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 minimize misfit between observations (e.g., topography) and predictions (e.g., synthetic landscapes). In the presence of natural or computational noise, widely used Euclidean measures of similarity can have complicated objective functions, obscuring the search for optimal models. Instead, we introduce the Wasserstein distance as a means to measure misfit between observed and theoretical landscapes. Our results come in two parts. First, we show that this approach can generate much smoother objective functions than Euclidean measures, simplifying the search for optimal models. Second, we show how locations and amplitudes of uplift can be accurately recovered from synthetic landscapes even when seeded with different noisy initial conditions. We suggest that this approach holds promise for inverting real landscapes for their histories.