Generalized L$_p$-norm joint inversion of gravity and magnetic data using cross-gradient constraint

A generalized unifying approach for $L_{p}$-norm joint inversion of gravity and magnetic data using the cross-gradient constraint is presented. The presented framework incorporates stabilizers that use $L_{0}$, $L_{1}$, and $L_{2}$-norms of the model parameters, and/or the gradient of the model parameters. Furthermore, the formulation is developed from standard approaches for independent inversion of single data sets, and, thus, also facilitates the inclusion of necessary model and data weighting matrices that provide, for example, depth weighting and imposition of hard constraint data. The developed efficient algorithm can, therefore, be employed to provide physically-relevant smooth, sparse, or blocky target(s) which are relevant to the geophysical community. Here, the nonlinear objective function, that describes the inclusion of all stabilizing terms and the fit to data measurements, is minimized iteratively by imposing stationarity on the linear equation that results from applying linearization of the objective function about a starting model. To numerically solve the resulting linear system, at each iteration, the conjugate gradient algorithm is used. The general framework is then validated for three-dimensional synthetic models for both sparse and smooth reconstructions, and the results are compared with those of individual gravity and magnetic inversions. It is demonstrated that the presented joint inversion algorithm is practical and significantly improves reconstructed models obtained by independent inversion.

💡 Research Summary

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The paper presents a unified framework for the joint inversion of gravity and magnetic data that integrates generalized Lp‑norm regularization (with p ranging from 0 to 2) and a cross‑gradient structural constraint. The authors begin by discretizing the subsurface into a set of rectangular prisms, each characterized by two physical parameters: density (gravity) and magnetic susceptibility (magnetics). These parameters are stacked into a single model vector, while the observed data vectors for gravity and magnetics are similarly stacked. Linear forward operators G₁ and G₂ map the model parameters to the observed data, and the combined forward matrix G is block‑diagonal.

The objective function P(α,λ) consists of three terms: (1) a data‑misfit term weighted by a diagonal matrix W_d that incorporates measurement uncertainties; (2) a regularization term weighted by W D, where D is the identity and W is a block‑diagonal matrix that can embed depth weighting, hard constraints, and an Lp‑norm weighting matrix W_Lp; and (3) a cross‑gradient term ‖t(m)‖₂², where t(m)=∇m₁×∇m₂ measures the structural similarity between the density and susceptibility models. The scalar α controls the overall strength of the regularization, while λ controls the influence of the cross‑gradient constraint.

The Lp‑norm regularization is approximated through an iteratively re‑weighted least‑squares (IRLS) scheme. Specifically, the weighting matrix W_Lp is defined element‑wise as 1/