Conversion of IP data and its uncertainty from time-domain to frequency-domain using Debye decomposition

The time-domain (TD) induced polarization (IP) method is used as an extension to classical DC resistivity measurements to capture information on the ability of the subsurface to develop electrical polarization, which is closely coupled to petrophysical parameters of relevance in hydrogeophysical characterization. In a TD IP measurement, the transient voltage decay between two electrodes is measured after the termination of an injected current between two other electrodes. TD IP measurements are typically analyzed in terms of chargeabilities, while in the frequency domain (FD) polarization responses are measured as complex-valued impedances. The latter can be inverted into a subsurface model of complex electrical resistivity by means of existing tomographic inversion algorithms. In order to apply these FD inversion algorithms to TD IP measurements, the necessity of TD to FD data conversion arises. A suitable conversion approach must transform the measured decay curve into a FD impedance and, preferably, also propagate the corresponding measurement uncertainty from TD into FD. Here we present such an approach based on a Debye decomposition (DD) of the decay curve into a relaxation-time distribution (RTD). Since equivalent formulations of the DD exist in TD and FD, it is possible to compute the FD response from the RTD inverted from the TD response. The corresponding FD data error can be obtained by applying error propagation through all these steps, assuming that the errors on the underlying parameters are normally distributed. To accomplish the DD we implemented a non-linear Gauss-Newton inversion scheme which automatically tunes the regularization strength to achieve a stable FD estimate and FD uncertainty. We test the performance of the inversion scheme in a synthetic study and demonstrate its application to field data on a tomographic TD IP data set measured on the Maletoyvaemskoie field of altered rocks (Kamchatka, Russia), which features epithermal gold deposits of high sulfidation type. The converted tomographic TD IP data set is inverted into subsurface models of complex electrical resistivity at frequencies of 1 Hz and 20 Hz. The proposed conversion approach yields accurate impedance data for relaxation processes which are resolved by the TD measurements. The error propagation scheme provides a reasonable FD uncertainty estimate, as revealed by a Monte-Carlo analysis of the underlying parameter distributions. Propagated FD errors are in agreement with previously established FD error models. The presented methodology to convert TD to FD IP data allows to invert and analyze field data collected with widely used TD instruments in the frequency domain, where the diagnostic potential of electrical impedance spectroscopy can be fully exploited for an improved interpretation.