Seismic imaging, optimal transport, and remarks on deep learning
The purpose of exploration seismology is to find geophysical properties, such as wave velocity and location of reflecting sub-layers in the earth from measurements of seismic waves at the surface. A recently popular computational technique for seismic imaging is Full Waveform Inversion (FWI), which is formulated as PDE constrained minimization where the miss-match between measured and computed signals plays an important role. The geophysical properties are given by unknown variable coefficients in the PDE. We propose using optimal transport and the Wasserstein metric for this miss-match in order to reduce the risk of only finding local minima in the PDE constrained minimization. The optimal transport can be given by the gradient of the solution to a Monge–Ampère equation. Analysis of convexity properties and numerical examples comparing these new techniques with the classical L2 miss-match will be presented. Seismic imaging is data-driven and data-intensive so it is natural to comment on the emerging application of deep learning in seismic exploration. We will also remark on the striking similarities of FWI and the structure of deep learning.
Björn Engquist received his Ph.D. in numerical analysis from Uppsala University in1975. He has been professor of mathematics at UCLA, and the Michael Henry Stater University Professor of Mathematics and Applied and Computational Mathematics at Princeton University. He was director of the Research Institute for Industrial Applications of Scientific Computing and of the Centre for Parallel Computers at the Royal Institute of Technology, Stockholm. At Princeton University, he was director of the Program in Applied and Computational Mathematics and the Princeton Institute for Computational Science.