Stability analysis of deep-learning methods for seismic data interpolation

Deep-learning algorithms are applied to solve various geophysical problems, which draw significant concerns over their stability. How to evaluate the stability of artificial neural networks has been a challenging problem for a long time. We focus on seismic data interpolation. The stability of artificial neural networks is evaluated using three criteria: tiny worst-case perturbations, small structure recovery, and hallucinations. Tiny perturbations to the input may lead to large interpolation errors. The perturbation that results in the largest error can be optimized and measured as a stability criterion. Small structure recovery measures how effectively a network can reconstruct detailed structures. Hallucinations are structures that have been falsely added by the network. The stability of three existing deep-learning algorithms for seismic data interpolation are evaluated. Quantitative analysis shows that instability is commonly observed in all networks. Different network architectures, even trained with the same data set, exhibit varied sensibility to small perturbations from the input. A UNet structure can better recover small structures within the seismic data than the other network structures. With mathematical proof and tests on numerical data, we show that an improperly prepared training set may cause hallucinations during the testing stage, especially when the network reaches a low training loss during the training stage. The developed metrics for stability analysis can provide valuable information for evaluating network performances and thus are potentially useful in other seismic data processing routines.