On the Detection of Statistical Heterogeneity in Rain Measurement

The 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) (WSS), i.e. 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. Up to now, there has been little attention to physically based methods independent of statistical models for evaluating whether a set of data in rain is WSS or not. This work provides one such method.

An index of statistical heterogeneity (IXH) is developed for rain such that as its value approaches zero, the more acceptable is the assumption of WSS and the more acceptable is the use of the Wiener-Khintchine theorem. Several examples from real rain measurements demonstrate the potential applicability of this approach to actual observations in rain. In particular, an example of a 17 hour rain event clearly shows that visual inspection of the data alone is insufficient for determining whether or not data are WSS. 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.