Abstract
Abstract. We analyze the scaling behaviors of two field-scale log permeability data sets showing heavy-tailed frequency distributions in three and two spatial dimensions, respectively. One set consists of 1-m scale pneumatic packer test data from six vertical and inclined boreholes spanning a decameters scale block of unsaturated fractured tuffs near Superior, Arizona, the other of pneumatic minipermeameter data measured at a spacing of 15 cm along three horizontal transects on a 21 m long and 6 m high outcrop of the Upper Cretaceous Straight Cliffs Formation, including lower-shoreface bioturbated and cross-bedded sandstone near Escalante, Utah. Order q sample structure functions of each data set scale as a power ξ(q) of separation scale or lag, s, over limited ranges of s. A procedure known as extended self-similarity (ESS) extends this range to all lags and yields a nonlinear (concave) functional relationship between ξ(q) and q. Whereas the literature tends to associate extended and nonlinear power-law scaling with multifractals or fractional Laplace motions, we have shown elsewhere that (a) ESS of data having a normal frequency distribution is theoretically consistent with (Gaussian) truncated (additive, self-affine, monofractal) fractional Brownian motion (tfBm), the latter being unique in predicting a breakdown in power-law scaling at small and large lags, and (b) nonlinear power-law scaling of data having either normal or heavy-tailed frequency distributions is consistent with samples from sub-Gaussian random fields or processes subordinated to tfBm or truncated fractional Gaussian noise (tfGn), stemming from lack of ergodicity which causes sample moments to scale differently than do their ensemble counterparts. Here we (i) demonstrate that the above two data sets are consistent with sub-Gaussian random fields subordinated to tfBm or tfGn and (ii) provide maximum likelihood estimates of parameters characterizing the corresponding Lévy stable subordinators and tfBm or tfGn functions.
Highlights
Many earth and environmental variables exhibit power-law scaling of the following type
Whereas the literature tends to associate extended and nonlinear power-law scaling with multifractals or fractional Laplace motions, we have shown elsewhere that (a) extended self-similarity (ESS) of data having a normal frequency distribution is theoretically consistent with (Gaussian) truncated fractional Brownian motion, the latter being unique in predicting a breakdown in power-law scaling at small and large lags, and (b) nonlinear power-law scaling of data having either normal or heavy-tailed frequency distributions is consistent with samples from sub-Gaussian random fields or processes subordinated to tfBm or truncated fractional Gaussian noise, stemming from lack of ergodicity which causes sample moments to scale differently than do their ensemble counterparts
We focus here on parameter estimates obtained by Riva et al (2012) using a maximum likelihood (ML) approach applied to a log characteristic function ln eiφX = iμφ − σ α |φ|α 1 + iβ sign (φ) ω (φ, α)
Summary
Many earth and environmental (as well as physical, ecological, biological and financial) variables exhibit power-law scaling of the following type. In addition to the classic case of turbulent velocities (Chakraborty et al, 2010), these examples include geographical (e.g. Earth and Mars topographic profiles), hydraulic (e.g. river morphology and sediment dynamics), atmospheric, astrophysical, (e.g. solar quiescent prominence, low-energy cosmic rays, cosmic microwave background radiation, turbulent boundary layers of the Earth’s magnetosphere), biological (e.g. human heartbeat temporal dynamics), financial time series and ecological variables (see Guadagnini and Neuman (2011), Leonardis et al (2012) and references therein) In virtually all these examples, ESS yields improved estimates of ξ (q) and shows it to vary in a nonlinear fashion with q, a finding commonly taken to imply that the variables are multifractal. Our analysis (a) demonstrates that the two data sets are statistically and theoretically consistent with sub-Gaussian random fields subordinated to tfBm or truncated fractional Gaussian noise (tfGn) and (b) provides maximum likelihood estimates of parameters characterizing the corresponding Levy stable subordinators and tfBm or tfGn functions
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