Neuman Seminar on multi-fractals

Today Wednesday March 10 Dr. Shlomo Neuman will be giving the department seminar on "On Fractal and Multifractal Scaling of Hydrological and Geophysical Variables" at 4PM in Harshbarger 206. The topic is one with which Dr. Neuman is very familiar and has spent a signficiant portion of his career pursuing. We can all look forward to an elightening seminar that steers us clear of mistakes in the use of fractals and multi-fractals in hydrologic applications.

For a taste of what Dr. Neuman will cover you can read this article from the Reviews of Geophysics in 2003 that Dr. Neuman co-authored with Vittorio Di Federico.

Abstract -
Space-time fluctuations in hydrologic variables generally depend on scale. There has been a growing tendency to treat such variables as samples from self-affine (monofractal) or multifractal random fields (or processes) with spatial (or temporal) increments having exceedance probability tails that decay as powers of -α where, in most reported cases, 1 < alpha <= 2. The literature considers self-affine and multifractal modes of scaling to be fundamentally different, the first arising from additive and the second from multiplicative phenomena. Among recent advances is a theoretical demonstration that samples from additive fractional Brownian motion (alpha = 2) yield square (or absolute) increments which behave as if the field was multifractal when in fact it is monofractal. Samples from additive fractional Lévy motions ( 1 < alpha < 2) tend to exhibit spurious multifractality. Deviations from apparent multifractal behavior at small and large lags are due to nonzero data support and finite domain size, unrelated to noise or undersampling (causes cited for such behavior in the literature). These findings are based on an earlier advance, the formal decomposition of anisotropic (when alpha = 2) into
a continuous hierarchy of statistically independent and homogeneous random fields, or modes, which captures the above behavior in terms of only E + 3 parameters where E is Euclidean dimension. The decomposition has precise spectral and wavelet analogues. It has been used successfully to elucidate the reason why apparent autocorrelation scales of many variables increase with domain size; explain why apparent longitudinal dispersivities of subsurface tracer plumes increase with mean travel distance or time; provide a reason as to why the rate of this increase diminishes with increased resolution of plume details; derive expressions for scale-dependent effective permeabilities of self-affine geologic media; recognize and quantify the uncertain nature of such effective parameters; develop multiscale relationships between length scales, apertures, densities and permeabilities of natural rock fractures; derive ensemble analogues of Horton’s scaling laws for river networks; relate statistical moments of river network attributes to arbitrary lower and upper cutoff scales that may (but need not) be taken to represent data support and maximum watershed size; provide a theoretical basis for the previously
unexplained observation that transverse fluctuations of basin boundaries and main channels, having a common Hurst scaling exponent, are larger in the former than in the latter; upscale and downscale statistics of data collected on disparate support scales; provide a way to condition these statistics on data measured at specific space-time locations; and create a blueprint for the propagation of corresponding data and parameter uncertainties through hydrologic models. These broad interpretive and analytical powers of the approach are sure to expand in the future.