showed the lowest average of the number of modes is

1.424, slightly over unimodality, giving credence to the assertion that multimodality is

prevalent in the data.



W < 1

1 <= W < 2

2 <= W < 3

3 <= W < 5

5 <= W < 8

8 <= W < 11

11 <= W < 14

14 <= W < 17

17 <= W < 21

21 <= W < 240

1

2

3

4

5

Light rain (R < 2mm/h) Moderate (2 <= R < 10) Heavy/V Heavy (R >= 10 mm/h) All Rain Rates

Wind speed (m/s)

Mod

e

To establish a relationship between the measured parameters (rain rates and wind

speeds) with the number of measured modes, a multiple linear regression was fitted to

the data in Table 5.4, and results showed that given the rain rate, R, and the wind speed,

W, the number of modes, Nm, can be predicted as

Nm=1.3096+0.0543 ∙R+0.0456 ∙W (5.1)

The result of equation (5.1) is presented with three significant digits for the second and

third part of the equation based on the small contributions of the measured parameters

to the predicted number of modes. This is to allow for more precision. It should be

pointed out that even though the computations in this thesis assumes a non-integral

number of modes, the practical applications would, however, require that the usable

number of modes be rounded to the nearest integer. Using non-integral modes in this

thesis, however. ensures that much information is not lost by rounding the number of

modes to the nearest integer.

From the results as presented, a correlation was drawn between the rain rates and the

number of modes as well as the wind speeds and the number of modes. Results show a


Fig. 5.26: Average number of modes for different rain rates and wind speeds.


strong correlation (0.9) each between the average number of modes and wind speeds as

well as the rain rates.

Equation (5.1) shows the individual contribution of each of the quantities to the total

number of modes. Fig. 5.5 shows the DSD for 26th July 2007 at 14:10 with a fit of the

GMM using the observed and predicted number of modes. The figure shows an

observation of one mode, whereas two modes are predicted from equation (3.18). The

top panel shows the individual Gaussians that make up the GMM (one for observed

number of modes, two for the predicted number of modes), whereas the bottom panel

compares the GMMs for both observed and predicted number of modes.

1 1.5 2 2.50

100

200

300

400

500Observation

N(D

) (m

m-1

m-3

)

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

100

200

300

400

500

26-Jul-2007, 14:10 - Rain rate = 1.23 mm/h, Wind speed = 5.47 m/s, Modes: Observation = 1, Prediction = 2

Drop diameter (mm)

N(D

) (m

m-1

m-3

)

1 1.5 2 2.50

100

200

300

400

500Prediction

N(D

) (m

m-1

m-3

)

To determine the agreement of the prediction with the observed data, a root mean

square percentage error (rmspe) was computed between the observed number of modes

and the predicted number of modes, where the rmspe is given as:

rmspe=√[ 1N ∑

∀ i ( M pi−M oi

M oi

∙ 100)2] (5.2)


-o- Observation

-x- Prediction

Fig. 5.27: GMM fits for the new model based on the predicted number of modes


where Mp and Mo represent predicted and observed number of modes respectively, and

N is the sample size. Result gave an rmspe of 35.4%, a rather high deviation, but still

marginally acceptable.

Table 5.5 shows the average values of i, i and wi in each of the rain rate regimes. It

can be seen that the spread of cluster centres (i.e. the range from 1 to 4 for example)

increases with rain rate. Furthermore, the clusters are approximately equally weighted.

It should be noted that although Table 5.4 shows a trend of increasing number of modes

with increasing rain rate, there were only 622 four-mode DSDs at high rain rate, which

represents only ~3% of the dataset, and given the number of samples here, may

therefore be considered too small a sample size for a reliable conclusion to be drawn.



Table 5.14: Average i, i and wi for GMM fits for different rain rates

Rain rates: 0.10 mm/h to 2.00 mm/h (light rain) Average μis Average σis Average Weight

# modes μ1 μ2 μ3 μ4 σ1 σ2 σ3 σ4 w1 w2 w3 w4

1 0.8143 - - - 0.168

8 - - - 1.0000 - - -

2 0.8017 1.1503 - - 0.122

2 0.1894 - - 0.5060 0.4940 - -

3 0.7744 1.0234 1.379

5 - 0.0987 0.1409 0.170

1 - 0.3305 0.3661 0.3034 -

4 0.7567 0.9632 1.207

5 1.539 0.0891 0.1227 0.151

3 0.1653 0.2526 0.2705 0.2472 0.2296

Rain rates: 2.00 mm/h to 10.00 mm/h (moderate rain)

Average μis Average σis Average Weight# modes μ1 μ2 μ3 μ4 σ1 σ2 σ3 σ4 w1 w2 w3 w4

1 0.9677 - - - 0.278

7 - - - 1.0000 - - -

2 0.8566 1.3087 - - 0.159

3 0.246 - - 0.4834 0.5166 - -

3 0.8336 1.1673 1.561

7 - 0.1309 0.1806 0.225

8 - 0.3240 0.3471 0.3289 -

4 0.8086 1.0863 1.406

8 1.8119 0.1189 0.1649 0.186

2 0.2295 0.2441 0.2750 0.2579 0.2230



Rain rates: 10.00 mm/h to 100.00 mm/h (heavy/v. heavy rain) Average μis Average σis Average Weight

# modes μ1 μ2 μ3 μ4 σ1 σ2 σ3 σ4 w1 w2 w3 w4

1 1.2212 - - - 0.342

7 - - - 1.0000 - - -

2 1.0382 1.6921 - - 0.203

6 0.2629 - - 0.4458 0.5542 - -

3 0.9765 1.4806 2.036

0 - 0.1817 0.2135 0.233

1 - 0.3182 0.3603 0.3215 -

4 0.9895 1.3736 1.707

1 2.2871 0.1888 0.2096 0.194

6 0.2201 0.2361 0.2570 0.2607 0.2463



Based on the results in Table 5.5, regression analyses were carried out to arrive at linear

relationships between rain rate and the GMM model parameters i, i and wi. The

resulting model equations are presented in Table 5.6.

Table 5.15: Model equations for i, i and wi at various number of modes

Mode s σs Weights1 μ1 = 0.0207 R + 0.8420∙ σ1 = 0.0079 R + 0.2027∙ w1 = 1

2 μ1 = 0.0127 R + 0.8008 ∙μ2 = 0.1651 R + 2.1149∙

σ1 = 0.0040 R + 0.1309∙σ2 = 0.0030 R + 0.2935∙

w1 = – 0.0031 R + 0.5019∙w2 = 1 – w1

3μ1 = 0.0106 R + 0.7793∙μ2 = 0.0560 R + 1.8921∙μ3 = 0.1320 R + 2.6679∙

σ1 = 0.0042 R + 0.1048∙σ2 = 0.0066 R + 0.1604∙σ3 = 0.0008 R + 0.2629∙

w1 = – 0.0006 R + 0.3287∙w2 = – 0.0003 R + 0.3606∙w3 = 1 – (w1 + w2)

4

μ1 = 0.0126 R + 0.7548∙μ2 = 0.0415 R + 1.7732∙μ3 = 0.0781 R + 2.2031∙μ4 = 0.1522 R + 3.0179∙

σ1 = 0.0052 R + 0.0920∙σ2 = 0.0044 R + 0.1796∙σ3 = 0.0029 R + 0.2267∙σ4 = – 0.0007 R + 0.2473∙

w1 = – 0.0014 R + 0.2367∙w2 = – 0.0009 R + 0.2664∙w3 = 0.0013 R + 0.2495∙w4 = 1 – (w1 + w2 + w3)

Based on the definition of a multimode in equation (3.18) and the predicted number of

modes given in equation (5.1), this thesis went on to investigate further how the

distributions will be distributed based on seasons, (spring: March-May, summer: June-

August, autumn: September-November, and winter: December to February).

Table 5.16: Seasonal unimodal and multimodal distributions for observed and predicted Nm

Season PeriodObserved Predicted

Unimodes

Multimodes

Unimodes

Multimodes

Winter Dec - Feb 130 4,699 272 4,557Spring Mar –

May127 5,756 687 5,196

Summer

Jun - Aug 288 5,317 341 5,264

Autumn

Sep - Nov 148 1,970 254 1,864

The result (Table 5.7) shows that 98% of the samples were multimodes in spring (the

highest) for the observed number of modes, and 93% for autumn (the lowest). No

discernible pattern emerged here.



Using rain regimes boundary demarcations a specified earlier (light rain with R ≤ 2

mm/h, moderate rain with 2 ≤ R ≤ 10, heavy rain with 10 ≤ R ≤ 50, and very heavy rain

with R > 50 mm/h), this thesis investigated how the multimodal distributions will be

distributed in the different rain regimes.

Table 5.17: Unimodal and multimodal distributions for both observed and predicted Nm

Rain ClassRain Rate

(mm/h)

Observed Predicted

Unimodes Multimodes Unimodes Multimodes

Light rain ≤ 2 563 9,060 1,397 8,226Moderate rain 2 – 10 129 8,263 157 8,235

Heavy rain 10 – 50 1 417 0 418V. heavy rain > 50 0 2 0 2

Results are shown in Table 5.8. As expected, most of the distribution were multimodal,

with 98% of the captured rainfall data exhibiting multimodality for both observed and

predicted number of modes. Data for very heavy rain is however too small to draw any

meaningful conclusion. This is consistent with Chilbolton being in the temperate

region, where most rainfall experienced will be classed light or moderate.

Again, borrowing from the earlier wind speed classification, the work investigated the

effect of wind speeds on the multimodality of the rainfall drop size distribution.



Table 5.18: Wind speeds unimodal and multimodal distributions for Nm

Wind classWind speeds

(m/s)Observed Predicted

Unimode MultimodeUnimode Multimode

Calm < 1 12 385 344 53Light air 1 – 2 43 1,173 775 441

Light breeze 2 – 3 62 1,503 435 1,130Gentle breeze 3 – 5 192 4,520 0 4,712

Moderate breeze 5 – 8 221 4,749 0 4,970Fresh breeze 8 – 11 124 3,023 0 3,147

Strong breeze 11 – 14 30 1,332 0 1,362Near gale 14 – 17 8 734 0 742

Gale 17 – 21 1 302 0 303Severe gale 21 – 24 0 21 0 21

Results (Table 5.9) show that most of the sample fell between light breeze (2 – 3 m/s)

strong breeze (11 – 14 m/s), with more multimodality predicted in the higher wind

speeds than observed. With Chilbolton lying in the temperate region, the small sample

size for the high wind speeds can be explained.

The study also looked at the effect of ground-measured temperature on multimodality,

with temperature classification given earlier. The results are shown in Table 5.10 below.

Table 5.19: Unimodal and multimodal distributions for both observed and predicted Nm

Temp (K)Observed Predicted

Unimode

Multimode

Unimode

Multimode

< 270 0 0 0 0270 – 275

0 58 29 29

275 – 280

46 3,384 271 3,159

280 – 285

220 8,608 770 8,058

285 – 290

395 5,268 458 5,205

290 – 31 424 26 429



300> 300 1 0 0 1

The result shows that the data exhibited more multimodality than unimodality, with

most of the temperature measured mostly between 275 K and 290 K (1.85 °C and

16.85 °C). Again, this is expected for a temperate region.

The fundamental result here is that multimodality does occur significantly often,

particularly at higher rain rates and increases with rain rates and wind speeds, and the

number of modes can be predicted by equation (3.18). The set of equations given in

Table 5.6 serves as a tool for the prediction of the basic parameters of the GMM model.

