Efficient Uncertainty Quantification for 3-D MT Inversion via Variational Inference

We present an efficient method for uncertainty quantification in 3-D magnetotelluric (MT) inversions based on variational inference principles and the variational autoencoder (VAE) framework. In this approach, we replace the VAE’s encoder and decoder, respectively, with a forward modeling and a 3-D optimization module, where the latent variables represent updates in the constrained model space. Utilizing reparameterization, the gradient of the objective function can propagate back to the mean and variance of the latent variables, and thus restores the Gaussian distribution of resistivity models. By setting appropriate prior and initial distributions, our method explores the solutions with high log-variance to fit the data. To ensure a stable convergence, we apply a smooth constraint on the log-variance of latent variables and employ a modified adaptive moment estimation (Adam) optimizer. Numerical experiments confirm that our method can accurately estimate both the resistivity model and standard deviation at a time cost only twothree times that of conventional inversion, which provides a highly efficient solution for uncertainty analysis. The tests with varying regularization parameters reveal that the standard deviation estimates are sensitive to both the Kullback-Leibler (KL) divergence and log-variance smoothness terms, while the adaptive smoothing strategy can well balance these effects. The application to a dataset from USArray survey in the Northwestern USA demonstrates that the method can achieve robust inversion and uncertainty quantification.