SCALING OF RANDOM-FIELDS BY MEANS OF TRUNCATED POWER VARIOGRAMS AND ASSOCIATED SPECTRA

An interpretation is offered for the observation that the log hydrauli c conductivity of geologic media often appears to be statistically hom ogeneous but with variance and integral scale which grow with domain s ize. We first demonstrate that the power (semi)variogram and associate d spectra of random fields, having homogeneous isotropic increments, c an be constructed as weighted integrals from zero to infinity (an infi nite hierarchy) of exponential or Gaussian variograms and spectra of m utually uncorrelated homogeneous isotropic fields (modes). We then ana lyze the effect of filtering out (truncating) high- and low-frequency modes from this infinite hierarchy in the real and spectral domains. A low-frequency cutoff renders the truncated hierarchy homogeneous with an autocovariance function that varies monotonically with separation distance in a manner not too dissimilar than that of its constituent m odes. The integral scales of the lowest- and highest-frequency modes ( cutoffs) are related, respectively, to the length Scales of the sampli ng window (domain) and data support (sample volume). Taking each rerat ionship to be one of proportionality renders our expressions for the i ntegral scale and variance of a truncated field dependent on window an d support scares in a manner consistent with observations. The traditi onal approach of truncating power spectral densities yields autocovari ance functions that oscillate about zero with finite (in one and two d imensions) or vanishing (in one dimension) integral scales. Our hierar chical theory allows bridging across scales at a specific locale, by c alibrating a truncated variogram model to data observed on a given sup port in one domain and predicting the autocovariance structure of the corresponding multiscale field in domains that are either smaller or l arger. One may also venture (we suspect with less predictive power) to bridge across both domain scales and locales by adopting generalized variogram parameters derived on the basis of juxtaposed hydraulic and tracer data from many sites.