geometry-free analysis approaches for noises and … · computational algorithms of hardware delays...

GEOMETRY-FREE ANALYSIS

APPROACHES FOR NOISES AND

HARDWARE BIASES IN TRIPLE-

FREQUENCY GNSS SIGNALS

Yongchao Wang

B.E., Hebei University of Technology, China, 2002

M.E., Beijing University of Aeronautics and Astronautics, China, 2006

A dissertation submitted

in fulfilment of the requirements for the degree of

Doctor of Philosophy

School of Electrical Engineering and Computer Science

Science and Engineering Faculty

Queensland University of Technology

2017

i

Keywords

GNSS, Geometry Free, Ionosphere Free, Stochastic Analysis, Variance,

Covariance, Uncalibrated Signal Delay, Triple Frequency Signals

ii

Abstract

This PhD work deals with the two fundamental issues in Global Navigation

Satellite System (GNSS) data processing of triple frequency signals: models and

computational algorithms of hardware delays (biases) and covariance matrices of code

and phase measurements. The investigation efforts and contributions are made in

threefold.

Regarding the GNSS observational models, we systematically examine various

linear combinations, and classify all combinations into four types: Geometry-

Free/Ionosphere-Free (GFIF) combinations, Geometry-Free/Ionosphere-Present

(GFIP) combinations, Geometry-Based/Ionosphere-Free (GBIF) combinations and

Geometry-Based/Ionosphere-Present (GBIP) combinations. The complete set of

parameters in the original code and phase equations can be estimated with a set of

combinations that are equivalent to the original equations. However, the whole

network-based computation problem can be effectively decomposed into geometry-

free computing and geometry-based computing problems. The geometry-free

computing deals with the integer ambiguities, ionosphere delay, and the satellite- and

receiver-specific Uncalibrated Signal Delays (USDs). The geometry-based computing

deals with the estimation of satellite and receiver states, including troposphere delays

for each receiver.

We use the geometry-free models for computing the covariance matrices of

undifferenced triple-frequency GNSS measurements and analysing their impact on

positioning solutions. Four independent GFIF models formed from original triple-

frequency code and phase signals allow for effective computation of variance-

covariance matrices using real data. Variance Component Estimation (VCE)

algorithms are implemented to obtain the covariance matrices for three pseudorange

and three carrier-phase signals epoch-by-epoch. Covariance results from the triple

frequency BeiDou Navigation Satellite System (BDS) (C02, C06 and C14) and Global

Positioning System (GPS) (G01) data sets demonstrate that the estimated standard

deviation varies in consistence with the amplitude of GFIF error time series. The

single-point positioning results from GPS ionosphere-free measurements from 95

Multi-GNSS Experiment (MGEX) stations demonstrate an average improvement of

iii

about 10% relative to the results from a traditional elevation weighting model in terms

of root mean square statistics. This finding provides a preliminary confirmation that

adequate consideration of the variation of covariance lead to the improvement of

GNSS state solutions.

For the estimation of hardware delays of GNSS signals, we propose an approach

that rigorously decomposes the whole network-based GNSS computing problem into

Integer Ambiguity Resolution (IAR), Geometry-Free (GF) and Geometry-Based (GB)

processing problems. The IAR makes use of three GFIF models and three GBIF

models to determine the DD integers for all baselines and carriers. The GF processing

determines ionosphere-delays and phase and code hardware delays sequentially, using

five independent combinations from the GFIF and GFIP categories. Appending the

necessary Undifferenced (UD) and Single-Differenced (SD) integers as the integer

datum setting, Double-Differenced (DD) integers can be mapped to all line-of-sights.

The slant ionosphere-delays can also be obtained from the integer-fixed Line of Sight

(LOS) phase measurements in alignment with the integer datum settings. The hardware

delay term in the GBIF observable is set to zero as a boundary condition, so that the

hardware delays in six original code and phase signals can be obtained. The GB

computing determines the satellite and receiver states with the integer-fixed LOS

GBIF measurements. The six selected GFIF, GFIP and GBIF combinations are

equivalent to six original code and phase signals. After all the original code and phase

measurements are corrected for frequency-dependent quantities, they all have the

exactly same functional models for satellite and receiver states, but possess difference

noise terms. Experimental results from the data sets from the nine receivers that track

GPS PRN24 and PRN25, PRN06 and PRN09 over two 3.3-hour periods are analysed.

Eight DD narrow-lane ambiguities are fixed to their integers by integer rounding

within 400 independent samples. The results show that the satellite-specific hardware

delay estimates for original code and phase signals are determined to the precision of

a few centimetres and a few millimetres, respectively. The precision of slant

ionosphere-delay solutions purely depends on the phase noises, promising higher

satellite and receiver state estimation precision in the network-based processing and

single-receiver positioning.

iv

Table of Contents

Keywords .................................................................................................................................. i

Abstract .................................................................................................................................... ii

Table of Contents .................................................................................................................... iv

List of Figures ........................................................................................................................ vii

List of Tables ........................................................................................................................... ix

List of Abbreviations ................................................................................................................ x

Statement of Original Authorship ......................................................................................... xvi

Acknowledgements .............................................................................................................. xvii

Chapter 1: Introduction .................................................................................... 19

1.1 Research Background .................................................................................................. 19 1.1.1 Global Navigation Satellite Systems ................................................................. 19 1.1.2 GNSS Triple Frequency Signals ........................................................................ 23

1.2 GNSS Data Processing and Challenges ....................................................................... 24 1.2.1 GNSS Data Processing Modes .......................................................................... 24 1.2.2 Challenges in Stochastic Modelling and Biases Estimation .............................. 25

1.3 Objectives and Contributions ....................................................................................... 33 1.3.1 Research Objectives ........................................................................................... 33 1.3.2 Contributions ..................................................................................................... 34

1.4 Thesis Outlines ............................................................................................................. 36

Chapter 2: Literature Review ........................................................................... 39

2.1 Stochastic Modelling of GNSS Observations .............................................................. 39 2.1.1 Background ........................................................................................................ 39 2.1.2 Empirical Models ............................................................................................... 44 2.1.3 Measurement Based Stochastic Analysis ........................................................... 52

2.2 Differential Code Biases Estimation and Application ................................................. 58 2.2.1 DCB Definition and Correction ......................................................................... 59 2.2.2 DCB Estimation Methods .................................................................................. 66

2.3 Uncalibrated Phase Delay Estimation and Application ............................................... 80 2.3.1 UPD Definition and Correction ......................................................................... 80 2.3.2 UPD Estimation Methods .................................................................................. 81

2.4 Inter-System Bias among GNSS .................................................................................. 86 2.4.1 ISB Definition and Correction ........................................................................... 86 2.4.2 ISB Estimation Methods .................................................................................... 86

2.5 Summary and Implications .......................................................................................... 89

Chapter 3: Fundamentals of GNSS Observation Linear Combination ........ 91

3.1 Introduction .................................................................................................................. 91

3.2 Basic Equations ............................................................................................................ 92 3.2.1 Fundamental Equations ...................................................................................... 92 3.2.2 Delay/Bias and Noise Propagation .................................................................... 96

v

3.3 Four Types of Linear Combinations .............................................................................99 3.3.1 Geometry-Free and Ionosphere-Free Linear Combinations .............................100 3.3.2 Geometry-Free and Ionosphere-Present Linear Combinations ........................102 3.3.3 Geometry-Based and Ionosphere-Free Linear Combinations ..........................103 3.3.4 Geometry-Based and Ionosphere-Present Linear Combinations ......................104

3.4 Application Cases of All Type of Questions ..............................................................105

3.5 Summary .....................................................................................................................109

Chapter 4: Geometry-Free Approach for Stochastic Analysis .................... 111

4.1 Basic Stochastic Models .............................................................................................112

4.2 Geometry-Free/Ionosphere-Free Observation Equations ...........................................113

4.3 Stochastic Models for GFIF Observations..................................................................115 4.3.1 Multivariate Multiple Regression Equations ....................................................117 4.3.2 Estimation of Variance Components ................................................................120 4.3.3 Covariance Matrices for Combined Measurements .........................................123

4.4 Numberical Results .....................................................................................................125 4.4.1 GFIF Polynomial Fitting ..................................................................................126 4.4.2 Covariance Results ...........................................................................................127 4.4.3 SPP Results with Triple Frequency BDS Signals ............................................130 4.4.4 SPP Results with Dual-Frequency GPS Signals ..............................................131

4.5 Summary .....................................................................................................................132

Chapter 5: Geometry-Free Approach for USD Estimation ......................... 135

5.1 Combined Models for Decomposed Network Computing and the Processing Approach 136

5.1.1 Linear Combinations Selection ........................................................................136 5.1.2 Network Computing Approach ........................................................................139

