The GNSS integer ambiguities: estimation and validation

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Table 3.1: Overview of ambiguity resolution methods. Method Name References Least-Squares Ambiguity Search Technique LSAST Hatch (1990) Fast Ambiguity Resolution Approach FARA Frei and Beutler (1990) Modified Cholesky decomposition Euler and Landau (1992) Least-squares AMBiguity Decorrelation Adjustment LAMBDA Teunissen (1993) Null method Martín-Neira et al. (1995); Fast Ambiguity Search Filter FASF Fernàndez-Plazaola et al. (2004) Chen and Lachapelle (1995) Three Carrier Ambiguity Resolution TCAR Harris (1997) Integrated TCAR Vollath et al. (1998) Optimal Method for Estimating GPS Ambiguities OMEGA Kim and Langley (1999) Cascade Integer Resolution CIR Jung et al. (2000) 3.1.5 Other ambiguity resolution methods Besides LAMBDA, several other ambiguity resolution methods have been described in literature. Table 3.1.5 gives an overview of some well-known methods with references. Only TCAR and CIR are based on the bootstrapping estimator, all other methods are based on the ILS principle of minimizing the squared norm of residuals. The methods essentially differ in the way the search space is defined. In Kim and Langley (2000), some of the methods were conceptually compared. A comparison of LAMBDA with CIR, TCAR, ITCAR and the Null-method is made in Joosten and Verhagen (2003), Verhagen and Joosten (2004). 3.2 Quality of the integer ambiguity solution The uncertainty of parameter estimators is captured by their parameter distribution. For normally distributed data, the uncertainty is completely captured by the vc-matrix. However, the GNSS model contains integer parameters to which this does not apply. In this section, the distributional properties of both the real-valued and the integer ambiguity parameters are described. An important measure of the reliability of the fixed solution is the probability of correct integer estimation, the success rate. Once the parameter distribution of the integer estimator is given, the success rate can be determined. This will be the subject of section 3.2.2. 36 Integer ambiguity resolution
3.2.1 Parameter distributions of the ambiguity estimators The float ambiguities are assumed to be normally distributed, i.e. â ∼ N(a, Qâ), see appendix A.2. The probability density function (PDF) of â reads then fâ(x) = 1 |Qâ|(2π) 1 exp{−1 2 n 2 x − a2Qâ } (3.26) where | · | denotes the determinant. The joint PDF of the float and fixed ambiguities is given by: fâ,ǎ(x, z) = fâ(x)sz(x), x ∈ R n , z ∈ Z n (3.27) with sz(x) the indicator function of equation (3.7). For the proof see (Teunissen 2002). The marginal distributions of â and ǎ can be recovered from this joint distribution by integration or summation over respectively x and z. This gives the following distribution function of the fixed ambiguities: P (ǎ = z) = fâ,ǎ(x, z)dx = fâ(x)dx, z ∈ Z n R n Sz (3.28) It is a probability mass function (PMF), and is referred to as the integer normal distribution for the ILS estimator. The three top panels of figure 3.7 show the marginal and joint distributions of â and ǎ. The distribution of ǎ has some (desirable) properties, cf.(Teunissen 1998a; Teunissen 1998c). The first is that for all admissible integer estimators the distribution is symmetric about a: P (ǎ = a − z) = P (ǎ = a + z), ∀z ∈ Z n From this property it follows that all admissible estimators are unbiased: (3.29) E{ǎ} = zP (ǎ = z) = a (3.30) z∈Z n Another important property is that if a is the correct integer, the probability that this integer is obtained as solution is always larger than the probability that another integer is estimated: P (ǎ = a) > P (ǎ = z, z ∈ Z n \{a}) (3.31) Finally, the vc-matrix of the fixed ambiguities follows from equation (3.28) as: D{ǎ} = (z − a)(z − a) T P (ǎ = z) = Qǎ z∈Z n (3.32) Quality of the integer ambiguity solution 37
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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
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 22 and 23: d, δ instrumental delay for code a
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 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 78 and 79: 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
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
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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
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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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