Alternative to Multifractal Analysis of Scalable Random Variables Applied to Measured and Estimated Soil Properties at an Arizona Field Site

Abstract

Many earth, environmental, ecological, biological, physical, astrophysical and financial variables exhibit random space-time fluctuations; symmetric, non-Gaussian frequency distributions of increments characterized by heavy tails that often decay with separation distance or lag; nonlinear power-law scaling of sample structure functions (moments of absolute increments) in a midrange of lags, with breakdown in such scaling at small and large lags; extended power-law scaling at all lags; nonlinear scaling of power-law exponent with order of sample structure function; and pronounced statistical anisotropy. The literature has traditionally considered such variables to be multifractal. Previously we proposed a simpler and more comprehensive interpretation that views them as samples from stationary, anisotropic sub-Gaussian random fields or processes subordinated to truncated fractional Brownian motion or truncated fractional Gaussian noise. The variables thus represent mixtures of Gaussian components having random variances. We apply our novel approach to soil data collected at an Arizona field site and to corresponding hydraulic properties obtained by means of a neural network model and estimate their statistical scaling parameters by maximum likelihood. Our approach allows upscaling or downscaling statistical moments of such variables to fit diverse measurement or resolution and sampling domain scales.