5.2 Integer Ambiguity Resolutions in the Network-based Computing .............................140 5.2.1 DD Ambiguity Resolutions with GFIF and GBIF Models ...............................141 5.2.2 Mapping the DD Integers to LOS Directions ...................................................143 5.2.3 Computation of the Slant Ionosphere-Delays (SID).........................................145

5.3 USD Estimations ........................................................................................................146 5.3.1 GF Models for USD Parameters .......................................................................147 5.3.2 Network Adjustments .......................................................................................148 5.3.3 Conversion Between Combined and Original USDs .......................................150 5.3.4 Treatment of USDs Parameters in the Time Domain .......................................154

5.4 Clock and ZTD Estimation .........................................................................................156

5.5 Numerical Analysis and Results .................................................................................158 5.5.1 Assessment of GFIF DD Noise Levels and Integer Ambiguity Solutions .......160 5.5.2 Evaluation of Satellite-Specific USD Solutions Against Ionosphere-delay

Datum Settings .................................................................................................162 5.5.3 Consistence among Original Code and Phase Measurements after

Corrections for Biases and Ionosphere Delays .................................................164

5.6 Summary .....................................................................................................................165

Chapter 6: Conclusions ................................................................................... 169

6.1 Investigations ..............................................................................................................169

6.2 Findings ......................................................................................................................170

6.3 Contributions ..............................................................................................................171

vi

6.4 Further Investigations ................................................................................................ 172

Bibliography ........................................................................................................... 175

vii

List of Figures

Figure 1-1 GPS nominal constellation (ESA, 2016d) ................................................ 20

Figure 1-2 GLONASS nominal constellation (ESA, 2016c) ..................................... 21

Figure 1-3 BDS nominal constellation (ESA, 2016a) ................................................ 22

Figure 1-4 BDS service coverage by 2012 (Office, 2013a) ....................................... 22

Figure 1-5 Galileo nominal constellation (ESA, 2016b) ........................................... 23

Figure 2-1 A fully Populated Variance-Covariance Matrices (VCM) in the

original undifferenced phase observations ................................................... 42

Figure 2-2 DCB estimation with IF observations flowchart ...................................... 66

Figure 2-3 DCB estimation with ionosphere analysis flowchart ............................... 67

Figure 2-4 2-layer voxel model used in UPC ............................................................ 74

Figure 4-1 Illustration of 1

GFIF observables (blue) and 3-degree polynomial

fitting (red) ................................................................................................. 126



fitting (red) ................................................................................................. 126



fitting (red) ................................................................................................. 126

Figure 4-4 Illustration of WL


fitting (red) ................................................................................................. 127

Figure 4-5 Illustration of the residuals (blue) of four GFIF models for the

GMSD-C14 and G01 directions, against their ±2 STD curves (green). .... 127

Figure 4-6 Illustration of pseudo range and phase variances (blue, green and

red), plotted vs the elevation-dependent variance (cyan) at GMSD .......... 128

Figure 4-7 Illustration of pseudo range and phase variances (blue, green and

red), plotted vs the elevation-dependent variance (cyan) at JFNG ............ 129

Figure 4-8 Cross-correlation between signals for C02, C14 and G01 satellites

derived from the varying covariance matrices at GMSG and JFNG

stations ....................................................................................................... 129

Figure 4-9 The square root of the ratio of the phase variance coefficient θ2 and

pseudo range variance coefficient θ1, showing the variation with

respect to the rule-of-thumb assumption of 1/100. .................................... 130

Figure 4-10 Illustration of SPP RMS results (in metre) for UNE components at

stations GMSD, JFNG, MAYG and SEYG ............................................... 131

Figure 4-11 Illustration of RMS values of Scheme W and RMS differences

with respect to Scheme A for all the 95 stations in the Up component. .... 132

Figure 5-1 Diagram of a decomposed GNSS network-based processing ................ 140

viii

Figure 5-2 Illustration of the actual success rate versus the samples used for

both PRN24-PRN25 and PRN06-PRN09. ................................................. 161

Figure 5-3 Illustration of consistence between the ionospheric-delays computed

from the geometry-free narrow-lane and wide-lane signals, with green

and blue colours respectively. .................................................................... 161

Figure 5-4 Illustration of code biases for PRN24 and PRN25 plotted along with

their raw data after network adjustment ..................................................... 163

Figure 5-5 Illustration of phase biases for PRN24 and PRN25 plotted along

with their raw data after network adjustment ............................................. 163

Figure 5-6 Consistence between code and phase measurements after

corrections for estimated biases and ionosphere delays ............................. 165

Figure 5-7 Illustration of STD statistics of code and phase noises at all 9

stations ....................................................................................................... 165

ix

List of Tables

Table 1-1 GLONASS signals in current and future generation satellites .................. 24

Table 1-2 GNSS data processing modes overview .................................................... 25

Table 1-3 Treatment of parameters in GNSS data processing modes ....................... 27

Table 1-4 Parameter and observation statistics for ZD and DD dynamic and

kinematic POD (day 148 in 2001) ............................................................... 32

Table 2-1 Refined parameters for elevation-dependent stochastic models in

carrier phase case ......................................................................................... 48

Table 2-2 Refined parameters for SQ-dependent stochastic models in carrier

phase case..................................................................................................... 51

Table 3-1 Examples of GFIF combinations ............................................................. 101

Table 3-2 Examples of GFIP combinations ............................................................. 102

Table 3-3 Examples of GBIF combinations............................................................. 103

Table 3-4 Examples of GBIP combinations............................................................. 104

Table 3-5 Linear models employed in the thesis ..................................................... 109

Table 4-1 Contribution of pseudo range and phase noise terms in the GFIF

observables ................................................................................................. 116

Table 4-2 IGS MGEX stations for experimental analysis ....................................... 125

Table 4-3 Comparison of RMS statistics of UNE errors from 95 MGEX

stations between two schemes ................................................................... 131

Table 5-1 Examples of combined GF observables after integer and ionosphere

terms removed ............................................................................................ 148

Table 5-2 Information of receivers and data sets for experimental analysis............ 159

Table 5-3 RMS of DD GFIF models in metre and cycle units ................................ 160

Table 5-4 The offsets of the code-biases in DD measurements for PRN24-

PRN25 ........................................................................................................ 162

Table 5-5 Summary of the formal precision of satellite and receiver-specific

USD solutions ............................................................................................ 163

