theoretical analysis of non gaussian heterogeneity effects on subsurface flow and transport miur italian ministry of education university and research water jpi waterworks project we need water needs availability quality and sustainability european union 640979 university of arizona vanderbilt university under the consortium for risk evaluation with stakeholder participation cresp iii u s department of energy

fractional brownian motion and fractional gaussian noise in subsurface hydrology a review presentation of fundamental properties and extensions recent studies have shown that fractional brownian motion fbm and fractional gaussian noise fgn are useful in characterizing subsurface heterogeneities in addition to geophysical time series although these studies have led to a fairly good understanding of some aspects of fbm fgn a comprehensive introduction to these stochastic fractal functions is still lacking in the subsurface hydrology literature in this paper efforts have been made to define fbm fgn and present a development of their mathematical properties in a direct yet rigorous manner use of the spectral representation theorem allows one to derive spectral representations for fbm fgn even though these functions do not have classical fourier transforms the discrete and truncated forms of these representations have served as a basis for synthetic generation of fbm fgn the discrete spectral representations are developed and various implications discussed in particular it is shown that a discrete form of the fbm spectral representation is equivalent to the well known weierstrass mandelbrot random fractal function although the full implications are beyond the scope of the present paper it is observed that discrete spectral representations of fbm constitute stationary processes even though fbm is nonstationary a new and general spectral density function is introduced for construction of complicated anisotropic 3 d fractals including those characterized by vertical fgn and horizontal fbm such fractals are useful for modeling anisotropic subsurface heterogeneities but cannot be generated with existing schemes finally some basic properties of fractional levy motion and concepts of universal multifractals which can be considered as generalizations of fbm fgn are reviewed briefly

fractals in physics squig clusters diffusions fractal measures and the unicity of fractal dimensionality the three topics discussed in this paper are largely independent part 1 fractal squig clusters are introduced and it is shown that their properties can match to a remarkable extent those of percolation clusters at criticality physics on these new geometric shapes should prove tractable as background the author s theories of squig intervals and squig trees are reviewed and restated in more versatile form part 2 the notion of latent fractal dimensionality is introduced and motivated by the desire to simplify the algebra of dimensionality scaling noises are touched upon a common formalism is presented for three forms of anomalous diffusion the ant in the fractal labyrinth fractional brownian motion and levy stable motion the fractal dimensionalities common to diverse shapes generated by diffusion are given in table i as functions of the latent dimensionalities of the support of the motion and of the diffusion itself part 3 it is argued that every fractal point set has a unique fractal dimensionality but it is pointed out that many fractals involve diverse combinations of many fractal point sets such is in particular the case for fractal measures and for fractal graphs often called hierarchical lattices the fractal measures that the author had introduced in the early 1970s are described including new developments

a deterministic geometric representation of temporal rainfall results for a storm in boston the use of a deterministic fractal multifractal fm representation to model high resolution rainfall time series via projections of fractal interpolating functions weighed by multifractal measures is reported it is shown that the intrinsic shape and variability of an 8 hour boston storm recorded every 15 s on october 25 1980 may be encoded wholistically employing the fractal geometric methodology it is illustrated that the fm methodology provides very faithful descriptions of both major trends and small noisy fluctuations for this storm resulting in preservation of not only classical statistical characteristics of the records but also multifractal and chaotic properties present in them these results and those for other storms suggest that a stochastic framework for rainfall may be bypassed in favor of a deterministic representation based on projections

on the correlation structure of continuous and discrete point rainfall 1 this paper proposes new analytical results based on the theory of stochastic processes that establish a theoretical dependence of point rainfall variance on the scale of observation under somewhat general assumptions it is shown that power law scaling commonly assumed by analyses and models of point rainfall cannot hold over all scales of observation and that an inner regime exists for small sampling intervals where the variance is a quadratic function of interval length a power law scaling regime is shown to exist for larger aggregation intervals whose characteristics depend on the memory of the continuous rainfall process the presence of a transition regime between the inner and the scaling regime is also shown and may explain the deviations from a power law scaling behavior in observed rainfall reported in the literature furthermore the application of the theory to rainfall data from a representative variety of climate types shows that fractal and multifractal analysis techniques and models may indeed fail to capture observed rainfall properties over wide ranges of aggregation scales finally correlation structures with short and long term memory are considered and different theoretical relationships expressing rainfall variance as a function of aggregation interval are derived and tested against rainfall observed at different locations with aggregation timescales ranging from 15 min to 2 days the theoretical and observational analyses show that both finite and infinite memory rainfall processes may be observed in nature

