Page - 122 - 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

and∆λ=λ−λ0 this gives the equation system ∆λ=Ax∆x+eλ . (9.1.15) Equation (9.1.15) is a reformulation of the Gauß-Helmert problem eq. (9.1.6) as a Gauß- Markov problem with derived observations. Given covariance information for lind and ldep as Σll= [ Σdep Σcro ΣTcro Σind ] (9.1.16) the covariance matrix for the derived observationsλ follows from eq. (9.1.12) through variance propagation as Σλλ=FΣllF T . (9.1.17) The solution to eq. (9.1.15) is then simply that of eq. (2.2.3), ∆xˆ= ( ATxΣ −1 λλAx )−1 ATxΣ −1 λλ∆λ . (9.1.18) Neitzel and Petrovic, 2008; Reinking, 2008 assert that this gives the same solution as directly solving for the TLS problem formulated in eq. (9.1.4) using classical methods such as those presented by e.g. Golub and van Loan, 1980. Estimated Residuals and Observations Evaluating eq. (9.1.18) gives estimated residuals eˆλ in the derived observationsλ. From eq. (9.1.14) it is known thateλ=−Fe. Direct comparison with the identity eλ=ΣλλΣ −1 λλeλ =FΣllF TΣ−1λλeλ (9.1.19) allows one to identify that eˆ=−ΣllFTΣ−1λλeˆλ . (9.1.20) Having computed eˆ= [ eˆdep eˆind ]T using eq. (9.1.20), the estimated dependent and independent observations are lˆdep= ldep− eˆdep and lˆind= lind− eˆind . (9.1.21) The Taylor points for the observations in the next iteration of the adjustment are then lˆdep and lˆind. Note that the residuals are always added to the original observations, not the Taylor point of the current iteration. Chapter9 Co-Estimation of Orientation Parameters122