A Spectral-Domain Pseudo-Inverse Method for True 3D Gravity Inversion

Abstract

The main difficulty in 3D gravity inversion is that surface observations lack vertical wavenumber information, making the problem underdetermined and depth resolution poor. Building on the author's spectral-domain pseudo-inverse theory for unitary diagonalizable systems, this paper presents a true 3D inversion method. Surface data are analytically continued upward via Laplace's equation to form a 3D data volume, providing the vertical wavenumber sampling for the 3D Fourier transform. A general analytical expression for the half-space spectrum is derived, separating the horizontal spectrum from the vertical propagation kernel. The Green's function of the 3D Poisson equation is unitarily diagonalized, yielding the forward spectral response $λ(\kk) = -i 4πG k_z / K^2$, and a stable inverse filter $q_{\alpha }(\lambda )=\bar{\lambda }/(|\lambda {|}^2+\alpha )$ is constructed. This filter is proved to have bounded stability and consistency (reducing to the exact inverse as $\alpha \to 0^{+}$). Validation with a homogeneous sphere model shows correct recovery of the anomaly location and singular behavior at the source. The contributions are twofold: (1) upward continuation constructs a 3D volume from 2D surface data, mitigating underdetermination; (2) the inversion is reduced to one forward 3D Fourier transform, one spectral scaling, and one inverse transform.The proposed framework is not limited to gravity; it applies to any linear potential-field inverse problem with a translation-invariant forward operator, including magnetic inversion.