Comparison between two types of multifractal modeling

The interrelationships between two previously developed multifractal models are discussed. These are the Evertsz-Mandelbrot model developed on the basis of the multifractal spectrum f(alpha), and the Schertzer-Lovejoy model based on the codimension function C(gamma) where alpha and gamma Holder exponent and field order, respectively. It is shown how these two models are interrelated: they are identical for values of gamma within the range D - alpha(0) less than or equal to gamma less than or equal to D - alpha(min), where D is the Euclidean dimension. For D - alpha(max) less than or equal to gamma less than or equal to D - alpha(0), however, f(alpha) remains a continuous function of alpha whereas C(gamma) assumes constant value. In this respect, the fractal spectrum f(alpha) can provide more information about the multifractal measure than the codimension function C(gamma). The properties of the two models are illustrated by application to the binomial multiplicative cascade model.