Three-dimensional forward modelling of gravity field vector and its gradient tensor using the compact difference schemes

Abstract

Summary The traditional gravity forward modeling methods for solving partial differential equations (PDEs) only can yield second-order accuracy. When computing the gravity field vector and gradient tensor from the obtained potential, those numerical differentiation approaches will inevitably lose accuracy. To mitigate this issue, we propose an efficient and accurate three-dimensional (3D) forward modeling algorithm based on a fourth-order compact difference scheme. Firstly, a 19-point fourth-order compact difference scheme with general meshsizes in x −, y − and z − directions is adopted to discretize the governing 3D Poisson’s equation. The resulting symmetric positive-definite linear systems are solved by the preconditioned conjugate gradient algorithm. To obtain the first-order (i.e., the gravity field vector) and second-order derivatives (i.e., the gravity gradient tensor) with fourth-order accuracy, we seek to solve a sequence of tridiagonal linear systems resulting from the above mentioned finite difference discretization by using fast Thomas algorithm. Finally, two synthetic models and a real topography relief are used to verify the accuracy of our method. Numerical results show that our method can yield a nearly fourth-order accurate approximation not only to the gravitational potential, but also to the gravity field vector and its gradient tensor, which clearly demonstrates its superiority over the traditional PDE-based methods.

Fast 3D gravity and magnetic modelling using midpoint quadrature and 2D FFT

“…In this section, we examine the proposed method by presenting two different kinds of numerical experiments. At first, we use a benchmark scenario model 32,33 to verify the proposed algorithm and evaluate the performance of the proposed algorithm by comparing it with existing implementation algorithms. Then, we apply the proposed approach to a local real topographic model in New Zealand to demonstrate the practical application.…”

“…1, the benchmark model is used to verify the correctness and evaluate the performance of the gravity and magnetic forward modelling code. The model setup is identical to that used in Pan et al 32 , with a modelling domain of 100 km × 100 km × 100 km , which contains two cubic bodies. The model region is discretized into 128 × 128 × 128 cells with an equal interval of 781.25 m in three directions.…”

“…For the magnetic case, the maximum errors for the magnetic field B z and magnetic gradient T m zz (8) To evaluate the numerical performance of the proposed method, the forward results are compared with the space-wavenumber domain method 18 , which transforms the 3D integral equation methods into a 1D integral with different wavenumbers. In addition to the method mentioned 18 , we also compare our method with the compact difference scheme for solving the Posson's equation 32 .…”

“…In addition, the calculation time of the proposed method only required 0.25 s, whereas 19.48 s and 31.0 s were required for the method of Dai et al 18 and Pan et al 32 , respectively. Our method has a speedup ratio of 78 and 124 over the Dai et al 18 and Pan et al 32 methods, respectively. It can be seen that our proposed method has higher computational accuracy and efficiency.…”

Fictitious Point Technique Based on Finite-Difference Method for 2.5D Direct-Current Resistivity Forward Problem

“…The finite-difference numerical results indicate that our fictitious point technique can provide high accuracy for 2.5D direct-current resistivity modeling. The relative root-mean-square error was used to measure the overall accuracy of finite-difference scheme [33]:…”