The GNSS integer ambiguities: estimation and validation

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Table B.1: Time, location, number of satellites, ionospheric standard deviations and empirical ILS success rates for the different Qâ nn xx. nn xx σI [cm] success rate 80 60 40 20 0 270 −20 −40 −60 −80 02 01 0 0.869 Qâ = 4.9718 3.8733 3.8733 3.0188 , Qˆz = 06 01 0 0.818 10:05 am, 4 ◦ East, 52 ◦ North 06 02 1 0.442 4 visible satellites 08 01 0 0.999 10:15 am, 4 ◦ East, 52 ◦ North 08 02 1 0.874 5 visible satellites 10 01 1 0.989 17:00 am, 52 ◦ East, 4 ◦ North 10 02 2 0.774 6 visible satellites 10 03 3 0.477 300 240 210 330 25 0 −50 0 50 180 30 45 60 75 90 6 17 150 30 30 60 120 90 80 60 40 20 0 270 −20 −40 −60 −80 300 240 16 210 330 25 0 −50 0 50 180 30 45 60 75 90 6 17 150 30 30 60 120 90 80 60 40 20 0 −20 −40 −60 −80 0.0865 −0.0364 −0.0364 0.0847 0 −50 0 50 Figure B.1: Skyplots corresponding to examples 06 xx (left), 08 xx (center), and 10 xx (right). B.2 Examples The names assigned to the different vc-matrices of the float ambiguities all have the form Qâ nn xx and Qˆz nn xx. The number nn is equal to the number of ambiguities n, and xx is an identification number. The two-dimensional example (n = 2) corresponds to the double difference geometryfree model for dual-frequency GPS for two satellites. The higher-dimensional examples are all based on the single epoch geometry-based model for dual-frequency GPS. These models are all set up based on the Yuma almanac for GPS week 184, 19-FEB-2003; a cutoff elevation of 15 ◦ ; undiffenced code and phase standard deviations of 30 cm and 3 mm respectively. 152 Simulation and examples 270 300 240 31 210 330 3 16 30 14 45 180 60 75 90 25 150 30 15 60 120 90
Theory of BIE estimation C In chapter 4 the theory of Best Integer Equivariant estimation was developed. In this appendix a more detailed description of the derivations is given based on (Teunissen 2003f). C.1 Integer equivariant ambiguity estimation Starting point is the integer estimator of the unknown integer ambiguities a ∈ Z n . Let this estimator be S(â) with S : R n ↦→ Z n . The new class of integer estimators to be defined here will be larger than the class of admissible integer estimators of definition 3.1.1. Only the third condition of that definition will be considered, i.e. S(â − z) + z = S(â), ∀z ∈ Z n . This means that the result of integer estimation should not change when first an arbitrary integer z is removed from the float solution, then apply the integer estimator, and then restore the integer. So, the requirement is equivalent to: S(x + z) = S(x) + z ∀x ∈ R n , z ∈ Z n Estimators that satisfy this property will be called integer equivariant (IE). Lemma C.1.1 (C.1) S(x + z) = S(x) + z ∀x ∈ R n , z ∈ Z n ⇐⇒ S(x) = x + g(x) (C.2) with g(x) periodic: g(x + z) = g(x). Proof: If S(x) = x + g(x) with g(x + z) = g(x), then S(x + z) = S(x) + z. If S(x + z) = S(x) + z, then g(x) = x − g(x) is periodic and S(x) = x + g(x). C.2 Integer equivariant unbiased ambiguity estimation It follows that: a = E{S(â)} = E{â + g(â)} = a + E{g(â)} 153
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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
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: Simulation and examples B Throughou
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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