A Generalized Kolmogorov-Smirnov Statistic for

Detrital Zircon Analysis of Modern Rivers

Oscar M Lovera,  Marty Grove and Sara E. Cina

Abstract  The Kolmogorov-Smirnov (K-S) statistic is widely used to test the null hypothesis that two distributions drawn from the same population.  In detrital zircon provenance analysis of river systems, it is useful to have an equivalent statistical measure for comparisons involving composite samples that collectively represent the contributions of tributaries to regionally extensive river systems.  We present a generalized K-S statistic that depends on the proportional contribution and sample size of individual distributions as well as the nature of the respective individual populations.  Our generalized K-S statistic is designed to analyze mixtures of single independent variable data sets and can be calculated from either error weighted distributions (i.e., cumulative probability density functions) or raw data series (i.e., cumulative distribution functions).  Analytical expressions are provided for two limiting cases in which mixtures are derived entirely from either independent or identical populations.  Although intermediate cases must still be tested by numerical analysis, the limiting cases tightly bound all possible solutions and thus permit conservative evaluation of the null hypothesis.  We demonstrate how the generalized K-S statistic can be used with an example from the modern Marsyandi River (central Nepal Himalaya) river system.  Our results clarify the manner in which detrital zircon age distributions from tectonically active areas may be used to constrain key parameters such as erosion rates within the catchment of major rivers.