The GNSS integer ambiguities: estimation and validation

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0.6 0.4 0.2 0 −0.2 −0.4 −0.6 y −0.6 −0.4 −0.2 0 0.2 0.4 0.6 Figure 5.17: Aperture parameter for OIA estimation approximated using IAB estimation with Pf,IAB = 0.025. y is an element of the boundary of both the IAB pull-in region and the approximated OIA pull-in region. The dashed OIA pull-in region is the one for which Pf,OIA = 0.025. Since it is quite easy to determine the aperture parameter µ for a fixed fail rate in the case of IAB or EIA estimation, it would be interesting to investigate whether this parameter can be used to approximate the aperture parameter for e.g. Optimal IA estimation such that Pf,OIA(µ ′ ) ≈ Pf,IAB(µ) with µ ′ the approximated aperture parameter. One approach would be to determine the aperture parameter for IAB estimation using a root finding method. Then an ambiguity vector on the boundary of the bootstrapped aperture pull-in region Ω0,B is determined. From the definition of the bootstrapped pullin region in equation (3.12) it can be seen that S0,B is constructed as the intersection of half-spaces which always pass through the points ± 1 2 ci, with ci the canonical vector with a one as its ith entry. From equation (3.10) follows then that â = 1 2 cn will lie on the boundary of S0,B and Scn,B since ǎj,B = ǎn,B = =⇒ j−1 âj − σâjâ σ i|I i=1 −2 â (âi|I − ǎi) i|I ⎡ n−1 ⎢ ⎣ân − i=1 ǎB = 0 or ǎB = cn Now y is chosen as: σânâ i|I σ −2 ⎤ ⎥ â (âi|I − ǎi) ⎦ = [ân] = i|I =0 ∀i = 0, ∀j = 1, . . . , n − 1 1 2 y = µ · 1 2 cn, (5.57) since this will be an element of the boundary of the scaled bootstrapped pull-in region, i.e. the IAB pull-in region. It must then be checked if y ∈ S0,LS, which will generally 116 Integer Aperture estimation
µ 1.5 1.45 1.4 1.35 1.3 1.25 1.2 1.15 1.1 1.05 1 0 0.005 0.01 0.015 0.02 0.025 fail rate 0.03 0.035 0.04 0.045 0.05 Figure 5.18: Aperture parameter for OIA estimation as function of the fail rate. Black: determined with simulations; Grey solid: approximated using IAB estimation; Grey dashed: approximated using EIA estimation. be the case when the ambiguities are decorrelated, since then the bootstrapped and the ILS pull-in regions largely overlap. If that is the case the aperture parameter for OIA estimation is approximated as: µ ′ = fˇɛ(y) , (5.58) fâ(y) since then y is also an element of the boundary of Ω0,OIA. If the aperture pull-in regions of IAB and OIA would have a similar shape, this approximation may work well. However, it does not tell whether using µ ′ will give a larger or smaller fail rate than the fixed value that was used in order to determine µ ′ . This approach is illustrated for the 2-D example in figure 5.17. The IAB pull-in region for a fail rate of 0.025 is shown. The star shows y as determined with equation (5.57) 1 . The resulting OIA pull-in region is shown with a solid line; y is an element of the boundary of this pull-in region. This ’approximated’ OIA pull-in region is smaller than the one for which Pf,OIA = 0.025 (dashed line). Figure 5.18 shows the OIA aperture parameter as function of the fail rate, where the aperture parameter is determined with simulations (see section 5.7.2), or approximated using IAB estimation. It can be seen that the approximated aperture parameter is too low, which means that Pf,OIA(µ ′ ) < β. In other words, for this example the OIA pull-in regions are always smaller than necessary if the aperture parameter is approximated using IAB estimation. A similar approach can be followed by using EIA estimation. Then a vector on the 1 Note that if bootstrapping is applied, one starts with the first ambiguity which should be the most precise float ambiguity as was explained in section 3.1.2. For the example shown here, the second ambiguity is the most precise. Hence, y is permuted so that y = µ · 1 2 c1. Implementation aspects 117
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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
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 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: Table 5.1: Comparison of IA estimat
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
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
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Teunissen PJG (2003g). Towards a un
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Index adjacent, 33 ADOP, 42 alterna
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