multidimensional modeling of cumulative rainfall parameter estimation and model adequacy through a continuum of scales this paper presents the multidimensional representation of the cumulative rainfall process and derives a closed form expression for the mean and space time covariance functions a parameter estimation procedure is developed sensitivity analysis provides useful information for devising the parameter estimation strategy the cumulative rainfall model is applied to 8 years of digitized rain gage data from the walnut gulch experimental watershed the most useful feature of the model is its self consistency implying that even after using information from a given limited set of data it still leads to realistic parameters which reproduce historical statistics at several levels of time and space scales

definition of percolation thresholds on self affine surfaces we study the percolation transition on a two dimensional substrate with long range self affine correlations we find that the position of the percolation threshold on a correlated lattice is no longer unique and depends on the spanning rule employed numerical results are provided for spanning across the lattice in specified horizontal or vertical either or both directions

fractional fokker planck equation for nonlinear stochastic differential equations driven by non gaussian levy stable noises the fokker planck equation has been very useful for studying dynamic behavior of stochastic differential equations driven by gaussian noises however there are both theoretical and empirical reasons to consider similar equations driven by strongly non gaussian noises in particular they yield strongly non gaussian anomalous diffusion which seems to be relevant in different domains of physics in this paper we therefore derive a fractional fokker planck equation for the probability distribution of particles whose motion is governed by a nonlinear langevin type equation which is driven by a levy stable noise rather than a gaussian we obtain in fact a general result for a markovian forcing we also discuss the existence and uniqueness of the solution of the fractional fokker planck equation

the mathematical structure of rainfall representations 3 some applications of the point process theory to rainfall processes part 3 constitutes the final part of three parts devoted to the mathematical structure of rainfall the objective is to illustrate the scope of the tools developed in part 2 in the mathematical description of rainfall and rainfall driven processes a general overview of the three part series is given in part 1

heavy tailed log hydraulic conductivity distributions imply heavy tailed log velocity distributions 1 equations of contaminant transport describing non gaussian dispersion of solute in heterogeneous porous media have been developed by several authors e g berkowitz and scher 1995 1998 benson 1998 2001 berkowitz et al 2000 and baeumer et al 2005 the equations assume that solute particle movement in time and or space can be described by heavy tailed probability distributions e g levy stable distributions this assumption requires a mechanism for solute particles to move through heterogeneous porous media in a manner that can be approximated with heavy tailed probability distributions this work investigates whether a heavy tailed levy stable distribution of increments in log hydraulic conductivity results in a heavy tailed distribution of increments in log velocity a heavy tailed distribution of increments in log velocity would be a mechanism for solute particle movement in time or space that can be described by a heavy tailed probability distribution a computer algorithm was written to generate realistic hydraulic conductivity fields the algorithm modifies the spectral synthesis method for generating fractional brownian motion i e saupe 1988 and brewer and wheatcraft 1994 and produces self similar hydraulic conductivity fields with a heavy tailed levy stable distribution of increments in log hydraulic conductivity a monte carlo approach was adopted in which 100 hydraulic conductivity fields were generated for each of the following values of 0 8 1 1 1 4 and 1 7 the usgs finite difference groundwater code modflow was used to calculate the velocity fields and the increments in longitudinal log hydraulic conductivity and log velocity from the fields were analyzed to determine whether they were consistent with a heavy tailed levy stable probability density function pdf after determining that the distributions of increments in log hydraulic conductivity and log velocity were consistent with a levy stable pdf the tail parameters describing each pair of hydraulic conductivity and velocity fields were estimated with the nolan 1997 maximum likelihood estimator the relationship between the two tail parameters was then determined there are three conclusions of this research first mandelbrot and pp plots confirm that levy stable distributions of increments in log hydraulic conductivity give rise to levy stable distributions of increments in log velocity second the tail parameter for the increments in log velocity is typically smaller indicating a heavier probability tail than the corresponding tail parameter for the increments in log hydraulic conductivity third there is a positive statistical relation between the two tail parameters

