Stochastic Variational Inference for Electrical Resistivity Tomography and Time-Domain Induced Polarization with Adjoint-State Gradients

Electrical resistivity tomography (ERT) and time-domain induced polarization (TDIP) are widely used techniques for characterizing subsurface hydrogeological and mineralogical properties. Deterministic inversion yields a single model but no indication of which recovered features the data actually constrain, whereas characterizing the full posterior by Markov chain Monte Carlo (MCMC) is computationally prohibitive for the high-dimensional parameter spaces of PDE-constrained geophysical inversion. This study develops a stochastic variational inference (SVI) framework for the sequential inversion of ERT and TDIP data that occupies the middle ground between them. A mean-field Gaussian variational model is optimised by a reparameterized stochastic gradient scheme in which the mean update coincides with a deterministic gradient-based inversion. Gradients are computed by the adjoint-state method, with a single adjoint solution per iteration supplying both the mean and the covariance update and avoiding explicit Jacobian construction, so the covariance computation adds negligible cost. The recovered covariance is reported as the relative reduction of prior variance, a map of where the data constrain the model. This was validated on a reduced problem and its spatial pattern matches a Gauss–Newton resolution analysis. The forward and adjoint PDEs are solved by the spectral element method (SEM). The intrinsic chargeability for TDIP problems, which is physically bounded in (0, 1), is handled through a logit transformation, and the mean conductivity recovered from ERT is used in the TDIP forward and adjoint operators. The framework is applied to field data from the Hongqi Reservoir embankment, Chongqing, China, where it recovers the conductivity and chargeability structure near a known seepage outlet together with a map of which features the data constrain.