Hydrological objective functions and ensemble averaging with the Wasserstein distance

When working with hydrological data, the ability to quantify the similarity of different datasets is useful. The choice of how to make this quantification has a direct influence on the results, with different measures of similarity emphasising particular sources of error (for example, errors in amplitude as opposed to displacements in time and/or space). The Wasserstein distance considers the similarity of mass distributions through a transport lens. In a hydrological context, it measures the “effort” required to rearrange one distribution of water into the other. While being more broadly applicable, particular interest is paid to hydrographs in this work. The Wasserstein distance is adapted for working with hydrographs in two different ways and tested in a calibration and “averaging” of a hydrograph context. This alternative definition of fit is shown to be successful in accounting for timing errors due to imprecise rainfall measurements. The averaging of an ensemble of hydrographs is shown to be suitable when differences among the members are in peak shape and timing but not in total peak volume, where the traditional mean works well.