| I investigate the effects of domain and support scales on the multiscale properties of random fractal fields characterized by a power variogram using real and synthetic data. Neuman [1994] and Di Federico and Neuman [1997] have concluded empirically, on the basis of hydraulic conductivity data from many sites, that a finite window of length-scale L filters out all modes having integral scales λ larger than λ l = μL where μ ≃ 1/3. I confirm their finding computationally by generating truncated fBm (fractional Brownian motion) realizations on a large grid, using various initial values of μ, and demonstrating that μ ≃ 1/3 for windows smaller than the original grid. Synthetic experiments show that an fBm realization on a finite grid generated using a truncated power variogram yields more consistent sample variograms with theory than the realization generated using a power variogram. Wavelet interpretation of sample data from such a realization yields the comparable Hurst coefficient estimates with variogram analyses.; Di Federico et al. [1997] developed expressions for the equivalent hydraulic conductivity of a box-shaped volume, embedded in a log-hydraulic conductivity field characterized by a power variogram, under a mean uniform hydraulic gradient. I demonstrate that their expression and empirical value of μ ≃ 1/3 are consistent with a pronounced permeability scale effect observed in unsaturated fractured tuff at the Apache Leap Research Site (ALRS) near Superior, Arizona. I investigate the compatibility of single-hole air permeability data, obtained at the ALRS on a nominal support scale of about I m, with Min, fGn (fractional Gaussian noise), fLm (fractional Lévy motion), bfLm (bounded fractional Lévy motion) and UM (Universal Multifractals). I find the data become Gaussian from Lévy as the lag increases (corresponding to bfLm). Though this implies multiple scaling, it is inconsistent with the UM model, which considers a unique distribution. With a UM model, nevertheless, one obtains a very small codimension, which suggests that multiple scaling is minor. Variogram and rescaled range analyses of the log-permeability data yield comparable estimates of the Hurst coefficient. Rescaled range analysis shows that the data are not compatible with an fGn model. I conclude that the data are represented most closely by a truncated fBm model. |