multifractal scaling of the intrinsic permeability existing fractal studies dealing with subsurface heterogeneity treat the logarithm of the permeability k as the variable of concern we treat k as a multifractal and investigate its scaling and fractality using measured horizontal k data from two locations in the united states the first data set was from a shoreline sandstone near coalinga california and the second was from an eolian sandstone goggin 1988 by applying spectral analyses and computing the scaling of moments of various orders using the double trace moment method lavallee 1991 lavallee et al 1992 we found that k is multiscaling i e scaling and multifractal we also found that the so called universal multifractal um schertzer and lovejoy 1987 model essentially a log levy multifractal was able to reproduce the multiscaling behavior reasonably well the um model has three parameters a s and h representing the multifractality index the codimension of the mean field and the distance to stationary multifractal respectively we found a 5 1 7 s 5 0 23 h 5 0 22 and a 5 1 6 s 5 0 11 h 5 0 075 for the shoreline and eolian data sets respectively the fact that a values were less than 2 indicates that the underlying statistics are non gaussian we generated stationary and nonstationary multifractals and illustrated the role of the um parameters on simulated fields studies that treated log k as the variable of concern have pointed out the necessity for large data records especially when the underlying distribution is levy stable our investigation revealed that even larger data records are required when treating k as a multifractal because log k is less intermittent or irregular than k

new approximate analytical technique to solve richards equation for arbitrary surface boundary conditions a general approximation for the solution to the one dimensional richards equation is presented it applies to arbitrary soil properties and boundary conditions but only uniform initial conditions at current stage the result is very accurate within 2 when the diffusivity is constant suggesting that the present general formulation is reliable since the approximation becomes increasingly accurate as the soil water diffusivity approaches a delta function

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

spatial uniformity of power and the altitudinal geometry of river networks it is hypothesized that river networks adjust their geometries in the altitudinal space to achieve a state of maximum spatial uniformity of power a model developed to study the spatial variability of power shows that the notion of uniformity is linked to both minimum variance and maximum information theoretic entropy recent descriptions of link heights and streamflows gupta and waymire 1989 1990 are applied to formulate the power of a link in a network the hypothesis relates the scaling exponent of link heights to the scaling exponents of instantaneous flows it also predicts the superiority of the self similar model over alternative models of link heights in channel networks

flow in multiscale log conductivity fields with truncated power variograms in a previous paper we offered an interpretation for the observation that the log hydraulic conductivity of geologic media often appears to be statistically homogeneous but with variance and integral scale which grow with the size of the region window being sampled we did so by demonstrating that such behavior is typical of any random field with a truncated power semi variogram and that this field can be viewed as a truncated hierarchy of mutually uncorrelated homogeneous fields with either exponential or gaussian spatial autocovariance structures the low and high frequency cutoff scales l and u are related to the length scales of the sampling window and data support respectively we showed how this allows the use of truncated power variograms to bridge information about a multiscale random field across windows of different sizes either at a given locale or between different locales in this paper we investigate mean uniform steady state groundwater flow in unbounded domains where the log hydraulic conductivity forms a truncated multiscale hierarchy of gaussian fields each associated with an exponential autocovariance we start by deriving an expression for effective hydraulic conductivity as a function of the hurst coefficient h and the cutoff scales in one two and three dimensional domains which is qualitatively consistent with experimental data we then develop leading order analytical expressions for two and three dimensional autocovariance and cross covariance functions of hydraulic head velocity and log hydraulic conductivity versus h l and u examine their behavior and compare them with those corresponding to an exponential log hydraulic conductivity autocovariance our results suggest that it should be possible to bridge information about hydraulic heads and groundwater velocities across windows of disparate scales in particular when l u the variance of head is infinite in two dimensions and grows in proportion to l2 2h in three dimensions while the variance and longitudinal integral scale of velocity grow in proportion to l2h and l respectively in both cases