5.4 Conclusions Data captured at the Chilbolton Observatory from 2003 to 2009 were analysed, and

multimodality was established as present, and not just an instrumental artefact. Bins

were merged for better results and the effect of the loss of resolution due to the merge

minimised by the number of bins merged.

Since a large proportion (~50%) of the data did not fit into any of the standard

statistical distributions, this leads to the question of a search for what distribution would

best describe the captured data. This thesis attempted to fit the “orphaned” 50% with a

Gaussian Mixture Model, and the result arrived at showed an improvement on the

previous fits, hence a better model for the captured data. Different cluster sizes were

tried, and three (3) clusters gave the best result with 44% non-reject fits. However,

more work is needed in general to understand why the gamma and lognormal models

do not fit the data well, as well as a further exploration of the GMM and other models.

The results show that the number of modes increases with both the wind speeds and

rain rates, and a strong correlation (0.9) was found between the average number of

modes and rain rates as well as between the average number of modes and wind speeds.

The work goes on to define a model (equation (5.1)) determining the expected number

of modes, given the rain rate and the wind speed. Based on these, the parameters μ, σ

and weight of the GMM can be computed. It equally goes on to show the contributions



of the measured parameters of rain rates and wind speeds to the number of modes. This

shows that the rain rate contributes more than the wind speed in the determination of

the number of peaks.

Whilst the limitation of the measuring instrument is acknowledged, the effect of the

wind on the disdrometer is an issue that needs to be explored further. This study

eliminated drop diameters of less than 0.6 mm, as the drop counts for these sizes may

be unreliable.



Chapter 6 : SPECIFIC AND TOTAL RAIN

ATTENUATION



Chapter 6

SPECIFIC AND TOTAL RAIN ATTENUATION

6.1 IntroductionSpecific attenuation (measured in dB/km), as discussed in Section 2.8, is the attenuation

per unit distance, and this chiefly depends on the rain rate and the rainfall drop size

distribution, the temperature, shape, size and orientation of the raindrops as well as the

frequency, polarisation and the propagation direction of the incident wave. Signals

degrade more with higher frequencies as the wavelength to raindrop diameter ratio

reduces. This Chapter looks at the determination of the specific attenuation from the

various models studied in the previous Chapters, using the same data collected with the

disdrometer located at the Chilbolton Observatory.

On analyses, the rainfall DSD data sometimes suggests the presence of underlying

multimodal distribution. Standard statistical distributions like the lognormal and

gamma and the Gaussian Mixture Model (GMM) are used to model the rainfall data.

From these distributions, the specific attenuations were expressed as given in equation

(2.15) based on the theory as discussed in Section 2.8.

The later part of this Chapter equally draws from the work of previous Chapters. It

looks at the meteorological and satellite beacon signal measurements captured at the

Chilbolton Observatory in southern England. The meteorological instruments of

interest were; a ground-based radiometer, co-located rain gauges and a radiometer. The

satellite beacon signals of the Eutelsat Hotbird 13A satellite (formerly Hotbird 6) were

captured between 2010 and 2013.

This part of the Chapter investigates instantaneous rain attenuation aimed at providing

channel estimation to support adaptive fade mitigation strategies. It looks at the

determination of a zero dB reference template from captured data of both a sky-



pointing radiometer and the HotBird 13A satellite. These are analysed and rain

attenuation determined. Results are compared with the ITU-recommended attenuation

calculations for the location. How these attenuations vary with seasons, rain rates, wind

speeds and temperature are equally investigated.

6.2 Data Collection and ProcessingBuilding on the data collection and processing methods of Sections 4.4 and 5.2, this

Chapter looks at the data collected with the RD-69 JWD impact disdrometer, and a co-

located rain gauge from 2010 to 2013.

As in the previous Chapters, the ten-second samples collected with the disdrometer and

the rain gauge collected were merged into one-minute samples with the assumption that

the underlying distribution does not change radically over the one-minute period

considered. This study only considered rain events with rates greater than 0.1 mm/h,

with the drop velocities taken as in equation (3.2) and the rain rate (in mm/h) at time

instant t, (in seconds) was expressed as in equation (3.4).

The ITU-R P.838-3 suggested specific attenuation is the power relationship γ = aRb

(equation (2.7)), with R being the disdrometer-obtained rain rates (and results compared

to a co-located rain gauge) (equation (3.4)), and a and b given in equations (2.8) and

(2.9) respectively. Given the above, the a and b, as determined from the suggestions of

the stated in equation (2.7) were 0.0907 and 1.0229 respectively, and the melting layer

(or rain height) determined from the ITU R. P-839.4.

6.3 ProceduresGiven the doubt about the reliability of the smaller drops measurements [83, 84, 121],

especially in the RD-69 JWD impact disdrometer, this thesis considered only bins with

drop diameters greater than 0.6 mm. This consideration resulted in a sample of 93 bins,

with the boundaries determined by the ETH re-calibration given in McFarquhar and

List [80].



The method of Mätzler [123] was used to determine the extinction cross-section, Qt,

from the Mie coefficients at 19.7 GHz (the radio wave frequency), with the refractive

index of water taken as m = 6.867 + 2.630i [124].

For each one minute sample above the rainfall threshold, (taken here to be 0.1 mm/h as

before), the rain rate was computed from the disdrometer’s raindrop counts, and each

was compared to a co-located rain gauge to ensure accuracy.

As with previous Chapters, the spatial drop densities and the distributions (lognormal,

gamma (MoM), gamma (MLE) and the GMMs were fitted to each one minute spatial

drop density data sample, and the specific attenuation were computed for each of these

one-minute samples from each of the fitted models using equation (2.15), showing how

each drop diameter volume affects the signals. The results were compared with that

computed from the ITU-recommended specific attenuation calculations shown in

equation (2.7) using the disdrometer-obtained rain rates.

6.4 Results and Interpretations Based on the methodology described in Section 6.2, the specific attenuation over a 24-

hour period (4th of July, 2012) is shown in Fig. 6.1 and the rainfall DSD (shown in the

upper panel of Fig. 6.2) were used to calculate the specific attenuation for a one-minute

time slice. In fact, since b in equation (2.7) is 1.0229, the shape of the specific

attenuation graph matches the shape of the rain rate graph almost exactly.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (Hours)

0

0.5

1

1.5

2

2.5

dB/km

Specific Attenuation (dB/km)


Specific Attenuation

Fig. 6.28: Specific attenuation over a 24-hour period (4-Jul-2012)


The samples shown are representative of the dataset. The specific attenuations were

computed from the spatial drop densities, shown as bars in the lower panel of Fig. 6.2.

0.8 1 1.2 1.4 1.6 1.8 2 2.2Drop diams (mm)

0

1000

2000

3000

N(D

) (/m

m /m

3 )

04-Jul-2012/16:29

Drop densitiesLognormalGamma (MoM)Gamma (MLE)GMM: 4 mode(s)

0.8 1 1.2 1.4 1.6 1.8 2 2.2Drop diam (mm)

0

0.01

0.02

0.03

0.04

0.05

(D)

Disd Rainrate=15.737 mm/h; Wind speed=4.433 m/s; Temp=290.000 K; Drop Counts=1529 Sp Att (dB/km): logn=1.047; gamma (MoM)=0.982; gamma(MLE)=0.729; GMM=1.066; DropDens=0.988; ITU=1.603

Drop densitiesLognormalGamma (MoM)Gamma (MLE)GMM

The ITU estimates and the fitted distributions (lognormal, gamma (MoM), gamma

(MLE) and GMM), and the computed specific attenuations for 1st May, 2012 at 00:36,

4th July, 2012 at 16:29 and 23rd October, 2013 at 04:52 gave the results shown in Table

6.1 below.


Fig. 6.29: DSD and specific attenuation, with meteorological readings for 04-Jul-

2012@1629


Table 6.20: Summary of results for the specific attenuation

Date 01-May-12 04-Jul-12 23-Oct-13

Time 00:36 16:29 04:52

Drop count 490 1529 209

Disdrometer Rain Rate 21.10 15.74 2.26 mm/h

Wind speed 1.17 4.43 5.18 m/s

Temperature 283.60 290.00 286.43 K

Specific Attenuation

ITU 2.18 1.60 0.21 dB/km

Drop density 2.12 0.99 0.17 dB/km

Lognormal 2.16 1.05 0.12 dB/km

Gamma (MoM) 2.09 0.98 0.17 dB/km

Gamma (MLE) 2.09 0.73 0.16 dB/km

GMM 1.92 1.07 0.18 dB/km

From the above result in Table 6.1, it was observed that in sample 1 (1-May-

12@00:36), almost all the models give similar estimates of specific attenuation. In

sample 2 (4-Jul-12@16:29), the ITU estimate is 38% larger than that calculated directly

by aggregating the drop counts specific attenuations while in sample 3 (23-Oct-

13@04:52), the estimate from using a multimodal GMM is closer to the drop count

specific attenuation (out by 6%) compared to using ITU estimates (14%).

Given that each individual drop counts and densities are known, a further investigation

of the contribution of each bins to the one minute’s specific attenuation was carried out.

The specific attenuation component for the ith bin, γ (Di) is thus given as:

γ ( Di )=4.343× 103 ∙ N ( Di ) ∙Qt ( λ , m, Di ) ∙ ∆ Di (6.1)

where at a discrete time instant t, N(D) is the drop density in that channel (or bin). Qt is

the attenuation cross-section of a single rain drop as a function of the drop diameter Di,

with wavelength λ, and the refractive index of water as m, depending on both frequency



and temperature and Di is the central drop diameter of the ith channel, with bin width

ΔDi. The resulting contributions of each bin are shown in the lower panel of Fig. 6.2,

with the corresponding DSD shown in the upper panel.

The next task was to determine the typical contribution of each bin to the overall

specific rain attenuation to the one-minute time slice across the entire dataset. The

result, shown in Fig. 6.3 below shows the average contribution per bin over the entire

dataset.

1 1.5 2 2.5 3 3.5 4 4.5 5Bin diameter (mm)

0

1

2

3

4

5

6

7

(Di)

10-3 Average Bin Contributions

GMM (Predicted Nm)Drop densitiesGamma (MoM)LognormalGamma (MLE)

Results from the graph (Figs. 6.2 and 6.3) show that the medium-sized drops contribute

the most to the overall attenuation, as seen by the peak in the graphs. Given that the

larger drops are smaller in number, their contributions to the overall specific attenuation

seem to decrease with increasing drop size after the peak. The smaller drop sizes

contributions are equally noted as they do contribute to the overall specific rain

attenuation.

For each of the models under consideration, it was observed that the average

contribution peaks at the 51st of the 93 bins with a central diameter of 1.83 mm and bin

size of 0.0537 mm. The result here is as expected as the most contributions from the

middle-sized bins can be explained by the large number of medium-sized drops,

compared to the larger- and smaller-sized drops.


Bin Diameter (mm)

γ ( D i) ( d

Fig. 6.30: Average contribution of bins to the specific attenuation


The next investigation was how the specific attenuations compared with that calculated

using the ITU model for both unimodes (one mode distributions) and multimodes

(multiple modes distributions). Whilst Fig. 6.2 above show result for representative one

minute time slices, it is useful to know if these results will hold in the entire dataset.

Using the ITU model-obtained specific attenuation as the base reference in the

computation of the root mean square percentage error (rmspe) (used previously in

equation (5.2) of Chapter 5), the summary of the results for the entire dataset (2010 to

2013) is given in Table 6.2 below.

