The GNSS integer ambiguities: estimation and validation

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Determine the zero-crossing of the tangent line of the function at xk: xk+1 = xk − f(xk) f ′ , k = 0, 1, 2, . . . (xk) Compute the function value at xk+1, and evaluate whether or not: |f(xk+1)| < ε If yes, the root is found, otherwise continue. The Newton-Raphson method is in general much more efficient than other ’simple’ methods such as the bisection method, but it can only be used when it is possible to compute the derivative of f(x) at the reference point. It is, however, possible to simplify the method by replacing f ′ (xk) by a constant, but this may corrupt the speed of convergence. A.3.5 Matlab function fzero In this research the Matlab R○ function fzero was used in order to find the root of a function. The algorithm that is used is a combination of bisection, the secant method and inverse quadratic interpolation, and is based on the Algol 60 version described in (Brent 1973). 150 Mathematics and statistics
Simulation and examples B Throughout this work several examples are based on simulations for which the vc-matrix of the float solution is required as input. In this appendix the method of simulation and the underlying models for the input are described. B.1 Simulation Simulation is a useful method for evaluation of the success rate and fail rate. Although in practice it would be too time-consuming, the method is used in this research in order to evaluate the results of different ambiguity resolution methods. Simulation goes as follows. Since it is assumed that the float solution is normally distributed, the probabilities are independent of the mean, so one can use N(0, Q) and draw samples from this distribution. For example, if only float ambiguities need to be generated, the first step is to use a random generator to generate n independent samples from the univariate standard normal distribution N(0, 1), and then collect these in a vector s. This vector is transformed by means of â = Gs, with G equal to the Cholesky factor of Qâ = GG T . The result is a sample â from N(0, Qâ), and this sample is used as input for the integer estimator that is considered. If the output of this estimator equals the null vector, then it is correct, otherwise it is incorrect. This process can be repeated N number of times, and one can count how many times the null vector is obtained as solution, say Ns times, and how often the outcome equals a nonzero integer vector, say Nf times. The approximations of the success rate and fail rate follow than as: Ps = Ns N , Pf = Nf N In order to get good approximations, the number of samples N must be chosen sufficiently large, see (Teunissen 1998c). The random generator used in this research is the Matlab R○ routine mvnrnd from the Statistics toolbox. 151
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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 −
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: Choose the tolerance ε. Determine
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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