The GNSS integer ambiguities: estimation and validation

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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 0.2 0.4 0.6 0.8 1 ε 1.2 1.4 1.6 1.8 2 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 0.5 1 1.5 2 2.5 ε Figure 5.22: 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 10 01. Right: example 10 03. probability 1 0.99 0.98 0.97 0.96 0.95 0.94 10_01 06_01 10_02 0.93 0 0.1 0.2 0.3 fail rate 0.4 0.5 Figure 5.23: Probability P ( ¯ bx − bx ≤ ˆ bx − bx) as function of the fail rate. 124 Integer Aperture estimation 10_03 06_02
5.8.2 OIA estimation versus RTIA and DTIA The results in the preceding section have shown that the RTIA and the DTIA estimator both perform close to optimal for most examples. Therefore, in this section the relation between these estimators and the OIA estimator is investigated. Recall that with the ratio test the fixed solution is accepted if and only if: R1 ≤ µR R2 With the difference test the acceptance criterion is: R2 − R1 ≥ µD With OIA estimation the fixed solution is used if and only if: with fˇɛ(â − ǎ) ≤ µO fâ(â − ǎ) fˇɛ(â − ǎ) fâ(â − ǎ) = ∞ i=1 exp{− 1 2 Ri} exp{− 1 2 1 exp{− 2 = 1 + R1} R2} exp{− 1 + R1} 2 ∞ i=3 exp{− 1 2 Ri} exp{− 1 2 R1} (5.63) where the i refers to the solution that is best (1), second-best (2), etcetera. This expression follows from equations (3.26) and (3.48). An important difference with the ratio test statistic and the difference test statistic is that in this expression all Ri, i = 1, . . . , ∞ are taken into account and not just those that correspond to the best and second-best integer solution. From equation (5.63) follows that: fˇɛ(â − ǎ) 1 ≥ 1 + exp{− fâ(â − ǎ) 2 (R2 − R1)} (5.64) So, if Ω ′ 0 is the aperture pull-in region corresponding to: 1 + exp{− 1 2 (R2 − R1)} ≤ µO if follows that Ω0,O ⊂ Ω ′ 0. Equation (5.65) can be rewritten as: (5.65) R2 − R1 ≥ −2 ln(µO − 1) (5.66) Equation (5.66) is identical to the difference test. Hence, it follows that Ω0,O ⊂ Ω0,D if µD = −2 ln(µO − 1), and the fail rate and success rate of the DTIA estimator will be higher than with OIA estimation. If the precision as described by the vc-matrix Qâ is high the squared norms Ri, i > 1 will be large and the last term on the right-hand side of equation (5.63) will become small. Comparison of the different IA estimators 125
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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,
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
Page 149: probability probability 1 0.9 0.8 0
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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