On the Importance of Statistical Homogeneity to the Scaling of Rain

Scaling studies of rainfall are important for the translation of observations and numerical

model outputs among all the various scales. A common approach for determining scaling

relations is to calculate a correlation function and then to Fourier transform this function to

derive a variance (power) spectrum vis a vis the Wiener-Khintchine theorem.

This theorem, however, requires that the observations be wide sense statistically

stationary (WSS) in time or homogeneous in space. In general, though, it is not known whether a

set of observations are WSS or not so that the theorem is easily and often misapplied to

statistically heterogeneous data.

This can produce serious errors with regard to the identification of scaling relations. An

autocorrelation function can always be calculated for a set of rain observations even though it

does not exist for statistically heterogeneous data. Consequently, the Fourier transform of a

calculated autocorrelation function in statistically heterogeneous rain includes contributions from

cross-correlations among all of the contributing rainfall components. 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spectrum no longer represents a ‘universal’ scaling function beyond the observations.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) is provided for determining whether or not a

set rain data may be considered to be WSS. The greater IXH exceeds the null, the more likely the

Fourier transform of the observed correlation function should not be used for general scaling

studies. Several examples are provided to illustrate the magnitudes of these effects.