On the Detection of Statistical Heterogeneity in Rain Measurements

Abstract

AbstractThe application of the Wiener–Khintchine theorem for translating a readily measured correlation function into the variance spectrum, important for scale analyses and for scaling transformations of data, requires that the data be wide-sense homogeneous (stationary), that is, that the first and second moments of the probability distribution of the variable are the same at all times (stationarity) or at all locations (homogeneity) over the entire observed domain. This work provides a heuristic method independent of statistical models for evaluating whether a set of data in rain is wide-sense stationary (WSS). The alternative, statistical heterogeneity, requires 1) that there be no single global mean value and/or 2) that the variance of the variable changes in the domain. Here, the number of global mean values is estimated using a Bayesian inversion approach, while changes in the variance are determined using record counting techniques. An index of statistical heterogeneity (IXH) is proposed for rain such that as its value approaches zero, the more likely the data are wide-sense stationary and the more acceptable is the use of the Wiener–Khintchine theorem. Numerical experiments as well as several examples in real rain demonstrate the potential of IXH to identify statistical homogeneity, heterogeneity, and statistical mixtures. In particular, the examples demonstrate that visual inspections of data alone are insufficient for determining whether they are wide-sense stationary. Furthermore, in this small data collection, statistical heterogeneity was associated with convective rain, while statistical homogeneity appeared in more stratiform or mixed rain events. These tentative associations, however, need further substantiation.