Non-power-law scale properties of rainfall in space and time

The identification of general relationships linking statistical properties of rainfall aggregated at different temporal and spatial scales possesses clear theoretical and practical relevance. Among other properties it is important to characterize the scale dependence of rainfall variability, which may, for many purposes, be summarized by its variance. This paper presents theoretical results and observational analyses connecting the variance of temporal and spatial rainfall to the scale of observation under an assumption of secondorder stationarity. Previous results regarding the theoretical form of the temporal variance as a function of aggregation are refined and extended to the spatial case. It is shown that the variance of aggregated rainfall exhibits an overall nonscaling form within the range of scales of usual hydrological interest. It is further shown that the theoretically derived form of the variance as a function of scale is incompatible with power law relations usually assumed for the second-order statistical moment within multiscaling approaches to rainfall modeling. Predictions of the theoretical derivations are validated by use of observations representing a wide variety of resolutions (from 2 s to 1 hour in time and 4 km in space), climates, and measurement instruments. The validation shows a good agreement between observations and the theoretically predicted forms of the variance as a function of aggregation. The proposed theoretical framework thus provides a useful context for rainfall analysis, modeling, and downscaling in space and time and suggests careful reexaminations of usual multiscaling assumptions