Direct estimation of the distribution of relaxation times from induced-polarization spectra using a Fourier transform analysis

The analysis of low-frequency spectral induced polarization data involves the determination of the distribution of relaxation times either from time-domain or frequency domain measurements. The classical approach is to assume a simple transfer function (e.g., a Cole-Cole function) and to determine, by a deterministic or a stochastic fitting procedure, the parameters of this transfer function (for instance the four Cole-Cole parameters). Some other methods (based on optimization) have been developed recently avoiding the choice of a specific transfer function that can bias data interpretation. We have developed a new approach based on the Fourier transform also avoiding the use of a specific analytical transfer function. The use of the Fourier transform is a classical approach to retrieve the kernel of a Fredholm integral equation of the first kind (especially in potential field theory) and this corresponds exactly to the problem we want to solve. We adapt the Fourier transform approach to retrieve the distribution of the relaxation times (for instance to process low-frequency induced polarization data). Problems resulting from the use of this approach with noisy data are prevented by using Wiener filtering. As far as induced polarization is concerned, we found that it is necessary to fit the high-frequency dielectric contribution of the spectra and to remove this contribution from the quadrature conductivity data before inverting the distribution of the relaxation times. Our approach is benchmarked with analytical pair solutions and then tested by using synthetic and experimental data sets.