Probabilistic machine learning of relaxation time distributions in spectral induced polarization

Abstract

Debye decomposition methods are widely used to interpret spectral induced polarization (SIP) data and to recover the relaxation time distribution (RTD) of geomaterials. However, SIP interpretation remains challenging for heterogeneous data sets because conventional decomposition methods treat each spectrum independently and provide limited uncertainty quantification. A probabilistic machine learning method is introduced to infer continuous RTD directly from complex resistivity spectra, using a combined laboratory and field data set comprising 140 SIP measurements of granular mixtures, rock cores, field surveys, and cementitious materials. The approach relies on a conditional variational autoencoder (CVAE) that performs decomposition at the data set level and learns a shared inverse mapping from complex resistivity spectra to probabilistic RTD expressed as Gaussian mixtures. The CVAE reproduces measured spectra with global errors below 0.53% and 0.45% over the full frequency range for the real and imaginary components, respectively. Dominant relaxation modes are recovered consistently, and both the total chargeability and the mean relaxation time show strong correlations with polarizable grain content and grain size, respectively, with coefficients of determination up to 0.95. Jacobian-based sensitivity analysis shows that the placement, width, and relative weighting of relaxation modes contribute to approximately 89% of the decomposition process. In contrast, the total chargeability contributes to 10% and the resistivity scaling parameter less than 1%. Latent variables learned by the CVAE organize SIP data into a structured space where sample populations naturally cluster without supervision. Compared to the chargeability and relaxation time domain, a two-dimensional projection of the latent variables improves the Davies--Bouldin clustering index by nearly a factor of three.

Figures (14)

• Figure 1: Architecture of the CVAE used for DD. The encoder maps the normalized complex resistivity $\boldsymbol{\tilde{\rho}}$ to the parameters $\boldsymbol{\mu}_\phi$ and $\boldsymbol{\sigma}_\phi$ of a diagonal Gaussian distribution. Latent samples $\mathbf{z}$ are transformed by the decoder, conditioned on the measurement frequencies $\mathbf{f}$, into the parameters of a Gaussian mixture representation of the logarithmic RTD $\gamma_\theta(u)$. The RTD feeds the forward model in Equation \ref{['eq:cvae_output']} to reconstruct the normalized complex resistivity $\boldsymbol{\hat{\tilde{\rho}}}$.
• Figure 2: Model selection experiment with respect to the number of latent dimensions (a) and Gaussian mixture components (b) used by the decoder. Markers correspond to the NMAE averaged over the real and imaginary parts and three experiment repetitions. Error bars corresponding to the standard deviation of the results across repetitions are smaller than the markers.
• Figure 3: Heatmap of the modeled complex resistivity NMAE as a function of the number of hidden layers and the hidden layer dimension in the CVAE encoder. Each cell reports the mean and standard deviation of three experiment repetitions. Darker cells correspond to higher errors. The optimal model choice has two layers of 32 units.
• Figure 4: Learning curves of the selected CVAE. The plot shows the evolution of the negative log-likelihood term ($\mathcal{L}_\mathrm{NLL}$) and the Kullback--Leibler divergence ($D_\mathrm{KL}$) as a function of the optimization step. Training is stopped once the total loss ($\mathcal{L}_\mathrm{NLL} + D_\mathrm{KL}$) does not improve by more than 1 % in the last 1000 steps.
• Figure 5: Examples of CVAE reconstructions for four SIP spectra drawn from laboratory and field measurements. For each sample, the observed real and imaginary parts are plotted with their measurement error bars, and the mean CVAE prediction (solid line) is shown together with its $95\,\%$ posterior confidence interval (dotted lines). The examples span a wide range of spectral responses, including a pyrite--sand mixture, a graphite--sand mixture, and heterogeneous rock core and field measurements.