The GNSS integer ambiguities: estimation and validation

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Verhagen S, Teunissen PJG (2004c). PDF evaluation of the ambiguity residuals. In: F Sansò (Ed.), V. Hotine-Marussi Symposium on Mathematical Geodesy, International Association of Geodesy Symposia, Vol. 127, Springer-Verlag. Vollath U, Birnbach S, Landau H, Fraile-Ordoñez JM, Martín-Neira M (1998). Analysis of Three-Carrier Ambiguity Resolution (TCAR) Technique for precise relative positioning in GNSS-2. Proc. of ION GPS-1998, Nashville TN: 417–426. Wang J, Stewart MP, Tsakiri M (1998). A discrimination test procedure for ambiguity resolution on-the-fly. Journal of Geodesy, 72: 644–653. Wei M, Schwarz KP (1995). Fast ambiguity resolution using an iteger nonlinear programming method. Proc. of ION GPS-1995, Palm Springs CA: 1101–1110. Zandbergen R, Dinwiddy S, Hahn J, Breeuwer E, Blonski D (2004). Galileo orbit selection. Proc. of ION GPS-2004, September 21-24, Long Beach CA. 168 Bibliography
Index adjacent, 33 ADOP, 42 alternative hypothesis, 23 ambiguity, 9 ambiguity residual, 46 Anti-Spoofing, 6 aperture parameter, 115 aperture pull-in region, 88 aperture space, 88, 111 baseline increments, 16 Bayesian approach, 65 BIE estimation, 68, 153 bootstrapping, 31 χ 2 -distribution, 146 code observation, 7 cofactor matrix, 58 confidence region, 55 critical value, 23 decorrelation, 34 detection power, 23 DIA, 24 difference test, 62, 99, 125 discrimination, 60 disjunct, 29, 88 double difference, 18 elevation dependency, 21 ellipsoidal IA estimation, 92 external reliability, 24 fail rate, 89 false alarm rate, 23 F -distribution, 147 fixed fail rate, 91, 111, 115 fixed solution, 28 float solution, 28 Galileo, 6 gamma function, 146 Gauss-Markov model, 14 geometry-free, 12 GLONASS, 6 GNSS, 5 GPS, 5 IA estimation, 88 indicator function, 29 integer aperture estimation, 88 integer equivariant, 67 integer least-squares, 32 integer set, 47, 70 internal reliability, 24 ionosphere, 10 fixed, 17 float, 17 weighted, 17, 20 LAMBDA, 33 LDL T -decomposition, 31, 34 least-squares, 22 level of significance, 23 minimal detectable bias, 24 minimal detectable effect, 24 multi-modal, 54 multipath, 12 non-centrality parameter, 23 normal distribution, 145 null hypothesis, 23 optimal IA estimation, 111 parameter distribution, 36 penalized IA estimation, 107 phase observation, 9 169
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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
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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
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: Teunissen PJG (2003g). Towards a un
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