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 kz / K2, and a stable inverse filter qα(λ) = λ̅/(|λ|2 + α) is constructed. This filter is proved to have bounded stability and consistency (reducing to the exact inverse as α→ 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, which does not add physical observations but unfolds the 3D spatial variation information (data innovation) derived through the governing physical equation; (2) the inversion is reduced to one forward 3D Fourier transform, one spectral scaling, and one inverse transform. The framework extends to any linear potential-field inverse problem with a translation-invariant forward operator, including magnetic inversion.
Related
- Pseudo trans-dimensional 3d geometrical inversion: a proof of concept using gravity data
- Structure-Driven Inversion: A New Paradigm for Solving Inverse Problems
- Seismic Imaging of the Crust From Long‐Offset Ocean‐Bottom Seismometer Data Using Time‐Domain Extended‐Source Full‐Waveform Inversion: Application to the Eastern Nankai Trough, Japan
- Time domain spectral IP and FCM-constrained CSAMT inversions: An integrated approach to gold exploration
- Flow dependence of boundary-layer scheme parameters revealed by stochastic inversion
BibTeX
@misc{chen2026spectraldomain,
title = {{A Spectral-Domain Pseudo-Inverse Method for True 3D Gravity Inversion}},
author = {Shengchang Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.11014},
eprint = {2608.11014},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11014},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A Spectral-Domain Pseudo-Inverse Method for True 3D Gravity Inversion
AU - Shengchang Chen
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2608.11014
UR - https://arxiv.org/abs/2608.11014
AB - 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_α(λ) = λ̅/(|λ|^2 + α) is constructed. This filter is proved to have bounded stability and consistency (reducing to the exact inverse as α→ 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, which does not add physical observations but unfolds the 3D spatial variation information (data innovation) derived through the governing physical equation; (2) the inversion is reduced to one forward 3D Fourier transform, one spectral scaling, and one inverse transform. The framework extends to any linear potential-field inverse problem with a translation-invariant forward operator, including magnetic inversion.
ER -