Short Communication: The Wasserstein distance as a dissimilarity metric for comparing detrital age spectra, and other geological distributions

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.

Multidimensional Scaling of Varietal Data in Sedimentary Provenance Analysis

“…The W 1 and W 2 distances fulfill the metric requirements and can therefore be subjected to both classical and nonmetric MDS. Lipp and Vermeesch (2022) show that the W 2 -distance produces results that are often equivalent and sometimes better than those obtained by the KS-statistic. Thus, we could substitute the KS-statistic for the W 2 -distance in Sections 2 and 3.…”

“…Lipp and Vermeesch (2022) show that the W 2 ‐distance produces results that are often equivalent and sometimes better than those obtained by the KS‐statistic. Thus, we could substitute the KS‐statistic for the W 2 ‐distance in Sections 2 and 3.…”

The Zircon Story of the Niger River: Time‐Structure Maps of the West African Craton and Discontinuous Propagation of Provenance Signals Across a Disconnected Sediment‐Routing System

“…Forward mixing calculations of zircon‐age populations were made by both inverse Monte Carlo modeling with Kolmogorov‐Smirnov and Kuiper test statistics (Sundell & Saylor, 2017) and Wasserstein statistics, a refined method more sensitive to the tails of the distribution (Lipp & Vermeesch, 2023). The best fit is obtained for the synthetic mixing proportion yielding the minimum Wasserstein distance to the spectrum of the outlet sample.…”

The uncertain future of mountaintop-removal-mined landscapes 1: How mining changes erosion processes and variables

“…We also analyzed the Wasserstein Distance (W 2 ; Lipp and Vermeesch, 2022) between the pre-and post-mining distribution of elevation, slope, and area-slope product in each HUC-12 catchment. This is e↵ectively a cost function that measures the relative di culty of turning the pre-mining distribution into the post-mining distribution.…”

“…Figure 4: Comparisons between pre-and post-mining geomorphic characteristics of 88 HUC-12 watersheds with at least 90% coverage of pre-and post-mining elevation data.A-C show the influence of mining on the ratio of post-to pre-mining mean elevation, mean slope, and mean area-slope product, respectively. D-F show the Wasserstein distance(Lipp and Vermeesch, 2022) between the distributions of pre-and post-mining DEM pixels.Higher W 2 values indicate greater change. Inset plots show posterior distributions of the correlation coe cient found by Bayesian rank correlations(van Doorn et al, 2020).…”