x

List of Abbreviations

ABAS Aircraft-Based Augmentation System

ACF Autocorrelation Function

AR Ambiguity Resolution

ARMA Autoregressive Moving Average

ATT Atmospheric Turbulence Theory

BDS BeiDou Navigation Satellite System

BGD Broadcast Group Delay

BIUQE Best Invariant Unbiased Quadratic Estimation

C/A Coarse/Acquisition

CCF Cross-Correlation Function

CDMA Code Division Multiple Access

CHAMP CHAllenging Minisatellite Payload

CLKD Clock Determination

CNO Carrier-to-Noise power density ratio

CNAV Civil Navigation

CODE Centre for Orbit Determination in Europe

CRCSI Cooperative Research Centre Programme for Spatial

Information

CSNO China Satellite Navigation Office

DCB Differential Code Bias

DD Double-Differenced

ESA European Space Agency

ESOC European Space Operations Centre of European Space

Agency

EWL Extra-Wide Lane

xi

FAA Federal Aviation Administration

FCB Fractional Cycle Bias

FDMA Frequency Division Multiple Access

FOC Full Operational Capability

GAMIT A GPS Analysis package developed at MIT

GBAS Ground-Based Augmentation System

GB Geometry-Based

GBIP Geometry-Based/Ionosphere-Present

GBIF Geometry-Based/Ionosphere-Free

GEO Geostationary Orbit

GF Geometry-Free

GFIF Geometry-Free/Ionosphere-Free

GFIP Geometry-Free/Ionosphere-Present

GFPSR Geometry-Free linear combination of Phase-

Smoothed Range

GFZ Geo-Forschungs Zentrum, Potsdam, Germany

GIOVE Galileo In-Orbit Validation Element

GIM Global Ionosphere Map

GLONASS GLObal NAvigation Satellite System

GLS GNSS Landing System

GNSS Global Navigation Satellite System

GPS Global Positioning System

GRIM Ground Integrity Monitoring

GTSF Generalized Triangular Series Function

HPL Horizontal Protection Level

IAAC Ionospheric Associate Analysis Centres

IAR Integer Ambiguity Resolution

xii

IF Ionosphere-Free

IFB Inter-Frequency Bias

IFCB Inter-Frequency Clock Bias

IFPC Ionosphere Free Pseudorange Combination

IGG Institute of Geodesy and Geophysics, Wuhan, China

IGS International GNSS Service

IGSO Inclined Geosynchronous Orbit

ILS Integer Least Estimator

IOC Initial Operational Capability

IOV In-Orbit Validation

IPP Ionosphere Pierce Point

ISB Inter-System Bias

ISC Inter-Signal Correction

ISF Ionosphere Scale Factor

JPL Jet Propulsion Laboratory

KOD Kinematic POD

LAMBDA Least-square Ambiguity Decorrelation Adjustment

LC Linear Combination

LOS Line of Sight

LS Least Squares

LSB Lumped Signal Bias

LS-VCE Least Squares Variance Component Estimation

MEO Medium Earth Orbits

MF Mapping Function

MGEX Multi-GNSS Experiment

MINQUE Minimum Norm Quadratic Unbiased Estimators

xiii

MMR Multivariate Multiple Regression

MSLM Modified Single-Lay Model

MW Melbourne-Wübbena

NAV Navigation

NDOP Network Dilution of Precision

NGS National Geodetic Survey, NOAA, USA

NL Narrow-Lane

Network-RTK Network-Real Time Kinematic

PC Ionosphere-Free Pseudorange Linear Combination

PCO Phase Centre Offset

PCV Phase Centre Variation

PNF Phase Noise Factor

POD Precise Orbit Determination

PPP Precise Point Positioning

PPP-AR Precise Point Positioning-Ambiguity Resolutions

PRN Pseudorandom Noise

PVT Position, Velocity and Time

QC Quality Check

QUT Queensland University of Technology

QZSS Quasi-Zenith Satellite System

RAIM Receiver Autonomous Integrity Monitoring

RINEX Receiver Independent Exchange format

RMS Root Mean Square

RTCA Radio Technical Commission for Aeronautics

RTK Real Time Kinematic

SBAS Satellite-Based Augmentation System

xiv

SD Single-Differenced

SH Spherical Harmonic expansions

SID Slant Ionosphere-Delay

SNR Signal-to-Noise Ratio

SPP Single Point Positioning

SQ Signal Quality

STEC Slant Total Electron Content

STD Standard Deviation

SV Space Vehicle

TECU Total Electron Content Unit

TEQC Translation, Editing and Quality Checking

TGD Timing Group Delay

TNL Total Noise Level

UCD Uncalibrated Code Delay

UD Undifferenced

UPC Polytechnic University of Catalonia

UPD Uncalibrated Phase Delay

UPPP Un-combined Precise Point Positioning

USAF United States Air Force

USD Uncalibrated Signal Delay

VCE Variance Component Estimation

VCM Variance-Covariance Matrices

VPL Vertical Protection Level

VTEC Vertical Total Electron Content

WAAS Wide Area Augment System

WL Wide Lane

xv

ZTD Zenith Tropospheric Delay

ZD Zero-Differenced

xvi

Statement of Original Authorship

QUT Verified Signature

Introduction xvii

Acknowledgements

First of all, I wish to express my deep gratitude to my principal supervisor,

Professor Yanming Feng, for his continuous support, generosity and encouragement

through my PhD program. Professor Feng is an enthusiastic and dedicated mentor who

provided important perspectives about the critical issues presented in this thesis. His

careful guidance and thoughtful advice were essential for the completion of this work.

The experience of doing my PhD research with him, and the opportunities he has

offered to me, are the most valuable part of the whole PhD research process.

I especially appreciate the great support received from the Cooperative Research

Centre Programme for Spatial Information (CRCSI) and Queensland University of

Technology (QUT). These two great organisations offered me a valuable chance to

complete the PhD research work.

My research in Queensland University of Technology (QUT) also owes thanks

to QUT Adjunct Professor Matt Higgins and Dr John Hayes, Professor Chuang Shi,

Professor Yidong Lou, Dr. Xiaolei Dai in Wuhan University, Wuhan, China. They

offered lots of support when I stayed in Wuhan University and helped me to build the

foundation in GNSS precise processing and orbit determination.

I am grateful to Dr. Charles Wang and Dr. Lei Wang at QUT. These two

gentlemen have been providing generous help to me, not only in laboratory research,

but also in daily life such as travel and accommodation. I am honoured to know and

have them as enduring friends.

Finally, my deepest thanks are for all my beloved family members and friends

around me. Their backing drives me to stay focus, strong and never giving up.

Introduction 19

Chapter 1: Introduction

This chapter outlines the research background (Section 1.1) and challenges in

GNSS data processing (Section 1.2), followed by the overall objective and specific

aims in Section 1.3. The significance of the development is described in Section 1.3.

Finally, Section 1.4 outlines the remainder of the thesis.

1.1 RESEARCH BACKGROUND

1.1.1 Global Navigation Satellite Systems

A navigation satellite system transmits ranging signals from orbiting satellites

toward the earth’s surface, which allows a receiver to compute its geographic position.

Many Global Navigation Satellite Systems (GNSS) provides global signal coverage.

These systems are designed to provide signal services for terrestrial and airborne users

to compute their Position, Velocity and Time (PVT) information. A GNSS consists of

a space segment (the satellite constellation), a control segment (the control, monitoring

and uplink stations) and user segment (the receiver). To meet strict operational

requirements (ICAO, 2005), a GNSS is additionally augmented by Aircraft-Based

Augmentation System (ABAS), Satellite-Based Augmentation System (SBAS) and/or

Ground-Based Augmentation System (GBAS).

Current GNSS systems include the Global Positioning System (GPS), the

GLObal NAvigation Satellite System (GLONASS), the BeiDou Navigation Satellite

System (BDS) and the Galileo Satellite Navigation System (Galileo). These systems

are briefly described below.

1) GPS

GPS is operated by the United States Air Force (USAF) and has been providing

fully operational services since 1993 (DoD, 2008). The GPS constellation nominally

consists of 24 satellites deployed in six Medium Earth Orbits (MEO) uniformly as

shown in Figure 1-1. Each GPS satellite generates and transmits signals modulated in

L1 (1575.42MHz) and L2 (1227.6 MHz) L-band carriers, broadcasting three

Pseudorandom Noise (PRN) ranging codes: the precision (P) code, the Y code, and the

Coarse/Acquisition (C/A) code (DoD, 2008). The P code is the principal ranging code,

while the Y code is used to replace P code when the antispoofing mode is activated.

20 Introduction

The C/A code supports the acquisition of P or Y code and also is a civil ranging signal

(DoD, 2008).

Figure 1-1 GPS nominal constellation (ESA, 2016d)

Beyond the nominal design in GPS Performance Standards (DoD, 2008) , GPS

now effectively operates as a 27-slot constellation with worldwide improved coverage

after an "Expandable 24" configuration USAF completed in June 2011 (Space

Segment, 2016). Furthermore, the recently launched satellites provide new civil signal:

L5 (1176.45 MHz) which are designed to meet demanding requirements for safety-of-

life transportation and other high-performance applications together with L1 C/A and

L2C (ICWG, 2013; New Civil Signals, 2016).

2) GLONASS

GLONASS is operated by the Ministry of Defence of the Russian Federation. Its

nominal constellation has 24 operational satellites and two spares (ICAO, 2005;

Langley, 1997a). GLONASS satellites are designed to be located evenly in three

orbital planes with an altitude of around 19,100 km (10,310 nm), inclined at 64.8

degrees and spaced 120 degrees apart as illustrated in Figure 1-2. The theoretical orbit

period is 11 hours and 15 minutes. As of August 3, 2016, there are actually 27 satellites

in the constellation with 23 operational (Constellation Status, 2016; New Civil Signals,

2016). Similarly, to GPS, GLONASS satellites broadcast two binary codes: the C/A

code and the P-code, as well as the data message. The difference is that GLONASS

broadcast is based upon a Frequency Division Multiple Access (FDMA) technology.

The signals are transmitted in different carrier frequencies (L1 band: 1602.00 +

k× 9 16⁄ MHz; L2 band: 1246.00 + k×7

16⁄ MHz, where k is the integer frequency

Introduction 21

number (Estey & Meertens, 1999). A GLONASS receiver separates all the received

signals by assigning the tracking channels different frequencies (ICAO, 2005).

Figure 1-2 GLONASS nominal constellation (ESA, 2016c)

3) BDS

BDS is operated by the China Satellite Navigation Office (CSNO), and designed

to deploy 35 satellites in its Space Segment, including 5 Geostationary Orbit (GEO)

satellites, and 30 non-GSO satellites: 27 in Medium Earth Orbit (MEO) and 3 in

Inclined Geosynchronous Orbit (IGSO), as shown in Figure 1-3 (Office, 2013b).

