BOUNDED CASCADE MODELS AS NONSTATIONARY MULTIFRACTALS

We investigate a class of bounded random-cascade models which ark mult iplicative by construction yet additive with respect to some but not a ll of their properties. We assume the multiplicative weights go to uni ty as the cascade proceeds; then the resulting field has upper and low er bounds. Two largely complementary multifractal statistical methods of analysis are used, singular measures and structure functions yieldi ng, respectively, the exponent hierarchies D-q and H-q. We study in mo re detail a specific. subclass of one-dimensional models with weights 1+1(1-2p)r(n-1)(H) at relative scale r(n) = 2(-n) after n cascade step s. The parameter H > O regulates the degree of nonstationarity; at H=O , stationarity prevails and singular ''p-model'' cascades [Meneveau an d Sreenivasan, Phys. Rev, Lett. 59, 1424 (1987)] are retrieved. Our mo del has at once large-scale stationarity and small scale nonstationari ty with stationary increments. Due to the boundedness, the D-q all con verge to unity with increasing n; the rate of convergence is estimated and the results are discussed in terms of ''residual'' multifractalit y (a spurious singularity spectrum due to finite-size effects). The st ructure-function exponents are more interesting: H-q = min {H,1/q} in the limit n --> infinity. We further focus on the cases q = 1, related to the fractal structure of the graph, q = 2, related to the energy s pectrum, and q = 1/H, the critical order beyond which our multiplicati ve (and multiscaling) bounded cascade model can be statistically disti nguished from fractional Brownian motion, the corresponding additive ( and monoscaling) model. This bifurcation in statistical behavior can b e interpreted as a first-order phase transition traceable to the bound edness, itself inherited from the large-scale stationarity. Some geoph ysical applications are briefly discussed.