Page - 12 - 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

solutions. Here, a formalism for these operations based on polynomial interpolation is introduced. The derivations in this section follow those given by Mayer-Gu¨rr (2006). Let there be a series of n+1 values x(t0) . . .x(tn)with constant temporal sampling∆t. The coefficients of an n-th degree polynomial x(τ)= n ∑ k=0 akτk (2.7.1) defined by these supports can be computed by solving the system    x(t0+τ0) ... x(t0+τn)     ︸ ︷︷ ︸ x =     τ00 τ 1 0 · · · τn0 ... ... ... ... τ0n τ 1 n · · · τnn     ︸ ︷︷ ︸ A     a0 ... an     ︸︷︷︸ a , (2.7.2) for thevectora,withτj= j ·∆t.Thispolynomialcanthenbeevaluatedatanytimeτby inserting into eq. (2.7.1). A longer time series of k>n+1 values can be interpolated by a low degree polynomial applied to a moving segment of the time series, as illustrated in fig. 2.1. Support x(t0) x(t1) x(t2) x(t3) x(tk−3) x(tk−2) x(tk−1) x(tk). . . ... t0 t1 t2 t3 tk−3 tk−2 tk−1 tk ... Figure 2.1: Evaluation of an interpolation polynomial of degree n=2. Each polynomial is defined by three supports, and is evaluated at the times marked. The time series can be smoothed by computing a polynomial of degree d< n from n+1 supports in a least squares adjustment. The entries of the matrixAare constant, so it is possible to precompute the weights used for interpolation once for a specific combination of polynomial degree and sampling. For the not overdetermined case, the solution to eq. (2.7.2) is a=Wx , (2.7.3) Chapter2 Mathematical Theory and Notation12