Aapproximately 40 BDS navigation satellites are planned to be launched in total. The

Full Operational Capability (FOC) will provide a high-level PVT service with a

worldwide coverage by 2020 (Office, 2013b). Since the end of 2012, BDS has

consisted of 14 operational satellites, including 5 GEO satellites, 5 IGSO satellites,

and 4 MEO satellites. This provides FOC in the most of the region from 55°S to 55°N,

70°E to150°E as shown in Figure 1-4 (Office, 2013a, 2013b). From 2015, six extra

launches (3 IGSO and 3 MEO satellites) will result in 20 operational satellites. One

GEO satellite that was launched on 12 June 2016 is in commissioning (List of BeiDou

satellites, 2016). The BDS satellites transmit the signals in three frequencies: B1 =

1561.098 MHz, B2 = 1207.14 MHz, and B3 = 1268.52 MHz, as well as broadcasting

navigation message (CSNO, 2013; Li, Feng, Gao, & Li, 2015; Office, 2013b).

22 Introduction

Figure 1-3 BDS nominal constellation (ESA, 2016a)

Figure 1-4 BDS service coverage by 2012 (Office, 2013a)

4) Galileo

Europe's Galileo is another GNSS being developed by the European Space

Agency (ESA). It aims to offer a continuous, more flexible and precise positioning

service to different ranges of users. Galileo was originally planned to deploy a

complete constellation of 27 operational satellites and three reserves. The currently

plan involves fewer operational satellites plus six in-orbit spares (ESA, 2015; Galileo,

2015). The satellites will be stationed on three circular MEO at an altitude of 23,222

km and with an inclination of 56º to the equator, as in Figure 1-5 (ESA, 2013a). The

Introduction 23

signals are broadcast on five carrier frequencies: E1 (1575.42MHz), E6 (1278.75MHz),

E5 (1191.795MHz), E5b (1207.14MHz) and E5a (1176.45MHz) for commercial and

civilian use (Galileo, 2015).

Since 12 October 2012, Galileo has been in its In-Orbit Validation (IOV) phase.

It has an independent positioning capability (Steigenberger, Hugentobler, &

Montenbruck, 2013). The IOV phase is aims to qualify: the Galileo space, ground and

user segments (ESA, 2013b). Initial services will be made available by the end of 2016

(Svitak, 2016). Five satellites are now operational for users. Plans remain for the

deployment of the full Galileo constellation (30 orbiting satellites) and achieve full

operational capability (ESA, 2015; Svitak, 2016).

Figure 1-5 Galileo nominal constellation (ESA, 2016b)

1.1.2 GNSS Triple Frequency Signals

Broadcasting triple or multiple-frequency signals is an emerging trend in GNSS

technology evolution. As explained in Section 1.1.1, all BDS and Galileo satellites

transmit signals in three frequency bands – the Galileo E5a and E5b signals are part of

the E5 signal (Galileo, 2015). Since its Initial Operational Capability (IOC) in 1993,

all GPS satellites transmit signals in both L1 and L2 frequency bands (DoD, 2008).

Earlier, in 1999, the USA DoD declared their intention to provide a new GPS signal

called L5 centred in 1176.54 MHz (Spilker, 1999). As of 15 June 2016, there are 12

GPS satellites (Block IIF) broadcasting pre-operational L5 signals (New Civil Signals,

2016). The main feature of a GPS modernisation program planned for around 2024

24 Introduction

involves L5 signals being available from 24 GPS satellites (Block IIF, Block III and a

future type of space vehicle) in total (Dunn, 2013; Group, 2013; New Civil Signals,

2016). Traditionally, GLONASS satellites transmit navigational radio signals on two

frequency bands (L1: 1602 MHz and L2: 1246 MHz) and employing the FDMA

technique. The new generation GLONASS-K series (first launched on 2 February

2011), GLONASS satellite transmits a Code Division Multiple Access (CDMA)

signal in L3 band (L3OC with central frequency: 1202.025MHz) (Revnivykh, 2011;

Urlichich et al., 2011). In its signal implementation plan, future GLONASS satellite

generations will transmit CDMA signals in L1, L2, L3/L5 bands centred at 1600.995

MHz, 1248.060 MHz, 1202.025 MHz/1176.45 MHz, respectively, as shown in Table

1-1 (Mirgorodskaya, 2013; Revnivykh, 2011; Urlichich, et al., 2011).

Table 1-1 GLONASS signals in current and future generation satellites

Satellite L1 L2 L3 L1, L2 Future

GLONASS-M L1OF L2OF N/A N/A

GLONASS-K1 L1OF L2OF L3OC test N/A

GLONASS-K2 L1OF L2OF L3OC L1OC, L2OC

GLONASS-KM L1OF L2OF L3OC L1OC, L2OC L1OCM, L5OCM

FDMA signal CDMA signal

Note: N/A-Not available

1.2 GNSS DATA PROCESSING AND CHALLENGES

1.2.1 GNSS Data Processing Modes

To support various positioning services and applications, a diverse range of

GNSS data processing problems has been proposed. The problems fall into two

categories: single-receiver based processing and network-based processing. Single-

receiver based processing refers to cases in which the computations are done at a single

receiver end, regardless of whether it is a stationary or moving receiver. Network-

based processing deals with the data from multiple GNSS receivers in local, regional

or global networks. The processing results may include various satellite and receiver-

specific products, which may then be distributed via radio communication, internet, or

satellite-based datalink to users. An overview of GNSS data processing modes is

shown in Table 1-2.

Introduction 25

Table 1-2 GNSS data processing modes overview

Mode Main output Tag as

Single-receiver based processing

Single point positioning (SPP) Position, velocity and time (PVT) SPP

Precise point positioning (PPP) Precise position, velocity and time states PPP

Precise point positioning-

Ambiguity resolutions (PPP-AR)

Precise position and, velocity states PPP-AR

Real time kinematic (RTK) Precise position and velocity User-RTK

Reference station based computing Slant total electron content (STEC), zenith

tropospheric delay (ZTD), lumped biases STEC

Receiver autonomous integrity

monitoring (RAIM)

Horizontal protection level (HPL), vertical

protection level (VPL) RAIM

Stochastic modelling Computation of covariance matrices of code

and phase measurements (COV) COV

Network-based processing

Precise orbit determination (POD) Precise orbit and clock products, precise

user positon products, POD

Real time clock estimation Precise satellite clock products CLK

Network-real time kinematic

(Network-RTK)

baseline integers, precise ionosphere and

troposphere delays differential corrections Network-

RTK

Uncalibrated signal delay (USD)

estimation

Uncalibrated phase delay (UPD)/

fractional cycle bias (FCB),

uncalibrated code delay (UCD)

USD

Global/Regional ionosphere

modelling

Differential code bias(DCB), vertical total

electron content (VTEC) VTEC/ DCB

Ground integrity monitoring Satellite signal quality GRIM

Ground/Satellite based

augmentation system

(GBAS/SBAS)

Range corrections or correction model

parameters, integrity GBAS/SBAS

1.2.2 Challenges in Stochastic Modelling and Biases Estimation

In general, GNSS codes and phase measurements from station r ( r =1,2,…, n )

to satellite s ( s = 1,2,…, m ) can be represented by the following observation

equations:

, , , , ,

s s s s s s s

r i r r r r i r i i r i PP Clk Clk T I b b X X Eq. 1-1

, , , , , ,( )s s s s s s s s

r i r r r r i i r i i r i i r iClk Clk T I N B B X X Eq. 1-2

26 Introduction

where (default unit is metre):

i The index of frequency

,

s

r iP : Code pseudorange measurement

,

s

r i : Carrier phase measurement

rX : Position vector of station r

sX : Position vector of satellite s

rClk : Receiver clock error

sClk : Satellite clock error

s

rT : Tropospheric delay

,

s

r iI : The first order ionospheric group delay on carrier in the thi

frequency, , 2

ss rr i

i

qI

f

s

rq The first-order ionospheric delay parameter which is equal

to 40.3 Total Electron Content (TEC)

,r ib : Receiver-specific hardware delays in the code measurement

,

s

r iP ,which are known as receiver-specific Uncalibrated Code

Delay (UCD)

s

ib : Satellite-specific hardware delay in the code measurement

,

s

r i ,which are known as satellite-specific UCD

i : The wavelength of frequency i

,

s

r iN : Integer ambiguity of phase measurement in cycle

,r iB : Receiver-specific hardware delay (unit: cycle) in the phase

measurement ,s

r i ,which are known as receiver-specific

Uncalibrated Phase Delay (UPD)

s

iB : Satellite-specific hardware delays (unit: cycle) in the phase

measurement ,s

r i ,which are known as satellite-specific UPD

, ,

s

r i P : The receiver code noise and multipath error, plus higher

order ionospheric effects in ,s

r iP

Introduction 27

, ,

s

r i : The receiver phase noise and multipath error, plus higher

order ionospheric effects in ,s

r i

s : Satellite indicator

r : Receiver indicator

i : Carrier frequency indicator

In GNSS data processing problems, some of the above parameters need to be

estimated, while others can be identified using prior knowledge, or cancelled through

proper processing (e.g. double differencing, model correction). Table 1-3 summarises

the generic parameters treatment in various GNSS data processing modes.

