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

Saeed Vatankhah, Shuang Liu+3 more|arXiv (Cornell University)|Jan 10, 2020

Geophysical and Geoelectrical Methods20 references4 citations

TL;DR

This paper presents a unified Lp-norm joint inversion framework for gravity and magnetic data using a cross-gradient constraint to enhance structural similarity between density and susceptibility models. By incorporating L0, L1, and L2 stabilizers and employing iterative conjugate gradient optimization with adaptive regularization, the method produces improved, physically plausible reconstructions of subsurface structures—especially when compared to individual inversions—demonstrating effectiveness on 3D synthetic models with both smooth and blocky features.

ABSTRACT

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.

Motivation & Objective

• To develop a generalized, unified framework for joint inversion of gravity and magnetic data that integrates multiple stabilizers (L0, L1, L2) under a single optimization scheme.
• To incorporate the cross-gradient constraint to enforce structural similarity between density and susceptibility models without requiring petrophysical relationships.
• To enable flexible control over model characteristics—smooth, sparse, or blocky—through selection of the Lp-norm parameter and regularization weights.
• To improve inversion stability and accuracy by integrating depth weighting, hard constraints, and bound constraints into the joint inversion process.
• To validate the method on 3D synthetic models and demonstrate superior reconstruction performance over individual gravity and magnetic inversions.

Proposed method

• The framework formulates a nonlinear objective function combining data misfit, a general Lp-norm stabilizer for model parameters or their gradients, and a cross-gradient constraint to align structural features between models.
• The objective function is minimized iteratively by linearizing it about an initial model and solving the resulting linear system using the conjugate gradient (CG) algorithm at each step.
• Regularization parameters are adaptively adjusted during iterations, starting large and decreasing over time, with a balancing weight applied to the cross-gradient term.
• Depth weighting and hard constraint matrices are incorporated to improve resolution at depth and include known model values, while bound constraints are applied per iteration.
• The method supports various stabilizers (L0, L1, L2) via the choice of p in the Lp-norm, enabling control over smoothness, sparsity, or blockiness in the final model.
• The algorithm is validated on 3D synthetic models with known density and susceptibility distributions, comparing joint inversion results to those from independent inversions.

Experimental results

Research questions

• RQ1 Can a unified Lp-norm framework effectively integrate multiple stabilizers (L0, L1, L2) into a single joint inversion for gravity and magnetic data?
• RQ2 How does the inclusion of a cross-gradient constraint improve the structural consistency and accuracy of jointly inverted density and susceptibility models?
• RQ3 To what extent can the method reconstruct both smooth and blocky subsurface structures, and how does it compare to individual inversion results?
• RQ4 What is the impact of adaptive regularization parameter adjustment and depth weighting on inversion convergence and model fidelity?
• RQ5 Can the method reliably reconstruct subsurface structures even when only partial data support certain features, such as dipping dikes or vertical dikes?

Key findings

• The joint inversion with cross-gradient constraint successfully reconstructs both dipping and vertical dikes in synthetic models, with structural features in density and susceptibility models closely matching the true models.
• For the L1-norm stabilizer case, the reconstructed models showed high structural similarity between density and susceptibility distributions, with the algorithm correctly omitting non-supported structures (e.g., no susceptibility response for the vertical dike).
• The algorithm converged in 58 iterations for the L1-norm case and reached the maximum 100 iterations for the L2-norm of the gradient case, indicating robust convergence under different stabilizers.
• Reconstructed models using joint inversion showed significantly improved accuracy and structural coherence compared to individual gravity and magnetic inversions, especially in resolving complex 3D geometries.
• The adaptive regularization strategy—starting with large regularization parameters and decreasing them over iterations—proved effective in stabilizing convergence and improving model quality.
• The method successfully incorporated depth weighting and hard constraints, enabling better resolution of deep structures and inclusion of known values in the model.