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 his sand castles. 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
geoscience inverse modelling 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.