The GNSS integer ambiguities: estimation and validation

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µ success rate / undecided rate 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0 0.005 0.01 0.015 0.02 0.025 fail rate 0.03 0.035 0.04 0.045 0.05 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.005 0.01 0.015 0.02 0.025 fail rate 0.03 0.035 0.04 0.045 0.05 µ success rate 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.005 0.01 0.015 0.02 0.025 fail rate 0.03 0.035 0.04 0.045 0.05 0.005 0.01 0.015 0.02 0.025 fail rate 0.03 0.035 0.04 0.045 0.05 Figure 5.28: Comparison of IAB and IALS estimation, examples 06 01 (left) and 06 02 (right). Top: Aperture parameter µ as function of the fail rate for IAB (dashed) and IALS (solid) estimation. Bottom: Success rate (black) and undecided rate (grey) as function of the fail rate. µ 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 fail rate success rate / undecided rate 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 fail rate µ success rate / undecided rate 0.75 0.7 0.65 0.6 0.55 0.5 0 0.005 0.01 0.015 0.02 0.025 fail rate 0.03 0.035 0.04 0.045 0.05 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.005 0.01 0.015 0.02 0.025 fail rate 0.03 0.035 0.04 0.045 0.05 Figure 5.29: Comparison of IAB and IALS estimation, examples 10 01 (left) and 10 03 (right). Top: Aperture parameter µ as function of the fail rate for IAB (dashed) and IALS (solid) estimation. Bottom: Success rate (black) and undecided rate (grey) as function of the fail rate. 132 Integer Aperture estimation
Table 5.3: Comparison of IAB and IALS estimation. IAB IALS µ 0.5322 0.5322 0.5500 06 01 Pf 0.0010 0.0008 0.0010 Ps 0.1336 0.1458 0.1657 µ 0.7867 0.7867 0.8000 10 01 Pf 0.0012 0.0010 0.0012 Ps 0.8456 0.9023 0.9137 concluded that the DTIA, RTIA, and the IALS estimators perform almost as well as the OIA estimator. With the implementation as described in section 5.7.3 the DTIA and RTIA estimators have the advantage to be computationally simple and less demanding than the other two estimators. OIA estimation requires the approximation of fˇɛ(x) which may involve the determination of and summation over a very large integer set, see section 3.3. IALS estimation involves applying ILS estimation twice for each generated sample. In this section, examples are worked out which illustrate the benefits of IA estimation with a fixed fail rate for practical use. Besides the OIA estimator only the RTIA estimator will be considered, because of its attractive properties described above, and because it is well-known and already often used in practice, albeit based on other acceptance criteria. The vc-matrices of examples 06 02 and 10 03 are used. These vc-matrices correspond to a single epoch model. For the following epochs it is simply assumed that Qk = 1 k Q1, with k the epoch number. This relation is valid when the satellite geometry is not changed. The aperture parameters, success rates and fail rates as function of the number of epochs are determined using simulations. The following approaches are considered: • Optimal IA estimation, fixed fail rate Pf = 0.005; • Ratio Test IA estimation, fixed fail rate Pf = 0.005; • Ratio Test, fixed critical value µ = 1 3 ; • Ratio Test, fixed critical value µ = 1 2 and fixed solution is only accepted if Ps,LS ≥ 0.99; • fixed solution is only accepted if Ps,LS ≥ 0.995. If the ILS fail rate is smaller than 0.005, all solutions are accepted with the first two IA estimators, i.e. Ω0 = S0. Note that the last approach also guarantees that the fail rate will never exceed 0.005. Performance of IA estimation 133
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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
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Table 4.1: Probabilities P1 = P (|
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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.
Page 155 and 156: 0.6 0.4 0.2 0 −0.2 −0.4 −0.6
Page 157: three vc-matrices which are scaled
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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