The GNSS integer ambiguities: estimation and validation

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convex region symmetric with respect to the true baseline. It followed that the probability that the fixed solution is close to the true solution is always larger than for any of the other baseline estimators. At the same time, the fixed solution also has the highest probability of being far from the true solution. The opposite is true for the float estimator: smallest probability of being close to the true solution, but also small probability that it will be far off. The probabilities of both the BIE and the IA estimators generally are inbetween those of the fixed and float estimators. So, as could be expected the BIE and IA estimators take the best of the float and fixed estimators. Another probability considered here is the probability that a baseline estimator is better than the corresponding float estimator. Especially if IA estimation with a small fixed fail rate is used, this probability will be close to one and larger than for any of the other estimators. A problem is that the probabilities cannot be evaluated exactly, and no strict lower and upper bounds are available, so that simulations are needed. It would be interesting to have an easy-to-evaluate measure of the quality of the baseline solution. This is a point of further research. 6.3 Reliability of the results In this thesis simulations were used in order to assess the theoretical performance of different estimators and to make comparison possible, since one can be sure that the simulated data correspond to the mathematical model that is used. In practice, one cannot be sure that this will be the case, whereas the reliability of the ambiguity resolution results depends on the validity of the input. Only if one can assume that the input is correct, one can trust the outcome of integer estimation and validation. The input of the ambiguity resolution process consists of the float ambiguity estimates and their vc-matrix, and thus depends on the functional as well as the stochastic model that is used. Furthermore, it is important that the quality of the float solution is evaluated, such that errors due to for example multipath and cycle slips are eliminated. In practice the stochastic model is often assumed to be correct, although recent studies have shown that in many cases a refinement is possible. Possible refinements are the application of satellite elevation dependent weighting of the observations, and considering cross-correlation of the different observations. This means an appropriate stochastic model must be determined for the receiver at hand. Another assumption that was made is that the variance factor of unit weight, σ 2 , is known. If, on the other hand, it is assumed that this variance factor is not known, it must be estimated a posteriori by means of variance component estimation. It must be investigated how this affects the results presented in this thesis. Furthermore, it is assumed that the observations are normally distributed. The results in this thesis are all based on this assumption, although it would also be possible to apply the theory to parameters with different distributions. This is also an issue open for further research. 142 Conclusions and recommendations
6.4 Bias robustness In the preceding section it was stressed that it is important to eliminate errors in the float solution as much as possible. In that respect it would be interesting to investigate how ’common’ biases, such as multipath, outliers in the data, and cycle slips, propagate into the final solution if they are not corrected for or eliminated. It is namely possible that due to the bias the ambiguity is estimated to the wrong integer with IA estimation, which might result in a baseline estimate that has a larger bias than the float baseline estimate. Bias robustness 143
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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,
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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
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
Page 151 and 152: 5.8.2 OIA estimation versus RTIA an
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 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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The GNSS integer ambiguities: estimation and validation

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