Efficient spectral IP-modelling in the time domain

We show that the modelling of time-domain spectral induced polarization (SIP) data can be carried out using a DC resistivity code. The intrinsic resistivity at each point in space is calculated by Fourier transforming the complex, frequency-dependent resistivity, and a separate DC problem is solved for each time channel. This procedure constitutes an approximation, because the exact formulation involves a convolution of the time-dependent conductivities with the electric field. The approximation dramatically simplifies numerical modelling of time-domain SIP data. Implicitly, it is already in use in inversion routines where data are inverted independently for separate time channels. However, the accuracy has not been investigated in detail.We first formulate the underlying equations in terms of a maximum chargeability M. We expand the exact and the approximate solutions with respect to the maximum chargeability and show that the differences are at least second order in M. Next, we investigate the reflection factor for a simple two-layer model, where the second layer resistivity is assumed to exhibit Cole–Cole behaviour. We calculate exact and approximate solutions analytically and show that the early- and late-time limits are identical. In the intermediate time range the error of the approximation is limited to a few percent. Finally, we calculate approximate and exact apparent resistivities for dipole–dipole configurations over a layered and a full 3-D model and show that the differences are far below the typical measurement noise level. The conclusion is that the approximation is excellent and will work in all practical situations.