The GNSS integer ambiguities: estimation and validation

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0.5 0 −0.5 0.5 0 −0.5 0.5 0 −0.5 −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5 Figure 5.15: Aperture space Ω0 as function of the penalties, for Qˆz 02 01. Top: PDF of â (within pull-in region); Center: PDF of ˇɛ; Bottom: Ω0 as function of µ. 110 Integer Aperture estimation 2 1.5 1 0.5 2 1.8 1.6 1.4 1.2 1 0.8 0.6 2.5 2 1 µ 1.5
5.6 Optimal Integer Aperture estimation The method of penalized IA estimation as described in section 5.5 is some way regulates the probabilities of success and failure. As mentioned in the beginning of this chapter, this is why the new class of integer estimators was defined. A disadvantage of PIA estimation is that the user has to choose the values for the penalties that will be assigned to the three cases success, failure and undecided. A user would like to choose the penalties in such a way that he can be sure that the fail rate is below a certain value. All other IA estimators described in previous sections are not optimal in the sense of maximization of the success rate. Therefore, still another approach could be based on the optimization problem defined as: max Ps subject to: Pf = β (5.49) Ω0⊂S0 where β is a chosen value for the fail rate, (Teunissen 2003g). In words, this means that the aperture space is defined such that the success rate is maximized subject to a fixed fail rate. The solution of the optimization problem is given by: Ω0 = {x ∈ S0 | fâ(x + z) ≤ µfâ(x + a)} (5.50) z∈Z n The proof was given in Teunissen (2004d), and is as follows. Lemma 5.6.1 Let f(x) and g(x) be integrable functions over R n . Then the region ˆΩ = {x ∈ R n | f(x + z) ≥ λ g(x + z), λ ∈ R} (5.51) z∈Z n z∈Z n solves the constrained maximization problem: max Ω⊂Rn Ω subject to: Ω⊂R n f(x)dx (5.52) g(x)dx = c Ω = Ω + z, ∀z ∈ Z n if λ ∈ R is chosen such that the integral constraint of equation (5.53) is satisfied. (5.53) Note the similarity with the Neyman-Pearson lemma, cf. (Rao 1965). The proof of lemma 5.6.1 is as follows. Optimal Integer Aperture estimation 111
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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
Page 86 and 87: always larger than one. This shows
Page 88 and 89: where χ2 α(m−p, 0) is the criti
Page 90 and 91: Table 3.5: Overview of all test sta
Page 92 and 93: with the weight function wz(â) = 1
Page 94 and 95: 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: µ 200 100 10 2 1 1 1.1 1.2 1.3 1.4
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 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
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1. Choose fixed fail rate: Pf = β
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Bibliography Abidin HA (1993). Comp
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Jung J, Enge P, Pervan B (2000). Op
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Teunissen PJG (2003g). Towards a un
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Index adjacent, 33 ADOP, 42 alterna
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The GNSS integer ambiguities: estimation and validation

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