On the importance of statistical homogeneity to the scaling of rain

Abstract

Scaling studies of rainfall are important for the conversion of observations and numerical model outputs among all the various scales. Two common approaches for determining scaling relations are the Fourier transform of observations and the Fourier transform of a correlation function using the Wiener-Khintchine (WK) theorem. In both methods, the observations must be wide-sense statistically stationary (WSS) in time or wide-sense statistically spatially homogeneous (WSSH) in space so that the correlation function and power spectrum form a Fourier transform pair. The focus here is on developing an explicit understanding for the requirement. Statistically heterogeneous (either in space or time) data can produce serious scaling errors. This work shows that the effects of statistical heterogeneity appear as contributions from cross correlations among all of the distinct contributing rainfall components using either method so that the correlation function and its FFT do not form a transform pair. Moreover, the transform then also depends upon the time and location of the observations so that the ''observed'' power spectrum no longer represents a ''universal'' scaling function beyond the observations. An index of statistical heterogeneity (IXH) defined in previous work provides a way of determining whether or not a set of rain data may be considered to be WSS or WSSH. The greater IXH exceeds the null, the more likely the derived power spectrum should not be used for general scaling. Numerical simulations and some observations are used to demonstrate all of these findings.