self similarity in diffraction by a self similar fractal screen using the sierpinski gasket as an example it has been shown here that the paraxial fraunhoffer zone diffracted field of a self similar fractal screen also exhibits self similarity this also establishes that fractal structures can be used with great profit in problems involving array syntheses recently it has been shown that the sierpinski gasket is itself a member of a much wider class of gaskets and the potential for the use of fractal structures in electromagnetic em problems may be vast indeed

how number line estimation skills relate to neural activations in single digit subtraction problems the number line nl task requires judging the relative numerical magnitude of a number and estimating its value spatially on a continuous line children s skill on this task has been shown to correlate with and predict future mathematical competence neurofunctionally this task has been shown to rely on brain regions involved in numerical processing however there is no direct evidence that performance on the nl task is related to brain areas recruited during arithmetical processing and that these areas are domain specific to numerical processing in this study we test whether 8 to 14 year old s behavioral performance on the nl task is related to fmri activation during small and large single digit subtraction problems domain specific areas for numerical processing were independently localized through a numerosity judgment task results show a direct relation between nl estimation performance and the amount of the activation in key areas for arithmetical processing better nl estimators showed a larger problem size effect than poorer nl estimators in numerical magnitude i e intraparietal sulcus and visuospatial areas i e posterior superior parietal lobules marked by less activation for small problems in addition the direction of the activation with problem size within the ips was associated with differences in accuracies for small subtraction problems this study is the first to show that performance in the nl task i e estimating the spatial position of a number on an interval correlates with brain activity observed during single digit subtraction problem in regions thought to be involved in numerical magnitude and spatial processes

self similarity versus self affinity the sierpinski gasket revisited a hilbert curve is constructed on the sierpinski gasket it is shown that despite the fact that the sierpinski gasket is rigorously self similar the hilbert curve is merely self affine

flexible scaling model for use in random field simulation of hydraulic conductivity a new fractal scaling model is proposed for use in aquifer heterogeneity modeling and simulation the new model which is obtained by subordination of fractional brownian motion provides a unifying framework for scaling models of heterogeneity in that it includes previous scaling models as special cases and can also be tuned continuously between these models choosing the subordinator to be a lognormal distribution results in a non gaussian scaling model with a moderate degree of variability that matches data better than previous fractal scaling models thus providing more accurate heterogeneity representations for use in stochastic flow and transport predictions two new stochastic simulation algorithms are constructed from the new models one based on sequential simulation algorithms and the other based on probability field simulation both algorithms can be made conditional on available measurements

new scaling model for variables and increments with heavy tailed distributions many hydrological as well as diverse earth environmental ecological biological physical social financial and other variables y exhibit frequency distributions that are difficult to reconcile with those of their spatial or temporal increments y whereas distributions of y or its logarithm are at times slightly asymmetric with relatively mild peaks and tails those of y tend to be symmetric with peaks that grow sharper and tails that become heavier as the separation distance lag between pairs of y values decreases no statistical model known to us captures these behaviors of y and y in a unified and consistent manner we propose a new generalized sub gaussian model that does so we derive analytical expressions for probability distribution functions pdfs of y and y as well as corresponding lead statistical moments in our model the peak and tails of the y pdf scale with lag in line with observed behavior the model allows one to estimate accurately and efficiently all relevant parameters by analyzing jointly sample moments of y and y we illustrate key features of our new model and method of inference on synthetically generated samples and neutron porosity data from a deep borehole

long range anticorrelations and non gaussian behavior of the heartbeat we find that the successive increments in the cardiac beat to beat interals of healthy subjects display scale invariant long range anticorrelations up to 10 4 heart beats furthermore we find that the histogram for the heartbeat intervals increments is well described by a l evy stable distribution for a group of subjects with severe heart disease we find that the distribution is unchanged but the long range correlations vanish therefore the different scaling behavior in health and disease must relate to the underlying dynamics of the heartbeat

