A Factorized Green-Operator Framework for Efficient Wavenumber-Domain Inversion of Magnetic Anomalies and Gradient Tensors

Summary

Magnetic inversion is a key tool for imaging subsurface geological structures, but conventional 3-D magnetic inversion in the spatial domain is often limited by the computational and memory cost of large dense kernel matrices. Existing transformed-domain approaches improve efficiency, yet pseudo-3D implementations still rely on layer-by-layer accumulation and repeated Fourier transforms. In this study, we develop a unified wavenumber-domain framework for the forward modelling and inversion of total-field magnetic anomalies and magnetic gradient-tensor data. For regularly discretized rectangular prisms beneath a planar observation surface, the wavenumber-domain Green operator is reformulated into a factorized representation consisting of two explicitly stored diagonal/block-diagonal spectral factors and one implicitly applied separable horizontal operator. This implementation avoids repeated vertical layer superposition and reduces the forward evaluation to a single FFT/IFFT pair together with structured spectral multiplications. The factorized forward operator is then embedded in a Tikhonov-regularized inversion and solved through a Sherman-Morrison-Woodbury (SMW) reduced system. The transformed-domain data term is defined as an unweighted complex-valued least-squares residual, and its relation to the spatial-domain least-squares formulation is stated under the corresponding padding and truncation assumptions. Synthetic examples show that the method reproduces conventional spatial-domain responses and recovers the principal features of prescribed magnetization models under 5% Gaussian noise. For a 200×200×100 model, the forward modeling and core inversion times are 0.172 s and 31.73 s, respectively, on a standard laptop. Application to field data is used as a practical feasibility test and shows a data-consistent recovered magnetization distribution, but it should not be regarded as an independent geological validation of the recovered model. The current implementation assumes a planar observation surface, a regular FFT-compatible grid, and a spatially uniform magnetization direction. It does not yet address strong remanence, spatially variable magnetization, irregular topography, irregular acquisition geometries, depth weighting, focusing stabilizers, or geological constraints. Under these assumptions, the proposed framework provides an efficient, memory-economical, and scalable alternative for large-scale magnetic anomaly interpretation.