Apparent multifractality and scale‐dependent distribution of data sampled from self‐affine processes

It has been previously demonstrated theoretically and numerically by the author that square or absolute increments of data sampled from fractional Brownian/Lévy motion (fBm/fLm), or of incremental data sampled from fractional Gaussian/Lévy noise (fGn/fLn), exhibit apparent/spurious multifractality. Here, we generalize these previous development in a way that (a) rigorously subordinates (truncated) fLn to fGn or, in a statistically equivalent manner, (truncated) fLm to fBm; (b) extends the analysis to a wider class of subordinated self‐affine processes; (c) provides a simple way to generate such processes and (d) explains why the distribution of corresponding increments tends to evolve from heavy tailed at small lags (separation distances or scales) to Gaussian at larger lags. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical investigation of apparent multifractality of samples from processes subordinated to truncated fBm

We investigate numerically apparent multi‐fractal behavior of samples from synthetically generated processes subordinated to truncated fractional Brownian motion (tfBm) on finite domains. We are motivated by the recognition that many earth and environmental (including hydrological) 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. It has been demonstrated theoretically by one of us that square or absolute increments of samples from Gaussian/Lévy processes subordinated to tfBm exhibit apparent/spurious multifractality at intermediate ranges of separation lags, with breakdown in power‐law scaling at small and large lags as is commonly exhibited by real data. A preliminary numerical demonstration of apparent multifractality by the same author was limited to Gaussian fields having nearest neighbor autocorrelations and led to rather noisy results. Here, we adopt a new generation scheme that allows us to investigate apparent multifractal behaviors of samples taken from a broad range of processes including Gaussian with and without symmetric Lévy and log‐normal (as well as potentially other) subordinators. Our results shed new light on the nature of apparent multifractality, which has wide implications vis‐a‐vis the scaling of many hydrological as well as other earth and environmental variables. Copyright © 2011 John Wiley & Sons, Ltd.     

Impact of fractional probability distributions on statistics of hydraulic conductivity,dynamics of groundwater flow and solute transport at a low-permeability site

The fractional Brownian motion (fBm) and fractional Lévy motion (fLm) can easily describe the geometry and the statistical structure of hydraulic conductivity (K) for real-world. However, the fBm and fLm models have not been systematically evaluated when building the K field for a low-permeability site. In this study, both the fBm and fLm are used to simulate the low-K field at NingCheGu (NCG), Tianjin, China. Groundwater flow and solute transport are then computed using MODFLOW and MT3DMS, respectively, and the influence of the fBm/fLm models for K on groundwater flow and solute transport is discussed. Results show that the fLm fits better the statistics of the low-K medium than fBm, and the random logarithmic K (LnK) field generated by fLm is more stable because the resultant LnK field captures more of the measured properties at the field site than that generated by fBm. In contrast, the LnK generated by fBm is more likely to form both high-K channels and low-K barriers. The fBm therefore predicts more extreme behaviours in flow and transport, including the preferential flow, low-concentration blocks and solute retention. The overall groundwater renewal period and solute travel time for the fLm simulation are slightly shorter than those for fBm. The impacts of the fLm and fBm models on the statistics of the resultant LnK fields and the dynamics of groundwater flow and solute transport revealed by this study shed light on the selection and evaluation of the fractional probability distribution models in capturing the K fields for low-K media.     

Multiscale flow and transport model in three-dimensional fractal porous media

This paper proposes a multiscale flow and transport model which can be used in three-dimensional fractal random fields. The fractal random field effectively describes a field with a high degree of variability to satisfy the one-point statistics of Levy-stable distribution and the two-point statistics of fractional Levy motion (fLm). To overcome the difficulty of using infinite variance of Levy-stable distribution and to provide the physical meaning of a finite domain in real space, truncated power variograms are utilized for the fLm fields. The fLm model is general in the sense that both stationary and commonly used fractional Brownian motion (fBm) models are its special cases. When the upper cutoff of the truncated power variogram is close to the lower cutoff, the stationary model is well approximated. The commonly used fBm model is recovered when the Levy index of fLm is 2. Flow and solute transport were analyzed using the first-order perturbation method. Mean velocity, velocity covariance, and effective hydraulic conductivity in a three-dimensional fractal random field were derived. Analytical results for particle displacement covariance and macrodispersion coefficients are also presented. The results show that the plume in an fLm field moves slower at early time and has more significant long-tailing behavior at late time than in fBm or stationary exponential fields. The proposed fractal transport model has broader applications than those of stationary and fBm models. Flow and solute transport can be simulated for various scenarios by adjusting the Levy index and cutoffs of fLm to yield more accurate modeling results.     

