Page - 68 - 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

Further, circular permutation of the product in the trace does not change the trace. Inserting the Cholesky factorization ofN, and then permuting the second trace in eq. (6.5.37) gives sn= trace ( Σ−1Vn ) − trace ( Σ−1AU−1U−TATΣ−TVn ) = trace ( Σ−1Vn ) − trace ( U−TATΣ−TVnΣ−1AU−1 ) . (6.5.38) At this point the rightmost trace can be approximated by using a Monte-Carlo trace estimator (Hutchinson, 1990). Hutchinson shows that for a P×P matrixX, trace(X)≈ 1 Z Z−1 ∑ z=0 zTzXzz , (6.5.39) for sufficiently large Z. Here eachzz is a random P×1 vector, with entries only 1 and −1, each with a probability of 0.5: zz= [ −1 1 . . . 1 −1 ]T (6.5.40) At this point, an evil maths trick is used in rewriting eq. (6.5.39). LetZ be a P×Z matrix of Monte Carlo vectorszz, then the sum in eq. (6.5.39) can be written as trace(X)≈ 1 Z trace ( ZTXZ ) . (6.5.41) Here, the diagonal elements of the matrix product are exactly the component sums from eq. (6.5.39). The trace operator performs the sum over these diagonal components, neglecting the off-diagonal products of all combinations of two non-identical Monte Carlo vectors. To clarify this approach, consider a simple example where Z=2. Here trace(X)≈ 1 2 trace ([ zT0 zT1 ] X [ z0 z1 ]) = 1 2 trace ([ zT0Xz T 0 z T 0Xz T 1 zT1Xz T 0 z T 1Xz T 1 ]) = 1 2 ( zT0Xz T 0+z T 1Xz T 1 ) = 1 2 1 ∑ z=0 zTzXzz , (6.5.42) which is exactly the result from eq. (6.5.39). This can be further simplified by nor- malizing the Monte Carlo vectors to the number of realizations with Z¯ = 1√ Z Z, giving trace(X)≈ trace ( Z¯TXZ¯ ) . (6.5.43) Applying this trick to the second trace in eq. (6.5.38) gives sn= trace ( Σ−1Vn ) − trace ( Z¯TU−TATΣ−TVnΣ−1AU−1Z¯ ) , (6.5.44) Chapter6 ITSG-Grace201668