Bayesian Uncertainty Estimation for Deep Learning Inversion of Electromagnetic Data

With the recent progress in deep learning (DL), DL inversion, which reconstructs subsurface physical properties from geophysical data using DL techniques, has been widely applied. For decision-making and risk management related to the application of DL inversion, assessing the reliability of a prediction is essential, and such assessment can be achieved through uncertainty estimation. However, most geophysical studies have focused on deterministic prediction that does not provide uncertainty estimates. In this letter, a practical uncertainty estimation method based on the Bayesian framework is introduced for DL inversion of electromagnetic data. More specifically, iterative estimation by a convolutional neural network with dropout provides epistemic and aleatoric uncertainties as well as a resistivity model. Using numerical tests, we observed that aleatoric uncertainty indicates the nonuniqueness of the inverse problem, showing which parts of the resistivity model are less sensitive to the data. In addition, we proposed an empirical criterion for determining whether new data are similar to training data using estimated epistemic and aleatoric uncertainties. Based on this criterion, out-of-distribution data were identified; these data showed larger data misfit, indicating that the predictions would be unreliable. The applicability of uncertainty estimation and the empirical criterion derived from uncertainties were demonstrated using field data. Bayesian uncertainty estimation and the criterion established here may help to achieve more reliable prediction via DL inversion.