Canonical vs. Microcanonical Cascades: Are they Interchangeable Notions?

Multifractal cascades, as the generic constructors of multifractal measures, have been widely used in hydrometeorology and other areas of geophysics. In order to parametrize the cascade models, the inverse problem in this case consists in finding the generator parameters, given a realization of the multifractal measure. From the direct measurement of a field or time series (e.g., rainfall), its decomposition into breakdown coefficients is immediate, producing a microcanonical (deterministically normalized) multiplicative cascade over a certain range of scales. However, the canonical (stochastically normalized) construction at underlying scales may generate statistical properties that are not easy to reproduce. This work analyzes those properties for the case of a one-dimensional, beta-lognormal discrete multiplicative cascade.