The GNSS integer ambiguities: estimation and validation

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Table 3.2: Two-dimensional example. Mean and maximum difference between success rate, Ps, based on simulations and the approximations. The success rate for which the maximum difference is obtained is given in the last row. LB bootstr. LB region UB ADOP UB region ADOP mean difference 0.005 0.018 0.001 0.018 0.004 max. difference 0.010 0.105 0.003 0.065 0.009 Ps 0.805 0.388 0.833 0.559 0.805 Table 3.3: Approximated success rates using simulations (sim.), the lower bounds based on bootstrapping (LB bootstr.) and bounding the integration region (LB region), the upper bounds based on ADOP (UB ADOP) and bounding the integration region (UB region), and the approximation based on ADOP (ADOP). example sim. LB bootstr. LB region UB ADOP UB region ADOP 2-D 1.000 0.999 1.000 1.000 1.000 0.999 06 01 0.818 0.749 0.698 0.848 0.942 0.765 06 02 0.442 0.410 0.118 0.475 0.626 0.417 10 01 0.989 0.976 0.988 0.999 0.992 0.978 10 03 0.476 0.442 0.126 0.681 0.661 0.525 though it is more useful to know how well the approximations work in practice. Therefore, simulations were carried out for several geometry-based models, see appendix B. The resulting lower and upper bounds are shown in table 3.3. The first row shows the results for the two-dimensional vc-matrix with f = 1. The results show that Kondo’s lower bound works very well for a high success rate, but in general the bootstrapped lower bound is much better. It can be concluded that Kondo’s lower bound seems to be useful only in a few cases. Firstly, to obtain a strict lower bound the precision should be high, so that the success rate is high. Even then, it depends on the minimum required success rate whether or not it is really necessary to use the approximation: if the bootstrapped success rate is somewhat lower than this minimum required success rate, Kondo’s approximation can be used to see if it is larger. The minimum required success rate could be chosen such that the fixed ambiguity estimator can be considered deterministic. In this case, the discrimination tests as used in practice can be used, see section 3.5. An advantage of the bootstrapped success rate is that it is very easy to compute, since the conditional variances are already available when using the LAMBDA method. The computation of Kondo’s lower bound is more complex, since for high-dimensional problems the number of facets that bound the pull-in region can be very large. It is difficult to say which upper bound is best. For the examples with only four visible satellites the ADOP-based upper bound is better than the one obtained by bounding the integration region, but in the examples with more satellites the latter is somewhat better. All bounds are best in the case of high precisions, i.e. high success rates. All in all, one can have a little more confidence in the ADOP-based bound, since its overall performance, based on all examples, is slightly better. An advantage of the ADOPbased upper bound is that it is easy to compute, whereas using the upper bound based 44 Integer ambiguity resolution
on bounding the integration region brings the problem of determining the n closest independent integers to the zero vector. The approximation based on ADOP performs quite well for the examples shown here. Except for example 10 03, this approximation is smaller than the empirical success rate, and better than the lower bound. The vc-matrix of example 10 03 is obtained for a relatively long baseline and its dimension is equal to 10. This results in conditional standard deviations of the transformed ambiguities that may differ considerably from the approximate average given by ADOP. This explains why the approximation is not so good in this case. 3.2.3 Bias-affected success rates In section 2.5.1 the minimal detectable effect (MDE) was introduced as a measure of the impact of a certain model error on the final solution. Using equation (2.77) the impact on the float solution can be computed, but for high-precision GNSS applications it would be more interesting to know the impact on the fixed solution. For that purpose, one could compute the effect of a bias in the float ambiguity solution, ∇â, on the success rate. This so-called bias-affected success rate can be computed once ∇â is known, see (Teunissen 2001a). The bootstrapped success rate in the presence of biases, and after decorrelation so that the bias is ∇ˆz = Z T ∇â, becomes: P∇(ˇzB = z) = = S0,B 1 |Qâ|(2π) 1 exp{−1 2 n F −1 (S0,B) 1 1 exp{−1 |D|(2π) 2 n 2 x − ∇ˆz2 Qˆz }dx 2 y − L−1 ∇ˆz 2 D}dy, (3.45) where the transformation F : y = Lx was used, so that the bootstrapped pull-in region is transformed as: F −1 (S0,B) = {y ∈ R n | |c T i y| ≤ 1 , i = 1, . . . , n} 2 with ci the vector with a one as ith entry and zeros otherwise. The transformed pull-in region equals an origin-centered cube with all sides equal to 1. Since D is a diagonal matrix, the multivariate integral can be written as: n 1 P∇(ˇzB = z) = √ exp{− σi|I 2π 1 yi − c 2 T i L−1 2 ∇ˆz }dyi σi|I = = i=1 |yi|≤ 1 2 n i=1 n i=1 1−2c T i L−1 ∇ˆz 2σ i|I − 1+2cT i L−1 ∇ˆz 2σ i|I Φ 1 √ 2π exp{− 1 2 v2 }dv 1 − 2c T i L −1 ∇ˆz 2σ i|I + Φ 1 + 2c T i L −1 ∇ˆz 2σ i|I − 1 (3.46) Quality of the integer ambiguity solution 45
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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
Page 19: gelegd door de keuze van de maximaa
Page 22 and 23: d, δ instrumental delay for code a
Page 25: Acronyms ADOP Ambiguity Dilution Of
Page 28 and 29: in practice, a user has to choose b
Page 31 and 32: GNSS observation model and quality
Page 33 and 34: Table 2.2: Signal and frequency pla
Page 35 and 36: 2.2.2 Phase observations The phase
Page 37 and 38: the propagation delay does not depe
Page 39 and 40: 2.2.9 The geometry-based observatio
Page 41 and 42: 2.3.2 Single difference models If o
Page 43 and 44: and difficult to predict. Therefore
Page 45 and 46: 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 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 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
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as follows: Ωz,R = ΩR ∩ Sz =
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3 2.5 2 1.5 1 0.5 0 −0.5 −1 −
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5.3.3 Difference test is an IA esti
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Figure 5.9: 2-D example for DTIA es
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Figure 5.11: 2-D example for PTIA e
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Figure 5.12: 2-D example for IAB es
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2-D example Figure 5.13 shows in bl
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µ 200 100 10 2 1 1 1.1 1.2 1.3 1.4
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5.6 Optimal Integer Aperture estima
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The optimization problem of (5.49)
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Table 5.1: Comparison of IA estimat
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µ 1.5 1.45 1.4 1.35 1.3 1.25 1.2 1
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In order to show that the fail rate
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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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The GNSS integer ambiguities: estimation and validation

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