The GNSS integer ambiguities: estimation and validation

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success rate percentage identical 1 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 1 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.9 DTIA RTIA OIA IALS EIA 0.002 0.004 0.006 fail rate 0.008 0.01 0.002 0.004 0.006 fail rate 0.008 0.01 success rate percentage identical 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 fail rate 1 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 DTIA RTIA OIA IALS EIA 0.9 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 fail rate Figure 5.20: Top: Success rate as function of the fail rate; Bottom: percentages of solutions identical to OIA. Left: example 10 01. Right: example 10 03. A user will be mainly interested in the baseline coordinates, therefore also the following probabilities are analyzed: P ( ˙ bx − bx ≤ ɛ) (5.61) where bx refers only to the three baseline coordinates. So, the distance to the true position is considered here. The results are shown in figures 5.21 and 5.22. For all examples the OIA estimates corresponding to the largest fail rate in table 5.2 is used. Note that the same probabilities were analyzed in section 4.3.3 for the float, fixed and BIE baseline estimators. The probability that ˇ b is close to the true b is highest for most examples. Note, however, that in practice ˇ b should only be used if the ILS success rate is close to one. This is the case for example 10 01, but then the baseline probability of ¯ b is comparable to that of ˇ b. Another disadvantage of the fixed estimator is that if ˇ b is not close to b, there is a high probability that it will be very far from the true solution. It follows that the baseline probabilities of ¯ b often are better than those of ˆ b. In the figures only the probabilities of the OIA estimator are shown; for the other IA estimators the probabilities are somewhat lower. If the aperture parameter would be chosen larger or smaller, the OIA baseline probability would be more like that of the fixed or float baseline probability respectively. 122 Integer Aperture estimation
probability probability 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 ε 4 5 6 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 12 ε probability probability 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ε 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 12 ε Figure 5.21: Top: P ( ˙ b−b 2 Q ˙b ≤ ɛ). Bottom: P ( ˙ bx −bx 2 ≤ ɛ). Dotted: ˙ b = ˆ b; Dashed: ˙b = ˇ b; Solid: ˙ b = ¯ b. Left: example 06 01. Right: example 06 02. Besides the probabilities as defined in equations (5.60) and (5.61), it is also interesting to know the probability that the IA baseline estimator is better than or identical to the float estimator, i.e. P ( ¯ bx − bx ≤ ˆ bx − bx) (5.62) As explained in section 3.4 the fixed baseline probabilities cannot be evaluated exactly; the same is thus true for the IA estimators. The probability is plotted in figure 5.23 as function of the fail rate for the different examples. The end points correspond to the ILS fail rate and therefore the probability is equal to P ( ˇ bx − bx ≤ ˆ bx − bx). Especially for low fail rates the probabilities are high, and P ( ¯ bx − bx ≤ ˆ bx − bx) > P ( ˇ bx − bx ≤ ˆ bx − bx). That is because then either the undecided rate will be high and thus there is a high probability that ¯ b = ˆ b, or the success rate is high so that there is a high probability that ˇ b is better than ˆ b if ¯ b = ˇ b. For examples 06 01 and 06 02 the probability for the IA baseline estimator is always better than for the fixed estimator; but that is not true for examples 10 02 and 10 03. Since a user will be interested in low fail rates, these probabilities show that the IA aperture estimator is to be preferred above the fixed estimator. Comparison of the different IA estimators 123
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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
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 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: success rate percentage identical 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
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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The GNSS integer ambiguities: estimation and validation

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