Solution of 2.5‐dimensional problems using the Lanczos decomposition

We consider the problem of electrical conduction in the context of geophysical prospecting and assume that the conductivity of the Earth is constant in a direction perpendicular to the probing plane. The resulting boundary value problem is reduced to two dimensions via a Fourier transform with respect to this direction. To date, the typical method of solution involves solving several of these two‐dimensional problems and computing the approximate inverse Fourier transform numerically. We propose a more efficient approach in which the inverse Fourier integral is taken analytically. This method involves the computation of an analytic function of the matrix approximation to a differential operator using its Lanczos decomposition. After deriving the method we present numerical verification of its validity and a discussion of its computational cost, which approaches that of two‐dimensional problems.