May 2023
The field of optimal transport is thought to have originated in the 19th century, when legend has it that Napoleon asked Gaspard Monge to rearrange some piles of sand. That started a 200 year story of discovery and re-discovery of the mathematics of how to map, or transport, one density function (or probability distribution on to another). Leonid Kantorovich reformulated Monge’s problem in terms of more familiar linear programming which contributed to his winning the 1975 Nobel prize for economics. Cedric Villani pioneered the modern mathematical treatment of the topic and was awarded the 2010 Fields medal.
What has all of this got to do with Geophysics? Here exploration geophysicists have led the way and shown how to exploit OT in Full seismic waveform inversion. It turns out that optimal transport may be used as an alternate to Least squares measures to create a new type of data misfit function. It has been demonstrated that it has significant potential in nonlinear inversion by reducing the presence of local minima in misfit functions which would otherwise by highly multi-modal. Over the past decade this has created a flurry of excitement and activity in Seismic Waveform inversion in exploration geophysics, and a gradual appreciation of the topic more broadly. This talk will introduce OT for geophysical inversion in a more general context, and also discuss some new ideas and open questions which, as always, take the form of how do we best exploit these `pure’ mathematical concepts in an effective manner for practical outcomes.