Using parallel computation to apply the singular value decomposition (SVD) in solving for large Earth gravity fields based on satellite data

Using satellite data only to estimate for an Earth gravity field introduces the problem of an ill-conditioned system of equations. This mathematical difficulty amplifies as the number of unknown gravity field parameters increases, requiring a stabilization of the inversion for solution. But the number of parameters to be estimated can also be too large to allow inversion using a sequential algorithm (one computer processor). Therefore the challenge is two-fold. A stabilized inversion must be performed with a parallel (multi-processor) algorithm. Thus, new code was developed in the parallel computing infrastructure of Parallel Linear Algebra Package (PLAPACK) to achieve the task of applying the Singular Value Decomposition (SVD) to invert for (and stabilize) very large gravity fields of well over 25,000 unknown parameters. This new code is given the name (Parallel LArge Svd Solver) PLASS. The choice of the SVD was made because it offers multiple opportunities of stabilization techniques. Poorly observed parameter corrections are removed from the culpable eigenspace of the normal matrix of CHAMP or the singular vector space of the upper R triangular matrix of GRACE. Solutions were stabilized based on the removal of either eigenvalues or singular values using four different standard optimization criteria: Inspection, Relative Error, Norm Norm minimization, trace of the Mean Square Error (MSE) matrix, and with a fifth method, independently introduced for this investigation, that optimizes removal of eigenvalues or singular values based on Kaula’s power rule of thumb. This method is given the name “Kaula Eigenvalue (KEV) or Kaula Singular Value (KSV) relation”. For the gravity fields of this investigation, orbital fits, geodetic evaluations and error propagations of the best of the resulting SVD gravity fields were performed, and shown to be comparable to the CHAMP solution obtained by the GeoForschungsZentrum (GFZ) and to the full rank GRACE solution obtained by the Center for Space Research (CSR).