Controlled source apparent resistivity tensors and their relationship to the magnetotelluric impedance tensor

The time-domain surface electric fields produced by a step current in collocated grounded sources can be represented by an apparent resistivity tensor defined by the relationship between the measured (time-varying) electric field and a reference field equal to the steady-state current density in a uniform half-space. A magnetic field response tensor is similarly defined for the horizontal components of the magnetic field. The 'magnetic field apparent resistivity tensor' is derived from a linear combination of the contractions of the outer product of the magnetic field response tensor. Frequency-domain apparent resistivity tensors are derived from the Laplace transforms of the corresponding time-domain electric and magnetic field response tensors. Both the frequency and time-domain tensors are independent of the source orientation where the sources can be approximated as (infinitesimal) dipoles. A simple combination of the frequency-domain (impulse response) tensors can be used to derive the controlled source magnetotelluric (CSMT) impedance tensor. The magnetic field apparent resistivity tensor is a useful representation of the conductivity structure only where the source-receiver offset is much less than the diffusion or skin depth and behaves in a similar way to the 'early-time' apparent resistivity traditionally used to represent time-domain electromagnetic data. Numerical modelling results demonstrate that the (horizontal) magnetic field apparent resistivity is insensitive to the localized 3-D conductivity structures that are typically the target of exploration surveys. In contrast, the electric field apparent resistivity tensor depends sensitivity upon the conductivity structure and is well behaved over the entire time or frequency range used in long-offset transient electromagnetic or CSMT measurements. By using the electric field apparent resistivity tensor, 'source effects' that hinder a conventional impedance tensor analysis of CSMT data can be avoided. Images of simple 3-D structures created from the invariants of the electric field apparent resistivity tensor (in either the time or frequency domain) provide a useful representation of the subsurface conductivity structure despite the simplicity of this imaging procedure. These images are independent of the coordinate system used to express the data and are, to a good approximation, independent of the current source orientations.