generalized scale invariance in the atmosphere and fractal models of rain in recent years there has been considerable interest in stochastic rain models by developing new ideas about scale invariance and intermittency we argue that the scope of such models can be greatly extended the notions of scale invariance intermittency and the associated idea of fractal dimension have lately gained considerable ground particularly in the context of extremely variable phenomena such as those found in the mesoscale and in hydrodynamic turbulence we review some relevant work and argue that the atmosphere respects a symmetry principle that we call generalized scale invariance in which the statistical properties of the small and large scale are related to each other by a magnification coupled with a differential stratification due to gravity and differential rotation due to the coriolis force we further argue that the extremely erratic intermittent nature of the atmosphere is characterized by scale invariant fractal measures leading to hyperbolic fat tailed probability distributions of the fluctuations the standard statistical methods based on exponential fall offs in both correlations and probability distributions are inappropriate when the variability is of this type we show how both the scaling and intermittency may be exploited to develop extremely variable stochastic models of rain although the models examined here are the simplest of a family of extremely variable processes depending on only two radar determined parameters they lead to realizations possessing many realistic features of the rainfield including complexity of form clustering and bands at all scales as well as differential stratification and rotation finally we point to weaknesses in the models in particular their monodimensional nature and suggest possible improvements

a fractal based stochastic interpolation scheme in subsurface hydrology real porosity and hydraulic conductivity data do not vary smoothly over space so an interpolation scheme that preserves irregularity is desirable such a scheme based on the properties of fractional brownian motion fbm and fractional gaussian noise fgn is presented following the methodology of hewett 1986 the authors test for the presence of fgn in a set of 459 hydraulic conductivity k measurements the use of rescaled range analysis strongly indicated the presence of fgn when applied to the natural logs of the k data and the resulting hurst coefficient h was determined to be 0 82 this h value was then used along with the methodology for successive random additions to generate a fbm k interpolation realization in the vertical cross section between two wells the results appeared realistic and the overall methodology presented herein may serve as an improved basis for a conditional simulation approach to the study of various transport processes in porous media copyright c 1993 american geophysical union

on generating conductivity fields with known fractal dimension and nonstationary increments 1 fractional brownian motion fbm is a stochastic process that has stationary increments with long range correlations and known fractal dimension we study a multiple dimensional extension of fbm with nonstationary increments that allows for trends in the statistical structure while maintaining the gaussian nature and fractal dimension of fbm two methods for simulating this extension are employed and described in detail one approach combines cholesky decomposition with a generalization of random midpoint displacement the other makes repeated use of the cholesky decomposition the resulting fields can be employed in various geophysical settings e g as log conductivity fields in hydrology and topographic elevation in geomorphology

multiscaling analysis and random cascade modeling of dye infiltration 1 we aimed at investigating whether the spatial variability of infiltration in field soils as visualized through dye infiltration experiments is characterized by a multiscaling behavior digitized high resolution dye images from three sites in an experimental catchment in tunisia were analyzed using three indicators of scaling empirical probability distribution functions power spectra and raw statistical moments the two former indicators suggested a general scaling behavior of the data which through the moments analysis was found to be of multiscaling type random cascade processes are frequently used to model multiscaling processes and we fitted the universal multifractal um model of schertzer and lovejoy 1987 to our data the um model closely reproduced the empirical k q functions and simulated fields reproduced key features in the observed ones the results indicate that multiscaling random cascade modeling is useful for statistically describing flow processes and solute transport under field conditions

the matching theorems and coincidence theorems for generalized r kkm mapping in topological spaces abstract in this paper we present some new matching theorems with open cover and closed cover by using the generalized r kkm theorems l deng x xia generalized r kkm theorem in topological space and their applications j math anal appl 285 2003 679 690 in the topological spaces with property h as applications some coincidence theorems are established in topological spaces our results extend and generalize some known results

numerical estimates of hausdorff dimension numerical methods for estimating hausdorff dimension useful in the analysis of turbulence are explained and applied to a specific example in particular methods involving rescaling and approximation by cantor sets are discussed

