The GNSS integer ambiguities: estimation and validation

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This means that the higher the precision, the more identical the aperture pull-in regions of the difference test and the optimal test become. It follows from equation (5.66) that then µO ≈ exp{− 1 2 µD} + 1 or µD ≈ −2 ln(µO − 1) (5.67) In figures 5.26 and 5.27 the aperture parameter µO is shown as function of the fail rate for different examples. The black solid lines show the µO that follow from the simulations, the dashed grey line show the µ ′ O as approximated with (5.67) as function of the DTIA fail rate. For example 10 01, the precision is high and the results in table 5.2 already showed that the DTIA estimator performs almost identical to the OIA estimator. It could therefore be expected that µO ≈ exp{− 1 2 µD} + 1, which is indeed shown in the figure. On the other hand, for the other examples the DTIA estimator performs worse than the OIA estimator. Because of the poorer precision the terms Ri, i > 3, and thus the last term on the right-hand side of (5.63), cannot be ignored. Not surprisingly µO is clearly different from µ ′ O . In the bottom panels of the figures, the OIA success rates as function of the ’fixed’ fail rate are shown in the case µO would be used, and in the case µ ′ O would be used. So, in the latter case the success rate is shown as function of the DTIA fail rate which was used to determine µ ′ O . As expected, only for example 10 01 approximately the same results are obtained. From equation (5.66) follows that it is not possible to rewrite the OIA estimator in the same form as the ratio test (3.78), but both tests depend on R1 and R2, which is the reason that similar results can be obtained with RTIA and OIA estimation if it would be possible to choose the critical value of the ratio test in an appropriate way. In practice µR is often chosen equal to a fixed value and the ratio test should only be applied if the precision is high. Figure 5.24 shows on the left the aperture parameter µR as function of the fail rate for the different examples. It can be seen that for the examples where n = 10 a small fail rate will be obtained when µR = 0.3; in the case of poorer precisions on the other hand, it is not clear how the appropriate µR can be chosen such that the fail rate will be low enough, but at the same time such that the test is not too conservative. The F -ratio test ê2 + R1 Qy ê2 ≤ µR + R2 Qy also looks at R1 and R2, but as explained this IA estimator is not really comparable with all others because it also depends on the float residuals ê. In practice the aperture parameter is often chosen equal to 1 2 or as 1/Fα(m − p, m − p, 0), see section 3.5. The panels on the right of figure 5.24 show the aperture parameter as function of the fail rate and as function of α in the case the F -distribution would be used to determine the aperture parameter. As explained, the latter choice is actually not correct. It follows that it is not obvious how α must be chosen, which confirms that it is better not to incorrectly use the F -distribution. With µR = 1 2 the fail rate will generally be quite low. The results of this section give an explanation for the fact that it is difficult to find a good criterion to choose the critical value for the ratio test and the difference test: 126 Integer Aperture estimation
µ µ µ 0.7 0.6 0.5 0.4 0.3 0.2 0.1 02_01 0 0 0.01 0.02 0.03 0.04 0.05 fail rate 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.01 0.02 0.03 0.04 0.05 fail rate 1 0.8 0.6 0.4 0.2 10_01 06_01 10_03 10_02 06_02 0 0 0.01 0.02 0.03 0.04 0.05 fail rate µ µ µ 1 0.8 0.6 0.4 0.2 02_01 0 0 0.1 fail rate / α 0.2 0.3 1 0.8 0.6 0.4 0.2 06_01 06_02 0 0 0.1 fail rate / α 0.2 0.3 1 0.8 0.6 0.4 0.2 10_01 10_02 10_03 0 0 0.1 fail rate / α 0.2 0.3 Figure 5.24: Aperture parameter µ as function of the fail rate. Left: RTIA; Right: F -RTIA. Top: example 02 01; Center: examples 06 xx; Bottom: examples 10 xx. Comparison of the different IA estimators 127
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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
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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
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 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: 5.8.2 OIA estimation versus RTIA an
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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