Using regularization for error reduction in GRACE gravity estimation

The Gravity Recovery and Climate Experiment (GRACE) is a joint National Aeronautics and Space Administration / Deutsches Zentrum für Luftund Raumfahrt (NASA/DLR) mission to map the time-variable and mean gravity field of the Earth, and was launched on March 17, 2002. The nature of the gravity field inverse problem amplifies the noise in the data that creeps into the mid and high degree and order harmonic coefficients of the earth's gravity fields for monthly variability, making the GRACE estimation problem ill-posed. These errors, due to the use of imperfect models and data noise, are manifested as peculiar errors in the gravity estimates as north-south striping in the monthly global maps of equivalent water heights. In order to reduce these errors, this study develops a methodology based on Tikhonov regularization technique using the L-curve method in combination with orthogonal transformation method. L-curve is a popular aid for determining a suitable value of the regularization parameter when solving linear discrete ill-posed problems using Tikhonov regularization. However, the computational effort required to determine the L-curve can be prohibitive for a large scale problem like GRACE. This study implements a parameter-choice method, using Lanczos bidiagonalization that is a computationally inexpensive approximation to L-curve called L-ribbon. This method projects a large estimation problem on a problem of size of about two orders of magnitude smaller. Using the knowledge of the characteristics of the systematic errors in the GRACE solutions, this study designs a new regularization matrix that reduces the systematic errors without attenuating the signal. The regularization matrix provides a constraint on the geopotential coefficients as a function of its degree and order. The regularization algorithms are implemented in a parallel computing environment for this study. A five year time-series of the candidate regularized solutions show markedly reduced systematic errors without any reduction in the variability signal compared to the unconstrained solutions. The variability signals in the regularized series show good agreement with the hydrological models in the small and medium sized river basins and also show non-seasonal signals in the oceans without the need for post-processing.