The decomposition of heterogeneous rain into homogeneous components: results from a statistical inversion of count data

Most variables in meteorology are statistically heterogeneous; that is, in the broadest sense, their statistics depend upon the location (temporal or spatial origin) of the observations. Yet often measurements of variables are gathered at widely disparate locations in space-time and are processed as though the data were fully characterized by just one pdf and one single set of parameters having one mean value. Is there, instead, a better way of treating the observations in a manner that is consistent with the actual statistical heterogeneity of the data? We address this question using a statistical inversion technique based upon Bayesian methodology. Two examples of disdrometer measurements in real rain reveal the presence of multiple mean values of the counts at all the different drop sizes. This immediately exposes the statistical heterogeneity of the two data sets, one 16 hours and the other three minutes long. Furthermore, the analyses reveal that in both cases the heterogeneous rain can be decomposed into five to seven statistically homogeneous components, each characterized by its own steady drop size distribution. Every observation can then be represented as a linear, weighted combination of a properly selected complete set of these components. Furthermore, rather imprecise concepts such as stratiform and convective rain can also be given more exact, objective meaning in terms of the contributions each component makes to the rain. In these data, for example, large numbers of smaller drops associated with one set of components found in the regions of light stratiform rainfall are sandwiched between those of more intense convective rainfall more closely associated with a different set of components having large drops. More important, however, this discovery allows for the incorporation of statistical heterogeneity explicitly and analytically into radar theory as illustrated in the other article at this conference.