The GNSS integer ambiguities: estimation and validation

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P fix 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2 4 6 8 10 number of epochs P fix 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 OIA RTIA µ= 1 / 3 µ= 1 / 2 ,P s ≥ 0.99 P s ≥ 0.995 2 4 6 8 10 number of epochs Figure 5.30: Probabilities that solution will be fixed as function of the number of epochs with 5 different tests. Left: 06 02, Right: 10 03. For (near) real-time applications it is important that the time to first fix is as short as possible. The probability that a fix is made at a certain epoch equals: Pfix = Ps + Pf (5.68) Figures 5.30 and 5.31 show Pfix and P s|ā=z obtained with the 5 approaches as function of the number of epochs. Obviously, IA estimation with a fixed fail rate result in the highest Pfix. Again, it follows that only with the fixed fail rate approach the performance of the RTIA estimator is close to optimal. Applying the ratio test with a fixed critical value of 1 3 (stars) may give comparable probabilities Pfix in the first epochs. Note that if the success rate is higher than the one obtained with the fixed fail rate, this implies that the corresponding fail rate is higher than 0.005. In later epochs the fixed critical value is far too conservative, so that the time to first fix may be unnecessarily long. For the first epoch the probability P s|ā=z is always low for the examples considered here. The reason is that the success rate Ps is very low, and thus the approximation (5.8) is not valid. For later epochs P s|ā=z becomes reasonably high with all approaches, and is highest for the fixed critical value approach. Note, however, that this is only the case because this approach is too conservative; Pfix is very low in the first epochs, so that if the solution is fixed it can be expected to be correct since the aperture space, and thus 134 Integer Aperture estimation
success rate if fixed 1 0.95 0.9 0.85 0.8 0.75 2 4 6 8 10 number of epochs success rate if fixed 1 0.95 0.9 0.85 0.8 0.75 OIA RTIA µ= 1 / 3 µ= 1 / 2 , P s ≥ 0.99 P s ≥ 0.995 2 4 6 8 10 number of epochs Figure 5.31: Success rates conditioned on ā = z as function of the number of epochs with 5 different tests. Left: 06 02, Right: 10 03. also the fail rate, is very small. With the two approaches where it is required that the ILS success rate exceeds a certain limit (x-signs and squares) no attempt is made to fix the solution in the first epochs. As soon as a fix is made, the probability P s|ā=z is high. With the OIA and the RTIA estimator with a fixed fail rate there is a very high probability that the solution will be fixed much faster than with the other approaches, and the RTIA estimator performs close to optimal. At the same time it is guaranteed that the fail rate will not exceed the user-defined threshold. This is not the case if the ratio test with a fixed critical value is used. It can be concluded that the performance of the ratio test with a fixed fail rate is much better than with a fixed critical value, especially when more epochs of data become available. The reason is that using a fixed critical value implies that the aperture pull-in region Ω0 does not change as function of the number of epochs. Recall that the success rate is given by: fâ(x + a)dx Ps = Ω0 So, if more epochs of data become available, the success rate only increases because fâ(x + a) becomes more peaked due to the improved precision. Using a fixed fail rate, on the other hand, means that Ω0 becomes larger when the precision improves. This Performance of IA estimation 135
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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
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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
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 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: Table 5.3: Comparison of IAB and IA
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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