The equivalent wavefield concept in multichannel transient electromagnetic surveying

Abstract

The equivalent wavefield concept relates diffusive electromagnetic propagation to non-diffusive propagation, provided equivalent source and boundary conditions are satisfied. This well-known concept is a special case of a more general theorem, proven here, relating the solutions of two systems of partial differential equations that have the same spatial, but different temporal derivatives. In the case we consider, the velocity of the equivalent wavefield is proportional to the square root of the resistivity of the diffusive electromagnetic medium. The use of the concept has two advantages: first, analytical results may be derived more easily in the equivalent wavefield domain; second, interpretation of data is easier after mapping to the equivalent wavefield domain. In general, electromagnetic fields have an irrotational component that is equivalent to Pwave propagation, and a solenoidal component that is equivalent to S-wave propagation. Using a moment tensor and a dipole moment to represent electromagnetic dipole sources allows comparison with seismic sources. For example, the equivalent wavefield of the magnetic field generated by an electric current dipole in a whole space is generated by a point source of torque, generating shear waves only. The electric field generated by a switch-off electric current dipole at the surface of a halfspace has an equivalent wavefield at the interface equal to a triangle with origin at the switch-off time, peak at the arrival time and zero thereafter. When graphed as a function of space versus time, arrivals in the equivalent wavefield lie on straight lines and can be interpreted using straightforward concepts from the seismic refraction method. Diffusive to propagative mapping of numerical data requires regularisation to stabilise whatever numerical inversion procedure is used. Approaches include matrix inversion and a new algorithm which uses deconvolution in log time. The latter approach is computationally inexpensive and permits analysis of the distortion of the recovered waveform which is caused by regularisation. Both approaches successfully extract several arrivals when these are well-resolved in the original diffusive synthetic. Diffusive to propagative mapping applied to synthetic electromagnetic responses calculated for a horizontal electric current dipole source over a uniform half-space or simple layered-Earth models yields equivalent wavefields which are interpretable after calibration for waveform regularisation