The GNSS integer ambiguities: estimation and validation

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error in pdf error in pdf error in pdf 10 −5 10 −10 10 −15 10 −20 10 −5 10 −10 10 −15 10 −20 10 −5 10 −10 10 −15 10 −20 10 −6 10 −6 10 −6 10 −8 10 −8 10 −8 4Q z 10 −12 10 −10 α Q z 10 −12 10 −10 α 0.25Q z 10 −12 10 −10 α 10 −14 10 −14 10 −14 10 −16 10 −16 10 −16 number of integer vectors number of integer vectors number of integer vectors 80 60 40 10 −6 20 20 15 10 10 −6 5 8 6 4 2 10 −6 0 10 −8 10 −8 10 −8 4Q z 10 −12 10 −10 α Q z 10 −12 10 −10 α 0.25Q z 10 −12 10 −10 α Figure 3.12: Approximation errors in fˇɛ(x). Top: 4Qˆz 02 01; Center: Qˆz 02 01; Bottom: 1 4 Qˆz 02 01. Left: mean errors (dashed) and maximum errors (solid); Right: the mean number (dashed) and maximum number (solid) of integer vectors in integer set. 52 Integer ambiguity resolution 10 −14 10 −14 10 −14 10 −16 10 −16 10 −16
Table 3.4: Order of the mean and maximum approximation errors in fˇɛ(x). Nz equals the minimum and maximum number of integers used. α λ mean maximum Nz 06 01 10 −16 10 −18 10 −16 10 −15 753-850 10 −12 10 −14 10 −12 10 −12 347-407 10 −8 10 −9 10 −8 10 −8 112-155 10 −6 10 −7 10 −6 10 −6 53-87 06 02 10 −16 10 −19 10 −17 10 −15 4730-4922 10 −12 10 −15 10 −12 10 −12 2203-2379 10 −8 10 −10 10 −8 10 −8 785-879 10 −6 10 −8 10 −6 10 −6 357-442 10 01 10 −16 10 −18 10 −16 10 −14 274-345 10 −12 10 −14 10 −12 10 −12 80-121 10 −8 10 −9 10 −8 10 −8 11-31 10 −6 10 −6 10 −6 10 −6 2-14 10 03 10 −16 10 −20 10 −17 10 −15 44323-45452 10 −12 10 −16 10 −12 10 −12 14535-15048 10 −8 10 −11 10 −8 10 −8 2959-3330 10 −6 10 −9 10 −6 10 −6 1007-1215 be presented. Furthermore, some baseline probabilities will be given that can be used to obtain a quality assessment of the fixed baseline solution. 3.4.1 Parameter distributions of the baseline estimators The float baseline solution is assumed to be normally distributed, i.e. ˆ b ∼ N(b, Qˆ b ). The PDF of ˆ b reads then fˆ b (x) = 1 |Qˆb |(2π) 1 exp{−1 2 p The conditional PDF is given by: fˆb|â (x|z) = 1 |Qˆ b|â |(2π) 1 2 2 x − b2 Qˆ b } (3.56) p exp{−1 2 x − b(z)2 Qˆ b|â } (3.57) with conditional mean b(z) = b − Qˆbâ Q −1 â (a − z), and conditional variance matrix Qˆb|â = Qˆb − Qˆbâ Q −1 â Qâˆb . The distribution function of the fixed baseline parameters, fˇ b (x), was derived in Teunis- Quality of the fixed baseline estimator 53
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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
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 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
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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
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µ 1.5 1.45 1.4 1.35 1.3 1.25 1.2 1
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In order to show that the fail rate
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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
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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.
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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
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Choose the tolerance ε. Determine
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Simulation and examples B Throughou
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Theory of BIE estimation C In chapt
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The first term on the right-hand si
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can be derived: L = {2h(x) T Q
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1. Choose fixed fail rate: Pf = β
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Bibliography Abidin HA (1993). Comp
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Jung J, Enge P, Pervan B (2000). Op
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Teunissen PJG (2003g). Towards a un
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Index adjacent, 33 ADOP, 42 alterna
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