Table 6.21: RMSPE variation of specific attenuation from the ITU’s aRb specific attenuation

RMSPE (%) All Unimodes MultimodesDrop Counts 40.84 74.68 39.08

Lognormal 51.55 83.46 50.02Gamma (MoM) 41.67 75.62 39.91

Gamma (MLE) 58.25 93.06 56.60GMM 41.80 77.34 40.01

The results show that the lognormal and the gamma (using the method of maximum

likelihood estimates)-obtained specific attenuations fared badly, deviating by ~52% and

~58% respectively when all the samples were considered. The least deviation of

40.84% was from the specific attenuation computed from the direct drop counts data.

The gamma (using the method of moments) and the GMM-obtained specific

attenuation deviated by 41.67% and 41.80% respectively. A variance of 41.8 % for the

GMM was observed using the entire dataset, while a 77.34 % variance was observed in

the GMM for unimodal distributions and a 40.01% for multimodal distributions

respectively. The deviation was worse for the unimodes, with as high as 93.06%

deviation from the ITU model-obtained specific attenuation for the gamma (MLE)-

obtained specific attenuation.

The GMM performs better than the lognormal and gamma (MLE) and compares to the

results of the gamma (MoM). The expectation, however, was that the gamma (MLE)

would fit better than the lognormal or the gamma (MoM), but since the GMM is being

fitted to multimodal data, this seems to work better than the other unimodal



distributions. The expectation would be that the direct data would be more accurate

than modelled estimates, it was observed that the specific attenuation computed from

the actual drop counts were about the same as that of the GMM. Noting the

performance of the GMM compared to that of the Gamma (MoM) for multimodes, it is

fair to say that the results of the GMM compares favourably with the other results.

As it was done in Chapter 5, this thesis investigated how the specific attenuation varied

with multimodality across the seasons, rain regimes, wind speeds, and ground-

measured temperature ranges. This was investigated across the different models used in

this study, including the proposed GMM, where equation (5.1) is used to predict the

number of modes and comparing that to the observed number of modes, and the

resulting specific attenuation from both models. This also looked at the specific

attenuation computed from the ITU-recommended specific attenuation aRb model

(equation (2.7)) as well as the specific attenuation that is computed from the raw count

of the rain drops, without fitting any model. Table 6.3 shows the dB/km values obtained

for all the models discussed.

For Table 6.3, the results computed from the GMM models compared favourably with

each other. The difference between the models cannot be said to be much, and no

pattern emerged for the seasons. The biggest rmspe deviation of the GMM with

predicted number of modes from the ITU mean measurements was in winter, with 35.4

dB/km, while the smallest was in summer with a 26.5 dB/km.

Table 6.4 shows that the specific attenuation rose with increasing rain rates in all the

models considered. This is as expected, as specific attenuation is dependent on rain

rates. Result however show that the rmspe deviation of the GMM with predicted

number of modes from the ITU standard decreased with increasing rain rates, the

biggest deviation of 47.1 dB/km being in very light rain, while the smallest deviation of

3,3 dB/km was in heavy rain.

Table 6.5 shows that the mean specific attenuation increases with wind speed in all the

models looked at. This is in line with equation (5.1). No pattern was discerned. The



rmspe deviation of the GMM with predicted number of modes varied between 15.1

dB/km in severe gale to 35.4 dB/km in several wind classes.

No pattern was discerned from Table 6.6, suggesting that temperature may have very

little effect, if any on specific attenuation. Valid results for the deviation of the GMM

with a predicted number of modes varied from 25.7 dB/km to 70.7 dB/km.



Table 6.22: Specific attenuation across seasons

Season Period Mea

n γ(

t) (I

TU)

Max

γ(t

) (IT

U)

Mea

n γ(

t)

(GM

M-O

bser

ved)

Max

γ(t

) (G

MM

-Obs

erve

d)

Mea

n γ(

t)

(GM

M-P

redi

cted

)

Max

γ(t

) (G

MM

-Pre

dict

ed)

Mea

n γ(

t)

(Act

ual D

rop

Coun

ts)

Max

γ(t

) (A

ctua

l Dro

p Co

unts

)

RMSP

E of

GM

M-

Pred

icte

d fr

om IT

U

Winter Dec-Feb 0.008

3.624 0.005 3.393 0.004 3.386 0.004 3.501 35.4

Spring Mar-May 0.009

5.094 0.005 7.33 0.005 7.33 0.005 4.897 31.4

Summer

Jun-Aug 0.008

6.371 0.005 5.93 0.005 5.942 0.005 6.101 26.5

Autumn Sep-Nov 0.005

4.639 0.003 4.568 0.003 4.544 0.003 4.585 28.3



Table 6.23: Specific attenuation across different rain regimes

Rain

clas

s

Rain

rate

(m

m/h

)

Min

γ(t

) (IT

U)

Mea

n γ(

t) (I

TU)

Max

γ(t

) (IT

U)

Min

γ(t

) (G

MM

-Obs

)

Mea

n γ(

t)

(GM

M-O

bs)

Max

γ(t

) (GM

M-

Obs

)

Min

γ(t

) (G

MM

-Pre

)

Mea

n γ(

t)

(GM

M-P

re)

Max

γ(t

) (G

MM

-Pre

)

Min

γ(t

) (D

rop

Coun

ts)

Mea

n γ(

t)

(Dro

p Co

unts

)

Max

γ(t

) (D

rop

Coun

ts)

RMSP

E of

GM

M-P

redi

cted

V. light < 2 0 0.003 0.184 0 0.001 0.193 0 0.001 0.199 0 0.001 0.204 47.1

Light 02-10 0.184 0.342 0.996 0 0.255 0.997 0 0.248 1.093 0.025 0.251 1.001 18.8

Moderate

10-50 0.998 1.637 5.094 0.004 1.45 7.33 0.486 1.442 7.33 0.491 1.453 4.897 7.9

Heavy >50 5.839 6.105 6.371 4.981 5.455 5.93 5.066 5.504 5.942 5.543 5.822 6.101 3.3



Table 6.24: Specific attenuation across wind classes

Wind class

Wind speeds (m/s) M

ean

γ(t)

(IT

U)

Max

γ(t

) (IT

U)

Mea

n γ(

t)

(GM

M-O

bs)

Max

γ(t

) (G

MM

-Obs

)

Mea

n γ(

t)

(GM

M-P

re)

Max

γ(t

) (G

MM

-Pre

)

Mea

n γ(

t)

(Dro

p Co

unts

)M

ax γ

(t)

(Dro

p Co

unts

)RM

SPE

of

GMM

-

Calm < 1 0.002 1.823 0.001 1.717 0.001 1.627 0.001 1.676 35. 4

Light air 1 – 2 0.003 3.724 0.002 3.597 0.002 3.602 0.002 3.81 23.6

Light breeze 2 – 3 0.004 5.839 0.002 4.981 0.002 5.066 0.002 5.543 35. 4

Gentle breeze 3 – 5 0.006 5.094 0.003 4.838 0.003 4.838 0.003 4.897 35. 4

Moderate breeze

5 – 8 0.009 6.371 0.005 5.93 0.005 5.942 0.005 6.101 31.4

Fresh breeze 8 – 11 0.024 4.073 0.014 7.33 0.014 7.33 0.014 3.941 29.5

Strong breeze 11 – 14 0.047 3.008 0.028 2.761 0.027 2.632 0.028 2.808 28.6

Near gale 14 – 17 0.125 2.68 0.088 2.363 0.086 2.343 0.086 2.423 22.1

Gale 17 – 21 0.201 3.624 0.154 3.393 0.152 3.386 0.15 3.501 17.9

Severe gale > 21 0.219 0.395 0.172 0.331 0.178 0.331 0.171 0.33 15.5



Table 6.25: Specific attenuation across temperature

Temp. (K) M

ean

γ(t)

(ITU

)

Max

γ(t

) (IT

U)

Mea

n γ(

t)

(GM

M-O

bs)

Max

γ(t

) (G

MM

-Obs

)

Mea

n γ(

t)

(GM

M-P

re)

Max

γ(t

) (G

MM

-Pre

)

Mea

n γ(

t)

(Dro

p Co

unts

)

Max

γ(t

) (Dr

op

Coun

ts)

RMSP

E of

GM

M-P

re

< 270 0 0.001 0 0 0 0 0 0 -270-275

0.001 0.431 0 0.398 0 0.349 0 0.397 70.7

275-280

0.007 5.094 0.004 4.838 0.004 4.838 0.004 4.897 30.3

280-285

0.011 4.419 0.007 4.218 0.007 4.222 0.007 4.558 25.7

285-290

0.009 6.371 0.005 7.33 0.005 7.33 0.005 6.101 31.4

290-300

0.002 2.475 0.001 2.26 0.001 2.275 0.001 2.207 35. 4

> 300 0 0.143 0 0.097 0 0.113 0 0.172 -



6.5 Rain AttenuationThis section of the Chapter investigates the calculation of attenuation figured from

measurements taken from a beacon signal located at the Chilbolton Observatory as well

as the use of the sky noise temperature measurements from a radiometer located in the

same vicinity in helping determine a rain threshold for clear sky days.

6.5.1 Data and procedure6.5.1.1 Data collectionRaw beacon measurements at 19.7 GHz from Hotbird 13A were collected at the

Chilbolton Observatory with an elevation angle of 29.9. The slowly varying clear air

reference level or template of this satellite beacon signal was established with the help

of a Radiometrics MP1516A Water Vapour Profiler that measured sky brightness

temperature at 21 different frequencies ranging from 22 to 30 GHz, which traverses the

water vapour absorption line. The radiometer specifications are given in Table 6.7

below.

Table 6.26: Radiometer specifications [7]

Specifications

Bandwidth of each channel:300 MHz (The signal is integrated over a band spanning the range 10-160 MHz below and above the centre frequency.)

3dB beamwidth: varies from 6.3 degrees to 4.9 degrees across the full frequency range

Brightness temperature accuracy per channel: 0.5K - 1K

Measurement frequency: approximately 2 minutesAccuracy of integrated water vapour (IWV) retrieval: ~1 - 2 kg/m2

Accuracy of total liquid water path (LWP) retrieval: ~15% in non-precipitating conditions.

In all, 578 days of beacon signal data were captured for the period under consideration.



6.5.1.2 ProcedureThis thesis looks at two different methods (Method 1 and Method 2) for the

determination of instantaneous attenuation. Method 1 uses entire 24-hour segments,

while method 2 builds the 24-hour segments in 10-minute pieces.

Method 1: Rain attenuation time series using 24-hours clear day level

In the first method (Method 1), the basic steps are as follows:

1. Extract the total attenuation from the captured beacon signals;2. Determine the zero dB reference level template;3. Subtracting the beacon level from the zero dB reference template gives the total

attenuation;4. Subtracting the gaseous attenuation gives the excess attenuation;5. The rain attenuation is arrived at by subtracting the scintillation level (the

distortion caused by the signal passing through an imperfect medium) from the excess attenuation.

Step 1: Extracting the total attenuation from the captured beacon signals.

The Hotbird satellite broadcasts are tracked through a satellite receiver which is tuned

to the carrier frequency. The amplitude of the raw beacon signals (in voltage) are

logged every second (at 1 Hz) and saved in daily files. The voltage levels are

represented as hexadecimals. A sample fragment of the captured daily file for 4th July

2012 is shown in Fig. 6.4 below.



The data (Fig. 6.4) shows the header containing the control and identification in rows 1

to 6. The logged data begins in row 7 showing the date, month, hour, minute, and

second. The beacon amplitude level is shown inside the curly bracket, e.g. on the 4 th

July 2012 at 00:00:09, the reading was {ALF000700004F7-}. The A indicates that the

equipment is receiving, while the LF indicates frequency. The next eight characters,

00070000, show the frequency (in kHz), here 70,000 kHz or 70 MHz, the carrier

frequency. The voltage amplitude level is next shown in hexadecimal, here 4F7. The

final hyphen represents the alarm status of the tracking receiver. There are other

quantities captured, but these are of no interest to this thesis [125].