Table 1-3 Treatment of parameters in GNSS data processing modes

Parameters

Modes sX rX

sClk rClk T I N b B e.g.

POD (DD) ○ ○ X X ○ ○ ○ X X ● ○1

POD (UD) ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ○2

CLK (UD) ● ● ○ ○ ○ X ○ ● ● ● ○3

SPP (UD) ● ○ ● ○ ● ●/ X N/A ● N/A ● ○4

PPP/PPP-AR ● ○ ● ○ ○ ○ ○ ○ ○ ● ○2

Network-RTK ● ● X X ○ ○ ○ X X ● ○5

User-RTK ● ○ X X ○ ○ ○ X X ● ○4

VTEC/DCB X X X X X ○ ○ ○ ○ ● ○6

USD (GB) ● ● ● ● ○ ○ ○ ○ ○ ● ○2

STEC ● ● ● ○ ○ ○ ○ ○ ○ ● ○4

RAIM ● ○ ● ○ X X N/A ○ N/A ○ ○7

COV X X X X X X ○ ○ ○ ○ N/A

GIMS ○ ● ○ ○ X ○ ○ X X ○ ○8

GBAS/SBAS ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ○9

28 Introduction

Note:

○ = to be estimated; ● = take as known X = cancelled

DD = double difference UD = un-differenced GB = Geometry-based

○1 : GAMIT ○2 : PANDA ○3 : International GNSS Service (IGS) Real-time

Service (RTS)

○4 : RTKLIB ○5 : Virtual reference station (VRS)

○6 : German Aerospace Centre (DLR)

○7 : generic RAIM functions based on RTCA*)

DO-229D in commercial

receivers, FAA** RAIM

prediction service, etc.

○8 : Ground Regional Integrity System (GRIMS)

○9 : GNSS Landing System (GLS), Wide Area Augment

System (WAAS)

T , I , N , b , B , Generic indicators of srT , ,s

r iI , ,s

r iN , ,r ib /s

ib , ,r iB /s

iB , , ,s

r i P /

, ,

s

r i

N/A Not applicable

* FAA: Federal Aviation Administration

** RTCA: Radio Technical Commission for Aeronautics

The noise term ( ) and the bias terms (b and B ) in navigation satellite signals

are of great concern to GNSS system operators, researchers and users. It is generally

agreed that adequate treatments or specific knowledges of noises and biases in these

signals play a key role in precise GNSS state estimations and applications.

The signal noises, including multipath effects, are receiver and/or location

dependent. Modelling and characterisation of random noises in GNSS signals involves

estimating covariance matrices of multivariate pseudo range and phase noise time

series. The variance and covariance of the GNSS code and phase measurements are

time varying, and depend on many factors such as elevation angles, observation

environment, receiver antenna, receiver hardware and software. Ideally, the full

covariance matrices for pseudo range and phase noises can be obtained for each line-

of-sight direction and can be updated from time to time. The resulting measurement

covariance matrices can then be used as part of the linear observational equation

system in the follow-on state estimation and quality assessment, such as SPP, PPP,

RTK, POD and RAIM.

Apart from UPD/FCB, DCB, signal biases also exist between different GNSS

systems (Inter System Bias, ISB), between code/phase signals at different frequencies

Introduction 29

(Inter Frequency Bias, IFB). Knowing the values and possible time variations of these

biases is a prerequisite for precise state estimation in GNSS processing problems,

which may use UD and/or DD code and phase measurements. For decimetre-level SPP

with single frequency code measurements, corrections for satellite-specific DCBs

must be applied, while receiver-specific DCBs can be absorbed by the receiver clocks.

In the PPP mode, the FCBs or UPDs in phase measurements have to be estimated as

part of ambiguity parameters or removed in order to resolve the integer ambiguities

(Bertiger et al., 2010; Collins, Bisnath, Lahaye, & Héroux, 2010; Ge, Gendt, Rothacher,

Shi, & Liu, 2008; Laurichesse, Mercier, Berthias, Broca, & Cerri, 2009). The

anticipated improvement in accuracy, availability and reliability from multi-frequency

and multi-system signals, depends to a large extent, on how well various biases

between different signals or systems are dealt with.

However, there are several critical challenges in current treatments of the

parameters listed in Table 1-3. Three fundamental challenges are described below.

1) Inadequate stochastic modelling for code and phase signals

All GNSS processing modes require information about the variance and

covariance of code and phase measurements - preferably covariance matrices for each

line of sight direction and in real time. However, the existing approaches only partially

address the above problems. The Translation, Editing and Quality Checking (TEQC)

software widely used by the IGS, can provide Standard Deviation (STD) of the

pseudorange multipath time series for each line-of-sight direction. However, no

covariance information is provided between code measurements, neither is variance

and covariance information provided about phase measurements. Another approach

uses SD or DD pseudo range and phase measurements to directly estimate the

covariance and correlation information over zero or short-baselines, based on dual or

triple frequencies. These analyses of SD or DD data over short- or zero-baseline are

suitable for understanding the overall performance characteristics of the certain type

of receivers. They do not directly provide the covariance knowledge for UD data

processing, nor do they provide real time covariance knowledge for any SD and DD

data processing (Bona, 2000; Cai et al., 2015; de Bakker, van der Marel, & Tiberius,

2009; Euler & Goad, 1991; Li, Shen, & Xu, 2008; Tiberius & Kenselaar, 2000; Wang,

Stewart, & Tsakiri, 1998; Yang et al., 2014).

30 Introduction

The most commonly adopted approaches for stochastic estimation of GNSS

signals in GNSS data processing depend on some empirical stochastic models for

weighting. Typically, the observation weight for code or phase measurements is

expressed as a function of elevation or Signal-to-Noise Ratio (SNR) (Brown, Kealy,

& Williamson, 2002; Collins & Langley, 1999; Dai, Ding, & Zhu, 2008; Jin & de Jong,

1996; Li, Dingfa, Meng, & Dongwei, 2015; Wang, et al., 1998; Wieser & Brunner,

2000). The empirical weight models do improve position estimation accuracy,

compared with a constant weighting (Collins & Langley, 1999). This generally proves

that the varying weights can more effectively reflect actual observational noise levels.

Nonetheless, the elevation-dependent or SNR-dependent empirical methods cannot

describe actual situation well enough, since code and phase multipath depend on not

only the elevation, but also on the real observational environment, antenna structure

and material, as well as other factors, such as high-frequency multipath interference

caused by dynamic reflecting surfaces (El-Mowafy, 1994; Ogaja & Satirapod, 2007).

The observation noises sometimes are sometimes independent of elevation angles.

Furthermore, the characteristics of pseudo range noises also depend on GNSS receiver

internal algorithms.

2) Incompetent mathematical modelling for satellite and receiver specific hardware biases

In the current GPS data processing, code and phase hardware biases are either

dealt with separately or within combined measurements. The DCBs, defined as the

inner delay differences between the two frequencies, are determined with geometry-

free code combinations. The DCBs, along with global ionosphere modelling, and

phase measurements, are smoothed over time. In this process, the DCB values for

satellite and receivers are assumed to remain unchanged typically over 24 hours, thus

being separated from the ionosphere delays through regional or global ionosphere

models. In addition, a 30-day moving averages of P1C1 and P1P2 DCBs are provided

by the IGS analysis centres, such as the Centre for Orbit Determination in Europe

(CODE), the Jet Propulsion Laboratory (JPL), ESA, and the Polytechnic University of

Catalonia (UPC) (Kouba, 2015; Sardon, Rius, & Zarraoa, 1994; Schaer, 1999), while

in fact the biases are changed daily at least (Choi, Cho, & Lee, 2011; Gu, Shi, Lou,

Feng, & Ge, 2013; Zhang & Teunissen, 2015). However, the time variation of the

DCBs is least studied and less well understood.

Introduction 31

The usage of the DCBs has caused much confusion. First of all, network-based

GNSS data processing, such as POD and real time clock estimation, does not involve

DCB estimation or corrections for DCB in code measurements. Dual-frequency based

precise point positioning also ignores the satellite-specific DCB correction. In cases

where users choose to use C/A code only, or P2-only pseudo range observations,

satellite-specific P1C1 DCB's are transformed C1 to P1. Precise single-frequency PPP

must also use the ionospheric delay corrections along with the corresponding satellite

(Kouba, 2015).

