The electromagnetic response is studied for a model three‐layer earth formed by constant conductivity in the first and the third layers and conductivity varying with depth in the second layer (i.e., the inhomogeneous transition layer). A generalization to the case of many constant or variable conductivity layers is presented, too. The model problem is addressed by numerically solving an initial value problem for an ordinary differential equation for the inhomogeneous transition layer. The applicability of the procedure proposed, which is limited by the numerical method used, is discussed. As an illustration, the computed apparent resistivities and phases are compared with the results of Kao and Rankin (1980). The technique presented is applied to the computation of the response in the very‐low frequency (VLF) method. Application to other methods employing the plane wave is similar.
One of the major advances of computer science in recent years is the introduction of parallel processors. The efficiency of such devices depends strongly on the symmetry of the algorithms implemented; probably the most efficient are fast Fourier transform‐type (FFT‐type) algorithms. It is possible to show that for these algorithms a SIMD (Single Instruction Multiple Data) processor reduces the number of operations from [Formula: see text] to [Formula: see text] (where N is the number of input data and P is the number of processors) provided that both N and P are integer powers of 2. It thus seems advantageous to reconsider the possible uses of the fast algorithms in geophysics, including those not successful on nonparallel computers (Bezvoda et al., 1986).
Numerical electromagnetic modeling by the finite‐difference or finite‐element methods leads to a large sparse system of linear algebraic equations. Fast direct methods, requiring an order of at most q log q arithmetic operations to solve a system of q equations, cannot easily be applied to such a system. This paper describes the iterative application of a fast method, namely cyclic reduction, to the numerical solution of the Helmholtz equation with a piecewise constant imaginary coefficient of the absolute term in a plane domain. By means of numerical tests the advantages and limitations of the method compared with classical direct methods are discussed. The iterative application of the cyclic reduction method is very efficient if one can exploit a known solution of a similar (e.g., simpler) problem as the initial approximation. This makes cyclic reduction a powerful tool in solving the inverse problem by trial‐and‐error.
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