The largest possible signal amplitude level is hexadecimal FFF, giving the amplitude

range from 0 to 4095, with a tracking range of ±10 V, a 20 V DC output range. Using a

2 dB/V slope factor, the maximum possible output range is 2 V × 20 dB/V = 40 dB. If

the maximum decimal range is 4096, the dB value per decimal level is 40 dB/4096 =


Fig. 6.31: Beacon signal data


0.00976 dB, meaning that a ±1 DC change results in 0.00976 dB change in the received

signal. hereq

The tracking receiver output can be adjusted to 0 V for input power levels between -20

dBm and -60 dBm, this allows for the definition of the 0 V, depending on the

experimental requirements. The received power is thus computed as:

BdBm=RL−30+(BDC × 0.00976 dB) (6.2)

where:

BdBm = Received beacon signal level (in dBm);

RL = User dependent 0 V input reference power level (in dBm); (set to -60 dBm in this

thesis);

BDC = Received beacon signal Level in DC values (converted from hex to decimal) [125,

126], with a -30 conversion factor converting dB to dBm.

Do however note that even though the satellite data was logged per second, it was

necessary to convert it to one-minute values by determining the mean of sixty

successive readings. This was done to match the one-minute disdrometer-obtained rain

rates used in the previous Chapters.



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (Hours)

-78

-77.5

-77

-76.5

-76

-75.5

-75

-74.5

-74

-73.5

dBm

Beacon Signals and Rain Rates

27-Oct-2011Nearest clear day: 23-Oct-2011

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240

1

2

3

4

5

6

7

Rai

n R

ate

(mm

/h)

Fig. 6.5 shows the captured beacon signal levels (measured in dBm) for the current day

being processed (27th October 2011; shown in green) and the nearest rain-free (or clear

sky) day (23rd October 2011; shown in red) compared with the rain rates (measured in

mm/h) for the day under consideration (shown in black).

Step 2: Determining the zero dB reference template.

Given the diurnal movement of satellites due to telemetry, tracking, housekeeping,

power conservation and other factors, and its attendant effect on the variation of a

satellite signal’s attenuation, it is necessary to establish a daily zero dB reference

template (sometimes called the clear sky template) [126]. This not only gives a picture

of attenuation during clear skies, but it also helps to determine the attenuation that is

caused by other factors other than rain (gases, and other constituents of the

atmosphere). On a rain-free day, the attenuation encountered can be set as the zero dB

reference (or clear sky) template, which may be subtracted from that of a rainy day to

determine attenuation caused only by rain. The determination of rain-free days is best

done with the help of a radiometer, as its range of accuracy decreases with the amount


27-Oct-2011 Nearest clear day: 23-Oct-2011 Rain rate (mm/h)

dB m

Rai

n ra

tes

(mm

/h)

Time (Hours)

Fig. 6.32: Beacon signal data and rain rates


of precipitation in the atmosphere. The sky noise temperatures are collected for both

rainy days and rain-free days, and the measurements for rain-free days from the

disdrometer readings are used to determine the threshold of the sky noise temperature

for rain.

This thesis looks at a novel method of determining the zero dB reference level, by

fitting a ninth degree polynomial to the radiometer-obtained total attenuation based on

clear sky measurements. The method proposed here is easier as it involves lesser

computation.

Using the recommendations of ITU R. P 676-6 [24], the gaseous attenuation of both the

nearest clear sky day and the day under process were determined. Where the day under

consideration is a clear sky day, the data for the day is used. The ITU gaseous

attenuation determination process code downloaded from the ITU resources website

requires as input the frequency (19.7 GHz), the ground-measured temperature (between

-20° C and 40° C), the relative humidity, the elevation angle (29.9°), the consideration

of whether the rain or gaseous attenuation is desired and the water vapour or oxygen

option for the computation of the gaseous attenuation.

The clear day template is taken as the sum of the beacon signal and the gaseous

attenuation of the same clear day. Next, a fit of a ninth-order polynomial to the

computed data, and this fit is taken to be the zero dB reference template.

Steps 3 to 5:

These steps determine the rain attenuation time series. Based on the result of the zero

dB reference template arrived at in the previous step, the total attenuation is gotten by

subtracting the 24-hours extracted signal beacon level from the zero dB reference

template. Next subtract the 24-hours gaseous attenuation from the total attenuation to

get the excess attenuation. The scintillation level is last subtracted from the excess

attenuation to arrive at the rain attenuation or attenuation due to rain only.



However, do note that in this thesis, the scintillation level was not determined, hence

not subtracted from the excess attenuation. The averaging of sixty successive readings

of the beacon signals to achieve a one-minute interval is assumed to have excluded the

fast fluctuations due to scintillation. This is on the basis that averaging smoothes the

results. No further filtering is, therefore, necessary to exclude scintillation beyond this

one-minute averaging; hence the excess attenuation is considered here as the final step

in the derivation of the rain attenuation. Do however observe that the accuracy of this

has not been measured in this work, and may be determined in further works.

Method 2: Rain attenuation time series using ten-minute clear sky segments

As shown earlier, the determination of the zero dB reference level is best done with the

help of a radiometer, as its range of accuracy decreases with the amount of precipitation

in the atmosphere. This thesis looked at a novel method (referenced as Method 2) of

determining the zero dB reference level, by fitting a ninth-degree polynomial to the

radiometer-obtained total attenuation based on clear sky measurements.

The method for the determination of the rain attenuation time series is laid out below.

As in the previous method, if the day under consideration is a clear day, the 24-hours

sky noise temperature as the data was used for the determination of the zero dB

reference level.

The determination of the zero dB template is done by determining a sky noise

temperature threshold, where if the sky noise temperature exceeds the threshold, one is

bound to find rain. Equations (2.5) and (2.6) give the relationship in determining the

attenuation from measured sky noise temperature. Given that the determination of the

gaseous attenuation from ITU R. P 676-6 gives the attenuation due to only the gaseous

components of the atmosphere, it is reasoned that the maximum gaseous attenuation on

a clear day will be a safe threshold to determine the rain boundary. Based on this

reasoning, the maximum gaseous attenuation on the nearest clear sky day was set as the

attenuation threshold, while equations (2.5) and (2.6) were used to determine the clear

sky temperature threshold, Kcls.



Based on the determined threshold, Kcls, the process searched (above and below the

current date considered) for ten-minute segments of neighbouring days until a suitable

segment that has sky noise temperature less than the threshold was found. The segment

found will now form part of the clear day template. This is repeated for each ten-minute

segment for the entire twenty-four-hour period. The pieced-together segments will be

the zero dB clear sky template.

The final determination of the clear sky template is the fitting of a ninth order

polynomial to the zero dB clear sky template results.

This is slightly different from the methods of Ventouras, et al [25] (who used a Fourier

fit for the clear air template based on neighbouring rain-free days) and Stutzman, et al

[26] (who used a sixth-order polynomial fit for the clear air template based on the

nearest twenty-four-hour rain-free day). This was found to be simpler, but equally

appropriate as and results are comparable with the other methods. This method fits the

zero dB template to the sky noise temperature before converting to dB values, whereas

the methods mentioned fitted it to the dB values of the beacon signals.

Next compute the effective temperature, Teff, as given by equation (2.5), with the fitted

template taken as Kcls, and determine the total attenuation from equation (2.5) from the

measured sky noise temperature of the current day, Tsky, with the cosmic background

temperature taken as T0 = 2.7 K.

As in the previous method, the beacon signal was subtracted after conversion from the

sky noise temperature values using:

clearSkyTemplatedB=10 log10(clearSkyTemplateK

290+1) (6.3)

from the zero dB (clear day) template before removing the gaseous attenuation from the

total attenuation to arrive at the excess attenuation, and the rain attenuation is arrived at

when the scintillation level is subtracted (not done in this thesis as stated earlier) from

the excess attenuation.



6.6 Results and possible interpretationsRecall that Method 1 uses entire 24-hours rain-free day as the zero dB template while

Method 2 (the preferred method), uses a zero dB template built in ten-minute segments

from neighbouring clear sky equivalent time slots from the threshold determined using

the maximum gaseous attenuation. Fig. 6.6 shows a typical 24-hour period. The top

panel shows the rain rate (black – in mm/h), the gaseous attenuation for the 5 th

September 2011 and that of the nearest clear sky day (1st October 2011). The middle

panel shows zero dB template for both methods under consideration.

The red dotted line for Method 1 (24-hour clear air method) and the blue for Method 2

(24-hours template built from neighbouring 10-minute segments method with the help

of the radiometer). The bottom panel shows the rain attenuation computed from both

methods (blue for the Method 1 and green for the Method 2). Both methods appear to

produce a near-identical result for the date considered.

For Fig. 6.6, the ninth-order polynomial fitted with the help of MATLAB® function

fit for Method 2 given as (Table 6.8):

g ( x )=∑i=1

10

pi x10−i (6.4)

where the pi coefficients are:

Table 6.27: Ninth-order polynomial fit

Coefficient Value 95% confidence bounds

p1 2.679e-07 (1.895e-07, 3.464e-07)

p2 -2.783e-05 (-3.631e-05, -1.936e-05)

p3 0.001195 (0.0008099, 0.001579)

p4 -0.02732 (-0.03684, -0.0178)

p5 0.3579 (0.2183, 0.4975)

p6 -2.686 (-3.919, -1.454)

p7 10.93 (4.55, 17.31)

p8 -21.01 (-38.89, -3.136)



p9 15.49 (-7.503, 38.48)

p10 47.84 (38.24, 57.45)



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (hours)

0.6

0.7

0.8

0.9A

tten.

(dB

)

Gaseous Attenuation

Gaseous atten for: 05-Sep-2011Gaseous atten for nearest clear day: 01-Oct-2011

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240

10

20

Rai

n R

ate

(mm

/h)

Rain Rate (mm/h)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (hours)

-74.1

-74

-73.9

-73.8

-73.7

Atte

n. (d

B)

Clear sky template fit (dB) (Method 1): 01-Oct-2011

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

60

80

100

Sky

Tem

p (K

)

Clear sky template fit (K) (Method 2): 05-Sep-2011

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (hours)

0

5

10

Atte

n. (d

B)

Day's Rain Attenuation: 05-Sep-2011Atten from Mehod 1Atten from Method 2


Fig. 6.33: Gaseous attenuation, zero dB templates and attenuation for both beacon signal and radiometer-obtained data for 5th September 2011


As a test of the accuracy of the two methods, Fig. 6.7 below shows the cumulative

exceedance for the attenuation computed using the two methods (computed attenuation

using an entire 24-hour data (Method 1) and computed attenuation using the

radiometer data to establish ten minute clear segments (Method 2)) for the period 2010

to 2013.

The result shows that the attenuation computed from Method 2 (shown in blue) closely

matches that of the ITU’s expected attenuation levels (from European Space Agency’s

calculations, using parameters for this thesis’s experiment). The failure of the Method 1

to give as good a result as Method 2 may be attributed to the fact that the zero dB clear

air template is taken from the nearest 24 hour rain-free day, and this sometimes may be

too far away from the date considered, meaning that the satellite parameters may have

varied so much for the result to be useful. Whereas the Method 2 builds the zero dB

clear air template from ten-minute segments sourced from the nearest neighbouring

days.

As in previous Chapters, this thesis seeks to establish a pattern by looking at how the

parameters differ across seasons, rain regimes, wind speeds and temperature range.


