Page - 125 - in Contributions to GRACE Gravity Field Recovery - Improvements in Dynamic Orbit Integration, Stochastic Modelling of the Antenna Offset Correction, and Co-Estimation of Satellite Orientations
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More compactly, this
is[
∆ldep
∆lind ]
= [
A Find
0 I ][
∆x
∆lind ]
+ [
edep
eind ]
. (9.1.39)
It is important to note that the two occurrences of∆lind in eq. (9.1.39) refer to two
separate variables. The first is the linearisation of the independent variable about the
Taylor point,∆lind= lind−g(lind,0). The second is the parameter to be estimated in the
LSA. The independent variable does not depend on the parametersx. The covariance
matrix of the combined observation vector is the same as in the previous model, given
in eq. (9.1.16). The weight matrix for the combined observation vector is then
P=Σ−1ll = [
Σdep Σcro
ΣTcro Σind ]−1
= [
Pdep Pcro
PTcro Pind ]
. (9.1.40)
Using the matrix inversion lemma (see e.g. Bernstein, 2009, Proposition 2.8.7), the
elements ofP are:
Pdep=Σ −1
dep+Σ −1
depΣcro (
Σind−ΣTcroΣ−1depΣcro )−1
ΣTcroΣ −1
dep
= (
Σdep−ΣcroΣ−1indΣTcro )−1
(9.1.41)
Pcro=−Σ−1depΣcro (
Σind−ΣTcroΣ−1depΣcro )−1
=− (
Σdep−ΣcroΣ−1indΣTcro )−1
ΣcroΣ −1
ind (9.1.42)
PTcro=− (
Σind−ΣTcroΣ−1depΣcro )−1
ΣTcroΣ −1
dep
=−Σ−1indΣTcro (
Σdep−ΣcroΣ−1indΣTcro )−1
(9.1.43)
Pind= (
Σind−ΣTcroΣ−1depΣcro )−1
=Σ−1ind+Σ−1indΣTcro (
Σdep−ΣcroΣ−1indΣTcro )−1
ΣcroΣ −1
ind (9.1.44)
Normal Equation System
The normal equation for this Gauß-Markov model
is[
N11 N12
NT12 N22 ]
= [
AT 0
FTind I ][
Pdep Pcro
PTcro Pind ][
A Find
0 I ]
= [
ATPdep A
TPcro
FTindPdep+P T
cro F T
indPcro+Pind ][
A Find
0 I ]
= [
ATPdepA A
TPdepFind+A
TPcro
FTindPdepA+P T
croA Pind+P T
croFind+F T
indPcro+F T
indPdepFind ]
.
(9.1.45)
9.1 Uncertainties in Independent Variables 125
Contributions to GRACE Gravity Field Recovery
Improvements in Dynamic Orbit Integration, Stochastic Modelling of the Antenna Offset Correction, and Co-Estimation of Satellite Orientations
- Title
- Contributions to GRACE Gravity Field Recovery
- Subtitle
- Improvements in Dynamic Orbit Integration, Stochastic Modelling of the Antenna Offset Correction, and Co-Estimation of Satellite Orientations
- Author
- Matthias Ellmerr
- Publisher
- Verlag der Technischen Universität Graz
- Location
- Graz
- Date
- 2018
- Language
- English
- License
- CC BY 4.0
- ISBN
- 978-3-85125-646-8
- Size
- 21.0 x 29.7 cm
- Pages
- 185
- Keywords
- Geodäsie, Gravitation, Geodesy, Physics, Physik
- Categories
- Naturwissenschaften Physik
- Technik