The GNSS integer ambiguities: estimation and validation

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in practice, a user has to choose between two undesirable situations: • Integer validation is based on wrong assumptions, since the randomness of the fixed ambiguities is incorrectly ignored; • No attempt is made to fix the ambiguities because of the low success rate, although there is still a probability that the ambiguities can be fixed correctly. Some integer validations tests are defined such that the invalid assumption of a deterministic fixed ambiguity estimator is avoided. However, these tests lack a sound theoretical foundation. Moreover, often fixed critical values are used based on experience. But that implies that in many situations the tests will either be too conservative or too optimistic. 1.2 Objectives and contribution of this work Obviously, validation of the integer ambiguity solution is still an open problem. In order to deal with the above two situations, two approaches are investigated: • A new class of ambiguity estimators is used, which results in estimators that in some sense are always superior to their float and fixed counterparts. This estimator is the Best Integer Equivariant (BIE) estimator. It is best in the sense that it minimizes the mean squared errors of the estimators. This can be considered a weaker performance criterion than that of the integer least-squares estimator, the maximization of the success rate. This might be an advantage if the success rate is not high, since then a user will generally not have enough confidence in the fixed solution, and stick to the float solution. It is investigated how BIE estimation can be implemented. The performance of this estimator is then compared with the fixed and float estimators. • A new ambiguity estimator is used by defining an a priori acceptance region, or aperture space. This new ambiguity resolution method is the so-called integer aperture estimation. The problem of ambiguity validation is incorporated in the estimation procedure with this approach. Instead of distinguishing only the probability of success and of failure, also the probability of not fixing (undecided rate) is considered a priori. An integer acceptance region is then defined by putting constraints on the three probabilities; only if the float ambiguities fall in the acceptance region, the ambiguities will be fixed using one of the well-known integer estimators. The theoretical aspects of all approaches are considered, as well as implementation aspects. Thereby it is avoided to go into too much detail with respect to proofs and derivations; these are only included if they contribute to the understanding of the theory. The methods are implemented in Matlab R○. Furthermore, it is investigated how the various methods will perform. For that purpose, Monte-Carlo simulations are used, since then the ’true’ situation is known and it is possible to compare the performance of different estimators and validators. 2 Introduction
The contribution of this thesis is that a clear overview of the traditional and the new ambiguity resolution methods will be given, and it will be investigated why and when a certain approach is to be preferred – from a theoretical point of view as well as by comparing the performance of the methods. Furthermore, the implementation aspects will be discussed, as well as the open issues in that respect. 1.3 Outline The outline of this thesis is as follows. In chapter 2 a brief introduction to GNSS positioning is given. The observations models are presented, as well as the general theory of quality control for GNSS applications. Chapter 3 is devoted to the problem of ambiguity resolution. Three integer estimators are described: integer rounding, integer bootstrapping, and integer least-squares. Furthermore, the probabilistic properties of the float and fixed estimators are analyzed. Finally, the existing methods for integer validation are described and evaluated. The pitfalls of these methods, both theoretical and practical, will be discussed. Integer least-squares estimation is generally accepted as the method to be used for integer ambiguity estimation. A disadvantage is, however, that it is not guaranteed that the resulting baseline estimator is closer to the true but unknown baseline than the original least-squares (float) estimator. Therefore, it is investigated in chapter 4 how the best integer equivariant estimator performs, which is defined such that the resulting baseline estimator always outperforms the least-squares solution in a certain sense. The conclusions on the existing integer validation methods reveal the need for an integer testing method, which is based on a sound theoretical criterion. For that purpose, an alternative ambiguity resolution method, namely integer aperture estimation, is developed in chapter 5. It is shown that integer estimation in combination with the well-known ambiguity validation tests result in estimators, which belong to the class of integer aperture estimators. Furthermore, some alternative integer aperture estimators are presented, among them the optimal integer aperture estimator. It is defined such that the maximum success rate is obtained given a fixed fail rate. Finally, the conclusions and recommendations are given in chapter 6. Outline 3
Page 1: The GNSS integer ambiguities: estim
Page 4 and 5: The GNSS integer ambiguities: estim
Page 7 and 8: Contents Preface v Summary vii Same
Page 9 and 10: 4 Best Integer Equivariant estimati
Page 11: Preface It was in the beginning of
Page 14 and 15: • No attempt is made to fix the a
Page 17 and 18: 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 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 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
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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,
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variane ratio 1.4 1.2 1 0.8 0.6 0.4
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ambiguity residual 0.5 0.4 0.3 0.2
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Table 4.1: Probabilities P1 = P (|
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Table 4.2: Probabilities Ps = P (ǎ
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Table 4.4: Probabilities that float
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probability probability 1 0.9 0.8 0
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Integer Aperture estimation 5 In se
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can be distinguished: â ∈ Ωa s
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and the hybrid distribution of ā i
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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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