Learning relaxation time distributions from spectral induced polarization data with a complex-valued variational autoencoder

Abstract

Spectral induced polarization (SIP) is a geophysical method used to characterize subsurface materials. It measures the frequency-dependent complex resistivity of rocks and soils through the application of a small alternating current in the subsurface or in laboratory samples. Debye decomposition (DD) is a standard method for analyzing and interpreting SIP data, as it allows estimation of the relaxation time distribution (RTD) of geomaterials. However, conventional DD approaches treat measurements independently, work in real-valued spaces despite the complex-valued nature of SIP data, and provide limited uncertainty quantification. These limitations reduce the effectiveness of conventional DD on heterogeneous datasets. We reformulate DD as an unsupervised machine learning problem and introduce a conditional variational autoencoder (CVAE) that learns a shared mapping from resistivity spectra to continuous RTDs. The model is validated on a dataset comprising 140 laboratory and field SIP measurements of granular mixtures, mineralized rocks, and cementitious materials. The CVAE operates in complex-valued data space and achieves reconstruction errors of 0.45 % and 0.24 % for the imaginary and phase components of resistivity, respectively, with statistically significant improvements over conventional methods (p-values of 4x10^-6 and 2x10^-3). The inferred RTDs are stable and physically consistent, and their total chargeability and mean relaxation time correlate with polarizable grain content and grain size, respectively, with coefficients of determination up to 0.98. An additional contribution of the proposed method is the learned latent representation, which organizes SIP spectra into a structured space. Unsupervised clustering in a two-dimensional projection of this space improves the Davies--Bouldin index by nearly a factor of three relative to conventional RTD parameters.