The GNSS integer ambiguities: estimation and validation

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2.4.3 Cross-correlation and time correlation In equation (2.56) the assumption was made that there is no frequency dependent correlation. Several studies have shown, however, that this assumption may not be realistic, see e.g. (Bona 2000; Bona and Tiberius 2001; Liu 2002). It was shown that the stochastic model can be refined by performing receiver tests or variance component estimation, in order to determine the cross-correlation. It was also assumed that no time correlation is present, although the observations collected at different epochs may be correlated when the sampling frequency is high. In Borre and Tiberius (2000) significant time correlation was found for 5 Hz data. If, on the other hand, the sampling interval is larger than 30 s, according to Bona (2000) time correlation can be neglected. So, time correlation does not play a role for instantaneous applications. 2.5 Least-squares estimation and quality control In sections 2.3 and s:stochastic, the GNSS functional models and stochastic models were presented. In order to solve the observation model, a least-squares adjustment is generally used, since it is simple to apply and, by definition, it has important optimality properties. For a model written as equation (2.26) the least-squares estimator is defined as: ˆx = arg min x y − Ax 2 Qy (2.65) and its solution is given by: ˆx = (A T Q −1 y A) −1 A T Q −1 y y ˆy = Aˆx (2.66) ê = ˆy − y The GNSS observations are generally considered to be normally distributed, and then least-squares estimation is equivalent to maximum-likelihood estimation, and to best linear unbiased estimation if the model is linear, (Teunissen 2000a). The estimator ˆx is a random vector since it is functionally related to the random vector y. Because of the linear relationship between the two vectors, ˆx will also have a normal distribution like y. Therefore the distribution of the estimator can be characterized by its mean and dispersion, which are given as: E{ˆx} = x (2.67) D{ˆx} = Qˆx = (A T Q −1 y A) −1 (2.68) This shows that the estimator is unbiased, i.e. that the expectation of the estimator equals the unknown and sought for parameter vector x. The vc-matrix describes the precision of the estimator, which is independent of the observations, so that the precision is known without the need for actual measurements. 22 GNSS observation model and quality control
The quality of the solution can then be measured by the precision of the estimators. However, the precision does not give information on the validity of the model, so that the unbiasedness of the estimators cannot be guaranteed. It is therefore important to use statistical tests in order to get information on the validity of the model. 2.5.1 Model testing Statistical testing is possible if there is a null hypothesis, H0, that can be tested against an alternative hypothesis, Ha. These hypotheses can be defined as: H0 : E{y} = Ax; D{y} = Qy Ha : E{y} = Ax + C∇; D{y} = Qy (2.69) where C is a known m × q-matrix that specifies the type of model error, and ∇ an unknown q-vector. It is assumed that y is normally distributed, see appendix A.2. H0 can be tested against Ha using the following test statistic: T q = 1 q êT Q −1 y C C T Q −1 y QêQ −1 y C −1 C T Q −1 y ê (2.70) The test statistic T q has a central F -distribution with q and ∞ degrees of freedom under H0, and a non-central F -distribution under Ha: H0 : T q ∼ F (q, ∞, 0); Ha : T q ∼ F (q, ∞, λ) (2.71) with non-centrality parameter λ: λ = ∇ T C T Q −1 y QêQ −1 y C∇ (2.72) The test is then given by: reject H0 if T q > Fα(q, ∞, 0), (2.73) where α is a chosen value of the level of significance, also referred to as the false alarm rate since it equals the probability of rejecting H0 when in fact it is true. Fα(q, ∞, 0) is the critical value such that: ∞ α = fF (F |q, ∞, 0)dF (2.74) Fα(q,∞,0) where fF (F |q, ∞, 0) is the probability density function of F (q, ∞, 0), see appendix A.2.3. The value of λ = λ0 can be computed once reference values are known for the level of significance α = αq, and the detection power γ = γ0, which is the probability of rejecting H0 when indeed Ha is true: γ = ∞ Fα(q,∞,0) fF (F |q, ∞, λ)dF (2.75) Least-squares estimation and quality control 23
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 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: (appendix A.1): Qy = C sd pφ ⊗ D
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 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
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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
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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
Page 193 and 194:
Teunissen PJG (2003g). Towards a un
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Index adjacent, 33 ADOP, 42 alterna
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