Abstract
We have recast the forward pass of a multilayered convolutional neural network (CNN) as the solution to the problem of sparse least-squares migration (LSM). The CNN filters and feature maps are shown to be analogous, but not equivalent, to the migration Green’s functions and the quasi-reflectivity distribution, respectively. This provides a physical interpretation of the filters and feature maps in deep CNN in terms of the operators for seismic imaging. Motivated by the connection between sparse LSM and CNN, we adopt the neural network version of sparse LSM. Unlike the standard LSM method that finds the optimal reflectivity image, neural network LSM (NNLSM) finds the optimal quasi-reflectivity image and the quasi-migration Green’s functions. These quasi-migration Green’s functions are also denoted as the convolutional filters in a CNN and are similar to migration Green’s functions. The advantage of NNLSM over standard LSM is that its computational cost is significantly less and it can be used for denoising coherent and incoherent noise in migration images. Its disadvantage is that the NNLSM quasi-reflectivity image is only an approximation to the actual reflectivity distribution. However, the quasi-reflectivity image can be used as an attribute image for high-resolution delineation of geologic bodies.
Highlights
Deep convolutional neural networks (CNNs) have been recently used for solving geophysical problems, such as seismic first-arrival picking (Lu and Feng, 2018; Yuan et al, 2018; Hu et al, 2019), seismic interpretation (Shi et al, 2019; Wu et al, 2019a, 2019b), and seismic imaging and inversion (Xu et al, 2019; Kaur et al, 2020; Sun et al, 2020)
The merit is that trial-and-error with different architecture parameters is likely to give excellent results for a particular data set, but it might not be the best one for a different data set. This shortcoming in using empirical tests for parameter selection largely results from the absence of a rigorous mathematical foundation (Papyan et al, 2016, 2017a) for neural networks in general, and CNN in particular. To partly mitigate this problem for CNN-based imaging algorithms, we present a physical interpretation of CNN filters and feature maps in terms of physics-based operators for seismic imaging
The advantages of neural network LSM (NNLSM) over standard least-squares migration (LSM) are that its computational cost is significantly less than that for LSM
Summary
Deep convolutional neural networks (CNNs) have been recently used for solving geophysical problems, such as seismic first-arrival picking (Lu and Feng, 2018; Yuan et al, 2018; Hu et al, 2019), seismic interpretation (Shi et al, 2019; Wu et al, 2019a, 2019b), and seismic imaging and inversion (Xu et al, 2019; Kaur et al, 2020; Sun et al, 2020). The CNN filters and feature maps are shown to be analogous, but not equivalent, to the migration Green’s functions (Hessian) and the reflectivity distribution. We show the connection between the multilayer neural network and the solution to the multilayer NNLSM problem This is followed by numerical examples with synthetic models and field data from the North Sea. The theory of standard image-domain LSM is first presented to establish the benchmark solution in which the optimal reflectivity function minimizes the image misfit under the L2 norm. The corresponding stacked coefficient images, known as feature maps, are shown in Figure 7f–7h, which give the quasi-reflectivity distributions. After reconstruction from the learned filters and feature maps, the migration image is shown in Figure 13d with less noise.
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