Efficient microcomputer‐based finite‐difference resistivity modeling via Polozhii decomposition

Finite‐difference modeling for direct current resistivity problems can be cast in a form allowing rapid and efficient solution on a microcomputer. Almost a million nodes can be accommodated on a microcomputer with only 256 kilobytes of memory. The Polozhii decomposition procedure (Polozhii, 1965) allows a three‐dimensional (3-D) problem with no more than a one‐dimensional (1-D) property variation to be transformed into a series of decoupled and independent one‐dimensional problems. The decomposition utilizes matrix transforms analogous to analytic spatial Fourier and Hankel transforms. These matrix transforms, along with recursive formulas for the solution of the 1-D problems, yield a process which can be thought of as a continuation operator when compared with current popular methods of potential field continuation using the Fourier transform. The result is that only a single plane of node potentials needs to be determined fully. By setting up more complicated models as separate regions with differing 1-D models the above procedures can be used to create a problem where the unknown potentials along the boundary planes between the regions are cast in terms of known outer boundary potentials and known applied source terms. Once all interior boundary potentials have been calculated, the solution on any arbitrary plane of nodes in any region can be quickly calculated using the standard Polozhii decomposition procedure. Execution times for a variety of grid sizes composed of two such regions show much slower growth with increasing numbers of nodes than conventional finite‐difference solution schemes. Test cases for comparison with analytic calculations yield errors of 5 percent or less for problems composed of 756 000 nodes. The choice of boundary condition, however, has a significant influence on the accuracy obtained.