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

revisiting rainfall clustering and intermittency across different climatic regimes 1 one of the vexing questions in rainfall research is the role of intermittency and its nonuniversal signature in anomalous scaling functions whether this lack of universal behavior is due to the bursting patterns in rainfall intensity or the alternation between long dry periods and highly clustered wet phases or both remains an open issue to progress on a narrower scope of this problem the effects of intermittency originating from rainfall occurrence are first separated from rainfall intermittency induced by intensity variability across five climatic regimes considered here it was shown that the rainfall occurrence process op exhibits 1 a near constant spectral slope 2 a near constant clustering exponent and 3 a probability density function of dry phases displaying a power law behavior with an exponent 1 5 consistent with other studies for timescales commensurate with frontal and storm systems also for the op the scaling exponents of the normalized higher order structure functions reveal an extensive monofractal scaling at all five climatic regimes when taken together these intersite results are suggestive that rainfall intensity modulations are the main cause of the nonuniversal anomalous scaling and not the clustering properties associated with the support the nature of these modulations is markedly different when comparing rainfall to a familiar and often interrelated process such as scalar turbulence in the case of turbulence amplitude variability of scalar dissipation rates appear to mitigate the intermittency effects connected with anomalous scaling while for rainfall series intensity fluctuations seem to amplify them

a simple scaling model for extreme rainfall the simple scaling hypothesis is applied to the intensity duration frequency idf description of rainfall it is shown that the cumulative distribution function for the annual maximum series of mean rainfall intensity has a simple scaling property over the range 30 min to 24 hours and in some instances to 48 hours this behavior is demonstrated through an examination of the scaling properties of the moments and the scaling of the parameters of an extreme value distribution fitted to the data a simple analytical formula for the idf relationship is proposed which embodies the scaling behavior once the scaling parameter has been obtained for a gauge or set of gauges in a region this formula enables the calculation of rainfall amounts of a chosen return period and duration shorter than a day directly from the information obtained from the analysis of daily data

possible sets of autocorrelations and the simplex algorithm the problem of imposing a set of correlations of any order on binary sequences is addressed the entropy of infinitely long sequences obeying such a given set was calculated in previous works using the saddle point method and it was observed that a finite fraction of sets are characterized by a non extensive entropy in this paper the region of finite entropy the allowed region of sets of correlations is found to be a convex hyper polygon in the space of correlation sets using the simplex algorithm outside of this region the simplex solution indicates that sequences obeying the correlations cannot be found therefore the entropy is in particular the boundaries of the allowed region for c1 cm are presented at the boundaries the entropy drops in a first order phase transition fashion and this drop can be explained from a combinatorial point of view finally we observe that the fraction of the volume occupied by allowed correlation sets drops exponentially with the number of correlations imposed and a qualitative explanation of this scaling phenomenon is provided

aquifer operator scaling and the effect on solute mixing and dispersion 1 since aquifer parameters may have statistical dependence structures that are present across a huge range of scales the concepts of fractional brownian motion fbm have been used in both analytic and numerical stochastic settings most previous models have used isotropic scaling characterized by a single scalar hurst coefficient any real world anisotropy has been handled by an elliptical stretching random k field we define a d dimensional extension of fbm in which the fractional order integration may take on different orders in the d primary possibly nonorthogonal scaling directions and the degree of connectivity and long range dependence is freely assigned via a probability measure on the unit sphere this approach accounts for the different scaling found in the vertical versus horizontal directions in sedimentary aquifers and allows very general degrees of continuity of k in certain directions it also allows for the representation of fracture networks in a continuum setting the eigenvectors of the scaling matrix describe the primary fracture scaling directions and discrete weights of fractional integration represent fracture continuity that may be limited to a small number of directions in a numerical experiment the motion of solutes through 2 d operator fractional gaussian fields depends very much on transverse hurst coefficients transverse scaling in the range of fractional gaussian noise engenders greater plume mixing and a transition to a fickian regime higher orders of integration in the range of fractional brownian motion are associated with thicker layering of aquifer sediments and more preferential unmixed transport therefore direct representation of the unique directional scaling properties of an aquifer is important for realistic simulation of transport

