The modeling of the marine electromagnetic (EM) problem requires fine gridding to account for the seafloor bathymetry and to model complicated targets. This makes the computational cost of the problem large by using conventional Finite-Difference (FD) solvers. To circumvent these problems, we employ a volume Integral Equation (IE) approach to arrive at a preconditioning operator. Such an approach significantly reduces the condition number and the size of the stiffness matrix. The preconditioner is divergence free and is based on a magnetic field formulation. The cost of constructing the preconditioning operator is much less than the one used to construct the operator in a standard IE approach. Further a homogenization technique that allows grids to be non-conformal to the inhomogeneity interfaces is used. The optimal grid technique is used to extend the boundaries of the simulation domain to infinity. Some numerical results are presented in order to illustrate the accuracy and the effectiveness of the method.
Owing to an increased interest in marine resources, a variety of numerical methods have been developed to map conductivity distribution of the subsurface. For hydrocarbon exploration and development, surface EM methods are arguably one of the important geophysical technologies for imaging below the seafloor since the emergence of the three-dimensional (3D) reflection seismology some twenty-five years ago. Therefore, an effective and accurate solution of the marine electromagnetic (EM) problems is very important. However most of the existing simulation codes are either slow or not accurate. These marine EM problems have challenges that distinguish them from other types of EM simulation problems. In these problems, we usually deal with a layered background medium with several inhomogeneities embedded in it.
One type of anomalies consists of hydrocarbon-bearing layers located under the seafloor. The second type is due to topological variation of the seafloor, the so-called bathymetry. In this paper we present our efforts to develop a fast and robust numerical algorithm for the solution of 3D marine EM problems.
For discretization, we employed the Lebedev grid so that we can accurately take into account the full anisotropy of the conductivity. Inside the domain of interest (where the sources, receivers, anomalies and bathmetry are located) we use a uniform grid. This domain of interest is extended in y-and z-direction using optimal grids; see Davydycheva et al. (2). The combination of Lebedev’s grid and optimal grids allows the truncation of error for approximation in infinite domains and near the source and receiver. In order to use optimal grid (which are non-uniform) without sacrificing accuracy, we use a specific homogenization method called nodal homogenization; see Moskow etal. (3). This technique is based on a continuous and discrete energymatching condition for certain class of functions approximating the exact solution of our problem. The resulting linear system of equations (the stiffness matrix) is iteratively solved using an IE-type preconditioner based on the 1D background medium mentioned above. It eliminates the background from the iterative process; i.e., all iterations are performed only within the anomaly in the x-direction.