The wide-lane UPDs are computed with the geometry-free Melbourne-Wübbena

(MW) (Wübbena, 1985) model and the narrow-lane UPDs are estimated with the GB

ionosphere-free phase observables where the satellite receivers and clocks are kept

fixed (Ge, et al., 2008; Gu, Lou, Shi, & Liu, 2015). The wide-lane UPD contains the

effects of both code and phase hardware biases, while the narrow-lane UPD contains

the effects of phase biases. However, the narrow-lane UPD are determined in the

geometry-based models along with many other parameters. The effects of inaccurate

clocks and atmosphere delays are composed with the UPD parameters. It is very

difficult to separate the narrow-lane UPD with other error sources. As a result, the

determined UPDs are only applicable to PPP processing with the same types of Wide-

Lane (WL) and Narrow-Lane (NL) observables. It is often the case that UPD products

determined by one family of software may not yield the same performance when the

products are used by other families of PPP software.

There exist some inherent dependencies between the models and processing

strategies within present GNSS data processing methodologies. The physical models

for various state/propagation delay/bias parameters are usually complicated.

Inaccurate modelling of one type of states will affect the modelling others. For instance,

the residuals caused by insufficient ionosphere correction models contributes errors to

single receiver states determination, or even worse to DCB estimation in a large

network - which affects any end users of DCB data. In the same way, an inaccurate

orbit model may result in imprecise orbit information, resulting poor estimation of

receiver states, satellite clock, TEC, etc. Additionally, clock and phase hardware

delays depend on satellite and receiver combinations, which are difficult to separate.

The wide-lane UPD computed with the geometry-free Melbourne-Wübbena model

contains effects of both code and phase biases. The hardware biases in original code

32 Introduction

and phase signals cannot be recovered from UPD and DCB results. In practice, the

above separation problem and rank-deficient equation can be solved by adding

artificially constraints to satellite DCBs. Thus, the estimated DCBs depends on both

the precision of the ionosphere map and the observables, and appears to be independent

of constraint selection (Xie, Chen, Wu, & Hu, 2014) .

3) Inefficient network-based GNSS computation

In current network-based processing, especially with a large-scale POD, CLK,

and DCB, significant quantities of long-arc data are required from hundreds of

worldwide stations. The computation load is considerable and time consuming. In

addition to the orbit parameters, a large number of unknown model coefficients require

to be identified, such as 9 solar radiation pressure model, Saastamoinen tropospheric

model, gravity model, etc (Jing-nan & Mao-rong, 2003; Scharroo & Visser, 1998;

Visser & Van den Ijssel, 2000). A simple table (Table 1-4) is provided to show the

number of parameters and observations for Zero-Differenced (ZD) and DD dynamic

and Kinematic POD (KOD) for CHAllenging Minisatellite Payload (CHAMP)

(Švehla & Rothacher, 2003).

Table 1-4 Parameter and observation statistics for ZD and DD dynamic and kinematic POD (day 148

in 2001)

Solution ZD Dynamic ZD KOD DD Dynamic DD KOD

Ambiguities 450 450 13200 13200

Orbit

parameters

300 N/A 300 N/A

Kinematic

coordinates

N/A 8640 N/A 8640

Satellite

clocks

2880 2880 N/A N/A

Total

parameter

number

3630 11700 13500 21840

Total

observation

number

18400 18400 340000 340000

Introduction 33

1.3 OBJECTIVES AND CONTRIBUTIONS

1.3.1 Research Objectives

Addressing the above challenges in present GNSS data processing requires

theoretical GNSS model and algorithm development. The overall objective in this

research is to develop a set of models and algorithms to deal with the above issues in

the context of triple frequency signals processing. This involves the computation of

hardware delays (biases) and covariance matrices of code and phase measurements,

based on a Geometry-Free (GF) approach. The covariance and bias information are

compiled for each frequency and line of sight measurement, and are updated epoch by

epoch. As a result, the network-based computation can be decomposed into GF

processing and Geometry-Based (GB) processing problems. The computational

efficiency problem is also largely resolved.

The specific objectives contained in the overall one are:

1) Overview of research developments

A review of relevant GNSS literature is conducted. The review covers stochastic

analysis and GNSS bias/delay estimation (DCB, UPD, UCD) and ISB. The concepts,

estimation methods, applications and limitations are summarised, which provides

valuable insights into the approach proposed in this thesis.

2) Investigation of multi-frequency signal combinations and desirable signals for estimation of different types of parameters

The first aim is to systematically examine various combinations for specific

estimation problems, such as ambiguity resolution, hardware delays estimation and

covariance analysis. The models provide the basis for the follow-up geometry-free

studies.

3) Investigation of linear combinations fundamentals

The fundamentals of linearly combing triple-frequency GNSS signals are

investigated. Most frequently used types of linear combinations are analysed, with

some illustrative examples. The application cases of the combinations are outlined to

solve different questions: ambiguity resolution, ionosphere estimation, USD

estimation, state estimation and covariance estimation.

34 Introduction

4) Development of geometry-free stochastic analysis approach

This section develops a general stochastic analysis approach for characterising

line-of-sight triple-frequency GNSS signals in terms of code and phase variance,

covariance and cross correlation. To validate this new approach, the covariances are

incorporated into weighting parameters within a SPP application. The resulting

position accuracy performance is compared with that arising from a conventional

weighting strategy. It is demonstrated that the developed covariance weighting can

improve SPP performance, particularly in poor geometry scenarios.

5) Development of geometry-free USD estimation approach

A geometry-free approach is developed for efficient estimation of satellite and

receiver-specific USDs based on network-based triple frequency signals. In addition,

a fast method is developed for resolving USD processing ambiguities. The results of

experiments are presented in which the new USD estimation technique is applied to

code and phase measurements collected from a triple frequency signal GPS network.

It is demonstrated that this approach can estimate code bias with centimetre level

accuracy and phase bias with millimetre level accuracy.

1.3.2 Contributions

This thesis describes contributions in the following two areas.

1) Reclassifications of linear combinations for decomposed GNSS data processing

We systematically re-examine various linear combinations, and classify all

combinations into four types: Geometry-Free/Ionosphere-Free (GFIF) combinations,

Geometry-Free/Ionosphere-Present (GFIP) combinations, Geometry-

Based/Ionosphere-Free (GBIF) combinations and Geometry-Based/Ionosphere-

Present (GBIP) combinations. The complete set of parameters in the original code and

phase equations can be estimated with a set of combinations that are equivalent to the

original equations. However, the whole network-based computation problem can be

effectively decomposed into geometry-free computing and geometry-based computing

problems. The geometry-free computing deals with the integer ambiguities,

ionosphere delay, and the satellite- and receiver-specific USDs. The geometry-based

computing deals with the estimation of satellite and receiver states, including

troposphere delays for each receiver.

Introduction 35

2) A novel USD estimation methodology based on six GF/GB models

Regarding the estimation of hardware delays of GNSS signals, the thesis

presents a geometry-free approach for ambiguity resolution and estimation of the

satellite- and receiver-specific USDs in the network-based GNSS data processing with

triple frequency signals. For the first time, six combined GF/GB models that are

equivalent to the six original GB code and phase models are introduced in this research.

The approach based on the combined models allows the ambiguity resolution and

estimation of six USDs parameters to be performed separately from the estimation of

satellite and receiver states in the network-based processing. The results show that the

satellite-specific hardware delay estimates for original code and phase signals are

determined to the precision of a few centimetres and a few millimetres, respectively.

It is expected that a network consisting of more stations and satellites in view would

improve the USD estimates to higher precision.

3) A proven covariance matrices assessment based on an GF approach

The research provides an innovative method to assess the variation of covariance

metricises of undifferenced three pseudorange signals, as well three carrier-phase

GNSS signals epoch-by-epoch. In the proposed approach, four independent GFIF

models formed from original triple-frequency code and phase signals allow for

effective computation of variance-covariance matrices using real data. Experiments

implemented using triple frequency signals of both BDS and GPS demonstrate that the

estimated standard deviation varies in consistence with the amplitude of GFIF error

time series. The SPP results based on BDS at four MGEX stations have shown an

effective improvement in three dimensions respectively by using the estimated

covariance matrices based weighting, as compared to a traditional elevation-dependent

weighting method. Especially when the geometry is worse, the estimated covariance

brings higher accuracy gain. The SPP results based on GPS from 95 MGEX stations

demonstrate an average improvement of about 10% relative to the results from a

traditional elevation weighting model in terms of root mean square statistics. This

finding provides a preliminary confirmation that adequate consideration of the

variation of covariance lead to the improvement of GNSS state solutions. In particular,

in urban and Central Business District (CBD) areas where the GNSS geometry is

degraded by the skyscrapers, the proposed covariance assessment has more potentials

to improve the SPP performance.

36 Introduction

For convenience and brevity, the formulas and definitions related to frequency

are directly referred to the GPS triple frequency signals. However, the GF approaches

for hardware bias and noise analysis proposed in this research are applicable to all

other GNSS signals operating in triple frequencies, including BDS, Galileo and QZSS.

