Functional Box-Counting and Multiple Elliptical Dimensions in Rain

Abstract

Many physical systems that have interacting structures that span wide ranges in size involve substantial scale invariant (or scaling) subranges. In these regimes, the large and small scales are related by an operation that involves only the scale ratio. The system has no intrinsic characteristic size. In the atmosphere gravity causes differential stratification, so that the scale change involves new elliptical dimensions (d_el). Fields that are extremely variable, such as rain, involve multiple scaling and dimensions that characterize the increasingly intense regions. Elliptical dimensional sampling and functional box-counting have been used to analyze radar rain data to obtain both the multiple dimensions of the rain field and the estimate d_el = 2.22 ± 0.07.