Page - 7 - 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

other set of observations l. The parameters are connected to the observations through a functional model l= f(x). Exhaustive descriptions of this algorithm can for example be found in Koch (1997) and Niemeier (2008). The algorithm prescribes the computation of a designmatrix, denotedA, as the Jacobian of the observations with regard to the parameters A= ∂ f(x) ∂x ∣∣∣∣ x0 (2.2.1) atsomeinitialvaluefor theparametersx0.Withthis linearisation,anequationsystem ∆l=A∆x+e (2.2.2) is set up, where e is the vector of residuals, the misfit of the observations with the prediction made by the model, and∆l= l− f(x0) the reduced observations. Given a matrix of observation weightsP , the least squares solution of eq. (2.2.2) is ∆xˆ= ( ATPA )−1 ATP∆l . (2.2.3) The adjusted vector of parameters∆xˆ is that solution to eq. (2.2.2) that minimizes the weighted square sum of residualseTPe. The desired parameters are then xˆ=x0+∆xˆ. This process of linearisation and adjustment must be iterated until the additions to the parameters∆xˆare small, signalling convergence. In case the functional relationship f(x)between observations and parameters is linear, no differences have to be formed and the adjusted parameters can directly be computed as xˆ= ( ATPA )−1 ATPl . (2.2.4) The estimated residuals of the observations are eˆ= l−Axˆ . (2.2.5) The weight matrixP is the inverse of the covariance matrix of the observationsΣll: P=Σ−1ll (2.2.6) Equation (2.2.4) can be written as Nxˆ=n (2.2.7) with the normal equation of the system N=ATPA (2.2.8) and the right hand side n=ATPl . (2.2.9) 2.2 Least Squares Adjustment 7