The GNSS integer ambiguities: estimation and validation

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Notation and symbols Mathematical notation and matrices Ip identity matrix of order p ep p-vector with ones 0 vector or matrix with zeros |Q| determinant of matrix Q x2 Q squared norm, with positive definite Q: x2 Q = xT Q−1 [x] x x is rounded to the nearest integer Cɛ z ellipsoidal region centered at z and with radius ɛ Signals c velocity of light in vacuum: c = 299792458 m s −1 f frequency λ wavelength: λ = c/f GNSS observation model s refers to satellite s r refers to receiver r ,j refers to frequency j m number of satellites f number of frequencies k number of epochs p, φ code and phase observation respectively ts transmission time from satellite s tr reception time at receiver r τ signal travel time e, ε code and phase noise respectively dt, δt clock error of code and phase observations respectively xv
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 23: Parameter distributions fˆx(x) pro
Page 27 and 28: Introduction 1 1.1 Background In th
Page 29: The contribution of this thesis is
Page 32 and 33: Table 2.1: Signal and frequency pla
Page 34 and 35: of reception diminished with the si
Page 36 and 37: 2.2.3 Signal travel time In the pre
Page 38 and 39: 2.2.6 Multipath Ideally, a GNSS sig
Page 40 and 41: Table 2.3: Availability and accurac
Page 42 and 43: with ep a p-vector with ones, Ip an
Page 44 and 45: The single difference models can no
Page 46 and 47: The single difference approach can
Page 48 and 49: 2.4.3 Cross-correlation and time co
Page 50 and 51: So, λ0 = λ(αq, γ0, q). From thi
Page 53 and 54: Integer ambiguity resolution 3 The
Page 55 and 56: Figure 3.2: Left: Pull-in regions t
Page 57 and 58: 2 1 0 −1 −2 −2 −1 0 1 2 Fig
Page 59 and 60: 2 1 0 −1 −2 −2 −1 0 1 2 Fig
Page 61 and 62: matrix D. From equation (3.21) the
Page 63 and 64: 3.2.1 Parameter distributions of th
Page 65 and 66: have to be used. The most important
Page 67 and 68: Then Ua = {â ∈ R n | p ∩ i=1 |
Page 69 and 70: success rate 1 0.9 0.8 0.7 0.6 0.5
Page 71 and 72:
on bounding the integration region
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for any R, it follows that the PDF
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5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 σ=0
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2 1 0 −1 −2 −2 −1 0 1 2 0.4
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Table 3.4: Order of the mean and ma
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3.4.2 Baseline probabilities The qu
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probability probability 1 0.9 0.8 0
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Since H3 ⊂ H1 ⊂ H2, starting po
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where F (n, m − n − p, λi) den
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The test is then defined by the con
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validation criteria. The fundamenta
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Best Integer Equivariant estimation
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where y ∈ R m with mean E{y} = Aa
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difference between the BIE estimato
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elative frequency relative frequenc
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estimator 1.5 1 0.5 0 −0.5 −1
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probability 1 0.8 0.6 0.4 0.2 0 200
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0.6 0.4 0.2 0 −0.2 −0.4 −0.6
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probability 1 0.9 0.8 0.7 0.6 0.5 0
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Note that since ˜ɛ 2 Qˆz ≤ ˇ
Page 111:
squared norm of ambiguity residuals
Page 114 and 115:
5.1 Integer Aperture estimation The
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0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05
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1.5 1 0.5 0 −0.5 −1 −1.5 −1
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2-D example Figure 5.3 shows in bla
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1 0.5 0 −0.5 −1 −1 −0.5 0 0
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Figure 5.7: 2-D example for F -RTIA
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1 0.5 0 −0.5 −1 −1 −0.5 0 0
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1 0.5 0 −0.5 −1 −1 −0.5 0 0
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success rate and fail rate. From eq
Page 132 and 133:
Figure 5.13: 2-D example for IALS e
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Therefore, the average penalty also
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0.5 0 −0.5 0.5 0 −0.5 0.5 0 −
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Clearly, the region ˆ Ω in (5.51
Page 140 and 141:
Figure 5.16: 2-D example for OIA es
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0.6 0.4 0.2 0 −0.2 −0.4 −0.6
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oundary of the aperture pull-in reg
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Table 5.2: Comparison of IA estimat
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success rate percentage identical 1
Page 150 and 151:
probability probability 1 0.9 0.8 0
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This means that the higher the prec
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in order to guarantee a low fail ra
Page 156 and 157:
µ success rate 1.4 1.35 1.3 1.25 1
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µ success rate / undecided rate 0.
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P fix 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3
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implies that the success rate incre
Page 165 and 166:
Conclusions and recommendations 6 6
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• The discrimination tests - rati
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6.4 Bias robustness In the precedin
Page 172 and 173:
Let x be normally distributed with
Page 174 and 175:
The following notation is used: x
Page 176 and 177:
Determine the zero-crossing of the
Page 178 and 179:
Table B.1: Time, location, number o
Page 180 and 181:
So that: 0 = E{g(â)} = g(x)fâ(x
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C.4 Best integer equivariant unbias
Page 185 and 186:
Implementation aspects of IALS esti
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Curriculum vitae E Sandra Verhagen
Page 190 and 191:
Euler HJ, Goad C (1991). On optimal
Page 192 and 193:
Teunissen PJG (1995). The least-squ
Page 194 and 195:
Verhagen S, Teunissen PJG (2004c).
Page 196:
probability density function, 37 pr
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The GNSS integer ambiguities: estimation and validation

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