Towards inverse modeling of landscapes using the Wasserstein distance

Extricating histories of uplift and erosion from landscapes is crucial for many branches of the Earth sciences. An objective way to calculate such histories is to identify calibrated models that minimise misfit between observations (e.g. topography) and predictions (e.g. synthetic landscapes). A challenge is to make use of entire (two-dimensional) landscapes to identify optimal models. In the presence of natural or computational noise, for example arbitrary noise introduced to force channelisation, the widely used Euclidean measures of similarity (e.g. root mean square differences between elevations) unfortunately can have very complicated objective functions. Such complexity obscures the search for optimal models. Instead, we introduce the Wasserstein distance as a means to measure misfit between observed and theoretical landscapes. We first demonstrate its use with a one-dimensional topographic transect. We then show how it can be used to identify optimal uplift histories from synthetic landscapes in the presence of noise.