ABSTRACT
The literature indicates that many hydrologic (as well as other) variables exhibit isotropic and directional dependencies on scales of measurement, observation, sampling window, spatial correlation, and spatial resolution. Attempts to explain such scale dependencies have generally focused on observed and/or hypothesized power-law behaviors of corresponding structure functions. Doing so has revealed a tendency of such functions to exhibit nonlinear power-law scaling in a midrange of separation lags, breakdown in power-law scaling at small and large lags, extension of power-law scaling to all lags via a procedure known as Extended Self Similarity (ESS), apparent lack of compatibility between sample frequencies of data and their increments, and decay of increment sample frequency tails with increased separation scale or lag. Existing scaling models capture some but not all of these phenomena in a consistent manner. We describe a new scaling model that does so within a unified, self-consistent theoretical framework. The framework is based on the notion of sub-Gaussian fields (or processes) Subordinated to truncated fractional Brownian motion (tfBm) with heavy tailed subordinators such as log-normal or Levy. As tfBm is a truncated version of additive, self-affine, monofractal fractional Brownian motion (fBm), corresponding nonlinear power law scaling is not an indication of multifractality (as commonly assumed in the literature) but an artifact of sampling. We illustrate our new approach to scaling on synthetically generated as well as published laboratory and field scale log permeability data.
