The GNSS integer ambiguities: estimation and validation

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The single difference approach can be seen as a kind of compromise. The amount of observations is reduced with respect to the undifferenced approach, but not as much as with the double difference approach. The same holds for the unknown parameters. An advantage of the single difference approach is that in the case of a single baseline model (two receivers) no correlation between the observations is introduced, see section 2.4. As explained before, for high-precision applications it is important that the carrier phase ambiguities are resolved as integers. It has been shown that the parameterization of the unknown ambiguities is such that these parameters are known to be integers in the single difference and double difference approach. 2.4 GNSS stochastic model The stochastic model describes the precision of the observations. As a starting point the following assumptions are made: E{e s r,j(t)e s r,j(t)} = σ 2 p,j no satellite dependent weighting E{e s r,i (t)esr,j (t)} = 0 no frequency dependent correlation (2.56) E{e s r,j (t1)e s r,j (t2)} = 0 no time correlation Often, the code standard deviation is chosen identical for all frequencies, σp,j = σp ∀j. The same assumptions are made for the undifferenced phase observations, with the phase standard deviation denoted as σφ. The vc-matrix of the undifferenced observations of one satellite and one receiver becomes then: Cpφ = diag(σ 2 p,1, . . . , σ 2 p,f , σ 2 φ,1, . . . , σ 2 φ,f , σ 2 I ) (2.57) where σ2 I is the variance of the ionospheric pseudo-observation, which has to be included if the ionosphere-weighted model is considered. For the single differenced observations, the vc-matrices of the observations of both receivers must be added according to the propagation law of variances, see appendix A.2.1: C sd pφ = Cpφ,q + Cpφ,r (2.58) If the standard deviations are assumed equal for both receivers, all variances must simply be multiplied by 2. The single difference vc-matrix corresponding to the functional models in equations (2.45) and (2.46) becomes: Qy = C sd pφ ⊗ Im (2.59) The vc-matrix of the double difference models in equations (2.54) and (2.55) is obtained again with the propagation law of variances and the properties of the Kronecker product 20 GNSS observation model and quality control
(appendix A.1): Qy = C sd pφ ⊗ D T D (2.60) with the transformation matrix D T as defined in (2.48). 2.4.1 Variance component estimation The stochastic model as specified by the vc-matrix of the observations can be given as: Qy = σ 2 Gy (2.61) with σ 2 the variance factor of unit weight, and Gy the cofactor matrix. If the variance factor is assumed to be unknown, it can be estimated a posteriori. For that purpose equation (2.61) can be generalized into: Qy = p α σ 2 αGα (2.62) Hence, the stochastic model is written as a linear combination of unknown factors and known cofactor matrices, (Tiberius and Kenselaar 2000). The factors σ 2 α can represent variances as well as covariances, for example variance factors for the code and phase data, a variance for each satellite-receiver pair, and the covariance between code and phase data. The problem of estimating the unknown (co-)variance factors is referred to as variance component estimation. 2.4.2 Elevation dependency Until now the standard deviations of a certain observation type were chosen equal for all satellites, although it might be more realistic to assign weights to the observations of different satellites depending on their elevation. That is because increased signal attenuation due to the atmosphere and multipath effects will result in less precise observations from satellites at low elevations. Therefore, Euler and Goad (1991) have suggested to use elevation-dependent weighting: σ 2 p s r = (qs r) 2 σ 2 p (2.63) with σp the standard deviation of code observations to satellites in the zenith, and q s r = 1 + a · exp{ −εsr }, (2.64) ε0 where a is an amplification factor, ε s r the elevation angle, and ε0 a reference elevation angle. The values of a and ε0 depend on the receiver that is used, and on the observation type – the function q in equation (2.64) may be different for code and phase observations as well as for observations on different frequencies. Experimental results have shown that an improved stochastic model can be obtained if elevation dependent weighting is applied, see e.g. (Liu 2002). GNSS stochastic model 21
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: The double difference observation v
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 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
Page 147 and 148:
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
Page 175 and 176:
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
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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