The GNSS integer ambiguities: estimation and validation

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5.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6 Conclusions and recommendations 139 6.1 Integer estimation and validation . . . . . . . . . . . . . . . . . . . . . 139 6.2 Quality of the baseline estimators . . . . . . . . . . . . . . . . . . . . . 141 6.3 Reliability of the results . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.4 Bias robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 A Mathematics and statistics 145 A.1 Kronecker product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 A.2 Parameter distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 145 A.2.1 The normal distribution . . . . . . . . . . . . . . . . . . . . . . 145 A.2.2 The χ 2 -distribution . . . . . . . . . . . . . . . . . . . . . . . . 146 A.2.3 The F -distribution . . . . . . . . . . . . . . . . . . . . . . . . 147 A.2.4 Student’s t-distribution . . . . . . . . . . . . . . . . . . . . . . 147 A.3 Numerical root finding methods . . . . . . . . . . . . . . . . . . . . . 148 A.3.1 Bisection method . . . . . . . . . . . . . . . . . . . . . . . . . 148 A.3.2 Secant method . . . . . . . . . . . . . . . . . . . . . . . . . . 148 A.3.3 False position method . . . . . . . . . . . . . . . . . . . . . . . 149 A.3.4 Newton-Raphson method . . . . . . . . . . . . . . . . . . . . . 149 A.3.5 Matlab function fzero . . . . . . . . . . . . . . . . . . . . . . 150 B Simulation and examples 151 B.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 B.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 C Theory of BIE estimation 153 C.1 Integer equivariant ambiguity estimation . . . . . . . . . . . . . . . . . 153 C.2 Integer equivariant unbiased ambiguity estimation . . . . . . . . . . . . 153 C.3 Best integer equivariant unbiased ambiguity estimation . . . . . . . . . 154 C.4 Best integer equivariant unbiased baseline estimation . . . . . . . . . . 156 D Implementation aspects of IALS estimation 159 E Curriculum vitae 161 Bibliography 163 Index 169 iv Contents
Preface It was in the beginning of the nineties that the first major steps for solving the problem of GPS integer ambiguity resolution were made. Ten years later part of the problem could be considered solved, as efficient algorithms were available for the estimation of the integer ambiguities with the highest possible success rate. However, as a geodesist one cannot be content with this result, since parameter resolution means estimation and validation. The latter problem turned out to be very complex due to the non-standard distributions of the parameters involved. Prof. Peter Teunissen has made a major contribution to the solution of the integer estimation problem with the introduction of the LAMBDA method in 1993. He made me aware of the open problem with respect to integer validation, and - as I started to work on this problem - has given me indispensable theoretical input. This thesis starts with a description of the GNSS observation models, and then outlines the theory on integer estimation and validation as it was available when I started to work on this topic. This part does contain some relatively new results on the parameter distributions, and some important comments on the validation procedures as proposed in the past and used in practice. The remaining chapters are all devoted to ’new’ estimation and validation algorithms. In this place I would of course like to thank all of my (former) colleagues at MGP for their support and contribution to a good research environment. A special thanks goes to the following people: my promotor Peter Teunissen for the supervision, input, feedback, comments and discussions; Kees de Jong, my supervisor in the first two years; Dennis Odijk for proofreading this thesis; Peter Joosten for the computer support; Ria Scholtes for the administrative support; the members of the examination committee for their critical reviews and useful comments. Finally, I would like to thank René for his help with the layout and graphics and of course his overall support. v
Page 1: The GNSS integer ambiguities: estim
Page 4 and 5: The GNSS integer ambiguities: estim
Page 7 and 8: Contents Preface v Summary vii Same
Page 9: 4 Best Integer Equivariant estimati
Page 14 and 15: • No attempt is made to fix the a
Page 17 and 18: Samenvatting (in Dutch) De GNSS geh
Page 19: gelegd door de keuze van de maximaa
Page 22 and 23: d, δ instrumental delay for code a
Page 25: Acronyms ADOP Ambiguity Dilution Of
Page 28 and 29: in practice, a user has to choose b
Page 31 and 32: GNSS observation model and quality
Page 33 and 34: Table 2.2: Signal and frequency pla
Page 35 and 36: 2.2.2 Phase observations The phase
Page 37 and 38: the propagation delay does not depe
Page 39 and 40: 2.2.9 The geometry-based observatio
Page 41 and 42: 2.3.2 Single difference models If o
Page 43 and 44: and difficult to predict. Therefore
Page 45 and 46: The double difference observation v
Page 47 and 48: (appendix A.1): Qy = C sd pφ ⊗ D
Page 49 and 50: The quality of the solution can the
Page 51: is maximum. The degrees of freedom
Page 54 and 55: * â Figure 3.1: An ambiguity pull-
Page 56 and 57: 2 1.5 1 0.5 0 −0.5 −1 −1.5
Page 58 and 59: estimator is given by: n Sz,B = x
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7 6 5 4 3 2 1 0 −1 −2 −3 −4
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Table 3.1: Overview of ambiguity re
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1 0.5 0 −2 −1 0 1 z 2 1 0 −1
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and has units of cycles. It was int
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chosen equal to half the number of
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Table 3.2: Two-dimensional example.
