A Recursive Regularized Solution to Geophysical Linear Ill-Posed Inverse Problems

Linear ill-posed models are widely encountered in various problems in geophysics and remote sensing. The regularization technique can significantly improve the accuracy of the estimates since the biases introduced by the regularization are much smaller than the errors reduced by regularization. However, from the spectral point of view, certain low-frequency terms with large singular values might become over-regularized, whereas other high-frequency terms with small singular values might be insufficiently regularized for a given regularization parameter. For this reason, we developed a recursive regularization approach to further improve the regularized solution via additional regularization of some high-frequency terms and restricted regularization of some low-frequency terms. The analytical conditions to determine the terms to be further regularized are derived based on the criterion that the introduced biases should be smaller than the reduced errors; in other words, the mean squared error (mse) should be reduced. Furthermore, the universal form of the recursive regularized solution is derived. Two examples from remote sensing are designed to demonstrate the performance of the developed approach. The first example involves solving the Fredholm integral equation of the first kind, a fundamental mathematical model used in many inverse problems in remote sensing. The results indicate that the proposed method outperforms the ordinary Tikhonov regularization, partial regularization, and adaptive regularization, with roots of mse reduced by 25.8%, 14.5%, and 8.1%, respectively. Subsequently, we apply the proposed method to estimate regional mass anomalies based on the mascon modeling using the Gravity Recovery and Climate Experiment (GRACE) time-variable gravity field models. The results demonstrate that the proposed method preserves more signal than conventional regularization methods.