  We 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 propose the neural
network version of sparse LSM. Unlike the standard LSM method that finds the
optimal reflectivity image, neural network LSM (NNLSM) finds both 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 a superresolution attribute image for
high-resolution delineation of geologic bodies.
