Finite‐difference calculations of the transient field of an axially symmetric earth for vertical magnetic dipole excitation

Abstract

A finite‐difference formulation of the coaxial‐loop or wire‐loop transient electromagnetic (EM) prospecting systems is used to model the fields from a buried cylindrical conductor whose axis is coincident with that of the field system. Solutions are obtained directly in the time domain. The formulation is implicit and two‐dimensional (2-D) in space. The variable‐directions method reduces each advance of one step in time from one 2-D problem to a large number of one‐dimensional (1-D) problems. The result is a reduction in computational effort. In order to avoid including the air in the finite‐difference grid, an integral equation approach is used to formulate the surface boundary condition. Thus, two sets of 1-D finite‐difference solutions and one Fredholm integral equation solution are required for each step forward in time. Comparison with analytical solutions shows excellent agreement in the case of a four‐layer earth. All computations were carried out for a perfectly conducting basement, but the method can be used for finitely conducting basement as well. If the basement is an insulator, an additional integral equation solution is required on the lower boundary. Results for a buried cylindrical conductor show that there is a high degree of sensitivity to conductor size. Inversion of transients to a stratified model can be useful if the effect of finite conductor size is taken into account. For cylindrical conductors with lateral extent comparable to or larger than the source‐receiver separation, the inversion results are valid. For conductors with lateral extent small compared with source‐receiver separation, the inversion will yield a stratified model which shows better agreement between actual and inverted thicknesses than resistivities. The involved resistivities are somewhat higher than those actually present in this case.