ELF propagation parameters for the ground-ionosphere waveguide with finite ground conductivity

In the extremely low frequency (ELF) range, the space between the ground and ionosphere forms a spherical waveguide. When studying the ELF radio-wave propagation on Earth using analytical models, it is commonly assumed that the ground has perfect conductivity. In some planetary applications, the ground conductivity is so low that it has a significant influence on the radio-wave propagation in the ELF range and it has to be included in the model. This issue has not been resolved in a satisfactory way yet. In this paper, we analyze the penetration of the ground by an electromagnetic wave traveling along the surface and we propose a solution that allows including ground parameters in the frequently used 2-D analytical model of the ground-ionosphere waveguide. The presented calculations show that in most cases on Earth, the differences with such a model are small, up to a few percent for phase velocity