It is worth noting that the frequencies should meet the condition 1 2 5f f f , in order

to directly use the related equations. For instance, with BDS signals, we need to set B1

as 1f , B3 as 2f and B2 as 5f , where B1=1561.098 MHz, B2=1207.14 MHz and

B3=1268.52 Hz.

1.4 THESIS OUTLINES

The remainder of this thesis is structured as follows.

1) Chapter 2: Literature Review

Chapter 2 provides an overview of the roles of GNSS multi-frequency signal

combinations. The status of GNSS stochastic modelling research is summarised. The

discussion describes both conventional single/dual-frequency GNSS signal processing

methods and the more recent triple-frequency measurement techniques. The IFB, ISB,

DCB, UPD, UCD estimation methodologies are defined. The performance and

computational loads of different methods are discussed.

2) Chapter 3: Fundamentals of GNSS Observation Linear Combination

Fundamental equations of linear combination are described, as well as

ionosphere delay, bias and error propagation. Four main types of linear combinations

are introduced, including GF and GB signals, with typical examples. The application

cases of all types of questions are listed to guiding subsequent data processing with

linear combinations, e.g. covariance and USD estimation investigated in this thesis.

3) Chapter 4:Geometry-Free Approach for Stochastic Analysis

This chapter assesses the variation of covariance matrices of undifferenced

triple-frequency GNSS measurements and their impact on positioning solutions. Four

independent GFIF models constructed from original triple-frequency code and phase

signals are proposed for the Multivariate Multiple Regression (MMR) problem. VCE

algorithms are implemented to obtain covariance matrices and covariance components

based on the MMR model. Next, the variance propagation for combined or differenced

Introduction 37

observables is discussed with the ionosphere-free combination and DD observation

examples. Experiments using BDS (C02, C06 and C14) and GPS (G01) data sets

demonstrate that the estimated standard deviation varies in consistently with the

amplitude of GFIF error time series. Single-point positioning results from GPS

ionosphere-free measurements and 95 MGEX stations are presented which

demonstrate an average improvement of about 10% relative to the results from a

traditional elevation weighting model in terms of root mean square statistics.

4) Chapter 5: Geometry-Free Approach for USD Estimation

Chapter 5 is organised as follows. After a short discussion of grouping linear

combinations for USD estimation, ambiguity resolution of the DD integer ambiguities

of all independent baselines is performed with three GFIF models. The necessary UD

and SD ambiguities are appended, and the fixed DD ambiguities are then propagated

to all the UD directions. After that, with integer-fixed UD GFIF and GFIP observables,

the receiver- and satellite-specific USD solutions are adjusted in the network domain.

A constraint between the USD parameters is set to overcome the rank deficiency in the

network adjustment. Appending the boundary conditions, the raw measurement time

series for each original code and phase signal can be obtained are filtered in the time

domain with the polynomial functions of different degrees. With the data sets from the

nine receivers that track GPS PRN24, PRN25 and PRN06, PRN09 over two 3.3-hour

periods, the penultimate section demonstrates the results about the performance of DD

narrow-lane integer ambiguity solutions over eight independent baselines of up to

1,700 km in length, and the assessment of UD ambiguity resolutions and USD

solutions. Last section summarises the findings of the USD research.

Chapter 5 is organised as follows. Firstly, the linear models are selected from

GFIF, GFIP and GBIF combination and the decomposed procedures are outlined. Then,

selected GFIF observables are analysed for resolution of two wide-lane and one

narrow-lane ambiguities and the slant ionosphere-delays are computed with a GFIP

model. The next section describes the estimation of satellite- and receiver-specific

USDs from four GFIF and one GFIP observations over a data period, with two datum

conditions set for the GFIP and GBIF models. Following USD estimation, the clock

and ZTD estimation procedures with the GBIF models is outlined briefly. Finally, the

last section presents the numerical results about the performance of DD narrow-lane

integer ambiguity solutions over eight independent baselines of up to 1700 km in

38 Introduction

length, USD solutions and consistence between measurements after all corrections are

applied. The findings in this part is summarized at the end.

5) Chapter 6: Conclusions

The main findings and conclusions of the thesis are summarised in the final

chapter. Recommendations are stated for investigations that could follow on from the

new techniques developed herein. The potential applications and benefits of the

developments are also discussed.

Literature Review 39

Chapter 2: Literature Review

In this chapter, we review the pre-existing investigations on relevant topics:

stochastic modelling and hardware bias estimation. The review covers not only basic

concepts, substantive findings, theoretical and methodological contributions, but also

an analysis of their limitations, as well as implications for developing an improved

methodology for noise and hardware bias analysis with GF approaches and triple

frequency signals.

More specifically, this chapter addresses the following.

➢ Stochastic Modelling of GNSS Observations (Section 2.1)

Reviews methods for stochastic analyses of noise and multipath interference

together with their applications and limitations

➢ Differential Code Biases Estimation and Application (Section 2.2)

Investigates and summarises existing strategies for bias estimation, their

performance and limitations.

➢ Uncalibrated Phase Delay Estimation and Application (Section 2.3)

Summarises the UPD estimation methods and limitations.

➢ Inter-System Bias among GNSS (Section 2.4)

Reviews the ISB concept and estimation methods.

The literature reviews are summarised in the last section. The discussions

highlight the implications of previous work and develops the conceptual framework

for the studies presented in the chapters that follow.

2.1 STOCHASTIC MODELLING OF GNSS OBSERVATIONS

2.1.1 Background

A functional model and a relevant stochastic model are essential in most typical

GNSS applications, especially those using a Least Squares (LS) method or a Kalman

filter (Jin, Wang, & Park, 2005). The functional model, which is known as a

mathematical model with a series of linear or nonlinear equations (see Eq. 1-1 and Eq.

40 Literature Review

1-2) for a fundamental description, or Eq. 2-1 and Eq. 2-2 for traditional PPP

functional models), presents the mathematical relationship between the measurements

and the unknowns to be estimated. The unknowns include satellite orbital and user

position states, ambiguity parameters and propagation delays (Ren & Lu, 2012).

2 2

1 ,1 2 ,2

, , , ,2 2

1 2

s s

r rs s s s

r IF r r IF r IF P

f P f PP p b

f f

Eq. 2-1

2 2

1 ,1 2 ,2

, , , , ,2 2

1 2

s s

s r r s s s s

r IF r IF r IF IF r IF r IF

f fp N B

f f

Eq. 2-2

here (default unit is metre):

,

s

r IFP : Ionosphere-free code pseudorange measurement

,

s

r IF : Ionosphere-free carrier phase measurement

s

rp The sum of non-dispersive terms: position, clock,

tropospheric delay, s s sr r rp Clk Clk T

s

rX X

,

s

r IFb Hardware delays in the ionosphere-free code measurement

, ,

s

r IF P Noise in the ionosphere-free code measurement

IF Wavelength in the ionosphere-free phase measurement

,

s

r IFN Ambiguity in the ionosphere-free phase measurement

(nondimensional)

,

s

r IFB Hardware delays in the ionosphere-free phase measurement

in cycle

, ,

s

r IF Noise in the ionosphere-free phase measurement

The stochastic model describes the observations statistical properties (Leick,

Rapoport, & Tatarnikov, 2015; Rizos, 1997). It is usually presented in the form of a

variance-covariance matrices (VCM) characterising the accuracy and correlations by

the main and off-diagonal elements respectively. It also describes the expectation of

multivariate pseudo range and phase noise time series (Ren & Lu, 2012; Schön &

Brunner, 2008b). Here, the variance is defined to be the dispersion of measurements.

The covariance represents the spatial correlation between different channels, the cross

correlation among carrier frequencies and the temporal correlation between different

epochs. The spatial correlation is caused by the similar observational conditions of the

measurements from one receiver to different satellites or from different receivers to


one satellite. Intuitively, the closer spatially the observations are, the stronger the

correlation is. Previous analyses of time series data have shown that correlations exist

between GPS L1 and L2 observations. Depending on the types of the receivers, the

cross-correlation coefficients can reach up to 0.8 between the phase observations

(Amiri-Simkooei, Teunissen, & Tiberius, 2009; Tiberius & Kenselaar, 2003).

However, the correlation is negligible for code and phase observations (Bona, 2000).

