Key Takeaways & Executive Findings
- •• Introduces an efficient iterative solver (GMRES with incomplete LU preconditioning) for 3D finite-difference gravitational field modeling, overcoming memory and time limitations of direct solvers. • Employs central finite-difference discretization on right rectangular prismatic grids and high-degree Lagrange interpolation for accurate partial derivatives of gravitational potential. • Validates the algorithm using three density models, demonstrating high accuracy, reliability, and flexibility for 3D forward modeling. • Provides a practical solution for large-scale 3D gravity data inversion and geological interpretation.
Abstract
With the evolution of geophysical surveys from traditional two-dimensional (2D) to three-dimensional (3D) models, the resulting large data volumes pose significant challenges to inversion, particularly when resolving large-scale 3D structures. A direct solver for solving an ill-conditioned linear system resulting from the finite-difference approximation of a boundary value problem requires more memory and time than iterative solvers. To overcome this limitation, an efficient iterative solver for 3D finite-difference approach is introduced to calculate the 3D gravitational potential and the associated gravitational field. Firstly, the boundary value problem associated with 3D gravitational potential is discretized using central finite-difference technique based on right rectangular prismatic grids. The resulting large unsymmetric sparse systems are then solved using the generalized minimal residual algorithm (GMRES) iterative solver in combination with incomplete LU factorization. Secondly, to obtain high-accuracy partial derivatives of gravitational potential, a high-degree Lagrange interpolation scheme is employed. Finally, three density models are applied to test the accuracy, reliability, and flexibility of our 3D finite-difference algorithm. All computational results demonstrate that our method provides an accurate approximation of the gravitational field and is applicable to 3D forward modeling.
1. Introduction
The gravity method is one of the oldest applicable geophysical techniques used to deduce the density distribution of subsurface media, facilitating the study of the Earth's interior. This method is commonly utilized to identify the locations of subsurface targets at depth and to delineate geological subsurface models with improved resolution [1, 2]. Recently, the applications of gravity data, including the gravitational field, have grown substantially in diverse areas, such as searching for oil and gas [3, 4], exploring mineral resources [5−7], mapping deep geological formations [8, 9], and delineating geological structures in engineering and the environment [10, 11]. By utilizing gravimeter instruments, geophysicists can capture signals related to the vector gravitational field. These measured gravity data correlate directly with subsurface density distributions, which can be inverted using established inversion algorithms [12]. Therefore, the development of efficient gravity methods for regional geological investigations has emerged as a significant research priority. The 3D forward-modeling scheme appears to be an important tool for improving inversion accuracy and the quality of geological interpretation [13].
Forward modeling approaches for the gravitational field exhibit a fundamental dichotomy: spatial domain methodologies versus frequency domain techniques. In the spatial domain, computational strategies are divided into analytical solutions that utilize integral expressions, and numerical procedures that employ differential operators. The analytical solutions are available for various bodies and density distributions. These solutions subsequently enable computation of the gravitational field at arbitrary spatial coordinates [14−18]. Nevertheless, analytical approaches face significant constraints, including complex mathematical formulations, prohibitive computational expenses for whole-space simulations, and limited flexibility for sources exhibiting variations in physical properties. In contrast, Fourier domain representations of potential field feature compact mathematical forms, allowing for efficient forward simulations via fast Fourier transform algorithms [19, 20]. Regrettably, substituting continuous Fourier transform with discrete implementations substantially degrades computational precision in frequency-domain approaches [21].
Forward modeling of the gravitational field fundamentally requires solving the 3D boundary value problem via spatial-domain numerical methods—a critical procedure for enabling accurate geophysical data interpretation [22]. This computational procedure...
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TONG Xiao-zhong, XIE Wei, MA Hui-ying, WEN Xin-yue, ZHU Wen-di, ZHANG Chen (2026). 3D finite-difference numerical simulation of the gravitational field using a preconditioned GMRES iterative solver. Journal of Central South University. https://doi.org/10.1007/s11771-026-6214-5
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Frequently Asked Questions
What is the main contribution of this paper?
The paper introduces an efficient iterative solver (GMRES with incomplete LU preconditioning) for 3D finite-difference gravitational field modeling, which reduces memory and computational time compared to direct solvers, making large-scale 3D forward modeling feasible.
How does the proposed method achieve high accuracy?
The method uses central finite-difference discretization on right rectangular prismatic grids and a high-degree Lagrange interpolation scheme to compute partial derivatives of gravitational potential, ensuring high accuracy in the computed gravitational field.
What are the advantages of using an iterative solver over a direct solver?
Iterative solvers require less memory and computational time for large sparse systems, making them more suitable for large-scale 3D problems, whereas direct solvers become prohibitive in terms of resources.
How is the method validated?
The method is validated using three density models, demonstrating its accuracy, reliability, and flexibility for 3D forward modeling of the gravitational field.
What are potential applications of this work?
The method can be applied to 3D gravity data inversion and geological interpretation, including oil and gas exploration, mineral resource exploration, and mapping deep geological structures.
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