Multifractal versus monofractal analysis of wetland topography

The land surface elevation distribution will serve as fundamental input data to any wetland flow model. As an alternative to the traditional smooth function approach to represent or interpolate elevation data, we explore the use of Levy monofractals and universal multifractals as a means for defining a statistically equivalent topography. The motivation behind this effort is that fractals, like natural topography, are irregular, they offer a way to relate elevation variations measured at different scales, and the relationships are of a statistical nature. The study site was a riparian wetland near Savannah, GA, that contained beavers, and a total of four elevation transects were examined. The elevation increments showed definite non-Gaussian behavior, with parameter values, such as the Hurst coefficient and Lévy index (α), depending on the question of presence of beaver activity. It was obvious that the data were highly irregular, especially the transects influenced by beavers. Significantly different α values were obtained depending on whether the entire data set or just the tails were examined, which demonstrated inability of the monofractal model to reflect fully the irregularity of wetland data. Further analysis confirmed definite multifractal scaling, and it is concluded that the multifractal model is superior for this data set. Universal multifractal parameters are calculated and compared to those obtained previously for more typical terrain. Although it is difficult to consider a unique universal multifractal parameter α for the entire wetland, multifractal-like scaling was evident in each transect as reflected by the nonlinear behaviors of the scaling functions. We demonstrate a good agreement between theory and measurements up to a critical order of statistical moments, q _D , close to 3.5 and obtain realistic unconditioned simulations of multifractal wetland topography based on our parameter estimates. Future work should be devoted to conditioning multifractal realizations to data and to obtaining larger data sets so that the question of anisotropy may be studied.     

On the Occurence of Extreme Events in Long-term Correlated and Multifractal Data Sets

We review recent studies of the statistics of return intervals (i) in long-term correlated monofractal records and (ii) in multifractal records in the absence (or presence) of linear long-term correlations. We show that for the monofractal records which are long-term power-law correlated with exponent γ, the distribution density of the return intervals follows a stretched exponential with the same exponent γ and the return intervals are long-term correlated, again with the same exponent γ. For the multifractal record, significant differences in scaling behavior both in the distribuiton and correlation behavior of return intervals between large events of different magnitudes are demonstrated. In the absence of linear long-term correlations, the nonlinear correlations contribute strongly to the statistics of the return intervals such that the return intervals become long-term correlated even though the original data are linearly uncorrelated (i.e., the autocorrelation function vanishes). The distribution density of the return intervals is mainly described by a power law.     

Multifractal modelling and spectrum analysis: Methods and applications to gamma ray spectrometer data from southwestern Nova Scotia, Canada

Multifractal theory was developed for handling scale invariant fields instead of geometry only[1―4]. From a multifractal point of view, some fractal models, ordinary physical processes and relevant probability distribution types can be considered as special cases of multifractal models which provides new insight into the interrelationships between systems and subjects. For example, the low order moment exponents τ (0), τ (1), τ (2) or τ ″(1) obtained by means of the moment method determi…     

Prediction uncertainty for tracer migration in random heterogeneities with multifractal character

Travel-time statistics for non-reacting tracers in fractal and multifractal media are addressed through numerical simulations. The logarithm of hydraulic conductivity is modeled using fractional Brownian motion (fBm) and more recently developed multifractal model based on bounded fractional Levy motion (bfLm). These models have been shown previously to accurately reproduce statistical properties of large conductivity datasets. The ensemble-mean travel time increases nearly linearly with travel distance and the variance in the travel time increases nearly parabolically with travel distance. This is consistent with near-field analytical approximations developed for non-fractal media and suggests that these analytical results may have some degree of robustness to non-ideal features in the random-field models. The magnitudes of the travel-time moments are dependent on the system size. For fBm media, this size dependence can be explained using an effective variance that increases with increasing size of the flow system. However, the magnitudes of the travel-time moments are also sensitive to other non-ideal effects such as deviations from Gaussian behavior. This sensitivity illustrates the need for careful aquifer characterization and conditional numerical simulation in practical situations requiring accurate estimates of uncertainty in the plume position.     

