Slices of Multifractal Measures and Applications to Rainfall Distributions

Abstract

We discuss the relationship between the multifractal functions of a plane measure and those of slices or sections of the measure with a line. Motivated by recent mathematical ideas about the relationship between measures and their slices, we formulate the "slice hypothesis," and consider the theoretical limitations of this hypothesis. We compute the multifractal functions of several standard self-similar and self-affine measures and their slices to examine the validity of the slice hypothesis. We are particularly interested in using the slice hypothesis to estimate multifractal properties of spatial rainfall fields by analyzing rainfall data representing slices of rainfall fields. We consider how rainfall time series at a fixed site and slices of composite radar images can be used for this purpose, testing this on field data from a radar composite in the USA and on appropriate time series. Fractals 2000.08:337-348. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 02/09/15. For personal use only. Fractals 2000.08:337-348. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 02/09/15. For personal use only. Fractals 2000.08:337-348. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 02/09/15. For personal use only.

Fractional Generalized Random Fields of Variable Order

“…A way to characterize the multiscaling/multifractal behaviour of a random process or field is via the non-trivial singularity spectrum of its sample paths (Jaffard 20 ). A variety of multiplicative cascades and iterated function systems has been shown to generate multifractals (Mandelbrot 28 30 , Chigirinskaya and Marsan 10 , Pandey et al 32 , Marguerit et al 31 , Riedi et al 35 , Lammering 26 , Christakos 11 , Calvet and Fisher 9 , Anh et al 4 . This paper is concerned with a class of models for multifractals generated by pseudodifferential operators of variable order.…”