The slowly variant residual systematic errors may contribute temporal correlation

between the observations at different epochs. This temporal correlation relates to the

satellite geometry, the atmospheric conditions, the station-specific effects (e.g.,

multipath) and the receiver physical characteristics (Amiri-Simkooei & Tiberius, 2007;

Howind, Kutterer, & Heck, 1999; Kim & Langley, 2001; Nahavandchi & Joodaki,

2010; Schön & Brunner, 2008a, 2008b). In general, larger temporal separation result

in weaker temporal correlations. In addition to the factors mentioned above, the

solution type may also affect the temporal correlation level. For instance, PPP has a

smaller correlation, while the DD procedure could increase the correlation length (Luo,

2013; Nahavandchi & Joodaki, 2010). Figure 2-1 shows a fully populated Variance-

Covariance Matrices (VCM) for original undifferenced GPS phase observations from

one station (R) to four satellites ( j, k, l, r ) at two epochs (t1, t2) (Luo, 2013).


Figure 2-1 A fully Populated Variance-Covariance Matrices (VCM) in the original undifferenced

phase observations

Both the functional and stochastic models play critical roles in solution quality

(Jin, et al., 2005). However, in the existing geometry-based methods, the stochastic

model depends on the selected functional model (Tiberius & Kenselaar, 2000) and the

systematic errors (caused by the multi-path, atmosphere, orbit effects). The functional

model relies on knowledge of the physical phenomena that pertain to these errors


(Satirapod, 2013). In principle, an enhanced stochastic model can further improve the

accuracy and reliability of the final solution (Satirapod & Luansang, 2008), i.e. integer

ambiguity, position and troposphere parameters (Li, Dingfa, et al., 2015). It is well

known that statistical properties of GNSS codes and phase measurements are time

varying. They are affected by several factors such as satellite type, tracking loop

characteristics, elevation angles, observational environment, the receiver antenna,

receiver dynamics, the receiver hardware and software. Ideally, the full covariance

matrices for pseudo range and phase noises should be obtained for each line-of-sight

direction and updated from time to time, without dependence on the functional models.

The resulting measurement covariance matrices could then be used as part of the linear

observational equation system in follow-on state estimation and quality assessment.

Any misspecifications of the stochastic model may result in unreliable solutions. For

example, a simplified VCM that only possesses diagonal elements of variances, may

result in biased parameter estimates, resulting in over-optimistic accuracy measures

(Bischoff, Heck, Howind, & Teusch, 2005; El-Rabbany, 1994; Li, Shen, & Lou, 2011;

Luo, 2013).

In both the static and kinematic positioning applications, such as SPP, PPP, RTK,

POD and RAIM, the importance of stochastic model has been extensively investigated

(Amiri-Simkooei, Jazaeri, Zangeneh-Nejad, & Asgari, 2016; Brown, et al., 2002; Chan

& Pervan, 2009; Cross, Hawksbee, & Nicolai, 1994; Gao & Shen, 2002; Grejner-

Brzezinska, Da, & Toth, 1998; Han, 1997; He et al., 2015; Jin, et al., 2005; Julien,

Alves, Cannon, & Lachapelle, 2004; Kim & Langley, 2001; Lau & Mok, 1999;

Satirapod, 2001; Schön & Brunner, 2008b; Teunissen, 1998; Wieser & Brunner, 2002).

These comprehensive investigations conclude the followings.

1) The accuracy following from each observation (from different frequencies,

different receivers, different satellites, different direction, different epochs,

etc.) and the correlation between them varies. These differences should not

be neglected if a more precise solution is desired.

2) A precise observation should be assigned a higher weight and have more

contribution to unknown estimation. Outliers could be undetected and

contaminate the final results if improper weights are applied, or even worse,

rejecting true high-quality observations. In this case, having a large number


of redundant observations cannot prevent a considerable loss of accuracy

(Wieser, 2007).

Consider a simple stochastic model which assumes that all measurements are

uncorrelated and have identical variances. The use of this model can reduce baseline

estimation deviations with about 2 cm in the height direction. The simple satellite

elevation-based cosine function in the stochastic model can improve the baseline

estimations, including the vertical direction (Jin, et al., 2005). Studies into inaccurate

results caused by misspecifications of stochastic models are detailed in (Cannon &

Lachapelle, 1996; Satirapod, Wang, & Rizos, 2001; Wang, et al., 1998).

In the practice, complex relationships exist between observation weightings and

influencing factors, including tracking loop characteristics, receiver and antenna

hardware properties, signal strength, receiver dynamics, multipath and atmospheric

effects (Luo, 2013). In general, the data in VCM can be obtained from instrument

calibration, experience or GNSS-specific error budget analyses.

The classical stochastic models are reviewed in the sub-sections that follow.

2.1.2 Empirical Models

1) Independent and Identically Distributed Model

Many investigations of physical phenomena and experiments with real data have

revealed the variability and correlation of GNSS code and phase observations.

However, it is still commonly assumed that the error in the raw measurements (code

or phase) from the same GNSS (GPS, BDS, GLONASS) are independent and

identically distributed in both temporal and spatial domains (Ren & Lu, 2012; Wang,

Satirapod, & Rizos, 2002). This is called the Independent and Identically Distributed

(I. I. D) Model. In this model, the code pseudo range measurements (denoted by P )

have same variance (2

P ) and the carrier phase measurements (denoted by ) have

same variance (2

). The covariance matrices for all undifferenced observations at

single epoch m or in a session with t epochs are:

2

,m P PCov I ,2

,m Cov I Eq. 2-3

2

P PCov I ,2

Cov I Eq. 2-4

where:


,m PCov : Pseudo range covariance matrices at single epoch

2

P : Pseudo range variance

I Unit matrices

,m Cov Carrier phase covariance matrices at single epoch

2

Phase variance

PCov : Pseudo range covariance matrices in a session

Cov Carrier phase covariance matrices in a session

Eq. 2-3 and Eq. 2-4 indicate the spatial and temporal independence among the

observations.

Through the error propagation law (Eq. 2-5), the DD measurements result in a

time-invariant covariance matrices for pseudo range and carrier phase, respectively as

(Dai, et al., 2008):

T

B A B A Vec D Vec Cov D Cov D Eq. 2-5

2 2, T TP DD DD S R P DD P DD DD Cov D I D D D Eq. 2-6

2 2, T TDD DD S R DD DD DD Cov D I D D D Eq. 2-7

where:

AVec Data vector A

BVec Data vector B

D Mapping matrices from data vector A to B

TD The transpose of D

ACov Covariance matrices of data vector A

BCov Covariance matrices of data vector B

,P DDCov : DD pseudo range covariance matrices

DDD :

( 1) ( 1) ( )

1 1 0 1 1 0

0 1 1 0

1 0 0 1 0 1

DD

S R S R

D ,


( 1) ( 1) ( 1) ( 1)

4 2 2

2 4 2

2

2 2 4

T

DD DD

S R S R

D D

S Number of satellites

R Number of receivers

,DDCov DD carrier phase covariance matrices

The Independent and Identically Distributed Model offers simplicity and

availability so that it still can be used in some kinematic cases, and in some business

software (Ren & Lu, 2012). Ignoring the physical correlations between measurements

is an oversimplification and does not support accurate modelling of actual situations.

The unrealistic uncorrelated and homoscedastic signal assumptions render the I. I. D.

model inadequate for precise applications, especially in case of having the

observations with low elevations (Satirapod & Luansang, 2008; Wieser, 2007).

2) Elevation-Dependent Models

To model the heteroscedasticity in GNSS signals, satellite elevation angles are

often employed as a quality indicator for assessing one-way measurements. They are

also used in the construction of stochastic models in most software packages. Some

high-end software packages may also provide post-processing options (Tiberius &

Kenselaar, 2000). The basic assumption in the elevation-dependent weighting

approach is that measurements with lower elevations generally suffer more from

atmospheric delays and multipath effects, therefore they are noisier than those with

higher elevations (Jin, et al., 2005; (Luo, 2013). In other words, the measurements

from a low-elevation satellite has a larger standard deviation, than from a satellite close

to the zenith. Experimental results also support assumptions that the variance of the

code and phase observables per channel/satellite depends on the elevation and that

significant mutual correlation exists between either Cl and P2 code or L1 and L2 phase

observables (Tiberius & Kenselaar, 2000). It has also been reported that the elevation

dependence is valid only when satellite elevations are greater than about 40° or 55°

(Li, Dingfa, et al., 2015).

The elevation-angle dependent model assumes that different noise levels exist at

different elevation angles - no spatial correlation or temporal models are considered.

The elevation-angle dependent approach improves the ambiguity resolution efficiency


and should be used in the first instance, provided that the parameters in the model are

known or resolvable (Ren & Lu, 2012). In fact, elevation-dependent stochastic models

are used in many recent BDS/GPS receiver investigations, (Deng, Tang, Liu, & Shi,

2014; Odolinski, Odijk, & Teunissen, 2014; Odolinski, Teunissen, & Odijk, 2013;

Odolinski, Teunissen, & Odijk, 2014a, 2014b).

geometry-free analysis approaches for noises and … · computational algorithms of hardware delays...

Documents