Abstract
We investigate a class of one-dimensional bounded random cascade models which are multiplicative and stationary by construction but additive and non-stationary with respect to some, but not all, of their statistical properties. In essence, a new parameter H>0 is introduced to “smooth” standard (p-model) cascades, these well-studied processes being retrieved at H=0. The resulting ambivalent statistical behavior of the new model leads to a 1st order multifractal phase transition in the structure function exponents, i.e., there is a discontinuity in the derivative of ζq= min{qH, 1}. We interpret this bifurcation as a separation of the stationary and non-stationary “ingredients” of the model by lowering the multifractal “temperature” (1/q) below the critical value H. We also see exactly how the generalized dimensions Dq converge to one in the small scale limit for all q. We discuss this last finding in terms of “residual” multifractality, a singularity spectrum that is entirely traceable to finite size effects (to which we are never immune in data analysis situations). Finally, we locate the bounded and unbounded versions of the model in the “q=1 multifractal plane” where the coordinates are C1=1−D1 (a direct measure of “intermittency”) and H1=ζ1 (a direct measure of “smoothness”), both of which are normally in the interval [0, 1]. This provides us with a simple way of comparing the multiplicative models with their additive counterparts, as well as with different types of geophysical data.
Published Version
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