The GNSS integer ambiguities: estimation and validation

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Table 2.3: Availability and accuracy of GPS satellite positions, after (Neilan et al. 2000). Orbits Availability Accuracy Yuma almanacs Real-time or earlier ?? Broadcast Real-time 5 m IGS Ultra-rapid Near real-time 20 cm IGS Rapid 17 hours 10 cm IGS Final 10 days 5 cm 2.3 GNSS functional model In this section different GNSS functional models will be presented. Both the nonpositioning or geometry-free models and the positioning or geometry-based models are presented. Different GNSS models can also be distinguished based on the differencing that is applied. Differencing means taking the differences between observations from e.g. different receivers and/or different satellites. It is often applied in order to eliminate some of the parameters from the observation equations. 2.3.1 General mathematical model The functional model describes the relationship between the observations and the unknown parameters. The m observation equations can be collected in the system y = Ax + e, where y and e are random, e is the discrepancy between y and Ax. It is assumed that the mean E{e} is zero, since e models the random nature of the variability in the measurements, and this variability will be zero ’on the average’. This gives for the expectation of y: E{y} = Ax (2.26) The probability density function of y describes the variability in the outcomes of the measurements. For normally distributed data, it is completely captured by the dispersion: D{y} = E{ee T } = Qy (2.27) This is referred to as the stochastic model, with Qy the variance-covariance (vc-) matrix of the observations. This matrix describes the precision of the observations and it is needed in order to properly weigh the observations in the adjustment process, see section 2.5. In section 2.4, the stochastic models corresponding to the functional models described in this section will be given. The functional and stochastic model of (2.26) and (2.27) together are referred to as Gauss-Markov model. 14 GNSS observation model and quality control
2.3.2 Single difference models If observations from m satellites are observed simultaneously at two receivers it is possible to take the differences of these observations. In this way, some parameters will be eliminated from the model. Here, only the model for a single epoch will be derived and the index t will be left out for notational convenience. The set of single difference (SD) observation equations for a single satellite, a single frequency, and two receivers q and r is then given by: p s qr,j = p s r,j − p s q,j = ρ s qr + T s qr + µjI s qr + cdtqr,j + e s qr,j φ s qr,j = φ s r,j − φ s q,j = ρ s qr + T s qr − µjI s qr + cδtqr,j + λjM s qr,j + ε s qr,j with (·)qr = (·)r − (·)q. (2.28) Note that the lumped instrumental delays and clock errors of the satellite are eliminated. Furthermore, the initial carrier phase parameters of the satellite, which were contained in M s r and M s q , are cancelled out, so that M s qr only contains the differenced carrier phase ambiguity and the difference between the initial phases in the two receivers. The observations of m satellites and two receivers observing on f frequencies will now be collected in the geometry-free functional model. The vector of observations becomes: p 1 P qr,1 · · · p y = = Φ m 1 qr,1 · · · pqr,f · · · pm T qr,f 1 φqr,1 · · · φm 1 qr,1 · · · φqr,f · · · φm T (2.29) qr,f The parameter vectors are defined as: ρ = ρ1 qr · · · ρm T qr T = T 1 qr · · · T m T qr I = I1 qr · · · Im T qr dtqr = c · dtqr,1 T · · · dtqr,f T δtqr,1 · · · δtqr,f a = M 1 qr,1 · · · M m qr,1 · · · 1 Mqr,f · · · M m T qr,f (2.30) (2.31) (2.32) (2.33) (2.34) For notational convenience the Kronecker product is used, see appendix A.1, to arrive at the geometry-free model: µ E{y} = (e2f ⊗ Im) ρ + (e2f ⊗ Im) T + ⊗ Im I −µ (2.35) 0 + (I2f ⊗ em) dtqr + ⊗ Im a Λ GNSS functional model 15
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: 2.2.9 The geometry-based observatio
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 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
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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
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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
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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