Fig. 6.34: Cumulative Distribution for attenuations from both methods


On the investigation of the variation of attenuation across the seasons, with the seasons

defined as previously, the result shows that the highest mean attenuation was 0.5 dB

during autumn, with a corresponding maximum 17.8 dB. Spring had the least mean and

maximum attenuation, with 0.3 dB and 15.5 dB respectively. Detailed results across the

four seasons are presented in Table 6.9 below.

Table 6.28: Seasonal variation for attenuation

Season Period Mean A(t) (dB)

Max A(t) (dB)

Winter Dec - Feb 0.397 16.0Spring Mar - May 0.316 15.5Summer

Jun - Aug 0.488 16.4

Autumn Sep - Nov 0.537 17.8

Investigating the variation of attenuation with the rain rates, results show that as

expected, the attenuation tends to vary with the rain rates, with the mean attenuation

ranging from 0.42 dB for light rain to 3.86 dB for heavy rain. Data for the heavy

rainfall may have to be taken with caution as sample size may not have been enough to

draw a meaningful conclusion. Table 6.10 below presents the detailed attenuation

variation across the different rain classes, with the rain classes defined as in previous

Chapters.

Table 6.29: Variation for attenuation with rain rates

Rain classRain rate

(mm/h)

Mean A(t)(dB)

Max A(t)(dB)

V. light < 2 0.417 7.550Light 2-10 2.117 17.797

Moderate

10-50 3.682 15.364

Heavy > 50 3.861 5.871

The Table above suggests that attenuation is less during very light rain, with the most

attenuation expected as the rain rate increases. This is in line with expectations.



On the investigation of how attenuation varies with wind speeds, the mean attenuation

tends to generally increase with the wind speed, ranging from 0.4 dB for calm wind (W

< 1 m/s) to 2.7 dB for severe gale (W > 21 m/s). Detailed result is presented in Table

6.11 below.

Table 6.30: Variation for attenuation with wind speeds

Wind class Wind speeds(m/s)

Mean A(t)(dB)

Max A(t)(dB)

Calm < 1 0.372 4.260Light air 1 – 2 0.350 7.550

Light breeze 2 – 3 0.352 9.479Gentle breeze 3 – 5 0.419 14.961

Moderate breeze 5 – 8 0.496 17.797Fresh breeze 8 – 11 0.674 15.925

Strong breeze 11 – 14 1.015 15.364Near gale 14 – 17 2.056 15.041

Gale 17 – 21 2.961 12.396Severe gale > 21 2.729 4.372

The above Table suggests that attenuation tends to increase with wind speed, as the

mean attenuation shows a steady rise. As before, the data for severe gale (W > 21 m/s)

needs to be taken with caution due to the very little sample data available for a

meaningful conclusion to be arrived at. As high wind speeds go with high rain rates, the

results above are not surprising.

How does the ground-measured temperature vary with the computed attenuation? There

does not seem to be a clear pattern for either the mean or maximum attenuation as

shown by Table 6.12. Maximum attenuation ranged from 3.0 dB for temperatures

greater than 300 K to 17.8 dB for the temperature range of between 280 K and 285 K,

with corresponding mean temperatures of 0.43 dB and 0.52 dB respectively.



Table 6.31: Variation for attenuation with temperature

Temp (K) Mean A(t) Max A(t)

< 270 0.594 16.415270-275 0.170 7.305275-280 0.265 6.016280-285 0.524 17.797285-290 0.585 16.771290-300 0.467 7.062

> 300 0.434 3.025

Table 6.12 tends to suggest that temperature has very little role in the determination of

attenuation through free space. This, however, merits further investigation.

6.7 Rain AloftSince signals are affected by precipitation, and the rain aloft contributes to the total

attenuation, the challenge, as noted earlier, is that the signals slant path may encounter

precipitation as it travels between the satellite and the receiver along the Earth-space

path, even when the ground-based instruments may report no rain. This thesis looked at

this possibility, and attempts, based on available data to quantify the probability of this

scenario. Recall that the prediction of the fade level, A(t) is a function of R(t), it is also

useful to determine how probable that the A(t) prediction fails, given the R(t) was

recorded as zero.

Allowing for a 0.5 dB offset to cater for the zero dB template fit error, the data showed

that 35,806 (4.3 %) of the 832,320 minutes showed rain fade greater than the threshold

0.5 dB attenuation, even when the ground-based disdrometer reported 0 mm/h rain. Fig.

6.8 below clearly demonstrates this scenario.



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (Hours)

0

5

10

15

20

25

30

35Rain Rate (mm/h)Attenuation (dB)

Fig. 6.8 shows

that in the early hours of the day, the attenuation time series seem not to correspond

with the rain experienced by the ground-located instruments compared to the later

hours of the day. The possible explanation here is that the signals encountered rain

along the path of travel aloft, and this contributed to the higher than expected

attenuation, even when the ground-based instruments recorded no rain.

To ensure that the attenuation was due to rain and not just gaseous attenuation and

some other environmental factors, the attenuation threshold was set to 2 dB to ensure

significant fade. Result shows that even though 676,764 sample minutes (81.3 %) from

the 832,320 sample minutes reported no ground-measured rain, only 2,037 (0.2 %) of

these had attenuation above the 2 dB threshold.

To check how the proposed model performed, the probability of the attenuation

predicted from the model reporting zero attenuation based on a no rain record from the

ground-based instruments was investigated. Result shows that 799 samples minutes

(0.096 %) gave zero attenuation when R(t) = 0, even when the extracted attenuation

from the beacon signals reported greater than zero attenuation.


Fig. 6.35: Rain rates and rain attenuation for a 24-hour period


When the attenuation threshold was raised to 0.5 dB, allowing for errors in the zero dB

template fit, the result was 797 sample minutes (0.096%). To eliminate attenuations due

to gaseous and other atmospheric factors, the threshold raised to 2 dB gave just one

sample minute from the entire dataset, a fade of 7.93 dB on the 18th June, 2012 at 15:04.

While this goes on to show the reliability of the proposed model, there is however a

need for caution on the reliability of the results here. While the fade models rely on

ground-based point-measured rainfall, there is no accounting for the attenuation

contributed by rain aloft along the path travelled by the signals. Accounting for this

may involve the actual measurement of precipitation along the entire path, and this is

near impossible given the instruments used here. To overcome this will require

deploying measuring instruments at regular intervals along the signals’ path. What this

means is that the measurement here is essentially a path-averaged assumption of

precipitation along the signals slant path.

The application of these results in real-time FMTs will therefore have to recognise this

limitation, and build into the systems design the ability to mitigate this scenario when

encountered – allowance in the link budget for extra propagation power – however

small this may be to enable continuous provision of services to the consumer.

6.8 ConclusionsAccording to the Mie scattering model, it was noted that the medium-sized drops in the

DSD contribute the most to the overall attenuation of signals as it passes through the

Earth-space path. It also demonstrated that drop contributions peak in the 51st bin, and

larger drops contributions tend to decrease with increasing sizes, explained by the

decreasing number of drop counts as the drop size increases. Results equally suggest

that the Gaussian Mixture Model may sometimes model specific attenuation better than

unimodal statistical models, as the GMM gives a better fit to multimodality

encountered in rainfall DSD data.

While the dataset used in this study is small, there is a need to investigate further the

seasonal variation of the fit of the attenuation, as well as the variation of the specific



attenuation in different rain regimes, temperature ranges and maybe relative humidity

so as to establish the relationship between the quantities.

This Chapter also set out to obtain attenuation figures from meteorological and satellite

beacon data captured with instruments located at the Chilbolton observatory in southern

England. It has looked at the conversion of raw satellite DC signals to dBm and used

that to determine a zero dB reference level for both rainy days and clear sky days.

Gaseous attenuations were computed. The Chapter has also used radiometer data

captured in the same vicinity and uses these to determine clear sky days in a step to

obtain attenuation figures. These figures were compared with ITU-recommended long

term attenuation figures for the location.

The method of fitting a zero dB template from ten-minute clear sky segments sourced

from neighbouring days tend to agree more with ITU’s expected attenuation figures

more than using a 24-hour clear sky measurements. This may be explained as the

nearest rain-free day may be too far away from the date under consideration for it to be

useful.

The Chapter also investigated how these attenuation figures varied through the seasons,

how they varied across different rain classes, wind classes, and temperature ranges.

Results arrived at are validated, and are in line with ITU standards.

The Chapter concludes by looking at the possibility of encountering rain aloft even

when the ground equipment reports no rain. Adjustments were made for the fitting

errors of the zero dB clear day template, and the probability computed.



Chapter 7 : EFFECTIVE RAINY SLANT PATH LENGTH

MODELLING AND APPLICATION TO RAIN FADE

PREDICTION



Chapter 7

EFFECTIVE RAINY SLANT PATH LENGTH

MODELLING AND APPLICATION TO RAIN FADE

PREDICTION

7.1 Introduction

This Chapter, draws from the results arrived at in Chapter 6, presents results of the

modelling of an effective rainy slant path length, Leff, to allow future computation of

total instantaneous slant path rain attenuation A(t) as the product of Leff and a ground-

computed specific rain attenuation (t) (dB/km). Rain attenuation time series A(t)

averaged over one-minute intervals are extracted from measurements of the 19.701

GHz Eutelsat Hotbird 13A (formerly Hotbird 6) satellite beacon recorded at Chilbolton

Observatory in Southern England between 2010 and 2013 at a path elevation of 29.9

degrees.

Specific attenuation (t) at corresponding instants are computed using a theoretical

formulation that involves the attenuation cross-section of rain drops and their size

distributions at ground level. Both ground-based disdrometer-measured and standard-

modelled DSDs are employed and the results are validated by comparison to the ITU

aRb relation for at the beacon frequency. It goes on to investigate how this effective

rainy path length varies across seasons, rain regimes, wind speeds as well as

temperature ranges.

7.2 Effective rainy slant path length

As shown earlier in Section 2.9, the effective rainy slant path length Leff is defined as

the ratio between A(t) and (t) at each (one-minute) instant having ground-based rain-

gauge-verified rain rate R = R(t). Assuming a latitude-dependent fixed rain height, the



variability of instantaneous Leff(t) with R(t) is modelled, and the influence of prevailing

wind speed is investigated. The work also examines the variations in the correlation

between A(t) and (t) in rain events of various intensities.

Chapter 6 presented results for the computed specific and rain attenuations. There, the

specific attenuation was computed based on the ITU’s aRb model. Specific attenuations

were also computed based on the GMM using the observed number of modes as well as

the predicted number of modes, and finally, the specific attenuations were computed

based on the actual rainfall drop counts. Results there were comparable.

7.3 Results and InterpretationsThis thesis models the effective rainy path length, Leff, over time and attempts to

establish a relationship between the effective distance and the rain rates, R. It has tried

to model the effective slant path length, Leff, and looks at how this varies with both the

rain rates, R, and wind speeds, W. Determining the effective slant path length, the ratio

of A(t) with γ(t) gives an indication of the distance the signal of the particular frequency

and power may have to travel before being affected by precipitation.

Fig. 7.1 shows the variation of the effective rainy path distance over a 24-hour period,

here shown for 4th July 2012.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (Hours)

0

1

2

3

4

5

6

7

L eff (k

m)


Fig. 7.36: Effective rainy path length over a 24-hour period (4-Jul-2012)


Using data arrived at in Chapters 6 and 7 for the period April 2010 to December 2013,

where R(t) is taken to be the instantaneous rain rates (in mm/h), a power law, Leff = c Rm

fit (with c and m being constants) was made for the mean of the variation for the

distributions considered previously, including data from the actual drop counts, to

obtain a relationship between the quantities.

