The GNSS integer ambiguities: estimation and validation

More documents

Recommendations

Info

acceptable. Moreover, the bounds will only be tight if the success rate is high. Furthermore, it is also interesting to compare the ’quality’ of the float and the fixed baseline estimators. For that purpose, the probabilities P ( ˇ b ∈ Eb) and P ( ˆ b ∈ Eb) can be considered, with Eb identical in both cases. But then the choice of the region Eb is not so straightforward; with equation (3.65) it is not possible to give an exact evaluation of P ( ˆ b ∈ Eb). Another option is therefore to consider the probabilities: P ( ˆ b − b 2 Qˆ b ≤ β 2 ) and P ( ˇ b − b 2 Qˇ b ≤ β 2 ) (3.69) The first probability can be evaluated exactly since ˆb−b2 has a central χ Qˆb 2-distribution. The second probability, however, cannot be evaluated exactly. Until now the quality of the complete baseline estimator was considered. However, a user will be mainly interested in the baseline increments, whereas the baseline estimator may contain other parameters as well, such as ionospheric delays. Therefore, it is more interesting to look at the probabilities P ( ˆ bx − bx ≤ β) and P ( ˇ bx − bx ≤ β) (3.70) where the subscript x is used to indicate that only the baseline parameters referring to the actual receiver position are considered, i.e. bx = rqr from equation (2.39). So, ˆ bx − bx is the actual distance to the true receiver position. Examples It is possible to evaluate the baseline probabilities using simulations. Examples 06 01 and 06 02 from appendix B will be considered here. A large number of float baseline estimates was generated, and for each float sample the corresponding fixed baseline was computed. The vc-matrix of the fixed baseline estimator was determined from the samples. Figure 3.14 shows the various baseline probabilities. It can be seen that the bounds of equation (3.68) are generally not strict (shown in grey), and it is difficult to interpret the probabilities with Eb chosen as in equation (3.65). More examples and an evaluation of the results will be given in chapters 4 and 5, when the performance of the float and fixed estimators is compared to that of new estimators introduced in those chapters. 3.5 Validation of the fixed solution A parameter estimation theory cannot be considered complete without rigorous measures for validating the parameter solution. In the classical theory of linear estimation, the vc-matrices provide sufficient information on the precision of the estimated parameters. The reason is that a linear model applied to normally distributed (Gaussian) data, provides linear estimators that are also normally distributed, and the peakedness of the multivariate normal distribution is completely captured by the vc-matrix. 56 Integer ambiguity resolution
probability probability 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 β 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 β probability probability 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 200 400 β 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 200 400 β probability probability 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 β 20 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 β 20 Figure 3.14: Float (dashed) and fixed (solid) baseline probabilities for examples 06 01 (top) and 06 02 (bottom). Left: P ( ˆ b − b 2 Q ˆb ≤ β 2 ) and P ( ˇ b − b 2 Qˇ b ≤ β 2 ). Center: P ( ˆ b − b 2 Q ˆb|â ≤ β 2 ) and P ( ˇ b − b 2 Q ˆb|â ≤ β 2 ). Right: P ( ˆ bx − bx ≤ β) and P ( ˇ bx − bx ≤ β). Validation of the fixed solution 57
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 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 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
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 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
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 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
show all

The GNSS integer ambiguities: estimation and validation

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?