Operator-valued transmission lines as models of a horizontally layered earth under transient two-dimensional electromagnetic excitation

A new method of computing the transient behavior of a horizontally layered earth under two-dimensional electromagnetic excitation is presented. The earth's conductivity, magnetic permeability, and electric permittivity are assumed to vary only with depth, but those variations are allowed to be quite arbitrary. Displacement current is not neglected. After discretization of Maxwell's equations, the earth is modeled by infinite RLC electrical networks. No truncations of those networks are invoked during the computational procedure. Instead, those networks are viewed as infinite ladder networks of Hilbert ports, that is, as operator-valued lumped transmission lines, and the computations are based upon the recently devised theory of such systems. This analysis yields a new surface impedance operator, which provides a considerably more comprehensive representation of the earth's behavior than does the Tikhonov-Cagniard surface impedance. For example, the Tikhonov-Cagniard surface impedance is applicable only when the electric or magnetic fields vary slowly or linearly with respect to horizontal displacements. The surface-impedance operator derived herein does not require those restrictions; only quadratic summability and Laplace transformability are needed.