Determining the effective rainy path length using the specific attenuation computed

from the GMM with the predicted number of modes model, the result is presented in

Fig. 7.2.

5 10 15 20 25 30 35 40 45 50 55Rain Rates (mm/h)

2

4

6

8

10

12

14

L eff (k

m)

Mean Leff

Power Law Fit

The red dotted line which represents the fit of the blue scatter (the mean of the effective

rainy path length, Leff), is given as:

Leff =3.2102 R−0.9005 (7.1)

This was done for the different models under consideration, as well as for the drop

counts. Graph (Fig. 7.3) (and this is true for the other models not graphed) shows the fit

fails from about 25 mm/h of rain.


Fig. 7.37: Effective slant path length from GMM (Predicted Nm)’s specific attenuation


The resulting fits gave:

Table 7.32: Effective slant length model equations from different models

Leff from rain attenuation and specific attenuation from:

- ITU’s aRb model Leff = 2.6407 R-0.7557

- Gamma (MoM) Leff = 2.6299 R-0.9229

- Gamma (MLE) Leff = 2.2562 R-0.9101

- GMM (with observed Nm) Leff = 3.1905 R-0.8968

- GMM (with predicted Nm) Leff = 3.2102 R-0.9005

- Drop counts Leff = 3.2270 R-0.9134

The graph from Table 7.1 (Fig. 7.3) gives the resulting fits compared.

5 10 15 20 25 30 35 40 45 50 55 60Rain Rates (mm/h)

10-1

100

101

L eff (k

m)

ITUGamma (MoM)Gamma (MLE)GMM (Observed Nm)

GMM (Predicted Nm)

Drop Counts

Fig. 7.38: Effective slant path length variation with rain rates

All these imply that the effective rainy path length decreases with the intensity of

rainfall according to a power law. The Leff for the GMMs are much closer to that of the

actual drop counts than that calculated from the estimated ITU’s specific attenuation.



While there was a relationship between the effective rainy slant path length and the rain

rates, no discernible relationships were established between the effective slant path

length and the wind speed. There, however, was data for gale force winds (W > 20 m/s).

Fig. 7.4 shows the minimum, mean and maximum Leffs for wind speeds above 20 m/s.

20 20.5 21 21.5 22 22.5 23 23.5 24Wind Speed (m/s)

0

20

40

60

80

100

120

140

160

Effe

ctiv

e sl

ant d

ista

nce

(km

)

Effective Slant Path Length (ITU): 2010 to 2013

maximummeanminimum

This leads to the

preliminary conclusion that wind speed may not have a noticeable effect on effective

rainy path length, and the effect may only be noticeable at very high wind speeds. This

is consistent with the equation (5.1), which showed the little effect of the wind on the

number of modes presented in a multimodal DSD. No relationship was established

between the slant path length and temperature variation. There, however, is the need to

further explore the effect of wind speed and other parameters using a larger dataset.

The application of this work will be shown in how best the proposed model predicts

attenuation compared to the measured values. Since the effective rainy slant path

length, Leff, is the ratio of the rain attenuation, A(t) and the specific attenuation, γ(t),

computing the attenuation determined from the models is as follows: using the results

in Chapter 5, dividing the measured rain attenuation with the specific attenuations for

the different models gives the Leff for the model under consideration. Multiplying the


Fig. 7.39: Effective slant path length and wind speeds


Leff for the model with the ITU’s aRb specific attenuation gives us the estimated

attenuation for the model under consideration.

Leff model(t)=

Ameasu red (t )γ model (t)

(7.2)

and

Amodel ( t )=Leff model(t ) ∙ γ ITU (t) (7.3)

where:

Leff_model is the rainy slant path length for the model;

Ameasured is the measured attenuation;

γmodel is the model’s specific attenuation;

Amodel is the desired attenuation for the model; and

γITU is the ITU’s aRb specific attenuation.

It needs to be pointed out that a (inherent) condition for the validity of equation (7.2) is

that γmodel(t) > 0. And since the specific attenuation depends on the rain rate, a rain rate

of 0 mm/h (no rain) will give a specific attenuation of 0 dB/km, hence the effective

slant path is infinite as theoretically there is no rain to cause the signals to suffer

attenuation. In order not to have to resort to more sophisticated solutions for the

division by zero, the Leff corresponding to zero mm/h rain readings were all set to zero

km.

Using the above equations (7.2) and (7.3) for the computation of the attenuation based

on the desired model, Fig. 7.5 below shows the comparison of the measured attenuation

using the Method 2 discussed in Chapter 6, where the attenuation is computed using

clear air (zero dB) templates determined from the use of the radiometer for a fragment

of the twenty-four-hour period of 17th September, 2011, here a rain event, peaking at

about 13:06.



1300 1315 1330 1345 1400Time (Hours)

0

2

4

6

8

10

12

14

16

18

Atte

nuat

ion

(dB

)

GMM (Predicted Nm)

Measured

The red dashed line shows the measured attenuation, and the blue line shows the

computed attenuation using the GMM (with the predicted number of modes). The

results can be favourably compared.

To see how well the model varied from the measured attenuation in the overall data set,

the root mean square percentage error (rmspe) previously employed in Chapter 5

(equation (5.2)) was used. Using the measured attenuation as the base reference, the

results are presented in Table 7.2 below.

Table 7.33: Root mean square percentage errors for different models

Model RMSPE (%) Samples % of Total SampleLognormal 48.83 17,946 2.2Gamma (MoM) 40.68 18,366 2.2GMM (Observed Nm) 39.40 18,397 2.2GMM (Predicted Nm) 41.09 18,426 2.2Drop Counts 40.20 18,435 2.2


Fig. 7.40: Measured and model-obtained attenuations over a 1-hour period (17-Sep-

2011)


Given the undefined values returned as discussed above, the number of samples that

were used and the percentages of these from the overall sample of 832,320 minutes

from the 578 days’ useful data are shown in Table 7.2 above. Results above suggest a

variance of about 40% from the base reference for the GMM models. However, do

observe the closeness of the GMM variance values with that of the actual drop counts.

7.4 Application to Rain Fade PredictionThis thesis so far has looked at the measurement of the rainfall rate over time, R(t),

using a rain gauge and the derivation of the rainfall rate from the drop counts recorded

with a disdrometer, both averaged over one minute intervals and the disdrometer-

obtained rainfall rates are comparable to that measured with a co-located rain gauge.

Based on these, the work has modelled the rainfall drop counts data using standard

statistical distributions, as well as proposed the use of the GMM to cater for the

multimodality encountered in the data, which the unimodal distributions do not model

properly.

From these one-minute instantaneous rainfall rain rates, R(t), the instantaneous specific

attenuation, γ(t), based on the ITU’s aRb relation has been obtained. And from the

considered models’ DSDs, the γ(t) based on the integration involving the extinction

cross-section, Qt has also been obtained.

Using the γ(t) from these models, the instantaneous slant path attenuation, A(t), was

obtained as the product of γ(t) and Leff(t), the effective rainy slant path length, where the

Leff(t) is determined from the developed models as a function of the rain rates R(t).

A(t) in this case is the predicted slant path attenuation (shown in Fig. 7.6 below

compared to the beacon-measured attenuation).



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24Time (Hours)

0

2

4

6

8

10

12

14

16

18

Atte

nuat

ion

(dB)

GMM (Predicted Nm)

Measured

The cumulative distribution of the predicted attenuation for the entire dataset (2010 to

2013) was compared to ITU’s prediction for the location under consideration showed a

close match (Fig. 7.4). This thus shows that a statistical long-term fade prediction for

the proposed GMM model is equally reliable as the instantaneous prediction.

There is a need to clarify that prediction as used in this Chapter is the application of

rainfall rates, R(t), to determine or estimate or “predict” the current instantaneous fade,

A(t), as against the use of (immediate) past values, A(t – δt) to determine some current

A(t) or future A(t + δt).

Based on the discussion in Section 2.11, the proposed system is thus more adaptable to

the closed loop EIRP power control techniques, since the attenuation, predicted from

the ground-measured rain rate will be readily available, and not dependent on the

feedback from the receiving antenna.

7.5 Conclusions

This Chapter has investigated the effective rainy slant path length, Leff, the ratio of the

rain attenuation, A(t), and the specific attenuation, γ(t), and models presented with Leff

established it as a function of rain rates, R(t), as it decreases by a negative power law


Fig. 7.41: Predicted and measured attenuation for a 24-hour period


with R(t), when specific attenuation determined from different distributions, including

the actual drop counts and the ITU’s aRb model were used. It was found that rain rate

and effective rainy slant path length were strongly correlated, making some modelling

possible.

The Chapter also attempts to establish a relationship between Leff and wind speeds as

well as Leff and ground-measured temperature. No meaningful relationships were

arrived at, though it presented that only very high wind, gale forces and above (W(t) >

20 m/s) could have any effect on the effective rainy slant path length.

The Chapter also looked at how attenuation figures determined from the suggested

GMM models varied with measured values. Based on usable samples, it was discovered

that the attenuation determined from the GMM varied somewhat from the measured

attenuation, though maintaining a very close association with that determined from the

actual rainfall drop counts.



Chapter 8 : CONCLUSIONS



CHAPTER 8

CONCLUSIONS8.1 Introduction

In the search for the provision of better quality of service for the consumer, hence the

desire to design better fade mitigation techniques (FMTs), there is the need to have a

better prediction of precipitation intensity, since attenuation tends to increase with

precipitation in the Earth-space path.

8.2 Thesis Summary

This thesis has looked at the generation and propagation of electromagnetic signals in

the atmosphere of the Earth as well as the environment that the signal traverses and the

various factors that impede its transmission, chief amongst this being precipitation. The

relationship between the rainfall rain rates and attenuation is discussed.

This thesis looked at the physics of rainfall, analysing the formation of rainfall and

types, including the shape and sizes of raindrops as well as the velocity of rainfall

drops. The computation of rainfall rates was discussed, including the definition and the

reason for choosing of the rain classifications in this thesis.

Using several years’ data (2003 – 2013) captured by instruments (disdrometer, rain

gauges, radiometer) at the Chilbolton Observatory, southern England (51°08'42"N

1°26'18"W), this thesis models rainfall drop size distributions using standard classical

statistical distributions (lognormal and gamma), and determines how well the data

agrees with these models by doing goodness-of-fit tests using the Chi-square test.

Results show that although the method of moments for fitting both the lognormal and

the gamma distributions may produce reasonable rain rate predictions [46, 88, 105],

the empirically measured DSDs do not seem to be samples from either lognormal or



gamma distributions. The MLE method for fitting the gamma distribution appears to

give the best fit, confirming the theoretical prediction of Mallet and Barthes [98], and

more fits were found in spring and during light rain.

A search was made for a distribution that might fit the data better. This thesis has

proposed the Gaussian Mixture Model, a mixture of Gaussians, as a solution in the

modelling of rainfall DSD data, as this (unlike the unimodal distributions used earlier)

caters for the multimodality seen in the captured data.

By establishing the number of “modes” in the data, a suitable GMM was fitted. Results

in Chapter 4 show that this is a better fit for the data, as it modelled the multimodality

present in the data better than the unimodal models. To check that the multimodality

seen in the data is not just due to the error in the instrument, this thesis also analysed

data captured with a 2D Video Disdrometer (a more sophisticated disdrometer than the

JWD) located in Graz, Austria (47°4′N 15°26′E ) and it was confirmed that

multimodality could also be detected, furthermore establishing the presence of

multimodality in rainfall DSD data.

