Combining unsupervised deep learning and Monte Carlo dropout for seismic data reconstruction and its uncertainty quantification

Many methods, such as multichannel singular spectrum analysis (MSSA) and deep seismic prior (DSP), have been developed for seismic data reconstruction, but they do not quantify the uncertainty of reconstructed traces, relying on the subjective visual inspection of results. Our goal is to quantify the reconstructed uncertainty while recovering missing traces. We develop a framework including an unsupervised deep-learning-based seismic data reconstruction method and the existing Monte Carlo dropout method to achieve this goal. The only information required by our framework is the original incomplete data. A convolutional neural network trained on the original nonmissing traces can simultaneously denoise and reconstruct seismic data. For uncertainty quantification, the Monte Carlo dropout method treats the well-known dropout technique as Bayesian variational inference. This refers to the fact that the dropout technique can be regarded as an approximation to the probabilistic Gaussian process and thus can be used to obtain an approximate distribution (Bernoulli variational distribution) of the posterior distribution. The reconstructed result and uncertainty of the trained model are yielded through multiple Monte Carlo dropout simulations. The analysis of the reconstructed uncertainty quantifies the confidence to use reconstructed traces. Tests on synthetic and field data illustrate that our framework outperforms the MSSA and DSP methods on reconstructed accuracy and quantifies the reconstructed uncertainty as an objective benchmark to guide decision making.