Linearized gravity inversion for layer-constrained 3D density estimation using a series expansion-based discretization approach

Geophysical inversion has become a powerful tool for accurately estimating the model parameters that describe subsurface geological structures. In this study, we performed 3D inversion of gravity data to reveal the subsurface density distribution. Both the classical Gaussian least squares and the newly developed series expansion-based inversion methods were employed to solve the inverse problem, separately. Both techniques were evaluated using a comprehensive suite of synthetic experiments designed to investigate inversion performance under varying model complexities, regularization parameters, observation densities, and noise conditions in both the model and data spaces. In both cases, the inversion methods demonstrated stability and effectiveness in minimizing the misfit between observed and calculated data, ensuring reliable estimation of the model parameters, provided the problem is overdetermined. However, in an underdetermined inverse problem which is the most common case in practical applications where the number of model parameters exceeds the observed data, the Gaussian least squares method suffers from non-uniqueness often accompanied by numerical instability. Hence, reliable estimation of model parameters in such cases requires additional physical and mathematical constraints. In contrast, the series expansion-based inversion method, which reduces the need for explicit regularization constraints, approximates the spatial distribution of model parameters using polynomial coefficients of known basis functions. Instead of high number of density values, much less expansion coefficients are to be determined, which can significantly increase the data-to-unknowns ratio and help to transform the problem into a more stable overdetermined system. The proposed methodology was further applied to gravity data from the Shendi–Atbara Basin, Sudan, under both overdetermined and underdetermined inversion scenarios. These findings highlight the feasibility of the series expansion-based inversion method as a robust alternative to the Gaussian least squares method, particularly in case of fewer and sparsely distributed measured data points.