Based on the analysed Chilbolton data, an equation (equation (5.1)) was developed

predicting the number of modes in the rainfall DSD. This was found to be as reliable as

the visual determination of the number of modes (see Fig. 5.5). The equation, other

than predicting the expected number of modes, shows the contributions of both the rain

rate and the wind speed to the determination of the number of modes. It was determined

from this that the number of modes increases with increasing rain rates.

The contribution of individual drop sizes to the overall one-minute time-sliced specific

attenuation was investigated. Results show that the drops’ contributions peak at the 51st

bin. The larger drops’ contributions decrease with increasing drop sizes, explained by

the decreasing number of drop counts as the drop size increases. The larger drops do

however cause more attenuation so the decreased drop counts contributions to the

overall attenuation is balanced. As shown from the results in Chapter 6, the GMM is


https://tools.wmflabs.org/geohack/geohack.php?pagename=Graz&params=47_4_N_15_26_E_region:AT_type:city


also shown to be better when used to obtain the specific attenuation, as it describes the

multimodality found in the data better than the unimodal models.

The work looked at the use of radiometry in the measurement of sky noise temperature,

and the derivation of attenuation from these measurements. The attenuation present in

the satellite link measurements were determined from the collected beacon

measurements. This thesis utilised two different methods – the use of a 24-hour rain-

free day for the establishment of a clear day (zero dB) template and the use of 10-

minute segments from neighbouring equivalent clear period segments that is

determined by a rain threshold set with the use of the help of radiometer measurements

of sky noise temperatures. The result shows a close match for the second method with

the ITU-recommended long-term attenuation figures. The failure of the first method is

explained by the daily variations of the satellite, as the nearest clear sky day may be too

far away for the results to be useful.

The effective rainy slant path length, Leff was investigated, with the Leff determined from

the different models’ specific attenuation employed for the determination of the

attenuation. The Leff, was found to be a function of the rain rate and decreases

exponentially with the rain rates. The attenuation figures based on the GMM models

were comparable to those determined from the actual drop counts.

The work also looked at the possibility of rain being encountered aloft as the signal

passes along the Earth-space path, even when ground equipment report no rain. The

result shows that the chance of this happening is about 4.3%, based on analysed data.

The work has, in the final Chapter, looked at the application of the proposed

In summary, the following were highlighted by this thesis:

The standard classical models fail to model the rainfall DSD to 95% confidence.

This work went beyond the fitting of the model to the data by actually testing for

the goodness-of-fit of the model to the data using chi square test. The result

showed that this cannot be taken at 95 % confidence, as the unimodal models



failed as much as 50% of the time to fit the data. This led to the introduction of

the GMM;

The GMM is a better model for the multimodality encountered in the data. The

thesis fitted the data to the GMM, and results showed an improvement. This is to

be expected as the nature of the GMM enables it model multimodality

encountered in the data than the previously-used unimodal distributions;

The number of modes increases with rainfall intensity. Data and analyses, and

using correlation show that it is possible to actually predict the number of modes

in the DSD as a function of the rain rate and the wind speed. Result show that the

number of modes depends more on the rain rate and the wind speed contribution

to the number of modes is smaller;

The contributions of larger drops to the specific attenuation decreases with

increasing drop sizes because of the decreasing number of large drops with size

increase;

The effective rainy slant path length is related to the rain rate, and decreases with

a negative power law with rainfall intensity;

Precipitation aloft encountered by signals as they travel along the Earth-space

path contributes to the overall attenuation experienced, even when no

precipitation is experienced on the ground.

8.3 Work mapped against the initial objectives

The work had set out to achieve the following:

Aims: To carry out extensive and in-depth analysis of meteorological, satellite beacon

and radiometric measurements during rain in order to improve the modelling of the

rainfall drop size distribution, and apply these results to develop novel and improved

short-term prediction models for rain attenuation.

Major work breakdown were:

1. Analysis of disdrometer measurements at Chilbolton; modelling of DSD and

investigation of the variation of model parameters with time and the correlation of

any such variations with meteorological parameters such as wind speed and



direction; investigation of seasonal and (if possible, i.e. subject to availability of

tropical DSD measurements) climatic trends.

Outcome: Achieved.

2. Developing a novel model of DSD based on measurements in southern England.

This requires providing expressions for calculating model parameters as a

function of rain rate R (mm/h). The application of this model will be to allow

anyone to know the distribution of raindrops in a given rain of intensity (i.e. rain

rate) R in their (assumed temperate) locality. Note that for the model to be of

practical use, the only input parameters required to use the model should be ones

that are readily obtainable (e.g. rain rate and temperature, although the latter is

not expected to have any significant impact). To be more specific, a user should

not need disdrometer data in order to calculate model parameters.

Outcome: Achieved.

3. Use the model developed in (2) to compute specific rain attenuation. Purely for

comparison only, repeat this calculation of specific attenuation using actual

disdrometer data for N(D) recorded at Chilbolton between 2010 and 2013, and

using the lognormal and gamma DSD models. This thesis will require computing

the attenuation cross-section Q of rain drop of diameter D at the link frequency

(i.e. Eutelsat Hotbird 6 satellite beacon frequency) and ground-measured

atmospheric temperature and use this Q in a numerical integration involving

N(D).

Outcome: Achieved.

4. Pre-processing of the Hotbird 6 beacon data (recorded at Chilbolton between

2010 and 2013) to extract time series of rain attenuation along the path, averaged

over one-minute intervals in each rain event. Utilisation of a radiometer-based

algorithm for establishing a clear-air reference level for satellite beacon

attenuation measurements. Extraction of rain attenuation time series for the 3-

year dataset from Chilbolton. This will allow for the undertaking of a detailed

investigation into a number of relevant issues, including but not limited to the



following questions: How does Leff vary with rain rate R? What does this say

about the horizontal extent (i.e. cell diameter) of a rain cell of intensity R? How

does this Leff compare with that computed using ITU-R model? Comparing total

attenuation computed using different DSD models with measured beacon

attenuation in each one-minute rainy interval, what is the fit and what

improvements can be recommended or introduced into the attenuation

computation process?

Outcome: Achieved.

8.4 Limitations

Whilst the limitations of the instruments are acknowledged, the work would have been

much improved with the provision of a more diverse dataset. The acquisition of the 2D

Video Disdrometer data did help in the establishment of the presence of multimodality

in the data, and not just the instrumental error of JWD impact disdrometer. The use of

data from the more sophisticated 2DVD for the entire dataset may eliminate some of

the errors in the measurements.

The utilisation of radar data to check for rain aloft may have helped in the

determination and verification of the ground-measured rain rates. Here, it was

necessary to estimate drop speed to get spatial density of drops. A 2D Video

Disdrometer and radar would have avoided this.

The data, collected from one location may not give a true picture of the results arrived

at, hence may not be applicable in a different climatic region. Where data is available

from multiple sites in different climatic regions, more accurate and widely applicable

results would be reached.

8.5 Suggestions for further work

This report has shown that there is a clear need for further work with rainfall drop size

distributions and the derivation of specific and total attenuations, as well as the rainy

slant path length. Existing literature has done little testing of models for fit to the data.



Testing shows they do not fit particularly well. In particular, multimodal data has been

observed and tested. However, more work is needed in general to understand why the

gamma and lognormal models do not fit the data well, as well as a further exploration

and refinement of the GMM and other models.

Other models to be explored could include other mixture models (like the Weibull

distribution [127]) and not just a mixture of the Gaussians so as to have a better model

of the data. Other non-parametric models like neural networks (see [128]) may be

considered.

Based on the single site data, it may be difficult to draw the conclusion that the results

arrived at here will be valid for other locations, as earlier shown that different climatic

regions may need different models [94, 102, 103].

With the limitation of the measuring instrument in mind, the effect of the wind on the

disdrometer is an issue that needs to be explored further. This study eliminated drop

diameters of less than 0.6 mm, as the drop counts for these sizes may be unreliable.

This thesis, whilst it tried to establish relationships with the various quantities

considered can be expanded by exploring the effects of other quantities like

atmospheric pressure, relative humidity, solar radiation and wind direction. Some

careful analyses may reveal relationships between these and rain attenuation as well as

other quantities considered in this thesis, as well as how they contribute to the total

attenuation.

The inclusion of radar data may help in the study of rain cell dynamics; these will apply

in the measurement of the spatial and temporal variation of rain rate along a slant path,

and to the study of the structure and movement of rain cells. The retrieval of the DSD

using the radar can be investigated more (see [129]).

The disdrometer and rain gauges measurements are limited as they only measure rain in

the ground and within the immediate vicinity. In the absence of the radar data, if there



were a network of these instruments along the signals path, this would have helped in

ensuring a better and more accurate determination of the attenuation as the rain cells

rain rate variations would have been taken into consideration. These may then be

compared to the results determined from this thesis.

Equations (7.2) and (7.3) effectively scales the measured attenuation, Ameasured, as the

ratio of the ITU’s aRb specific attenuation, γITU and the model-obtained specific

attenuation, γmodel. This means that Amodel will be the same as Ameasured if γITU equals γmodel.

There is a need to further investigate this relationship, and determine the results

statistically when Ameasured is scaled up or down compared to other measured quantities

like the rain rate or wind speed.

This thesis may be applied and extended by using the Leff in the investigation and

estimation of rain cell movement (speed and direction) as a function of ground-

measured wind speed and direction, and maybe a possible measure of wind

measurements aloft. This may contribute to the understanding of the evolution of

rainfall distribution over time.

Based on the attenuation results arrived here, the investigation of the rain fade slope can

be undertaken to develop more reliable FMT strategies by developing rain fade slope

and rain fade duration models. Appendix VII gives a little insight into the fade data,

here over a 24-hour period.

There is also a suggestion to carry out simulation-based assessment of performance

improvement in proactive versus reactive FMT systems, where the proactive system

employs the short-term fade prediction method developed. There is equally room for

the development of an improved neural network-based algorithm for short-term fade

prediction.

Whilst this thesis uses data from just one location, the Chilbolton Observatory in

southern England, a temperate location, there is need to equally undertake similar

experiments in other climatic regions with the aim of comparing results. Even in the



same climatic region, there sometimes arises multi-site implementation of satellite

links. There is, therefore, the possibility of the collection of data from different sites so

as to optimise parameters involved in site diversity and time diversity applications.

The disdrometer and the rain gauge used in this thesis recorded data at 10 second

intervals. This thesis averaged six consecutive data samples to achieve a one minute (60

seconds) time series. This was done to achieve a smaller sample size, and this was

considered appropriate as the considered meteorological dynamics do not change that

much in one minute. A longer period may have yielded too coarse a data resolution for

it to be useful and reliable in the statistical analyses.

However, in some situations, especially where telecommunication service providers

may need to adjust power or other communication resource either proactively or

reactively based on prevailing conditions, a one minute interval may be considered too

long. There is the need to consider a shorter time duration, say 10 seconds (or even 1

second, depending on the capability of the equipment).

Using the results arrived at here, there is the need for the application of these in the

study and understanding of fade dynamics (fade durations, fade depth, interfade

durations, fade slopes) with a view to designing FMT models based on instantaneous

fade predictions.



