The GNSS integer ambiguities: estimation and validation

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Table 4.1: Probabilities P1 = P (| ˇ b − b| ≤ | ˆ b − b|), P2 = P (| ˜ b − b| ≤ | ˆ b − b|), and P3 = P (| ˜ b − b| ≤ | ˇ b − b|). 4.3.2 The 2-D case k P (ǎ = a) P1 P2 P3 1 0.249 0.557 1.000 0.443 2 0.346 0.583 1.000 0.417 5 0.522 0.635 0.642 0.365 10 0.685 0.707 0.716 0.296 20 0.844 0.820 0.826 0.240 30 0.918 0.885 0.887 0.301 60 0.986 0.937 0.937 0.458 75 0.994 0.939 0.939 0.480 110 0.999 0.933 0.933 0.552 200 1.000 0.913 0.913 0.982 A similar approach was followed to generate samples for the geometry-free dual-frequency GPS model for one satellite-pair. The double difference standard deviations were chosen such that success rates between 0.3 and 0.99 were obtained, see table 4.2. Figure 4.7 shows an example of the BIE and fixed ambiguity estimates that correspond to certain float ambiguity vectors (σp = 0.8m, σφ = 8mm). It can be seen that the BIE estimator is pulled in approximately the same direction as the ILS estimator, but the ambiguity residuals are always much smaller for this example. If the float solution falls close to an integer grid point, the BIE estimator is pulled in that direction. On the other hand, if the float solution falls close to the boundary of a pull-in region, which means that the distance to at least two integer grid points is approximately the same, the ambiguity residual is small and the BIE estimator is pulled in another direction. Figure 4.8 shows a scatter plot of simulated float and corresponding BIE ambiguity estimates (σp = 0.6m, σφ = 6mm). For higher precisions, the ’star’-shape of the distribution of the BIE ambiguity estimator becomes even stronger; for lower precisions the distribution becomes more ellipse-shaped like that of the float estimator. Table 4.2 shows the probabilities P1 = P (| ˇ b−b| ≤ | ˆ b−b|), P2 = P (| ˜ b−b| ≤ | ˆ b−b|), and P3 = P (| ˜ b−b| ≤ | ˇ b−b|) This table only shows the probabilities that the estimators will perform better than or equal to the other estimators. However, this does not say how much better, or how much worse in the instances that they do not perform better. Therefore, one should also consider the probabilities as shown in figure 4.9. From the table it follows namely that P (| ˇ b − b| ≤ | ˆ b − b|) is larger than P (| ˜ b − b| ≤ | ˆ b − b|). On the other hand, figure 4.9 shows that if ˇ b is not close to b, it will immediately be far from the true solution (because of the step-wise function), whereas for the BIE estimator that is not necessarily the case. 78 Best Integer Equivariant estimation
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 Figure 4.7: For different float ambiguity vectors the corresponding BIE and fixed estimator are determined. The arrows point from float to corresponding BIE estimator. Figure 4.8: Scatter plot of float ambiguities (black) and BIE ambiguity estimates (grey). Comparison of the float, fixed, and BIE estimators 79
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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
Page 54 and 55: * â Figure 3.1: An ambiguity pull-
Page 56 and 57: 2 1.5 1 0.5 0 −0.5 −1 −1.5
Page 58 and 59: estimator is given by: n Sz,B = x
Page 60 and 61: 7 6 5 4 3 2 1 0 −1 −2 −3 −4
Page 62 and 63: Table 3.1: Overview of ambiguity re
Page 64 and 65: 1 0.5 0 −2 −1 0 1 z 2 1 0 −1
Page 66 and 67: and has units of cycles. It was int
Page 68 and 69: chosen equal to half the number of
Page 70 and 71: Table 3.2: Two-dimensional example.
Page 72 and 73: The bias-affected success rate is a
Page 74 and 75: is close to one. This probability c
Page 76 and 77: 2 0 −2 0.5 0 −0.5 2 0 −2 −4
Page 78 and 79: error in pdf error in pdf error in
Page 80 and 81: 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 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 and 154: µ µ µ 0.7 0.6 0.5 0.4 0.3 0.2 0.
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0.6 0.4 0.2 0 −0.2 −0.4 −0.6
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three vc-matrices which are scaled
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Table 5.3: Comparison of IAB and IA
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success rate if fixed 1 0.95 0.9 0.
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Table 5.4: Overview of IA estimator
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IE IA I Figure 6.1: The set of rela
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convex region symmetric with respec
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Mathematics and statistics A A.1 Kr
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The expectation, E{x}, and the disp
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Choose the tolerance ε. Determine
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Simulation and examples B Throughou
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Theory of BIE estimation C In chapt
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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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