Em Coupling in Multifrequency Ip and a Generalization of the Cole‐cole Impedance Model*

The Cole‐Cole relaxation model has been found to provide good fits to multifrequency IP data and is derivable mathematically from a reasonable, albeit greatly simplified, physical model of conduction in porous rocks. However, the Cole‐Cole model is used to represent the mutual impedance due to inductive or electromagnetic coupling on an empirical basis: this use has not been similarly justified by derivation from any simple physical representation of, say, a half‐space, layered or uniform. A uniform conductive half‐space can be represented as a simple subsurface loop with particular resistive and inductive properties. Based upon this, a mathematical expression for the mutual impedance between the two pairs of electrodes of a dipole‐dipole array is derived and designated “model I”. It is seen that a degenerate case of model I is the Cole‐Cole model with frequency exponent c = 1. Model I is thus more general than the Cole‐Cole expression and must provide at least as good a fit to a set of field data. Provision for variation of c from unity could be made in model I equally well as for the Cole‐Cole model although, at present, this would be a purely empirical alteration. Model I contains four parameters, one of which is, in effect, the resistivity of the half‐space. Therefore only three parameters are involved in the model I expressions for normalized amplitude and for phase of the EM‐coupling mutual impedance. Model I is compared with previously published “standard” values for two different dipole separations. Under particular constraints, model I is shown to provide better fits than the Cole‐Cole model (with c = 1) over particular frequency ranges, specifically at very low frequencies and at moderately high frequencies where the model I phase curve follows the standard phase curve across the axis to positive values (negative coupling).