Effect of permeability variations on solute transport in highly heterogeneous porous media

The effect of aquifer heterogeneity on flow and solute transport in two-dimensional isotropic porous media was analyzed using the Monte Carlo method. The two-dimensional logarithmic permeability (ln K) was assumed to be a non-stationary random field with its increments being a truncated fractional Lévy motion (fLm). The permeability fields were generated using the modified successive random additions (SRA) algorithm code SRA3DC [http://www.iamg.org/CGEditor/index.htm]. The velocity and concentration fields were computed respectively for two-dimensional flow and transport with a pulse input using the finite difference codes of MODFLOW 2000 and MT3DMS. Two fLm control parameters, namely the width parameter (C) and the Lévy index (α), were varied systematically to examine their effect on the resulting permeability, flow velocity and concentration fields. We also computed the first- and second-spatial moments, the dilution index, as well as the breakthrough curves at different control planes with the corresponding concentration fields. In addition, the derived breakthrough curves were fitted using the continuous time random walk (CTRW) and the traditional advection-dispersion equation (ADE). Results indicated that larger C and smaller α both led to more heterogeneous permeability and velocity fields. The Lévy-stable distribution of increments in ln K resulted in a Lévy-stable distribution of increments in logarithm of the velocity (ln v). Both larger C and smaller α created sharper leading edges and wider tailing edges of solute plumes. Furthermore, a relatively larger amount of solute still remained in the domain after a relatively longer time transport for smaller α values. The dilution indices were smaller than unity and increased as C increased and α decreased. The solute plume and its second-spatial moments increased as C increased and α decreased, while the first-spatial moments of the solute plume were independent of C and α values. The longitudinal macrodispersivity was scale-dependent and increased as a power law function of time. Increasing C and decreasing α both resulted in an increase in longitudinal macrodispersivity. The transport in such highly heterogeneous media was slightly non-Gaussian with its derived breakthrough curves being slightly better fitted by the CTRW than the ADE, especially in the early arrivals and late-time tails.     

Nuclear magnetic resonance <Emphasis Type="Italic">T</Emphasis><Subscript>2</Subscript> spectrum: multifractal characteristics and pore structure evaluation

Pore structure characteristics are important to oil and gas exploration in complex low-permeability reservoirs. Using multifractal theory and nuclear magnetic resonance (NMR), we studied the pore structure of low-permeability sandstone rocks from the 4th Member (E_S4) of the Shahejie Formation in the south slope of the Dongying Sag. We used the existing pore structure data from petrophysics, core slices, and mercury injection tests to classify the pore structure into three categories and five subcategories. Then, the T₂ spectra of samples with different pore structures were interpolated, and the one- and three-dimensional fractal dimensions and the multifractal spectrum were obtained. Parameters α (intensity of singularity) and f (α) (density of distribution) were extracted from the multifractal spectra. The differences in the three fractal dimensions suggest that the pore structure types correlate with α and f (α). The results calculated based on the multifractal spectrum is consistent with that of the core slices and mercury injection. Finally, the proposed method was applied to an actual logging profile to evaluate the pore structure of low-permeability sandstone reservoirs.     

Temporal characteristics of seismicity in the Alborz and Zagros regions of Iran, using a multifractal approach

Multifractal characteristics of the temporal distribution of earthquakes in the Zagros and Alborz regions of Iran were analyzed using the fixed-mass method. The generalized multifractal dimensions, singularity spectrum, mass exponents, and the asymmetry factor were calculated for these regions. The results indicate that the temporal distributions of earthquakes in the Zagros and Alborz regions are likely to be chaotic and have multifractal structures. Although both of the study areas show heterogeneous structures, the D_q and f(α_q) spectra for the Zagros region indicate that densely populated time domains are as heterogeneous as the sparsely populated ones. On the other hand, the multifractal spectra of the Alborz region show that the densely clustered time domains are more heterogeneous than the sparsely populated ones. Such a multifractal spectrum shows that there are many more sparsely populated time domains (i.e. seismic gaps) within the multifractal structure than densely populated ones.     

Bracketed and normalized durations of earthquake ground acceleration

Multiple regression analyses of the duration of earthquake ground acceleration are presented. Two types of duration are considered, i.e. bracketed duration and normalized duration. The bracketed duration t_a is defined as the elapsed time between the first and last acceleration excursions greater than a [cm/s²], and the normalized duration Tα is defined as the elapsed time between the first and last acceleration excursions greater than α times (0 < α < 1) the peak acceleration. Employed are 394 components of horizontal strong motion acceleration records obtained at 67 free field sites in Japan. With the use of multiple regression analysis, the dependence of the bracketed and normalized durations on earthquake magnitude and epicentral distance is studied.     

