The GNSS integer ambiguities: estimation and validation

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d, δ instrumental delay for code and phase observations respectively da, δa atmospheric delay for code and phase observations respectively T tropospheric delay I ionospheric delay µj = λ2 j /λ21 dm, δm multipath effect for code and phase observations respectively M real-valued carrier phase ambiguity parameter N integer-valued carrier phase ambiguity parameter ρ satellite-receiver range r position vector u unit line-of-sight vector ψ mapping function for tropospheric delays P, Φ vector with code and phase observation respectively ρ vector with satellite-receiver ranges T vector with tropospheric delays I vector with ionospheric delays dt vector with lumped instrumental delays and clock errors a vector with carrier phase ambiguities Λ diagonal matrix with wavelengths as diagonal elements G matrix with line-of-sight vectors Ψ vector with mapping functions for tropospheric delays D double difference operator σp, σφ σI Estimation standard deviation of code and phase observation respectively standard deviation of ionospheric pseudo-observation E{·} expectation operator D{·} dispersion operator y m-vector of observations a n-vector with ambiguity parameters b p-vector with real-valued parameters: baseline, atmospheric delays, etc. e, ε noise parameters A, B design matrices: E{y} = Aa + Bb + e Qx variance-covariance matrix of x Qx|v conditional variance-covariance matrix: Qx|v = Qx − QxvQ−1 σ v Qvx 2 variance factor Gx cofactor matrix of x: Qx = σ2 Q = LDL Gx T LDLT -decomposition of Q with lower triangular matrix L, diagonal matrix D σi|I conditional standard deviation of ith entry of a parameter vector conditioned on the I = 1, . . . , i − 1 entries xvi Notation and symbols
Parameter distributions fˆx(x) probability density function of random variable ˆx f ˆx|ˆv(x|v) conditional probability density function of random variable ˆx ˆx ∼ N(x, Qx) random variable ˆx is normally distributed with mean x and variance-covariance matrix Qx ˆx ∼ χ 2 (n, λ) random variable ˆx has χ 2 -distribution with n degrees of freedom and non-centrality parameter λ ˆx ∼ F (m, n, λ) random variable ˆx has F -distribution with m and n degrees of freedom and non-centrality parameter λ P (ˆx = x) probability that ˆx will be equal to the mean x P (ˆx = x|ˆv = v) probability that ˆx will be equal to the mean x conditioned on ˆv = v α false alarm rate or level of significance γ detection power Ambiguity resolution ˆx float estimate of x (= least-squares estimate) ˇx fixed estimate of x ˇɛ ambiguity residuals â − ǎ ˜x best integer equivariant estimate of x ¯x integer aperture estimate of x Ps, Pf , Pu probability of success, failure, and undecided respectively Ps,LS, Pf,LS integer least-squares success and fail rate respectively Sz pull-in region centered at the integer z sz(x) indicator function: sz(x) = 1 ⇔ x ∈ Sz, sz(x) = 0 otherwise ˆΩ squared norm of residuals of float solution: ˆΩ = êT G−1 ˆσ y ê 2 float estimate of variance factor: ˆσ 2 ˇΩ ˆΩ = m−n−p squared norm of residuals of fixed solution: ˇΩ = ěT G−1 ˇσ y ě 2 fixed estimate of variance factor: ˇσ 2 = ˇ Ri Ω m−p squared norm of ambiguity residuals: R1 = (â − ǎ) T G −1 â (â − ǎ) = ˇ Ω − ˆ Ω R2 = (â − ǎ2) T G −1 Ωz â (â − ǎ2) aperture pull-in region centered at the integer z Ω aperture space: Ω = ∪ z∈Zn Ωz µ aperture parameter Notation and symbols xvii
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 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
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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
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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
Page 98 and 99:
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 (|
Page 106 and 107:
Table 4.2: Probabilities Ps = P (ǎ
Page 108 and 109:
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
Page 127 and 128:
Figure 5.9: 2-D example for DTIA es
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Figure 5.11: 2-D example for PTIA e
Page 131 and 132:
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
Page 145 and 146:
In order to show that the fail rate
Page 147 and 148:
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
Page 153 and 154:
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.
Page 163:
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
Page 175 and 176:
Choose the tolerance ε. Determine
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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
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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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