BTTB-based numerical schemes for three-dimensional gravity field inversion

Abstract

This paper presents efficient numerical schemes for 3-D gravity field inversion. We propose a 2-D multilayer model to approximate a 3-D density distribution, and prove that the solution of the multilayer model will converge to the discretized 3-D solution. Differed from the conventional fast Fourier transform (FFT) based methods in which FFT is applied to the kernel, the proposed approach directly generates the Block-Toeplitz Toeplitz-Block (BTTB) structure by discretizing the multilayer model and the BTTB matrix is embedded into a Block-Circulant Circulant-Block (BCCB) matrix such that FFT can be utilized. In this approach, both regularization and optimal pre-conditioning operator can be constructed in the form of BTTB matrix. Consequently, very efficient solvers can be developed, and tremendous reduction in storage requirement and computing time can be achieved. To validate the new approach, numerical simulations using synthetic and real field data are reported, and numerical analysis is carried out for the inversion problems. Based on this study, we conclude that the proposed methods are capable of performing large-scale 3-D density inversions with a modest computing resource.