The GNSS integer ambiguities: estimation and validation

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implies that the success rate increases more than with the fixed critical value approach, since fâ(x + a) is the same, but the integration region is larger. 5.10 Summary In practice validation of the integer ambiguity solution is taken care of by using test statistics, which lack a sound theoretical basis. The problem is then the choice of the critical value. As an alternative, in this chapter the concept of integer aperture estimation with a fixed fail rate was presented. The advantages are: • It is an overall approach: estimation and validation are not considered as separate problems. • An exact and overall probabilistic evaluation of the final solution is possible. • The user only needs to choose the maximum allowable fail rate, which is exactly the parameter on which a user wants to put restrictions. Moreover, this means that the size of the acceptance region is really based on the model at hand. Choosing a fixed critical value or a critical value based on incorrect assumptions on the distribution of the test statistic, as is common practice, will only work when the precision is high, and may even then result in a test that is either too conservative or too optimistic. • The discrimination tests – ratio tests, difference test, projector test – as used in practice belong to the class of integer aperture estimators, and the performance of the ratio test and difference test is shown to be close to optimal. So, based on the theory of Integer Aperture inference the theoretical foundation as well as the practical relevance of the ratio test and difference test have been shown. • Using the fixed fail rate approach implies that the critical value depends on the model (and thus the precision) at hand. This means that the time to first fix will be shorter, and at the same time it is guaranteed that the probability of incorrect fixing is below a user-defined threshold. Several new IA estimators have been presented; the optimal IA estimator is one of them. This estimator maximizes the success probability for a given fail rate. An overview of all IA estimators is given in table 5.4. Unfortunately, for most IA estimators it is difficult to determine the aperture parameter. Simulations were used to evaluate and compare the IA estimators. It follows that the ratio test IA estimator (RTIA), the difference test IA estimator (DTIA), and the IA least-squares estimator (IALS) have a close to optimal performance. In practice often the RTIA estimator is used, because it is known to work well if the success rate is high. From the comparison with the OIA estimator, it is now explained why this estimator and the DTIA estimator perform so well. The close to optimal performance of the IALS 136 Integer Aperture estimation
Table 5.4: Overview of IA estimators. The last column gives the methods that can be used for the determination of the aperture parameter µ in the case of fixed fail rate IA estimation. definition aperture space: determination µ EIA ellipsoid â − ǎ 2 Qâ ≤ µ2 root finding method RTIA ratio test F -RTIA F -ratio test â−ǎ 2 Qâ â−ǎ22 Qâ ê 2 Qy +â−ǎ2 Qâ ê2 Qy +â−ǎ22 Qâ DTIA difference test â − ǎ2 2 Qâ − â − ǎ2 Qâ PTIA projector test IAB bootstrapping 1 µ ˇɛB ∈ S0,B IALS least-squares 1 µ ˇɛLS ∈ S0,LS OIA optimal ≤ µ empirical/simulation ≤ µ empirical/simulation ≥ µ empirical/simulation (ǎ2−ǎ) T Q −1 â (â−ǎ) ≤ µ empirical/simulation ǎ2−ǎQâ root finding method = µB/simulation fˇɛ(ˇɛLS) fâ(ˇɛLS+a) ≤ µ simulation estimator is also not surprising, since it makes use of the scaled ILS pull-in region as aperture space, and the ILS estimator is known to be optimal in the class of admissible integer estimators. Which IA estimator to use? The RTIA and DTIA estimator are easy to apply and computationally simple, since the squared norms of the ambiguity residuals are already available after computing the ILS solution. However, these estimators are only close to optimal if an appropriate aperture parameter is used based on a fixed fail rate; and that is only possible by using simulations. The IALS estimator might be a good alternative: close to optimal performance, and easy to apply. The aperture parameter could be chosen equal to that of the IAB estimator for a given fail rate. Unfortunately, this will make the IALS estimator less optimal. If simulations are used for the determination of the aperture parameter, the computational efficiency of the IALS estimator will be much less than that of the RTIA and DTIA estimators, since ILS estimation has to be applied twice for each sample. In theory, the OIA estimator is of course to be preferred since it maximizes the success rate. However, computationally it is the most complex estimator because the probability density of the ambiguity residual must be computed. Furthermore, it is only possible to determine the aperture parameter for a given fail rate using simulations. If the aperture parameter is approximated using the IAB estimator, the resulting estimator will not be optimal. Summary 137
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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
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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
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 and 160: Table 5.3: Comparison of IAB and IA
Page 161: success rate if fixed 1 0.95 0.9 0.
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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