Apparent/spurious multifractality of data sampled from fractional Brownian/Lévy motions

Abstract

Abstract:Many earth and environmental variables appear to be self-affine (monofractal) or multifractal with spatial (or temporal) increments having exceedance probability tails that decay as powers of ˛where 1 <˛Ä 2. The literature considers selfaffine and multifractal modes of scaling to be fundamentally different, the first arising from additive and the second from multiplicative random fields or processes. We demonstrate theoretically that data having finite support, sampled across a finite domain from one or several realizations of an additive Gaussian field constituting fractional Brownian motion (fBm) characterized by˛D 2, give rise to positive square (or absolute) increments which behave as if the field was multifractal when in fact it is monofractal. Sampling such data from additive fractional Lévy motions (fLm) with 1 <˛< 2 causes them to exhibit spurious multifractality. Deviations from apparent multifractal behaviour at small and large lags are due to nonzero data support and finite domain size, unrelated to noise or undersampling (the causes cited for such deviations in the literature). Our analysis is based on a formal decomposition of anisotropic fLm (fBm when˛D 2) into a continuous hierarchy of statistically independent and homogeneous random fields, or modes, which captures the above behaviour in terms of only E C 3 parameters where E is Euclidean dimension. Although the decomposition is consistent with a hydrologic rationale proposed by Neuman (2003), its mathematical validity is independent of such a rationale. Our results suggest that it may be worth checking how closely would variables considered in the literature to be multifractal (e.g. experimental and simulated turbulent velocities, some simulated porous flow velocities, landscape elevations, rain intensities, river network area and width functions, river flow series, soil water storage and physical properties) fit the simpler monofractal model considered in this paper (such an effort would require paying close attention to the support and sampling window scales of the data). Parsimony would suggest associating variables found to fit both models equally well with the latter.

Extended power-law scaling of heavy-tailed random air-permeability fields in fractured and sedimentary rocks

“…Asymptotic tendency of β(q, q − 1) toward 1 then implies asymptotic tendency of ξ (q) toward a straight line. This commonly observed tendency, which the multifractal literature attributes to divergence of higher-order moments, is according to our theory (Neuman, 2010a;Guadagnini and Neuman, 2011) unrelated to such divergence, arising instead from the presence of an upper cutoff scale, λ u . Figure 4 includes two vertical broken lines demarcating a mid-range of lags within which log S 1 N appears to be quite unambiguously linear in log s. Fitting a straight line to the corresponding data by regression yields ξ (1) = 0.56 with a high coefficient of determination, R 2 = 0.97.…”

“…The consistency further implies that nonlinear scaling of both data sets, manifested in a nonlinear concave relationship between their power-law exponents ξ (q) and q, is not an indication of multifractality but an artifact of sampling as explained theoretically by Neuman (2010a) and Guadagnini et al (2012).…”

“…where the power or scaling exponent, ξ (q), is independent of s. When the scaling exponent is linearly proportional to q, ξ (q) = H q, Y (x) is interpreted to be a selfaffine (additive, monofractal) random field (or process) with3250 A. Guadagnini et al: Heavy-tailed random air-permeability fields in fractured and sedimentary rocks multifractal random fields or processes (Neuman, 2010a;Guadagnini et al, 2012). Nonlinear power-law scaling is also exhibited by fractional Laplace motions (Meerschaert et al, 2004;Kozubowski et al, 2006) recently applied to sediment transport data by Ganti et al (2009).…”

“…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. Yet computational analyses by Guadagnini and Neuman (2011) have shown that this need not be the case: they found signals constructed from sub-Gaussian processes subordinated to truncated (additive, self-affine, monofractal) fractional Brownian motion (tfBm) to display ESS scaling as well as typical symptoms of multifractality, such as nonlinear scaling and intermittency, even though the signals differ from multifractals in a fundamental way (Neuman, 2010a(Neuman, , b, 2011Guadagnini et al, 2012). Siena et al (2012) have pointed out that since multifractals and fractional Laplace motions do not capture observed breakdowns in power-law scaling at small and large lags, they cannot explain how and why ESS does.…”

On the nature of rainfall intermittency as revealed by different metrics and sampling approaches

“…For each station and range of scaling regime, two single values of the intermittency exponent were calculated, one for FS (indicated with µ FS 2 ) and the other for BS (µ BS 2 ), following the same approach illustrated for the other exponents. We highlight that previous studies (Veneziano and Iacobellis, 1999;Neuman, 2010aNeuman, ,b, 2012Guadagnini and Neuman, 2011) have demonstrated that techniques based on the gradient amplitude method, like the intermittency exponent, are not able to reveal presence of scaling and multifractal properties. Thus, the intermittency exponent was here only utilized to compare the intermittency characteristics of BS and FS series.…”