The GNSS integer ambiguities: estimation and validation

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GNSS observation model and quality control 2 In this chapter the GNSS observation model is presented. An observation model consists of the functional model, which describes the relation between the observations and the unknown parameters of interest, and of the stochastic model, which describes the stochastic properties of the observations. A great variety of GNSS models exists, which are used in applications like surveying, navigation and geophysics. An overview of these models can be found in textbooks like (Hofmann-Wellenhof et al. 2001; Leick 2003; Parkinson and Spilker 1996; Strang and Borre 1997; Teunissen and Kleusberg 1998). Here, first a brief overview of the current and future Global Navigation Satellite Systems is given. In section 2.2 the GNSS observation equations are presented. Also, a description of the different error sources is given. With this information it is then possible to set up the functional model. Different types of models can be distinguished. Firstly, positioning and non-positioning models. Secondly, depending on the differencing technique that is applied. The stochastic models corresponding to the different functional models, are presented in section 2.4. Finally, section 2.5 presents the general quality control theory applicable to GNSS models. 2.1 Global Navigation Satellite Systems In the last decade the Global Positioning System (GPS) has found widespread use in all kind of applications. It is the first Global Navigation Satellite System (GNSS) offering the accuracy nowadays needed for e.g. surveying, navigation, and geophysics. More or less synchronously to GPS, the former Soviet Union has been developing a similar system under the name GLONASS. A third Global Navigation Satellite System is still under development and should be operational in 2010. It is the European Galileo system. 2.1.1 Global Positioning System Currently, the nominal GPS constellation consists of 24 satellites, divided over six orbital planes, but more satellites are available at this moment (on average 28). The orbital inclination is 55 o and the orbital radius is 26,500 km. 5
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 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
Page 73 and 74: for any R, it follows that the PDF
Page 75 and 76: 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 σ=0
Page 77 and 78: 2 1 0 −1 −2 −2 −1 0 1 2 0.4
Page 79 and 80: 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
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5.1 Integer Aperture estimation The
Page 116 and 117:
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
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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
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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
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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
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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
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Conclusions and recommendations 6 6
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• The discrimination tests - rati
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6.4 Bias robustness In the precedin
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Let x be normally distributed with
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The following notation is used: x
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Determine the zero-crossing of the
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Table B.1: Time, location, number o
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So that: 0 = E{g(â)} = g(x)fâ(x
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C.4 Best integer equivariant unbias
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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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