Page - 121 - 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

9.1.1 Total Least Squares as Gauß-Markov Model with Derived Observations Reinking gives a derivation of the solution to the TLS problem in eq. (9.1.4) that relies on the removal of the distinction between dependent and independent variables. Rather, all variables are collected in one vector l= [ ldep lind ] . (9.1.5) Thissectionwillgiveasummaryof thealgorithmpublishedinReinking,2008.Building on this summary, the resulting normal equation system is analysed in preparation for later comparisons with the algorithm presented in section 9.1.2. Reinking’s algorithm begins by restating the problem eq. (9.1.1) in the Gauß-Helmert model as G ( xˆ, lˆ ) = lˆdep−f ( lˆind,xˆ ) =0 . (9.1.6) The functionalG is then linearised about both the parameters and observations, giving G(x0,l0)+ ∂G ∂x ∣∣∣∣ x0,l0 ∆x+ ∂G ∂l ∣∣∣∣ x0,l0 ∆l=0 (9.1.7) or, more compactly, G0+Ax∆x+F∆l=0 . (9.1.8) With∆l= lˆ−l0= l−e−l0, eq. (9.1.8) is G0+Ax∆x+F (l−e−l0)=0 (9.1.9) −Fl−(G0−Fl0)=Ax∆x−Fe (9.1.10) Here, it is important to note that the residuals e are actually residuals in both the dependent and independent variables: e= [ edep eind ] . (9.1.11) Reinking now defines the derived observationsλas λ=−Fl . (9.1.12) With λ0=(G0−Fl0) , (9.1.13) eλ=−Fe , (9.1.14) 9.1 Uncertainties in Independent Variables 121