Page - 70 - 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

trivial, as it is simply trace(R¯m), with no need to compute the product. For lags n>0 the trace used to compute the redundancy can be written as trace(R¯mVn)= trace ( R¯m ( V−n +V+n )) = trace ( R¯mV − n ) + trace ( R¯mV + n ) (6.5.49) whereV−n is only the lower half ofVn, and V+n the upper half. For e.g. Nmax = 5, these would be V−2 =         0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0         and V + 2 =         0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0         . (6.5.50) WithV+n =(V−n ) T and R¯being a symmetric matrix trace(R¯mVn)=2 · trace ( R¯mV − n ) . (6.5.51) AsVn is large and sparse, it is efficient to unroll the trace of the matrix product and directly write it as a sum trace ( R¯mV − n ) = Nm−n ∑ i=0 R¯m(i+n, i) . (6.5.52) The square sum of residuals can similarly be optimized by smartly computing eˆTΣ−Tll VnΣ−1ll eˆ . (6.5.53) First, let e˜=Σ−1ll eˆ=W−1W−Teˆ=W−1 ˆ¯e (6.5.54) which is again efficiently computed by solving a triangular system. Then eˆTΣ−Tll VnΣ−1ll e˜= e˜TVne˜ = e˜T ( V−n +V+n ) e˜ = { e˜Te˜ if n=0, 2 · e˜TV−n e˜ otherwise. . (6.5.55) This loop can again be unrolled, avoiding the product with the sparse matrixV−n , as e˜TV−n e˜= Nm ∑ i=n e˜(i) · e˜(i−n) . (6.5.56) Chapter6 ITSG-Grace201670