Bounded cascade models as nonstationary multifractals

Abstract

We investigate a class of bounded random-cascade models which are multiplicative by construction yet additive with respect to some but not all of their properties. We assume the multiplicative weights go to unity as the cascade proceeds; then the resulting field has upper and lower bounds. Two largely complementary multifractal statistical methods of analysis are used, singular measures and structure functions yielding, respectively, the exponent hierarchies ${\mathit{D}}_{\mathit{q}}$ and ${\mathit{H}}_{\mathit{q}}$. We study in more detail a specific subclass of one-dimensional models with weights 1±(1-2p)${\mathit{r}}_{\mathit{n}\mathrm{-}1}^{\mathit{H}}$ at relative scale ${\mathit{r}}_{\mathit{n}}$=$2^{\mathrm{-}\mathit{n}}$ after n cascade steps. The parameter H>0 regulates the degree of nonstationarity; at H=0, stationarity prevails and singular ‘‘p-model’’ cascades [Meneveau and Sreenivasan, Phys. Rev. Lett. 59, 1424 (1987)] are retrieved. Our model has at once large-scale stationarity and small scale nonstationarity with stationary increments. Due to the boundedness, the ${\mathit{D}}_{\mathit{q}}$ all converge to unity with increasing n; the rate of convergence is estimated and the results are discussed in terms of ‘‘residual’’ multifractality (a spurious singularity spectrum due to finite-size effects). The structure-function exponents are more interesting: ${\mathit{H}}_{\mathit{q}}$=min{H,1/q} in the limit n→∞. We further focus on the cases q=1, related to the fractal structure of the graph, q=2, related to the energy spectrum, and q=1/H, the critical order beyond which our multiplicative (and multiscaling) bounded cascade model can be statistically distinguished from fractional Brownian motion, the corresponding additive (and monoscaling) model. This bifurcation in statistical behavior can be interpreted as a first-order phase transition traceable to the boundedness, itself inherited from the large-scale stationarity. Some geophysical applications are briefly discussed.