Cross-Gradient Regularized Joint Seismic Full Waveform and ERT Inversion with Automatic Differentiation

Joint inversion of geophysical data leverages complementary sensitivities to improve subsurface imaging. In this study, we present a cross-gradient regularized joint inversion framework that combines full waveform inversion (FWI) and electrical resistivity tomography (ERT) using automatic differentiation. FWI is sensitive to elastic properties, while ERT resolves electrical resistivity variations, often linked to fluid content and lithology. Independently, each method faces limitations related to non-uniqueness and resolution. By structurally coupling them via the cross-gradient constraint—which promotes co-located spatial gradients—we enhance the geological consistency of the recovered models without requiring explicit petrophysical relationships. Both FWI and ERT forward solvers are implemented in PyTorch: FWI uses Deepwave for differentiable elastic wave propagation, and ERT is adapted from SimPEG as a differentiable Poisson solver. The entire inversion is expressed as a unified computational graph, enabling efficient gradient computation and optimization via L-BFGS. A synthetic example demonstrates that joint inversion improves feature recovery compared to individual inversions, particularly in regions where one modality alone lacks sensitivity. This flexible, autodiff-based approach offers a practical pathway for integrating multiple geophysical methods in structurally constrained inversion frameworks.