We analyze a strategy to optimize the computational effort of largescale electromagnetic (EM) modeling and particularly inversion, the latter usually requiring a large number of forward modeling solutions. While we are interested in finely gridded earth models to capture realistic structures, the forward modeling operator may act upon a coarser simulation grid, or a subsection of the model grid, thus providing significant potential for computational speed-up. After briefly outlining the methodology of the grid transfers and its implications on the inversion scheme, we demonstrate the method on a marine CSEM survey example.
Three-dimensional (3D) conductivity imaging of EM data has become increasingly important, owing to new exploration scenarios, as for example the marine environment, and new efforts of combining the capabilities of EM with those of more “traditional” seismic based methods. Realistic parameterizations of the earth’s conductivity may involve a large and fine mesh, leading to as many as 106 -107 grid cells, in order to correctly simulate highly structured 3D geology, bathymetry, or topography. However, depending on the excitation frequency, be it either from a natural or a controlled source, the characteristics of the EM field to be simulated may allow for a discretization with a lower degree of spatial sampling. Key to using this advantage is an appropriate mapping scheme to transfer between the vector based quantities on the simulation mesh and the scalar-type conductivity parameters on the modeling mesh.
We consider a modeling/inversion grid Wm of size (number of cells) M and a finite-difference (FD) simulation grid Ws of size N. Both grids are Cartesian with conformal grid axes, with Ws based on a staggered Yee (1966) grid. Wm defines the space of the model parameters, i.e. electrical conductivities sk, k =1,…,M, that are assigned to cell centers. The inversion domain will be either Wm or a subset. In the inverse problem, we want to determine the distribution of s over the inversion grid such that a set of observed measurements is reproduced. Solving the inverse problem is in principal accomplished by the iterative minimization of a cost functional F, denoting the misfit between observed and predicted data (e.g. Newman & Alumbaugh, 1997). This requires computing the gradient ÑF with respect to the model parameters sk, with its components ?F?sk . A full derivation of the gradient formulation for the 3D EM inverse problem, using a scattered-field formulation, is given by Newman & Alumbaugh (1997) and Newman & Hoversten (2000). Here, we only outline the differing methodology resulting from the case Wm ? Ws . The first linking point between the simulation grid and the grid defining the model parameter space occurs in the construction of the 3N x 3N FD stiffness matrix
of the linear system
=
, (1)
which denotes the forward modeling problem. See for example Alumbaugh et al. (1996) for details. In equation (1),
is the source vector that depends on the boundary conditions, source field polarization, and excitation frequency, and
is the electric field solution vector. Both vectors belong to the edge element space defined by Ws.
