The GNSS integer ambiguities: estimation and validation

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variane ratio 1.4 1.2 1 0.8 0.6 0.4 0.2 success rate fixed / float BIE / float fixed / BIE 0 0 20 40 60 80 100 120 140 160 180 200 number of epochs Figure 4.3: Variance ratios of: BIE and float estimator; BIE and fixed estimator; fixed and float estimator. Success rate as function of the number of epochs is also shown. 4.3.1 The 1-D case Monte Carlo simulations were carried out to generate 500,000 samples of the float range and ambiguity, using the geometry-free single frequency GPS model for k epochs, with vc-matrix: Qâ Q â ˆ bk Qˆ bkâ Qˆ bk = σ 2 p kλ2 (1 + ε) − σ2 p − σ2 p kλ 2 kλ2 σ 2 p 1 1+ε ( k with λ the wavelength of the carrier; σ2 p and σ2 φ are the variances of the DD code and phase observations respectively, and ε = σ2 φ /σ2 p. For all simulations, the standard deviations were chosen as σp = 30 cm and σφ = 3 mm. The number of epochs was varied. Note that in the one-dimensional case the fixed ambiguity estimator is obtained by simply rounding the float estimator to the nearest integer. + ε) Figure 4.2 shows the parameter distributions of all three estimators for k = 20, based on the simulation results. Note that the multi-modality of the distribution of the BIE range estimator is less pronounced than that of the fixed range estimator. For smaller k the distribution of the BIE ambiguity and range estimator would resemble the normal PDF of the float estimators. For larger k, and thus higher precision, the distribution of the BIE estimators would more and more resemble those of the fixed estimators. From equation (4.7) follows that the BIE baseline estimator has smallest variance, but in the limits of the precision the variance will become equal to the variance of the float and fixed estimator. This is illustrated in figure 4.3, where the variance ratio of the different estimators is shown as function of k. Also the success rate is shown. Indeed, for small k the variance of the BIE and the float estimator are equal to each other (ratio equals 74 Best Integer Equivariant estimation
estimator 1.5 1 0.5 0 −0.5 −1 −1.5 −1.5 −1 −0.5 0 float ambiguity 0.5 1 1.5 Figure 4.4: Ambiguity estimators as function of the float ambiguity. Solid lines: fixed estimator; Dashed line: float estimator; the BIE estimator will always fall in the grey area. one), and smaller than the variance of the fixed estimator. On the other hand, for large k the variances of the BIE and fixed estimator become equal to each other, and smaller than that of the float estimator. Only after 30 epochs, when the success rate is larger than 0.9, the variance of the fixed baseline estimator becomes lower than the variance of the float estimator. It can be shown that in the one-dimensional case |â − ã| ≤ |â − ǎ|, ∀â, σ 2 . First note that: |â − ã| = 0 if â = 1 z, z ∈ Z (4.21) 2 Furthermore, ã is monotonically increasing with â, since: dã 1 = dâ Γ2 z(z − x)σ z∈Z −2 exp{− 1 2σ2 (x − z)2 } exp{− z∈Z 1 2σ2 (x − z)2 }− − zσ z∈Z −2 exp{− 1 2σ2 (x − z)2 } (z − x)σ z∈Z −2 exp{− 1 2σ2 (x − z)2 } = 1 σ2Γ2 u(u − x) exp{− u∈Zz∈Z 1 2σ2 (x − u)2 } exp{− 1 2σ2 (x − z)2 } −u(z − x) exp{− 1 2σ2 (x − u)2 } exp{− 1 2σ2 (x − z)2 } = 1 σ2Γ2 (u 2 − uz) exp{− 1 2σ2 (x − u)2 } exp{− 1 2σ2 (x − z)2 } u∈Zz∈Z F (u,z) ≥ 0 (4.22) Comparison of the float, fixed, and BIE estimators 75
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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
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 98 and 99: 4.3 Comparison of the float, fixed,
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
Page 149 and 150: 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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The GNSS integer ambiguities: estimation and validation

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