The GNSS integer ambiguities: estimation and validation

More documents

Recommendations

Info

µ success rate 1.4 1.35 1.3 1.25 1.2 1.15 1.1 1.05 1 0 0.005 0.01 0.015 0.02 0.025 fail rate 0.03 0.035 0.04 0.045 0.05 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.005 0.01 0.015 0.02 0.025 fail rate 0.03 0.035 0.04 0.045 0.05 µ success rate 1.6 1.5 1.4 1.3 1.2 1.1 1 0 0.005 0.01 0.015 0.02 0.025 fail rate 0.03 0.035 0.04 0.045 0.05 0.1 0.05 0 0 0.005 0.01 0.015 0.02 0.025 fail rate 0.03 0.035 0.04 0.045 0.05 Figure 5.26: Aperture parameter µO (top) and success rate (bottom) as function of the fixed fail rate that is used to determine the aperture parameter. Black: µO determined from simulations; Dashed grey: µ ′ O = exp{− 1 2 µD} + 1; Solid grey: µ ′ O approximated with IAB estimation. Left: example 06 01. Right: example 06 02. µ success rate 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.001 0.0025 0.005 fail rate 0.0075 0.01 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.001 0.0025 0.005 fail rate 0.0075 0.01 µ success rate 1.6 1.5 1.4 1.3 1.2 1.1 1 0 0.005 0.01 0.015 0.02 0.025 fail rate 0.03 0.035 0.04 0.045 0.05 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 0.005 0.01 0.015 0.02 0.025 fail rate 0.03 0.035 0.04 0.045 0.05 Figure 5.27: Aperture parameter µO (top) and success rate (bottom) as function of the fixed fail rate that is used to determine the aperture parameter. Black: µO determined from simulations; Dashed grey: µ ′ O = exp{− 1 2 µD} + 1; Solid grey: µ ′ O approximated with IAB estimation. Left: example 10 01. Right: example 10 03. 130 Integer Aperture estimation
three vc-matrices which are scaled versions of each other: the RTIA aperture pull-in regions corresponding to situation b are all equal. The fixed fail rate approach works much better than using a fixed critical value for the ratio test, because the size of the aperture pull-in region is nicely adjusted in case of a fixed fail rate, as can be seen by comparing the results corresponding to situation c for the first three vc-matrices. In practice the choice of a fixed µ often works satisfactorily, but that is because either the value is chosen very conservative, cf. (Han 1997), or it is required that the ILS success rate is close to one before an attempt to fix the ambiguities is made. The aperture pull-in regions of the DTIA and OIA estimators become more and more identical the higher the precision is, so that the relations in equation (5.67) become valid, as can be seen by comparing these regions for the first three vc-matrices. 5.8.3 IAB and IALS estimation From table 5.2 it follows that IALS estimation may perform close to optimal, but IAB estimation does not perform so good. This can also be seen in the bottom panels of figures 5.28 and 5.29. For a certain fail rate, the success rates with IAB estimation are clearly lower than with IALS estimation. The advantage of IAB estimation, however, is that exact evaluation of the fail rate and success rate is possible. Moreover, the results in table 5.2 indicate that for a fixed fail rate, the aperture parameters of both estimators do not differ that much. This can also be seen in the top panels of figures 5.28 and 5.29. For all examples µB < µLS. This gives rise to the idea to choose a maximum allowed fail rate, i.e. Pf < β, and compute the aperture parameter µB such that Pf,IAB = β. Then apply IALS estimation with µLS = µB. For this choice it follows from equations (5.29) and (5.35) that the IALS success rate will always be larger than or equal to the IAB success rate. Based on the examples it can be expected that the IALS fail rate will then be smaller than β. However, this approach is not optimal since the corresponding success rate will also be lower than the one obtained with the IALS fail rate equal to β. This is illustrated with two examples in table 5.3. Moreover, it is not guaranteed that indeed Pf,IALS(µ) ≤ Pf,IAB(µ). In section 5.7.1 an approximation of the OIA aperture parameter was given which makes use of the IAB aperture parameter. For examples 06 01, 06 02, 10 01 and 10 03 the results are shown in figures 5.26 and 5.27. The approximation does not work very well, and the results are often too conservative: the aperture pull-in regions are smaller than needed. Only for low fail rates the approximated aperture parameter is sometimes too large, so that the actual OIA fail rate will be larger than allowed. 5.9 Performance of IA estimation The examples in the preceding section aimed at showing the potentials of IA estimation in general and comparing the different IA estimators. Based on the results it can be Performance of IA estimation 131
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 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
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: 0.6 0.4 0.2 0 −0.2 −0.4 −0.6
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

Create successful ePaper yourself

Delete template?

Save as template?