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The bias-affected success rate is a
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is close to one. This probability c
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2 0 −2 0.5 0 −0.5 2 0 −2 −4
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error in pdf error in pdf error in
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0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2 2.2
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acceptable. Moreover, the bounds wi
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Unfortunately, this relatively simp
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always larger than one. This shows
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where χ2 α(m−p, 0) is the criti
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Table 3.5: Overview of all test sta
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with the weight function wz(â) = 1
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IE IEU IU LU Figure 4.1: The set of
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4.2 Approximation of the BIE estima
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4.3 Comparison of the float, fixed,
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variane ratio 1.4 1.2 1 0.8 0.6 0.4
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ambiguity residual 0.5 0.4 0.3 0.2
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Table 4.1: Probabilities P1 = P (|
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Table 4.2: Probabilities Ps = P (ǎ
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Table 4.4: Probabilities that float
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probability probability 1 0.9 0.8 0
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Integer Aperture estimation 5 In se
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can be distinguished: â ∈ Ωa s
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and the hybrid distribution of ā i
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Figure 5.3: 2-D example for EIA est
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as follows: Ωz,R = ΩR ∩ Sz =
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3 2.5 2 1.5 1 0.5 0 −0.5 −1 −
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5.3.3 Difference test is an IA esti
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Figure 5.9: 2-D example for DTIA es
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Figure 5.11: 2-D example for PTIA e
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Figure 5.12: 2-D example for IAB es
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2-D example Figure 5.13 shows in bl
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µ 200 100 10 2 1 1 1.1 1.2 1.3 1.4
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5.6 Optimal Integer Aperture estima
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The optimization problem of (5.49)
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Table 5.1: Comparison of IA estimat
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µ 1.5 1.45 1.4 1.35 1.3 1.25 1.2 1
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In order to show that the fail rate
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success rate percentage identical 0
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probability probability 1 0.9 0.8 0
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5.8.2 OIA estimation versus RTIA an
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µ µ µ 0.7 0.6 0.5 0.4 0.3 0.2 0.
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0.6 0.4 0.2 0 −0.2 −0.4 −0.6
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three vc-matrices which are scaled
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Table 5.3: Comparison of IAB and IA
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success rate if fixed 1 0.95 0.9 0.
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Table 5.4: Overview of IA estimator
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IE IA I Figure 6.1: The set of rela
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convex region symmetric with respec
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Mathematics and statistics A A.1 Kr
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The expectation, E{x}, and the disp
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Choose the tolerance ε. Determine
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Simulation and examples B Throughou
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Theory of BIE estimation C In chapt
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The first term on the right-hand si
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can be derived: L = {2h(x) T Q
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1. Choose fixed fail rate: Pf = β
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Bibliography Abidin HA (1993). Comp
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Jung J, Enge P, Pervan B (2000). Op
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Teunissen PJG (2003g). Towards a un
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Index adjacent, 33 ADOP, 42 alterna
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