The GNSS integer ambiguities: estimation and validation

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in order to guarantee a low fail rate the critical value should somehow depend on the distance to all integer vectors in the neighborhood of â, where the distance is measured in the metric of Qâ. 2-D examples By comparing 2-D aperture pull-in regions of the RTIA, DTIA and OIA estimators, it is possible to get a better idea of the differences in performance of the three estimators. Therefore three aperture pull-in regions for are plotted for different vc-matrices in figure 5.25. The following vc-matrices are considered: 2 3 Qˆz 02 01, Qˆz 02 01, 4Qˆz 02 01, Qˆz 02 01 + The regions correspond to the following three situations: 0.8 0 0 0 a : µ chosen such that for each estimator the fail rate is obtained for which the percentage of RTIA / DTIA solutions identical to the OIA solution, Pid, is minimum; b : Ratio Test with fixed critical value of µ = 0.5, µ for DTIA and OIA chosen such that fail rates of all estimators are equal c : all estimators with fixed fail rate of Pf = 0.025 It follows that the first situation, smallest Pid for the same fail rate, in general occurs when the OIA pull-in region just touches the ILS pull-in region, and thus the aperture parameter is large. The reason is that for a large aperture parameter the shape of the RTIA and DTIA pull-in regions start to resemble the shape of the ILS pull-in region and is larger in the direction of the corner points of the ILS pull-in regions where little probability mass is located. The OIA pull-in region, on the other hand, touches the ILS pull-in regions there where more probability mass is located in Sa, since the PDF of â is elliptically shaped. The smallest Pid is not obtained for a large aperture pull-in region when the precision is so high that the PDF of â is peaked since then there is little probability mass close to the boundaries of the ILS pull-in region. If the aperture pull-in regions are chosen even larger than shown in figure 5.25, the OIA pull-in region is cut off by the ILS pull-in region. So, if µ is increased the OIA pull-in region cannot expand in all directions, and therefore the fail rate will not change much. The aperture pull-in regions of the RTIA and DTIA estimators are especially different from those of the OIA estimator in the direction of the vertices of the ILS pull-in region. That is because in those directions there are two integers, ǎ2 and ǎ3 with the same distance to â, but only â − ǎ22 is considered by the ratio test and difference test, Qâ whereas with OIA estimation the likelihood of all integers is considered. Using a fixed critical value for the ratio test (b) results in very different and sometimes large fail rates, depending on the vc-matrix. The reason is that the size of the aperture pull-in regions is not adjusted if the precision changes. This can be seen for the first 128 Integer Aperture estimation
0.6 0.4 0.2 0 −0.2 −0.4 −0.6 0.6 0.4 0.2 c b a −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0 −0.2 −0.4 −0.6 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 a b c −0.6 −0.4 −0.2 0 0.2 0.4 0.6 a −0.6 −0.4 −0.2 0 0.2 0.4 0.6 a b b c c −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 c b a −0.6 −0.4 −0.2 0 0.2 0.4 0.6 b c a −0.6 −0.4 −0.2 0 0.2 0.4 0.6 a b c −0.6 −0.4 −0.2 0 0.2 0.4 0.6 a b c −0.6 −0.4 −0.2 0 0.2 0.4 0.6 Figure 5.25: Aperture pull-in regions for: a) the µ that results in the smallest percentage of solutions identical to the optimal test; b) µR = 0.5; c) Pf = 0.025. Left: OIA (black) and RTIA (grey); Right: OIA (black) and DTIA (grey). Top to bottom: 2 3 Qˆz 02 01; Qˆz 02 01; 4Qˆz 02 01; Qˆz 02 01 + diag(0.8, 0). Comparison of the different IA estimators 129
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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
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 and 152: 5.8.2 OIA estimation versus RTIA an
Page 153: µ µ µ 0.7 0.6 0.5 0.4 0.3 0.2 0.
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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