The GNSS integer ambiguities: estimation and validation

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4.3 Comparison of the float, fixed, and BIE estimators Although it is known that the BIE estimator outperforms its float and fixed counterparts in terms of precision, it is not clear how large this difference will be under varying measurement scenarios. It follows that in the limits of the precision the following is true: lim σ→∞ ã = â and lim ã = ǎ (4.17) σ→0 with the vc-matrix factored as Qâ = σ2Gâ. The first limit corresponds to an integer grid size which is very small in relation to the size and extend of the PDF of the float ambiguities. This implies that the summation over all integers will be approximately equal to the integration over the space of reals, and then: R ã = n x exp{− 1 2â − x2Qâ }dx exp{− 1 2â − x2 n â(2π) 2 |Qâ| = }dx Qâ (2π) n = â (4.18) 2 |Qâ| R n This shows that if the precision is low the BIE estimator will approximate the float estimator. If, on the other hand, the PDF of the float ambiguities becomes very peaked the BIE estimator will approximate the fixed estimator. This can be shown as follows. Let the ILS pull-in region of z ∈ Z n be given as Sz = {x ∈ R n | x − z 2 Qâ ≤ x − u 2 Qâ , ∀u ∈ Zn }. Then Sz is independent of σ, and the weight wz(x) is equal to (see equation (4.8)): wz(x) = 1 exp{− 2x − z2Qâ } exp{− 1 2x − u2 Qâ } u∈Z n = 1 + u∈Z n \{z} 1 exp{− 1 2σ2 (x − u2 Gâ − x − z2 Gâ )} (4.19) It follows thus that lim wz(x) = 1 if x ∈ Sz, and since the sum of all weights must σ→0 equal one lim wu(x) = 0, ∀u = z if x ∈ Sz. So, wu(x) reduces to the indicator function σ→0 of the ILS pull-in region in the limit σ → 0. Obviously, the limiting cases in equation (4.17) imply that also the following is true: lim σ→∞ â − ã2Qâ = 0 and lim σ→0 â − ã2Qâ = â − ǎ2Qâ (4.20) It is now interesting to know how the BIE estimator performs in the intermediate cases compared to the float and fixed estimators. In this section the three estimators are therefore compared numerically. This also provides the possibility to compare their distributional properties. The results were presented in Verhagen and Teunissen (2003). 72 Best Integer Equivariant estimation
elative frequency relative frequency 4 3.5 3 2.5 2 1.5 1 0.5 fixed 0 −1.5 −1 −0.5 0 a 0.5 1 1.5 100 90 80 70 60 50 40 30 20 10 0 −0.25 −0.2 −0.15 −0.1 −0.05 0 b 0.05 0.1 0.15 0.2 0.25 Figure 4.2: Examples of distributions of the different estimators. Top: â (solid), ǎ (bars), ã (dashed); Bottom: ˆ b (solid), ˇ b (solid, multi-modal), ˜ b (dashed). Comparison of the float, fixed, and BIE estimators 73 BIE fixed BIE float float
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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
Page 47 and 48: (appendix A.1): Qy = C sd pφ ⊗ D
Page 49 and 50: The quality of the solution can the
Page 51: 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 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 and 148: success rate percentage identical 0
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probability probability 1 0.9 0.8 0
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5.8.2 OIA estimation versus RTIA an
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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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