Multifractal scaling of the kinetic energy flux in solar wind turbulence

The geometrical and scaling properties of the energy flux of the turbulent kinetic energy in the solar wind have been studied. Using present experimental technology in solar wind measurements we cannot directly measure the real volumetric dissipation rate, <varepsilon>(t), but are constrained to represent it by its surrogate the energy flux near the dissipation range at the proton gyro scale. There is evidence for the multifractal nature of the so defined dissipation field <varepsilon>(t), a result derived from the scaling exponents of its statistical moments. The generalized dimension D _q has been determined and reveals that the dissipation field has a multifractal structure, which is not compatible with a scale-invariant cascade. The related multifractal spectrum f(<alpha>) has been estimated for the first time for MHD turbulence in the solar wind. Its features resemble those obtained for turbulent fluids and other nonlinear multifractal systems. The generalized dimension D _q can for turbulence in high-speed streams be fitted well by the functional dependence of the p-model with a comparatively large parameter p ₁=0.87, indicating a strongly intermittent multifractal energy cascade. The experimental value for D _p/3 used in the scaling exponent s(p) of the velocity structure function gives an exponent that can describe some of the observations. The scaling exponent <mu> of the autocorrelation function of <varepsilon>(t) has also been directly evaluated, being 0.37. Finally, the mean dissipation rate was determined, which could be used in solar wind heating models.     

Hydraulic diffusivity and macrodispersivity calculations embedded in a geographic information system

Abstract

A numerical technique is presented whereby aquifer hydraulic diffusivities (D) and macrodispersivities (α) are calculated by linear equations rewritten from flow and solute transport differential equations. The approach requires a GIS to calculate spatial and temporal hydraulic head (h) and solute concentration gradients. The model is tested in Portugal, in a semi-confined aquifer periodically monitored for h and chloride/sulphate concentrations. Average D (0.46 m²/s) and α (1975 m) compare favourably with literature results. The relationship between α and scale (L) is also investigated. In this context, two aquifer groups could be identified: the first group is heterogeneous at the “macroscopic” scale (solute travelled distances <1 km), but homogeneous at the “megascopic” scale. The overall scale dependency in this case is given by an equation of logarithmic type. The second group is heterogeneous at the macroscopic and megascopic scales, with a scale dependency of linear type.

Citation Pacheco, F.A.L., 2013. Hydraulic diffusivity and macrodispersivity calculations embedded in a geographic information system. Hydrological Sciences Journal, 58 (4), 930–944.     

A multifractal formalism for vector-valued random fields based on wavelet analysis: application to turbulent velocity and vorticity 3D numerical data

Extreme atmospheric events are intimately related to the statistics of atmospheric turbulent velocities. These, in turn, exhibit multifractal scaling, which is determining the nature of the asymptotic behavior of velocities, and whose parameter evaluation is therefore of great interest currently. We combine singular value decomposition techniques and wavelet transform analysis to generalize the multifractal formalism to vector-valued random fields. The so-called Tensorial Wavelet Transform Modulus Maxima (TWTMM) method is calibrated on synthetic self-similar 2D vector-valued multifractal measures and monofractal 3D vector-valued fractional Brownian fields. We report the results of some application of the TWTMM method to turbulent velocity and vorticity fields generated by direct numerical simulations of the incompressible Navier–Stokes equations. This study reveals the existence of an intimate relationship between the singularity spectra of these two vector fields which are found significantly more intermittent than previously estimated from longitudinal and transverse velocity increment statistics.

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Multifractal of spatial distribution of seismicity in Liaoning area

Making use of multifractal theory and corresponding computational method and according to the feature of evolution of spatial distribution with respect to seismicity by earthquake data in Liaoning area, earthquake activity of the area has been studied in detail. The results show that the evolution of increase in seismicity and distributive process in space are a multifractal structure. Whole characteristic of evolution in fractal increasing process of seismicity is described by obvious variation in regard toτ(q)-q curve,f(α) spectrum and other parameters before and after moderate and strong earthquakes.     

