Operator-Valued Transmission Lines in the Analysis of Two-Dimensional Anomalies Imbedded in a Horizontally Layered Earth Under Transient Polarized Electromagnetic Excitation

A new method for computing the transient electromagnetic behavior of a horizontally layered earth containing a two-dimensional cylindrical anomaly is presented. The vertical extent of the anomaly is assumed to be finite. Above and below the anomaly the electrical parameters are assumed to vary only in the vertical direction, whereas at the levels of the anomaly the electrical parameters are allowed to vary in two dimensions. The electromagnetic excitation is assumed to be polarized in the axial direction. Also, displacement current is not neglected. Two infinite electrical grids model the finite-difference approximations to Maxwell’s equations, one for a transverse-electric polarization and the other for a transverse-magnetic polarization. Unlike virtually all computational methods for geophysical electromagnetic models, the present work does not truncate the earth except at the levels where the anomaly exists. Instead, the theory of operator-valued transmission lines is used to represent the earth above and below the anomaly by two driving-point admittance operators and a current-source vector, thereby contracting the computations to an analysis of only the levels where the anomaly appears. Once this is done, the electric and magnetic fields above and below the anomaly can be obtained very rapidly. Also, instead of using a time-stepping procedure, the Gaver–Stehfest algorithm is used to compute transients. Altogether, a computational method is, obtained for determining two-dimensional electromagnetic transients; that method is very different from prior procedures and quite fast. Moreover, this analysis yields a new surface admittance (or impedance) operator for a horizontally layered earth, 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 admittance operator derived herein does not require those restrictions; only quadratic summability and Laplace transformability are needed. Three examples of the procedure are presented: the first for a horizontally layered earth with no anomaly present and the earth’s conductivity varying continuously with depth, the second for a square cylindrical anomaly buried in a uniform earth, and the third for a uniform earth with a ridge on its surface.