The GNSS integer ambiguities: estimation and validation

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Figure 5.13: 2-D example for IALS estimation. Float samples for which ā = ǎ are shown as black dots. Left: Pf = 0.001; Right: Pf = 0.025. This shows that the IALS success rate can be computed in the same way as the ILS success rate by replacing Qâ with the up-scaled version 1 µ 2 Qâ. As explained in section 3.2.2 exact evaluation of the ILS success rate is too complex, but the up-scaled version of the vc-matrix can also be used to compute the lower and upper bounds of the IALS success rate as given in section 3.2.2. Lower and upper bounds for the IALS fail rate can be obtained by bounding the integration region with ellipsoidal regions. The lower bound can be obtained identically to the one for EIA estimation as described in section 5.2. The upper bound follows in a similar way by using µS0 ⊂ C ɛ 0 with C ɛ 0 an ellipsoidal region centered at 0 and with size governed by ɛ = µmax x∈S0 x Qâ . Summarizing, the following lower and upper bounds for the IALS success rate and fail rate can be used: Pf ≥ z∈Z n \{0} Pf ≤ Ps ≥ Ps ≤ P z∈Z n \{0} n [2Φ( µ i=1 P χ 2 (n, λz) ≤ 1 4 µ2 min z∈Zn \{0} z 2 Qâ P χ 2 (n, λz) ≤ µ 2 max x x∈S0 2 Qâ 2σ i|I (5.33) (5.34) ) − 1] (5.35) χ 2 (n, 0) ≤ µ2cn ADOP 2 (5.36) with λz = z T Q −1 â z, and cn and ADOP as given in section 3.2.2. Note that the lower bound of the success rate is equal to the IAB success rate. If IALS estimation with a fixed fail rate is applied, especially the upper bound of the fail rate is interesting. Unfortunately, equation (5.34) is not useful then, since max x x∈S0 2 Qâ must be known. 106 Integer Aperture estimation
2-D example Figure 5.13 shows in black all float samples for which ā = ǎ for two different fail rates. 5.5 Penalized Integer Aperture estimation Penalized Integer Aperture estimation (PIA) was introduced in Teunissen (2004e), and is based on the idea to assign a penalty to each of the possible outcomes of a decision process, and then to minimize the average penalty. In the case of the class of integer aperture estimators defined in equation (5.1) there are three outcomes and the corresponding penalties are denoted as ps for success, pf for failure, and pu for undecided. The expectation of the penalty is given by: E{p} = psPs + pf Pf + puPu (5.37) So, the average penalty is a weighted sum of the individual penalties, and depends on the chosen penalties, and on the aperture pull-in region Ω0 since this region determines the probabilities Ps, Pf and Pu as can be seen in equation (5.4). The goal is now to find the Ω that minimizes this average penalty, i.e. min Ω E{p} (5.38) This implies that the aperture space depends on the penalties. The solution is obtained by minimization of: ⎛ ⎞ ⎛ ⎞ E{p} = ps fâ(x + a)dx + pf ⎝ (fˇɛ(x) − fâ(x + a))dx⎠ + pu ⎝1 − fˇɛ(x)dx⎠ Ω0 Ω0 ⎛ ⎞ = (ps − pf ) fâ(x + a)dx + (pf − pu) ⎝ fˇɛ(x)dx⎠ + pu Ω0 Ω0 Ω0 (5.39) For the minimization of E{p} with respect to Ω only the first two terms on the righthand side of equation (5.39) need to be considered, i.e. Ω0 follows from the following minimization problem: min Ω0 Ω0 ((pf − pu)fˇɛ(x) + (ps − pf )fâ(x + a)) dx (5.40) F (x) It follows thus that Ω0 must be chosen such that all function values of F (x) are negative: Ω0,PIA = {x ∈ S0 | fˇɛ(x) ≤ pf − ps pf − pu µ fâ(x + a)} (5.41) Note that the aperture space still depends on the integer estimator that is used, since the pull-in region S0 is different for rounding, bootstrapping and integer least-squares. Penalized Integer Aperture estimation 107
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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
Page 82 and 83: acceptable. Moreover, the bounds wi
Page 84 and 85: 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: Figure 5.12: 2-D example for IAB es
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 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
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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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