Application of the contour transformation method to a vertical electric dipole over Earth

A contour transformation method is applied to the computation of the Sommerfeld integral associated with a vertical electric dipole above Earth. In this method the slowly varying part of the transformed Sommerfeld integrand is approximated by a few exponential terms using Prony's method. The resulting integrals are then carried out analytically, thereby, yielding incomplete Lipschitz-Hankel integrals. Therefore numerical integration of the Sommerfeld integral is unnecessary since the incomplete Lipschitz-Hankel integrals are computed using efficient Bessel series expansions. This method is compared with a direct numerical integration of the Sommerfeld integral and a contour deformation technique which also utilizes Prony's method. The results demonstrate that the contour transformation method enables the accurate and efficient computation of the Sommerfeld integral for all source and observation locations (i.e., near zone, far zone and even grazing). This technique can also be applied to problems involving multiple layers.