Green’s Tensors for the Diffusive Electric Field in a Vti Half-space

Explicit Green’s tensors for the diffusive electric field in a configuration with two homogeneous half spaces are of interest for primary-secondary formulations of frequency domain and time domain modeling schemes. We derive the explicit expressions for the Green tensor of the electric field generated by an electric dipole in space frequency and space time. Both source and receiver can have arbitrary positions in the vertical transverse isotropic (VTI) half space below a non conductive half space. Apart from their use in modeling schemes, the expressions can be used to understand the effect of the interface between the VTI and the non conducting half space. We show that the TE-mode refracts against the interface, and its effect in the VTI half space decays exponentially as a function of depth and is inversely proportional to horizontal distance cubed for horizontal source receiver distances larger than three times the source depth. In exploration geophysics, this part of the field is known as the “airwave”. The contribution from the “airwave” has a late time behavior that differs from the other contributions to the electric field. This makes time domain systems relevant for exploration geophysical applications.