The selective truncated singular value decomposition estimation for ill-posed geophysical observation equations

Ill-posed problems are prevalent in geodetic and geophysical data processing, significantly impacting the traditional least squares (LS) estimation. Consequently, the regularization estimation methods such as Tikhonov regularization estimation, and truncated singular value decomposition (TSVD) estimation have been developed. This paper studies and improves upon TSVD by proposing a new regularization method, which is referred to as the selective truncated singular value decomposition (STSVD) estimation. Firstly, from the perspective of the mean square error (MSE), the criteria for selecting the truncation parameter are analyzed, as are the limitations of TSVD estimation. Secondly, to more effectively reduce the MSE , we introduce the concept of bias-to-variance ratio and expanding the traditional notion of ‘truncation’. This leads to an optimized criterion for selecting the truncation parameter and the proposal of the STSVD estimation, whose theoretical superiority in terms of the MSE is proven. Thirdly, we present the estimation methods for estimating the unknown unit weight variance and squared bias required for the bias-to-variance ratio. Finally, the proposed method is applied to the solution of Fredholm integral equation of the first-kind and the simulation experiment involving downward continuation of airborne gravity measurement data, validating the effectiveness and superiority of the new estimation.