The GNSS integer ambiguities: estimation and validation

More documents

Recommendations

Info

So that: 0 = E{g(â)} = g(x)fâ(x)dx = g(x)fâ(x)dx Thus if the PDF fâ(x + a) is symmetric with respect to the origin, which is true in the case of normally distributed data, then g(−x) = −g(x) implies unbiasedness. C.3 Best integer equivariant unbiased ambiguity estimation The least mean square estimator will be considered as the best integer equivariant estimator. Theorem C.3.1 ˆS(x) = x − (x + z − a)fâ(x + z) = fâ(x + z) z∈Z n z∈Z n zfâ(x + a − z) fâ(x + a − z) z∈Z n z∈Z n (C.3) is the unique minimizer of S(x)−a 2 Qâ fâ(x)dx within the class of integer equivariant estimators, S(x + z) = S(x) + z Proof: Let · 2 = (·) T Q −1 â (·). Then: S(x) − a 2 fâ(x)dx = x − a + g(x) 2 fâ(x)dx = x − a 2 fâ(x)dx + {2g(x) T Q −1 â (x − a) + g(x)2 }fâ(x)dx 154 Theory of BIE estimation
The first term on the right-hand side of the last equation is independent of g(x). If the second term on the right-hand side is called K, the following can be derived: K = {2g(x) T Q −1 â (x − a) + g(x)2 }fâ(x)dx = {2g(x) T Q −1 â (x − a) + g(x)2 }fâ(x)dx z∈Zn Sz = = z∈Zn S0 {2g(y) T Q −1 â (y + z − a)fâ(y + z) + g(y) 2 fâ(y + z)}dy {2g(y) T Q −1 â (y + z − a)fâ(y + z) + g(y) z∈Z n S0 F (y) F (y) = {g(y) + G(y) S0 2 F (y) − G(y) ≥0 2 }dy independent of g(y) 2 z∈Z n fâ(y + z) }dy G(y)>0 where Sz are arbitrary pull-in regions, and use is made of the fact that if x ∈ Sz then y = x − z ∈ S0 and g(y + z) = g(y). This shows that K is minimized for g(y) = − F (y) G(y) is periodic. F (y) G(y) , which is an admissible solution since If fâ(x + a) is symmetric with respect to the origin, then ˆ S(−x) = − ˆ S(x) from which it follows that ˆ S(â) is unbiased. In that case the estimator ˆ S(â) is of minimum variance and unbiased. In practice, it is assumed that the float ambiguities are normally distributed. The BIE ambiguity estimator, ã follows then from equation (C.3) as: z∈Z ã = n z exp{− 1 2x − z2Qâ } exp{− 1 2x − z2 Qâ } (C.4) z∈Z n This shows that the BIE ambiguity estimator can be written in a similar form as equation (3.6): ã = zwz(â) (C.5) z∈Z n The weights wz(â) are given by equation (C.4), from which it follows that: 0 ≤ wz(â) ≤ 1 and wz(â) = 1 z∈Z n Best integer equivariant unbiased ambiguity estimation 155
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 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
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
Page 98 and 99:
4.3 Comparison of the float, fixed,
Page 100 and 101:
variane ratio 1.4 1.2 1 0.8 0.6 0.4
Page 102 and 103:
ambiguity residual 0.5 0.4 0.3 0.2
Page 104 and 105:
Table 4.1: Probabilities P1 = P (|
Page 106 and 107:
Table 4.2: Probabilities Ps = P (ǎ
Page 108 and 109:
Table 4.4: Probabilities that float
Page 110 and 111:
probability probability 1 0.9 0.8 0
Page 113 and 114:
Integer Aperture estimation 5 In se
Page 115 and 116:
can be distinguished: â ∈ Ωa s
Page 117 and 118:
and the hybrid distribution of ā i
Page 119 and 120:
Figure 5.3: 2-D example for EIA est
Page 121 and 122:
as follows: Ωz,R = ΩR ∩ Sz =
Page 123 and 124:
3 2.5 2 1.5 1 0.5 0 −0.5 −1 −
Page 125 and 126:
5.3.3 Difference test is an IA esti
Page 127 and 128:
Figure 5.9: 2-D example for DTIA es
Page 129 and 130: Figure 5.11: 2-D example for PTIA e
Page 131 and 132: Figure 5.12: 2-D example for IAB es
Page 133 and 134: 2-D example Figure 5.13 shows in bl
Page 135 and 136: µ 200 100 10 2 1 1 1.1 1.2 1.3 1.4
Page 137 and 138: 5.6 Optimal Integer Aperture estima
Page 139 and 140: The optimization problem of (5.49)
Page 141 and 142: Table 5.1: Comparison of IA estimat
Page 143 and 144: µ 1.5 1.45 1.4 1.35 1.3 1.25 1.2 1
Page 145 and 146: In order to show that the fail rate
Page 147 and 148: success rate percentage identical 0
Page 149 and 150: probability probability 1 0.9 0.8 0
Page 151 and 152: 5.8.2 OIA estimation versus RTIA an
Page 153 and 154: µ µ µ 0.7 0.6 0.5 0.4 0.3 0.2 0.
Page 155 and 156: 0.6 0.4 0.2 0 −0.2 −0.4 −0.6
Page 157 and 158: three vc-matrices which are scaled
Page 159 and 160: Table 5.3: Comparison of IAB and IA
Page 161 and 162: success rate if fixed 1 0.95 0.9 0.
Page 163: Table 5.4: Overview of IA estimator
Page 166 and 167: IE IA I Figure 6.1: The set of rela
Page 168 and 169: convex region symmetric with respec
Page 171 and 172: Mathematics and statistics A A.1 Kr
Page 173 and 174: The expectation, E{x}, and the disp
Page 175 and 176: Choose the tolerance ε. Determine
Page 177 and 178: Simulation and examples B Throughou
Page 179: Theory of BIE estimation C In chapt
Page 183: can be derived: L = {2h(x) T Q
Page 186 and 187: 1. Choose fixed fail rate: Pf = β
Page 189 and 190: Bibliography Abidin HA (1993). Comp
Page 191 and 192: 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
show all

The GNSS integer ambiguities: estimation and validation

Create successful ePaper yourself

Delete template?

Save as template?