The GNSS integer ambiguities: estimation and validation

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2.3 GNSS functional model . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.1 General mathematical model . . . . . . . . . . . . . . . . . . . 14 2.3.2 Single difference models . . . . . . . . . . . . . . . . . . . . . . 15 2.3.3 Double difference models . . . . . . . . . . . . . . . . . . . . . 18 2.4 GNSS stochastic model . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.1 Variance component estimation . . . . . . . . . . . . . . . . . . 21 2.4.2 Elevation dependency . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.3 Cross-correlation and time correlation . . . . . . . . . . . . . . 22 2.5 Least-squares estimation and quality control . . . . . . . . . . . . . . . 22 2.5.1 Model testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5.2 Detection, Identification and Adaptation . . . . . . . . . . . . . 24 2.5.3 GNSS quality control . . . . . . . . . . . . . . . . . . . . . . . 25 3 Integer ambiguity resolution 27 3.1 Integer estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1.1 Integer rounding . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1.2 Integer bootstrapping . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.3 Integer least-squares . . . . . . . . . . . . . . . . . . . . . . . . 32 3.1.4 The LAMBDA method . . . . . . . . . . . . . . . . . . . . . . 33 3.1.5 Other ambiguity resolution methods . . . . . . . . . . . . . . . 36 3.2 Quality of the integer ambiguity solution . . . . . . . . . . . . . . . . . 36 3.2.1 Parameter distributions of the ambiguity estimators . . . . . . . 37 3.2.2 Success rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2.3 Bias-affected success rates . . . . . . . . . . . . . . . . . . . . 45 3.3 The ambiguity residuals . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3.1 Parameter distribution of the ambiguity residuals . . . . . . . . 46 3.3.2 PDF evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Quality of the fixed baseline estimator . . . . . . . . . . . . . . . . . . 51 3.4.1 Parameter distributions of the baseline estimators . . . . . . . . 53 3.4.2 Baseline probabilities . . . . . . . . . . . . . . . . . . . . . . . 55 3.5 Validation of the fixed solution . . . . . . . . . . . . . . . . . . . . . . 56 3.5.1 Integer validation . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5.2 Discrimination tests . . . . . . . . . . . . . . . . . . . . . . . . 60 3.5.3 Evaluation of the test statistics . . . . . . . . . . . . . . . . . . 63 3.6 The Bayesian approach . . . . . . . . . . . . . . . . . . . . . . . . . . 64 ii Contents
4 Best Integer Equivariant estimation 67 4.1 The BIE estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 Approximation of the BIE estimator . . . . . . . . . . . . . . . . . . . 70 4.2.1 The integer set . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.2 Z-transformation . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3 Comparison of the float, fixed, and BIE estimators . . . . . . . . . . . . 72 4.3.1 The 1-D case . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3.2 The 2-D case . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3.3 The geometry-based case . . . . . . . . . . . . . . . . . . . . . 81 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5 Integer Aperture estimation 87 5.1 Integer Aperture estimation . . . . . . . . . . . . . . . . . . . . . . . . 88 5.1.1 Class of IA estimators . . . . . . . . . . . . . . . . . . . . . . . 88 5.1.2 Distribution functions . . . . . . . . . . . . . . . . . . . . . . . 89 5.1.3 IA estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2 Ellipsoidal integer aperture estimation . . . . . . . . . . . . . . . . . . 92 5.3 Ratio test, difference test and projector test . . . . . . . . . . . . . . . 94 5.3.1 Ratio test is an IA estimator . . . . . . . . . . . . . . . . . . . 94 5.3.2 F -Ratio test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.3.3 Difference test is an IA estimator . . . . . . . . . . . . . . . . . 99 5.3.4 Projector test is an IA estimator . . . . . . . . . . . . . . . . . 