Scaling of random fields by means of truncated power variograms and associated spectra

An interpretation is offered for the observation that the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale which grow with domain size. We first demonstrate that the power (semi)variogram and associated spectra of random fields, having homogeneous isotropic increments, can be constructed as weighted integrals from zero to infinity (an infinite hierarchy) of exponential or Gaussian variograms and spectra of mutually oncorrelated homogeneous isotropic fields (modes). We then analyze 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 modes. The integral scales of the lowest- and highest-frequency modes (cutoffs) are related, respectively, to the length scales of the sampling window (domain) and data support (sample volume). Taking each relationship to be one of proportionality renders our expressions for the integral scale and variance of a truncated field dependent on window and support scales in a manner consistent with observations. The traditional approach of truncating power spectral densities yields autocovariance functions that oscillate about zero with finite (in one and two dimensions) or vanishing (in one dimension) integral scales. Our hierarchical theory allows bridging across scales at a specific locale, by calibrating a truncated variogram model to data observed on a given support in one domain and predicting the autocovariance structure of the corresponding multiscale field in domains that are either smaller or larger. 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.