Page - 6 - 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

For vector by matrix derivatives, the vec()-operator is used, which stacks the columns of a matrix. For a k+1× l+1 matrix M=       m00 m01 · · · m0l m10 m11 · · · m1l ... ... ... ... mk0 mk1 · · · mkl       k+1×l+1 (2.1.5) this is vec(M)= [ m00 . . .mk0 m01 . . .mk1 · · · m0l . . .mkl ]T . (2.1.6) The vector by matrix derivative is then, equivalently to eq. (2.1.4), written as ∂x ∂vec(M) =               ∂x0 ∂m00 ∂x0 ∂m10 · · · ∂x0 ∂mkl ∂x1 ∂m00 ∂x1 ∂m10 · · · ∂x1 ∂mkl ... ... ... ... ∂xn ∂m00 ∂xn ∂m10 · · · ∂xn ∂mkl               n+1×(k+1)·(l+1) . (2.1.7) For differentiation of matrices by scalars, such as the often occurring time derivatives, Lagrange’s notation using parentheses may be used for brevity, such that ∂nM ∂tn =M(m) (2.1.8) is the m-th derivative and∫ M ∂tn=M(−m) (2.1.9) is the m-th antiderivative. For a time series of scaler or matrix values a variable with an explicit time point given, such asx(t), shall refer to that single epoch. The variable given without a specific time shall refer to the complete time series x= [ x(t0)T x(t1)T · · · x(tn)T ]T . (2.1.10) 2.2 Least Squares Adjustment The fundamental concept in this thesis is the least squares adjustment (LSA), an algorithm to determine the values of some set of parametersx that “best” fit some Chapter2 Mathematical Theory and Notation6