101 5.4 Integer Aperture Bootstrapping and Least-Squares . . . . . . . . . . . . 102 5.4.1 Integer Aperture Bootstrapping . . . . . . . . . . . . . . . . . . 103 5.4.2 Integer Aperture Least-Squares . . . . . . . . . . . . . . . . . . 104 5.5 Penalized Integer Aperture estimation . . . . . . . . . . . . . . . . . . 107 5.6 Optimal Integer Aperture estimation . . . . . . . . . . . . . . . . . . . 111 5.7 Implementation aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.7.1 Determination of the aperture parameter using root finding methods115 5.7.2 Determination of the aperture parameter using simulations . . . 118 5.7.3 Determination of the Integer Aperture estimate . . . . . . . . . 119 5.8 Comparison of the different IA estimators . . . . . . . . . . . . . . . . 120 5.8.1 Examples for the geometry-based case . . . . . . . . . . . . . . 120 5.8.2 OIA estimation versus RTIA and DTIA . . . . . . . . . . . . . . 125 5.8.3 IAB and IALS estimation . . . . . . . . . . . . . . . . . . . . . 131 5.9 Performance of IA estimation . . . . . . . . . . . . . . . . . . . . . . . 131 Contents iii
Page 1: The GNSS integer ambiguities: estim
Page 4 and 5: The GNSS integer ambiguities: estim
Page 7: Contents Preface v Summary vii Same
Page 11: Preface It was in the beginning of
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
Page 60 and 61:
7 6 5 4 3 2 1 0 −1 −2 −3 −4
Page 62 and 63:
Table 3.1: Overview of ambiguity re
Page 64 and 65:
1 0.5 0 −2 −1 0 1 z 2 1 0 −1
Page 66 and 67:
and has units of cycles. It was int
Page 68 and 69:
chosen equal to half the number of
Page 70 and 71:
Table 3.2: Two-dimensional example.
Page 72 and 73:
The bias-affected success rate is a
Page 74 and 75:
is close to one. This probability c
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2 0 −2 0.5 0 −0.5 2 0 −2 −4
Page 78 and 79:
error in pdf error in pdf error in
Page 80 and 81:
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
Page 96 and 97:
4.2 Approximation of the BIE estima
Page 98 and 99:
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 (|
Page 106 and 107:
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
Page 113 and 114:
Integer Aperture estimation 5 In se
Page 115 and 116:
can be distinguished: â ∈ Ωa s
Page 117 and 118:
and the hybrid distribution of ā i
Page 119 and 120:
Figure 5.3: 2-D example for EIA est
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as follows: Ωz,R = ΩR ∩ Sz =
Page 123 and 124:
3 2.5 2 1.5 1 0.5 0 −0.5 −1 −
Page 125 and 126:
5.3.3 Difference test is an IA esti
Page 127 and 128:
Figure 5.9: 2-D example for DTIA es
Page 129 and 130:
Figure 5.11: 2-D example for PTIA e
Page 131 and 132:
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
Page 139 and 140:
The optimization problem of (5.49)
Page 141 and 142:
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
Page 145 and 146:
In order to show that the fail rate
Page 147 and 148:
success rate percentage identical 0
Page 149 and 150:
probability probability 1 0.9 0.8 0
Page 151 and 152:
5.8.2 OIA estimation versus RTIA an
Page 153 and 154:
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.
Page 163:
Table 5.4: Overview of IA estimator
Page 166 and 167:
IE IA I Figure 6.1: The set of rela
Page 168 and 169:
convex region symmetric with respec
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Mathematics and statistics A A.1 Kr
Page 173 and 174:
The expectation, E{x}, and the disp
Page 175 and 176:
Choose the tolerance ε. Determine
Page 177 and 178:
Simulation and examples B Throughou
Page 179 and 180:
Theory of BIE estimation C In chapt
Page 181 and 182:
The first term on the right-hand si
Page 183:
can be derived: L = {2h(x) T Q
Page 186 and 187:
1. Choose fixed fail rate: Pf = β
Page 189 and 190:
Bibliography Abidin HA (1993). Comp
Page 191 and 192:
Jung J, Enge P, Pervan B (2000). Op
Page 193 and 194:
Teunissen PJG (2003g). Towards a un
Page 195 and 196:
Index adjacent, 33 ADOP, 42 alterna
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The GNSS integer ambiguities: estimation and validation

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