用于真三维重力反演的谱域伪逆方法
A Spectral-Domain Pseudo-Inverse Method for True 3D Gravity Inversion
浏览论文内容
中文总结 AI 辅助
针对三维重力反演欠定与深度分辨率差的问题，本文提出真三维反演方法，通过向上延拓构建三维数据体缓解欠定性，构造稳定逆滤波器简化反演，经均匀球体模型验证有效。
中文摘要 AI 辅助
三维重力反演的主要难点在于地表观测缺乏垂直波数信息，导致该问题欠定且深度分辨率差。基于作者针对酉可对角化系统的谱域伪逆理论，本文提出一种真三维反演方法：通过拉普拉斯方程将地表数据向上解析延拓以形成三维数据体，为三维傅里叶变换提供垂直波数采样；推导了半空间谱的通用解析表达式，将水平谱与垂直传播核分离；三维泊松方程的格林函数可酉对角化，得到正演谱响应λ(kk) = -i4πGkz/K²，并构造了稳定的逆滤波器qα(λ) = λ̄/(|λ|² + α)，该滤波器被证明具有有界稳定性和一致性（当α→0+时退化为精确逆）。通过均匀球体模型验证，结果显示异常体位置和源处奇异行为均被正确恢复。本文贡献有两点：（1）向上延拓从二维地表数据构建三维数据体，缓解欠定性；（2）反演简化为一次三维傅里叶正变换、一次谱缩放和一次逆变换。
英文摘要
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 $q_α(λ) = \barλ/(|λ|^2 + α)$ is constructed. This filter is proved to have bounded stability and consistency (reducing to the exact inverse as $α\to 0^+$). 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.
