The GNSS integer ambiguities: estimation and validation

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Table 5.2: Comparison of IA estimators for geometry-based examples. fail rate success rates EIA RTIA F -RTIA DTIA PTIA IAB IALS OIA 06 01 0.001 0.158 0.165 0.131 0.163 0.162 0.134 0.166 0.166 06 01 0.008 0.370 0.390 0.365 0.383 0.375 0.316 0.388 0.392 06 02 0.012 0.044 0.044 0.037 0.042 0.042 0.040 0.043 0.044 06 02 0.050 0.129 0.130 0.119 0.124 0.124 0.116 0.127 0.131 10 01 0.001 0.649 0.921 0.923 0.932 0.799 0.846 0.914 0.932 10 01 0.004 0.847 0.968 0.969 0.970 0.892 0.923 0.968 0.970 10 02 0.006 0.111 0.191 0.179 0.208 0.179 0.154 0.176 0.219 10 02 0.012 0.174 0.289 0.279 0.301 0.251 0.236 0.270 0.314 10 03 0.020 0.052 0.074 0.070 0.076 0.071 0.066 0.073 0.084 10 03 0.060 0.117 0.161 0.154 0.160 0.145 0.142 0.159 0.170 time, which is important in real-time applications, it has been investigated how much the results would differ with N much smaller so that the computation time becomes acceptable. It followed that for all models described in appendix B, already with a few thousand samples the results are quite good: same decision as with very large N in more than 99% of the cases. 5.8 Comparison of the different IA estimators In section 3.5 the integer validation tests as currently used in practice were described. If an admissible integer estimator is used in combination with one of the discrimination tests described in that section, the resulting estimator belongs to the class of integer aperture estimators as was shown in section 5.3 for the ratio test, difference test and projector test. It is now interesting to compare the performance of all the different IA estimators using the fixed fail rate approach. For that purpose simulations are used. It is then investigated which of the IA estimators perform close to optimal. Furthermore, the OIA estimator is conceptually compared with the ratio test and difference test, and with the IALS estimator. 5.8.1 Examples for the geometry-based case Simulations were carried out for several geometry-based models, see appendix B. Table 5.2 shows success rates obtained for two different fixed fail rates for all examples. From the results follows that OIA estimation indeed results in the highest success rates, but often the ratio test and the difference test give almost the same results. In section 5.8.2 these estimators are compared in more detail. Also the IALS estimator performs 120 Integer Aperture estimation
success rate percentage identical 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 fail rate 1 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 DTIA RTIA OIA IALS EIA 0.9 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 fail rate success rate percentage identical 0.12 0.1 0.08 0.06 0.04 0.02 0 0 0.005 0.01 0.015 0.02 0.025 fail rate 0.03 0.035 0.04 0.045 0.05 1 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 DTIA RTIA OIA IALS EIA 0.9 0 0.005 0.01 0.015 0.02 0.025 fail rate 0.03 0.035 0.04 0.045 0.05 Figure 5.19: Top: Success rate as function of the fail rate; Bottom: percentages of solutions identical to OIA. Left: example 06 01. Right: example 06 02. quite well. The EIA estimator only performs well for examples 06 01 and 06 02. The performance of the F -RTIA estimator seems to be much worse than the performance of the other estimators. The reason is that the F -RTIA estimator does not only consider the ambiguity residuals, but also the residuals of the float solution, so that the aperture space is not n-dimensional, but (m − p)-dimensional. Figures 5.19 and 5.20 show the success rates as function of the fail rate for those IA estimators that perform reasonably well; only low fail rates are considered. DTIA, RTIA and IALS estimation perform well for all examples: the success rates are almost equal to the OIA success rate, and the percentage of solutions identical to the OIA solution is high. For examples 06 01 and 06 02 the RTIA estimator performs best after OIA estimation; for examples 10 01 and 10 03 the DTIA estimator performs better than RTIA estimation. EIA estimation only performs well for example 06 02. Until now the performance of the IA estimators is solely based on the ambiguity success rates and fail rates. Of course, a user is more interested in the probability that the baseline estimate will be within a certain distance from the true baseline: P ( ˙ b − b 2 Q˙ b ≤ ɛ) (5.60) where ˙ b is either the float, fixed, or IA baseline estimator. The vc-matrices Qˇ b and Q¯ b can be determined from the estimates corresponding to the simulated float ambiguities. These probabilities are difficult to interpret, but will be considered for completeness, because they tell something about the distributional properties of the baseline estimators. Comparison of the different IA estimators 121
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The GNSS integer ambiguities: estim
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The GNSS integer ambiguities: estim
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Contents Preface v Summary vii Same
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4 Best Integer Equivariant estimati
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Preface It was in the beginning of
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• No attempt is made to fix the a
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Samenvatting (in Dutch) De GNSS geh
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gelegd door de keuze van de maximaa
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d, δ instrumental delay for code a
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Acronyms ADOP Ambiguity Dilution Of
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in practice, a user has to choose b
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GNSS observation model and quality
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Table 2.2: Signal and frequency pla
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2.2.2 Phase observations The phase
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the propagation delay does not depe
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2.2.9 The geometry-based observatio
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2.3.2 Single difference models If o
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and difficult to predict. Therefore
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The double difference observation v
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(appendix A.1): Qy = C sd pφ ⊗ D
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The quality of the solution can the
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is maximum. The degrees of freedom
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* â Figure 3.1: An ambiguity pull-
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2 1.5 1 0.5 0 −0.5 −1 −1.5
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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
Page 96 and 97: 4.2 Approximation of the BIE estima
Page 98 and 99: 4.3 Comparison of the float, fixed,
Page 100 and 101: variane ratio 1.4 1.2 1 0.8 0.6 0.4
Page 102 and 103: ambiguity residual 0.5 0.4 0.3 0.2
Page 104 and 105: Table 4.1: Probabilities P1 = P (|
Page 106 and 107: Table 4.2: Probabilities Ps = P (ǎ
Page 108 and 109: Table 4.4: Probabilities that float
Page 110 and 111: 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
Page 121 and 122: 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
Page 133 and 134: 2-D example Figure 5.13 shows in bl
Page 135 and 136: µ 200 100 10 2 1 1 1.1 1.2 1.3 1.4
Page 137 and 138: 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
Page 143 and 144: µ 1.5 1.45 1.4 1.35 1.3 1.25 1.2 1
Page 145: In order to show that the fail rate
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.
Page 155 and 156: 0.6 0.4 0.2 0 −0.2 −0.4 −0.6
Page 157 and 158: three vc-matrices which are scaled
Page 159 and 160: Table 5.3: Comparison of IAB and IA
Page 161 and 162: 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
Page 171 and 172: 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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