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• W2283313 abstract "A limited number of numerical solutions of the three-dimensional (3D) direct current (DC) resistivity problem have been discussed in the geophysical literature. These solutions have been obtained using integral equation, finite difference or finite element techniques. The integral equation method is most efficient for modeling one or a few inhomogeneous bodies in a homogeneous earth. Finite difference (FD) and finite element (FE) methods are better suited to model any arbitrarily complex 3D earth. Dey & Morrison (1979) developed a 3D finite-dif ference algorithm to evaluate the potential for a point current source. The equation of continuity is integrated over elemental volumes to obtain a system of self-adjoint difference equations. A mixed boundary condition was introduced, based on the asymptotic behavior of the potential field in a homogeneous medium. Spitzer (1995) reported a FD algorithm using conjugate gradient methods. A compact storage scheme was employed, which reduced the number of memory-resident coefficients and shortened the run time by avoiding unnecessary computational operations. Lowry et al. (1989) proposed a 3D integrated finite difference scheme using a singularity removal technique. This method actually models the anomalous potential which is due to conductivity contrasts. The anomalous potential is smoother than the total potential, therefore, the solution is generally more exact. Two modifications of the finite difference method were recently made by Zhao & Yedlin (1996). The first is a more accurate formula for the source singularity removal. The second is the analytic computation of the source terms that arise from the decomposition of the potential into the primary and secondary potential. Spitzer et al. (1999) presented a 3D DC and induced polarization (IP) finite difference code, which offers grid-independent electrode positioning and detaches both transmitters and receivers from grid nodes. The discussion of the finite element method is a bit more sparse, perhaps because finite elements are more complex to implement than finite differences. The application of the finite element method to the 2D resistivity problem was discussed by Coggon (1971). A finite element solution to the 3D resistivity problem was reported by Pridmore et al. (1981). Sasaki (1994) developed a 3D resistivity inversion algorithm using the finite element method. Recently, Zhou & Greenhalgh (2001) published a finite element solution to the 3D DC problem. The mixed boundary condition and a compact storage scheme were incorporated. However, they solved the governing equation of the total potential. In this paper, we revisited 3D FE resistivity modeling, but with the use of the singularity removal. Furthermore, a modified method of singularity removal is presented. First the theoretical basis of our FE solution for the secondary potential is developed. After that, the FE and FD schemes are compared in terms of accuracy and memory requirements. Finally, the effect of singularity removal is illustrated by three examples: a dike model, a cube buried in a two-layered earth and a cube buried near a vertical contact." @default.
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• W2283313 date "2001-01-01" @default.
• W2283313 modified "2023-09-27" @default.
• W2283313 title "Three-dimensional DC resistivity finite element and finite difference forward modeling in comparison" @default.
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