Recent advances in statistical and scaling analysis of earth and environmental variables

Many earth and environmental variables appear to be self-affine (monofractal) or multifractal with Gaussian or heavy-tailed distributions. The literature considers self-affine and multifractal types of scaling to be fundamentally different, the first arising from additive and the second from multiplicative random fields or processes. Recent work by the authors demonstrates theoretically and numerically that square or absolute increments of samples from truncated fractional Brownian motion (tfBm) exhibit apparent multifractality at intermediate ranges of separation lags, with breakdown in power-law scaling at small and large lags as is commonly exhibited by data. The same is true of samples from sub-Gaussian processes subordinated to tfBm with heavy-tailed subordinators such as lognormal or Lévy, the latter leading to spurious behavior. It has been established empirically that, in numerous cases, the range of lags exhibiting power-law scaling can be enlarged significantly, at both ends of the spectrum, via a procedure known as extended self-similarity (ESS). No theoretical model of the ESS phenomenon has previously been proposed outside the domain of Burger's equation. Our work demonstrates that ESS is consistent, at all separation scales, with sub-Gaussian processes subordinated to tfBm. This makes it possible to identify the functional form and estimate all parameters of corresponding models based solely on sample structure functions of the first and second orders. The authors' recent work also elucidates the well-documented but heretofore little-noticed and unexplained phenomenon that whereas the frequency distribution of log permeability data often seems to be Gaussian (or nearly so), that of corresponding increments (as well as those of many other earth and environmental variables) tends to exhibit heavy tails, which sometimes narrow down with increasing separation distance or lag.