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Appendix I

Drop velocities

Diameter (mm)

Fall speed(m/s)

Diameter(mm)

Fall speed(m/s)

0.10 0.27 2.60 7.57

0.20 0.72 2.80 7.82

0.30 1.17 3.00 8.06

0.40 1.62 3.20 8.26

0.50 2.06 3.40 8.44

0.60 2.47 3.60 8.60

0.70 2.87 3.80 8.72

0.80 3.27 4.00 8.83

0.90 3.67 4.20 8.92

1.00 4.03 4.40 8.98

1.20 4.64 4.60 9.03

1.40 5.17 4.80 9.07

1.60 5.65 5.00 9.09

1.80 6.09 5.20 9.12

2.00 6.49 5.40 9.14

2.20 6.90 5.60 9.16

2.40 7.27 5.80 9.17



Appendix IIETH Diameters, Di (mm)

Bin # Di Bin # Di Bin # Di Bin # Di

Bin #1 0.3130 Bin #33 0.6014 Bin #65 1.2877 Bin #97 2.5983Bin #2 0.3183 Bin #34 0.6160 Bin #66 1.3169 Bin #98 2.6558Bin #3 0.3240 Bin #35 0.6316 Bin #67 1.3450 Bin #99 2.7122Bin #4 0.3303 Bin #36 0.6484 Bin #68 1.3727 Bin #100 2.7693Bin #5 0.3357 Bin #37 0.6659 Bin #69 1.4003 Bin #101 2.8263Bin #6 0.3400 Bin #38 0.6839 Bin #70 1.4272 Bin #102 2.8838Bin #7 0.3472 Bin #39 0.7029 Bin #71 1.4548 Bin #103 2.9433Bin #8 0.3553 Bin #40 0.7226 Bin #72 1.4815 Bin #104 3.0089Bin #9 0.3620 Bin #41 0.7424 Bin #73 1.5090 Bin #105 3.0787Bin #10 0.3690 Bin #42 0.7618 Bin #74 1.5336 Bin #106 3.1505Bin #11 0.3762 Bin #43 0.7813 Bin #75 1.5589 Bin #107 3.2269Bin #12 0.3836 Bin #44 0.7997 Bin #76 1.5823 Bin #108 3.3082Bin #13 0.3907 Bin #45 0.8183 Bin #77 1.6068 Bin #109 3.3891Bin #14 0.3981 Bin #46 0.8360 Bin #78 1.6321 Bin #110 3.4709Bin #15 0.4080 Bin #47 0.8539 Bin #79 1.6568 Bin #111 3.5499Bin #16 0.4180 Bin #48 0.8714 Bin #80 1.6825 Bin #112 3.6284Bin #17 0.4279 Bin #49 0.8886 Bin #81 1.7134 Bin #113 3.7075Bin #18 0.4381 Bin #50 0.9053 Bin #82 1.7460 Bin #114 3.7857Bin #19 0.4488 Bin #51 0.9219 Bin #83 1.7878 Bin #115 3.8642Bin #20 0.4585 Bin #52 0.9389 Bin #84 1.8330 Bin #116 3.9448Bin #21 0.4686 Bin #53 0.9560 Bin #85 1.8867 Bin #117 4.0278Bin #22 0.4796 Bin #54 0.9747 Bin #86 1.9427 Bin #118 4.1236Bin #23 0.4898 Bin #55 0.9960 Bin #87 2.0025 Bin #119 4.2242Bin #24 0.5000 Bin #56 1.0198 Bin #88 2.0635 Bin #120 4.3317Bin #25 0.5106 Bin #57 1.0455 Bin #89 2.1242 Bin #121 4.4456Bin #26 0.5216 Bin #58 1.0761 Bin #90 2.1852 Bin #122 4.5634Bin #27 0.5315 Bin #59 1.1053 Bin #91 2.2466 Bin #123 4.6770Bin #28 0.5419 Bin #60 1.1357 Bin #92 2.3070 Bin #124 4.7922Bin #29 0.5529 Bin #61 1.1671 Bin #93 2.3655 Bin #125 4.9055Bin #30 0.5635 Bin #62 1.1998 Bin #94 2.4245 Bin #126 5.0190Bin #31 0.5751 Bin #63 1.2290 Bin #95 2.4825 Bin #127 5.1340Bin #32 0.5875 Bin #64 1.2594 Bin #96 2.5397



Appendix III

Seasonal variation of DSD fits (MFR/ETH_ALL/ETH_TRUNC)

Season Lognormal (%) Gamma (MoM) (%)

Gamma (MLE) (%) GMM (%)

Spring 33/31/33 22/21/27 48/53/60 51/54/50

Summe

r

24/22/26 15/14/20 36/38/47 47/46/46

Autumn 30/29/31 19/18/23 43/47/53 50/52/47

Winter 33/31/34 22/21/27 50/54/61 53/56/51

Overall 30/29/31 20/19/25 45/49/56 50/52/49



Appendix IVRain rate variation of DSD fits (MFR/ETH_ALL/ETH_TRUNC)

Rain type Rain rate(mm/h) Lognormal (%) Gamma

(MoM) (%)Gamma

(MLE) (%) GMM (%)

Light < 2 36/34/36 24/21/27 51/54/59 52/53/49

Moderate 2 – 10 16/19/23 10/14/20 32/40/52 48/53/51

Heavy 10 – 50 6/12/13 4/9/11 10/21/27 33/31/25

V. Heavy > 50 8/17/16 1/6/6 10/24/24 39/37/25

Appendix VWind speed variation of DSD fits (MFR/ETH_ALL/ETH_TRUNC)

Wind class Wind speeds Lognormal (%) Gamma (MoM)

(%)Gamma (MLE)

(%) GMM (%)

Calm < 1 m/s 20/17/24 14/11/17 32/33/46 40/40/43

Light air 1 – 2 m/s 22/20/26 15/12/19 36/38/50 46/45/46

Light breeze 2 – 3 m/s 20/16/23 14/11/18 38/39/50 46/45/47

Gentle breeze 3 – 5 m/s 22/18/77 15/13/19 40/42/50 48/47/46

Moderate breeze

5 – 8 m/s 29/27/29 20/18/24 47/51/57 52/55/49

Fresh breeze 8 – 11 m/s 42/43/41 26/26/31 51/58/62 56/61/54

Strong breeze 11 – 14 m/s 55/60/58 32/36/40 56/65/68 52/60/56

Near gale 14 – 17 m/s 58/70/69 36/49/53 56/70/71 41/45/43

Gale 17 – 21 m/s 58/76/78 35/57/58 54/77/77 33/28/31

Severe gale 21 – 24 m/s 71/100/100 71/100/100 86/100/100 29/17/14



Appendix VI

Sky Temperature (K)

Date Rain No Rain All Day

Max Min Mean Max Min Mean Max Min Mean01-Jul-12 300.28 299.51 299.90 300.80 293.19 297.59 300.80 293.19 297.6302-Jul-12 298.59 296.14 297.63 298.59 296.14 297.55 298.59 296.14 297.5503-Jul-12 300.90 297.91 298.74 300.89 297.91 298.92 300.90 297.91 298.9104-Jul-12 304.07 298.17 299.84 304.09 297.22 300.13 304.09 297.22 300.1805-Jul-12 297.74 297.58 297.68 305.91 296.93 301.33 305.91 296.93 301.3006-Jul-12 301.14 296.21 297.44 301.28 296.17 297.65 301.28 296.17 297.6707-Jul-12 299.23 296.27 297.33 299.23 296.27 297.43 299.23 296.27 297.4108-Jul-12 303.93 297.15 298.43 305.15 297.15 300.49 305.15 297.15 300.4609-Jul-12 302.80 298.16 300.20 302.80 298.10 299.68 302.80 298.10 299.7110-Jul-12 302.51 297.59 299.65 302.80 295.58 298.55 302.80 295.58 298.5911-Jul-12 301.81 297.62 298.92 303.83 294.67 298.70 303.83 294.67 298.7112-Jul-12 301.72 295.41 296.85 301.72 293.73 297.02 301.72 293.73 296.9413-Jul-12 302.55 296.70 298.61 303.51 296.70 299.49 303.51 296.70 299.3314-Jul-12 300.09 296.89 298.08 300.08 295.73 297.97 300.09 295.73 298.0015-Jul-12 - - - 301.96 295.63 298.35 301.96 295.63 298.3516-Jul-12 299.86 296.69 298.24 299.86 296.56 298.34 299.86 296.56 298.3417-Jul-12 305.13 300.92 303.01 306.69 297.58 301.44 306.69 297.58 301.5018-Jul-12 302.60 299.19 300.29 303.34 297.89 299.64 303.34 297.89 299.6919-Jul-12 300.90 298.20 299.18 303.75 295.22 299.51 303.75 295.22 299.5120-Jul-12 - - - 303.75 295.39 299.59 303.75 295.39 299.5921-Jul-12 - - - 305.51 295.88 299.93 305.51 295.88 299.9322-Jul-12 - - - 307.23 295.22 301.33 307.23 295.22 301.3323-Jul-12 - - - 310.79 294.41 302.96 310.79 294.41 302.9624-Jul-12 - - - 312.73 295.49 304.66 312.73 295.49 304.6625-Jul-12 - - - 313.68 296.83 305.91 313.68 296.83 305.9126-Jul-12 - - - 313.40 297.46 305.38 313.40 297.46 305.3827-Jul-12 302.59 302.25 302.50 309.37 299.14 303.95 309.37 299.14 303.9528-Jul-12 - - - 304.16 294.70 299.67 304.16 294.70 299.6729-Jul-12 302.61 299.46 300.52 302.73 293.52 297.95 302.73 293.52 298.0130-Jul-12 302.46 297.08 297.71 303.94 293.12 298.81 303.94 293.12 298.7231-Jul-12 302.73 295.57 297.64 304.30 295.57 300.18 304.30 295.57 300.05



Appendix VIIOutages at 2 dB

Outage Starts Outage Ends Duration (mins)01-Jul-2012/10:54 01-Jul-2012/10:59 501-Jul-2012/12:14 01-Jul-2012/12:20 603-Jul-2012/23:54 04-Jul-2012/00:00 604-Jul-2012/16:20 04-Jul-2012/16:42 2204-Jul-2012/16:44 04-Jul-2012/17:03 1906-Jul-2012/11:34 06-Jul-2012/11:36 207-Jul-2012/04:39 07-Jul-2012/04:46 707-Jul-2012/06:24 07-Jul-2012/06:27 307-Jul-2012/19:15 07-Jul-2012/19:17 208-Jul-2012/06:26 08-Jul-2012/06:31 508-Jul-2012/11:01 08-Jul-2012/11:06 508-Jul-2012/12:59 08-Jul-2012/13:08 908-Jul-2012/13:15 08-Jul-2012/13:18 310-Jul-2012/10:00 10-Jul-2012/10:01 110-Jul-2012/12:21 10-Jul-2012/12:30 910-Jul-2012/15:51 10-Jul-2012/15:55 411-Jul-2012/07:55 11-Jul-2012/07:59 411-Jul-2012/08:35 11-Jul-2012/08:36 113-Jul-2012/12:43 13-Jul-2012/12:44 113-Jul-2012/13:48 13-Jul-2012/13:51 313-Jul-2012/17:50 13-Jul-2012/17:54 413-Jul-2012/18:39 13-Jul-2012/18:41 213-Jul-2012/18:49 13-Jul-2012/18:56 714-Jul-2012/09:15 14-Jul-2012/09:22 714-Jul-2012/09:33 14-Jul-2012/09:59 2614-Jul-2012/10:10 14-Jul-2012/10:13 314-Jul-2012/11:28 14-Jul-2012/11:41 1315-Jul-2012/14:46 15-Jul-2012/14:47 118-Jul-2012/17:02 18-Jul-2012/17:03 118-Jul-2012/17:05 18-Jul-2012/17:07 218-Jul-2012/17:14 18-Jul-2012/17:19 518-Jul-2012/17:24 18-Jul-2012/17:25 129-Jul-2012/08:41 29-Jul-2012/08:46 529-Jul-2012/11:41 29-Jul-2012/11:44 3


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