Discrete complex images of a three-dimensional dipole above and within a lossy ground

The paper introduces a new method of discrete complex images, to evaluate the Sommerfeld integrals arising in the problem of a three-dimensional dipole above and within a lossy ground. The spectral domain quantity, which is independent of spatial co-ordinates, is accurately approximated by a short series of exponential functions. By use of the Prony method and non linear optimisation technique, and through the Sommerfeld equality, the Sommerfeld integrals can be analytically evaluated into simple closed forms of spatial complex images. Various dipoles (HED VED, HMD and VMD) above and within the lossy ground are studied. Numerical results show that for any parameters of the lossy ground, the closed-form Green function approaches the precision of the ´nearly exact´ results of Sommerfeld integrals