Abstract
Abstract. We use three methods to identify power-law scaling of multi-scale log air permeability data collected by Tidwell and Wilson on the faces of a laboratory-scale block of Topopah Spring tuff: method of moments (M), Extended Self-Similarity (ESS) and a generalized version thereof (G-ESS). All three methods focus on q-th-order sample structure functions of absolute increments. Most such functions exhibit power-law scaling at best over a limited midrange of experimental separation scales, or lags, which are sometimes difficult to identify unambiguously by means of M. ESS and G-ESS extend this range in a way that renders power-law scaling easier to characterize. Our analysis confirms the superiority of ESS and G-ESS over M in identifying the scaling exponents, ξ(q), of corresponding structure functions of orders q, suggesting further that ESS is more reliable than G-ESS. The exponents vary in a nonlinear fashion with q as is typical of real or apparent multifractals. Our estimates of the Hurst scaling coefficient increase with support scale, implying a reduction in roughness (anti-persistence) of the log permeability field with measurement volume. The finding by Tidwell and Wilson that log permeabilities associated with all tip sizes can be characterized by stationary variogram models, coupled with our findings that log permeability increments associated with the smallest tip size are approximately Gaussian and those associated with all tip sizes scale show nonlinear variations in ξ(q) with q, are consistent with a view of these data as a sample from a truncated version (tfBm) of self-affine fractional Brownian motion (fBm). Since in theory the scaling exponents, ξ(q), of tfBm vary linearly with q we conclude that nonlinear scaling in our case is not an indication of multifractality but an artifact of sampling from tfBm. This allows us to explain theoretically how power-law scaling of our data, as well as of non-Gaussian heavy-tailed signals subordinated to tfBm, are extended by ESS. It further allows us to identify the functional form and estimate all parameters of the corresponding tfBm based on sample structure functions of first and second orders.
Highlights
The literature indicates (Neuman and Di Federico, 2003) that hydrogeologic variables exhibit isotropic and directional dependencies on scales of measurement, observation, sampling window, spatial correlation, and spatial resolution
The finding by Tidwell and Wilson that log permeabilities associated with all tip sizes can be characterized by stationary variogram models, coupled with our findings that log permeability increments associated with the smallest tip size are approximately Gaussian and those associated with all tip sizes scale show nonlinear variations in ξ(q) with q, are consistent with a view of these data as a sample from a truncated version of self-affine fractional Brownian motion
After showing that our data behave as a sample from truncated fractional Brownian motion (tfBm) we demonstrate in Appendix A that this process is consistent with Eq (6) at all separation scales and with Eq (2) at intermediate scales, as are most of our data
Summary
The literature indicates (Neuman and Di Federico, 2003) that hydrogeologic variables exhibit isotropic and directional dependencies on scales of measurement (data support), observation (extent of phenomena such as a dispersing plume), sampling window (domain of investigation), spatial correlation (structural coherence), and spatial resolution (descriptive detail). Their work suggests that nonlinear variations in ξ(q) with q need not represent multifractal scaling but could instead be an artifact of sampling from tfBm or fields subordinated to tfBm. Power-law scaling is typically inferred from measured values of earth and environmental variables by the method of moments (M). Power-law scaling is typically inferred from measured values of earth and environmental variables by the method of moments (M) This consists of calculating sample structure functions Eq (1) for a finite sequence, q1, q2, ..., qn, of q values and for various separation lags. Noise-free signals subordinated to tfBm generated by Neuman (2010b) and Guadagnini et al (2011) show power-law breakdown at small and large lags even when sample sizes are large This breakdown is caused by cutoffs which truncate the fields at small lags proportional to the measurement and/or resolution scale of the data, and at large lags proportional to the size of the sampling domain, regardless of noise or undersampling. The same likely holds true for other Gaussian or heavy-tailed earth and environmental variables (such as those listed earlier) that scale according to Eq (2) at intermediate lags and according to Eq (3) over an extended range of lags, a possibility noted earlier by Guadagnini and Neuman (2011)
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