Comparing detrital age spectra, and other geological distributions, using the Wasserstein distance

Abstract. Distributional data such as detrital age populations or grain size distributions are common in the geological sciences. As analytical techniques become more sophisticated, increasingly large amounts of distributional data are being gathered. These advances require quantitative and objective methods, such as multidimensional scaling (MDS), to analyse large numbers of samples. Crucial to such methods is choosing a sensible measure of dissimilarity between samples. At present, the Kolmogorov-Smirnov (KS) statistic is the most widely used of these dissimilarity measures. However, the KS statistic has some limitations. It is very sensitive to differences between the modes of two distributions, and relatively insensitive to differences between their tails. Here we introduce the Wasserstein-2 distance (W₂) as an alternative to address this issue. Whereas the KS-distance is defined as the maximum vertical distance between two empirical cumulative distribution functions, the W₂-distance is a function of the horizontal distances (i.e., age differences) between individual observations. Using a combination of synthetic examples and a published zircon U-Pb dataset, we show that the W₂ distance produces similar MDS results to the KS-distance in most cases, but significantly different results in some cases. Where the results differ, the W₂ results are geologically more sensible. For the case study, we find that the MDS map that is produced using W₂ can be readily interpreted in terms of the shape and average age of the age spectra. The W₂-distance has been added to the R package IsoplotR, for immediate use in detrital geochronology and other applications. The W₂ distance can be generalised to multiple dimensions, which opens opportunities beyond distributional data.