3D finite-element forward modeling of electromagnetic data using vector and scalar potentials and unstructured grids

Abstract

We present a finite-element solution to the 3D electromagnetic forward-modeling problem in the frequency domain. The method is based on decomposing the electric field into vector and scalar potentials in the Helmholtz equation and in the equation of conservation of charge. Edge element and nodal element basis functions were used, respectively, for the vector and scalar potentials. This decomposition was performed with the intention of satisfying the continuity of the tangential component of the electric field and the normal component of the current density across the interelement boundaries, therefore finding an efficient solution to the problem. The computational domain was subdivided into unstructured tetrahedral elements. The system of equations was discretized using the Galerkin variant of the weighted residuals method, with the approximated vector and scalar potentials as the unknowns of a sparse linear system. A generalized minimum residual solver with an incomplete LU preconditioner was used to iteratively solve the system. The solution method was validated using five examples. In the first and second examples, the fields generated by small dipoles on the surface of a homogeneous half-space were compared against their corresponding analytic solutions. The third example provided a comparison with the results from an integral equation method for a long grounded wire source on a model with a conductive block buried in a less conductive half-space. The fourth example concerned verifying the method for a large conductivity contrast where a magnetic dipole transmitter-receiver pair moves over a graphite cube immersed in brine. Solutions from the numerical approach were in good agreement with the data from physical scale modeling of this scenario. The last example verified the solution for a resistive disk model buried in marine conductive sediments. For all examples, convergence of the solution that used potentials were significantly quicker than that using the electric field.

A finite-volume solution to the geophysical electromagnetic forward problem using unstructured grids

“…numerical results with the results of physical scale modeling (PSM) and the integral-equation (IE) method presented in Farquharson et al (2006) and with FE solutions by Ansari and Farquharson (2014). PSM is a laboratory study in which actual measurements are performed for idealized miniature geologic models.…”

“…Compared to FE methods, FV methods are less mathematically involved, they are simpler in idea, more general, and also physically meaningful (see, for example, Hirsch, 2007). Many different FE approaches have been proposed for solving EM problems, which in general solve either for the electric field or for the electric and magnetic potentials using nodal-or edgebased elements (see the reviews by Hou et al, 2006;Børner, 2010;Ansari and Farquharson, 2014). Alternatively, FV schemes have been proposed for general control volumes: Shankar et al (1990) and Remaki (2000) define all of the fields colocated at the centers of the control volumes, whereas Madsen and Ziolkowski (1990) use a staggered grid in which the full 3D vectors of the electric and magnetic fields are defined on the edges of the primary and dual cells, respectively.…”

Temporal variation in the resistivity structure of the first Nakadake crater, Aso volcano, Japan, during the magmatic eruptions from November 2014 to May 2015, as inferred by the ACTIVE electromagnetic monitoring system

“…For the source term in Eq. (1), we adopt a kind of Heaviside function, where i s has a constant value only on a straight line within an element (e.g., Ansari and Farquharson 2014). To include the actual topography, we adopt an unstructured tetrahedral mesh generated by Gmsh freeware (http://gmsh.info/).…”

A multi-domain decomposition-based Fourier finite element method for the simulation of 3D marine CSEM measurements

“…Direct matrix solvers for large geophysical 3D forward problems [33,28,30,26,24,14,19,8,35,2,15] can be computationally expensive, both in terms of memory and CPU time. A way to overcome this limitation is by reducing the dimensionality of the problem, namely from 3D modeling to a 2.5D [23], 2D [4] and/or 1D [20] approximations that are only suitable for particular geometries.…”