Page - 10 - 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

2.6 Variance Component Estimation Aproblemoftenencounteredwhensolvingforasetofparameters is that thecovariance of the observations Σll is not exactly known. Similarly, when combining several observation groups at the normal equation level, the relative weight of the observations must be known to compute the optimal solution xˆ. One approach to determine these unknown weights and correlations is to treat them as unknown parameters in the adjustment problem and co-estimate the weights in an iterative procedure. This is known as variance component estimation (VCE). The outline in this section follows the reasoning given in Niemeier (2008). The fundamental extension to eq. (2.2.2) is that the single vector of residuals is split into multiple vectors of residualsei: l=Ax+e0+e1+ · · ·+en. (2.6.1) Eachof theseresidualvectorsshallhaveitsowncovariancematrixΣi, eachaconstituent of the complete covariance of the observations Σll= n ∑ i=0 Σi . (2.6.2) The structure ofΣi is given by a known cofactor matrixQ, which is then scaled by an unknown variance factorσ2i . Equation (2.6.2) is then Σll= n ∑ i=0 σ2iQi . (2.6.3) Using this scheme, an arbitrary covariance matrixΣll can be formed by choosing the right cofactor matrices and scaling them appropriately. Given an initial guess for the variance factors the optimal values are determined iteratively by introducing weights α2i =1, writing Σll= n ∑ i=0 α2i ( σ2iQi ) . (2.6.4) After computing the least squares solution of eq. (2.6.1) using this initial covariance matrix, the estimated weights are αˆ2i = Ω s , (2.6.5) with Ω= eˆTΣ−Tll ΣiΣ−1ll eˆ (2.6.6) and s= trace(RΣi) , (2.6.7) Chapter2 Mathematical Theory and Notation10