Attractor reconstruction from the time series of information entropy of seismic kinetics process

The attractor is reconstructed from the time series of the information entropy of the seismic kinetics process. It is shown that the seismic kinetics process is governed by three order parameters and is characterized by a strange attractor in the three-dimensional phase space. The D_q-spectrum of the multifractal measure induced by the attractor, which describes the topological structure of the latter, is obtained. The monofractal dimension of the attractor is D_q(0) = 2.31…, and the correlation dimension is D_q(2) = 2.16…. The estimate of the largest Lyapunov exponent of the attractor λ1 = 0.331…. The positive signature of the largest Lyapunov exponent suggests that the attractor is chaotic and the behavior of the phase trajectory is unpredictable.     

CORRECTIONS FOR CONDUCTIVITY ESTIMATES IN INDUCTION PROSPECTING OF SULPHIDE DYKES IN A LAYERED ENVIRONMENT*

In a weathered environment estimates of depth and conductance of metallic sulphide dykes from conventional anomaly index diagrams for a vertical half-plane in air have to be corrected, besides the usual corrections, for: 1. moderate conductivity of the host rocks, and 2. finiteness of strike length S and depth extent D. Model experiments have been carried out to evaluate the response variation of a vertical planar conductor with varying depth extent and strike length for both insulating and conductive surroundings. The results indicate: 1. A conductor with finite depth extent (D/L < 2.5) or strike length (S/L < 5.0) in an insulating medium yields a lower estimate of conductance (mineralization) and a greater depth. 2. A moderately-conductive host rock enhances the anomaly and rotates the phase so that the conductor appears to be more resistive (less mineralized) and shallower. The results have practical significance since in weathered surroundings a highly-mineralized body of finite size could be missed, or misjudged, because of low estimates of conductivity and depth.     

Correlation of atmospheric occurences over the United States with some physical and chemical parameters. Part III: Frequencies of precipitation

Summary Correlation between some physical and chemical variables, measured at 28 stations of the United States Weather Bureau Network, and seasonal and annual frequencies of precipitation, has been attempted with the aim of gaining insight into the bearing of such variables, on occurrence of precipitation. — Concurrent trends of frequencies with local temperature functions, altitude parameters, precipitable water vapour increments, and some chemical species have been found.Contribution of the «Centro Nucleazione Aerosoli» of the National Research Council of Italy, Via Vettore 4 (Monte Sacro),Roma.     

Multifractal and Chaotic Analysis of Vrancea (Romania) Intermediate-depth Earthquakes: Investigation of the Temporal Distribution of Events

The Vrancea seismic region contains an isolated cluster of events beneath the Carpathian Arc Bend in Romania, dipping to about 200 km depth. Seismic activity mainly occurs at intermediate depths (h > 60 km). The main goal of the paper is to perform an in-depth, complex analysis of the occurrence times of these intermediate-depth events. We also try to show the versatility of the methods used to characterize different aspects of the seismicity evolution and to offer a user-friendly software toolbox to do most of the related computations. The earthquake catalog used in this study spans from 1974 to 2002 and includes only the intermediate-depth events. In the first part of the paper, we analyze the multifractal characteristics of the temporal distribution of earthquakes. The study reveals two distinct scaling regimes. At small scales we found a clear nonhomogeneous, multifractal pattern, while at large scales the temporal distribution of events shows a monofractal, and close to Poissonian (random), behavior. The multifractal behavior at small scales (minutes-hours) is shown to be clearly an effect of the short aftershock sequences that occurred after some major Vrancea earthquakes. In the second part of the paper we analyze whether our temporal series shows a persistent (or anti-persistent) long-term behavior, by using the Detrended Fluctuation Analysis (DFA) method. The results suggest that the analyzed temporal series of Vrancea earthquakes is a non-correlated process. In part three of the paper we seek to determine whether the dynamics of our earthquake system (described by the occurrence time of Vrancea earthquakes) is deterministically chaotic, deriving from a rather simple evolution law, or whether it is stochastic and is generated by a system that possesses many degrees of freedom. The results suggest that our signal is stochastic (probably does not possess an attractor). The limited time-span of the catalog and the analysis performed in this paper cannot rule out the emergence of an interesting, quasi-deterministic and low-dimensional structure in the case of major Vrancea earthquakes.Acknowledgement One of the authors (BE) is grateful to the Japanese Ministry of Education for providing him a Monbusho scholarship for studying in DPRI, Kyoto University. We thank Z.R. Struzik, M Holschneider, J. Mori and D. Kaplan for their useful comments, and acknowledge the support of the staff of DPRI, Kyoto University and the National Inst. for Earth Physics, Bucharest. We thank the two reviewers, M.B. Geilikman and M. Anghel, for their useful suggestions which improved the quality of this work.