Multiscaling properties of spatial rainfall and river flow distributions

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Abstract
Two common properties of empirical moments shared by spatial rainfall, river flows, and turbulent velocities are identified: namely, the log‐log linearity of moments with spatial scale and the concavity of corresponding slopes with respect to the order of the moments. A general class of continuous multiplicative processes is constructed to provide a theoretical framework for these observations. Specifically, the class of log‐Levy‐stable processes, which includes the lognormal as a special case, is analyzed. This analysis builds on some mathematical results for simple scaling processes. The general class of multiplicative processes is shown to be characterized by an invariance property of their probability distributions with respect to rescaling by a positive random function of the scale parameter. It is referred to as (strict sense) multiscaling. This theory provides a foundation for studying spatial variability in a variety of hydrologic processes across a broad range of scales.

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