GNGTS 2023 - Atti del 41° Convegno Nazionale

Session 3.3 ______ ___ GNGTS 2023 dataset. In order to circumvent the training dataset construction, Zhang et al. (2020) proposed to use a natural image dataset to train convolutional neural network (CNN) denoisers and then plug the trained denoisers into the project onto convex set (POCS) framework. Wang et al. (2020) used the acquired common shot gathers to train a U-net for interpolation of missing shots. Besides using particular training datasets, Kong et al. (2020) proposed an unsupervised strategy based on the deep image prior (DIP) (Ulyanov et al., 2018). Here, to circumvent the training pairs construction, we borrow the idea of using equivariant imaging for the natural and medical image processing (Chen et al, 2021) and present an end-to-end self-supervised deep learning method for regular seismic data interpolation. During the training stage, the observed undersampled seismic data itself is the CNN input, and the output is the interpolated seismic data. For the loss function construction, besides the measurement consistency, the equivariance of seismic data with respect to shift and undersampling is also utilized. Synthetic and field data examples are used to prove the effectiveness and validity of the proposed method. METHOD Problem definition Without loss of generality, undersampled seismic data d can be modelled as (1) where m is the vector of fully sampled seismic data, and S is the sampling operator which includes the information of trace locations of d . The goal of seismic interpolation is to reconstruct the fully sampled seismic data m from undersampled measurement d , which is an ill-posed inverse problem. Therefore, it needs prior information. The straightforward model-based way to solve the interpolation inverse problem is to minimize the cost function (2) where is the data fidelity term, also known as the measurement consistency, is a regularization term imposing the desired prior, and is a tuning parameter. Differently from model-based strategies, the end-to-end deep learning CNN based strategies aim at learning a reconstruction mapping by training a CNN on corresponding pairs of fully sampled/undersampled data. Then the trained network is used to obtain (3) The performances of these methods strongly depend on training data. However, collecting high-quality training datasets is a time consuming operation and might not be available in real-world scenarios.