The GNSS integer ambiguities: estimation and validation

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• 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. Examples of integer validation tests proposed in literature, are the ratio test, the projector test, and the difference test. Some of these integer validation tests are defined such that the invalid assumption of deterministic fixed ambiguities 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. Obviously, validation of the integer ambiguity solution is still an open problem. In order to avoid the above two situations, two approaches are investigated here. The first method uses a new ambiguity estimator, which is in some sense always superior to its 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 is 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 would not use the fixed solution, but stick to the float solution. The BIE ambiguity estimator is equal to a weighted sum over all integer vectors, and the weights depend on the probability density function of the float ambiguities. Therefore, the BIE estimator can be considered as a compromise between the float and fixed solution; those are approximated in the extreme cases of very bad or very good precisions, respectively. The disadvantages of BIE estimation are that an exact probabilistic evaluation of the solution is still not possible, and the procedure is computationally complex. Moreover, it was shown that the BIE estimator significantly outperforms the float or fixed solution only in a limited number of cases. Therefore, another new class of integer estimators is investigated: the class of Integer Aperture (IA) estimators. IA estimation is an overall approach of integer estimation and validation. This is the case because three a priori probabilities are distinguished: the success rate, the fail rate, and the undecided rate. Around each integer an identically shaped acceptance region, referred to as the aperture pull-in region, is defined, with the size determined by the choice of a maximum allowed fail rate. With respect to the shape of the regions different choices are possible. Examples are ellipsoids and scaled versions of the integer bootstrapping or integer least-squares pull-in regions. It is also shown that the acceptance regions corresponding to the ratio tests, the projector test and the difference test are examples of aperture pull-in regions. Finally, the optimal IA estimator is defined such that the success rate is maximized for a fixed fail rate. IA estimation has several important advantages. The first is that it is an overall approach to the integer estimation and validation problem, which makes exact probabilistic evaluation of the final solution possible. Furthermore, the acceptance region is determined by the choice of the maximum allowed fail rate, which is exactly what a user wants. This means that the region depends on the model at hand, and thus on the precision. With the traditional approaches this is not the case, so that in general the tests are either too conservative or too optimistic. In the first case this leads to a unnecessarily long time to first fix, in the second case the fail rate will be unacceptably high. viii Summary
Finally, it is shown that the popular ratio test and the difference test perform almost as good as the optimal IA estimator if the fixed fail rate approach is used. So, a theoretical justification is given for these tests, such that an appropriate choice of the critical values is possible. Also the IA least-squares estimator, based on using downscaled integer least-squares pull-in regions, performs well. This estimator is identical to the integer least-squares estimator combined with integer hypothesis testing, but with the advantage that the acceptance region is determined by the choice of the fail rate and not by the false alarm rate. Summary ix
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 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 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
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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,
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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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