Universal multifractal theory and observations of land and ocean surfaces and of clouds

摘要

The extreme variability of geophysical fields can be characterized by scale invariant (sensor resolution independent) 'codimension' functions, which are exponents characterizing the probability distribution. These codimension functions form a three parameter universality class. The parameter H measures the degree of nonstationarity of the process, C$-1$/ characterizes the sparseness/inhomogeneity of the mean of the process, $alpha characterizes the degree of multifractality; $alpha $EQ 0 is monofractal, $alpha $EQ 2 is the maximum. We review the properties of these multifractal processes and describe the 'double trace moment' technique that is the first data analysis technique specifically designed to estimate these parameters. The technique is then applied to digital elevation maps of Deadman's Butte, to the topography of France, to a pair of aircraft photos of the ocean surface, and to a visible satellite image of a cloud field.