Image and Exact Models of a Vertical Wire Penetrating a Two-Layered Earth

In this paper, we investigate the validity of image theory in modeling a vertical wire penetrating a uniform or two-layered earth. First, a rigorous full-wave electromagnetic model is described that may be used as a standard for comparison. The model is based on the method of moments (MoM) and exact Green functions for the layered earth that involve Sommerfeld-type integrals. Next, we derive the image model from the exact model and analyze the approximations. We compare the current along a vertical wire as computed by the MoM using the exact Green functions, with the current computed using the image Green functions. The image model is accurate at dc, and the error is generally low at low frequencies but rises at high frequencies. We show that the image model leads to smaller errors when the wire is embedded in one layer. However, the error is larger in cases of penetration through different layers. We also show a strong dependence of the error on the resonant frequencies.