Page - 13 - 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

withW=A−1. Inserting eq. (2.7.3) into eq. (2.7.2), the evaluation of the interpolation polynomial at a specific time t can be written as a vector-matrix-vector product x(τ)=τWx (2.7.4) withτ= [ τ0 . . .τn ] andτ= t− t0. This has the important property that for a constant valueτ, meaning always evaluating the interpolation polynomial at the same fraction of its length, e.g. always at the centre support, the n+1 weights w=τW (2.7.5) are constant and only have to be computed once. This is an expression which can be differentiated and integrated analytically. An arbitrary derivative or integral for a time series can be computed using these interpolation polynomials. The m-th time derivative of eq. (2.7.4) is ∂mx(τ) ∂τm =w(m)x=τ(m)Wx . (2.7.6) The m-th integral is equivalently∫ x(τ)∂τm=w(−m)x=τ(−m)Wx , (2.7.7) plus some integration constants. Both differentiation and integration can thus be written as a linear operator on a time seriesx of arbitrary length k, by populating a matrix of dimensions k×k with the appropriate entries of the respective weight matrices. This would yield a matrix structure similar to what is illustrated in fig. 2.1. 2.7 Interpolation, Numerical Integration, and Numerical Differentiation 13