some results on r kkm mappings and r kkm selections and their applications first some intersection theorems based on r kkm mappings are obtained and then their application to minimax inequalities is given

on a generalized class of minimax inequalities abstract some results on a generalized class of minimax inequalities based on the r i g h kkm mapping theorems in a g h space setting are presented the r i g h kkm mappings represent a new class of kkm mappings in g h spaces as well as in the interval spaces

a spectral theory of rainfall intensity at the meso β scale the available empirical descriptions of extratropical cyclonic storms are employed to formulate a physically realistic stochastic representation of the ground level rainfall intensity field in space and time the stochastic representation is based on three component stochastic point processes which possess the general features of the embedding of rain cells within small mesoscale areas within large mesoscale areas within synoptic storms certain scale idealizations and assumptions on functional forms which qualitatively reflect the physical features lead to a closed form expression for the covariance function i e the real space time spectrum of the rainfall intensity field the theoretical spectrum explains the empirical spectral features observed by zawadzki almost a decade ago of particular interest and importance in this connection is an explanation of the empirical observation that the taylorian propogation of the fine scale structure via a transformation of time to space through the storm velocity holds only for a small time lag and not throughout the results here indicate the extent of this lag in terms of the characteristic scales associated with cell durations cellular birthrates and velocities etc

numerical ordering and symbolic arithmetic share frontal and parietal circuits in the right hemisphere a prominent proposal in numerical cognition states that our mental calculation abilities are grounded in the approximate number system ans recently it was proposed that this association is mediated by numerical ordering abilities as a first step in elucidating the neural correlates of this link this study tested which areas in the human brain carry information common to both calculation and numerical ordering while lying in an mr scanner 17 healthy participants a decided whether or not a given number triplet was presented in numerically ascending order and b solved simple addition and subtraction problems standard general linear model analyses revealed a largely overlapping network in fronto parietal regions for both tasks by analyzing the spatial information over voxels using a whole brain searchlight algorithm we identified a right hemispheric network comprising areas along the intraparietal sulcus and in the inferior frontal cortex which was similarly involved in order judgments and symbolic arithmetic functional and anatomical characteristics of this network make it a candidate for linking the ans to mental arithmetic

scale invariant fluctuations at different levels of organization in developing heart cell networks in the last few years it has been realized that the intervals of spontaneous spiking events in the intact heart exhibit coexisting scale invariant count fluctuations and anticorrelated interspike intervals fluctuations here we show experimentally that this feature is an intrinsic property of single isolated heart cells which is preserved when the cells couple into networks we present a model explaining this behavior at both the single cell and network levels

the mathematical structure of rainfall representations 2 a review of the theory of point processes part 2 constitutes the second of three parts devoted to the mathematical structure of rainfall an expository yet current treatment of those features of point process theory which are essential to descriptions of the hydrologic processes is developed a general overview of this three part series is given in part 1

the mathematical structure of rainfall representations 1 a review of the stochastic rainfall models this is the first of a three part series on the mathematical structure of rainfall models several important attempts at modeling rainfall are reviewed special attention is given to the mathematical structures that arise in the rainfall descriptions a general overview of the three part series is given as preface to this part

stochastic interpolation of aquifer properties using fractional levy motion this paper describes a new technique for conditional simulation of aquifer properties in unobserved regions the technique utilizes a recently introduced model for heterogeneity based on the levy stable family of probability distributions to achieve a higher degree of realism than is possible with gaussian based techniques specifically permeability and porosity variations are modeled using fractional levy motion a new algorithm for producing fractional levy motion conditioned to match known data is introduced two dimensional simulations using actual permeability measurements as input demonstrate that this algorithm produces random fields with the specified properties simulations using a subset of a detailed permeability map of an outcrop as input illustrate how the new method can be used to estimate important flow and transport properties and to quantify uncertainties in these estimates

time evolution of stochastic processes with correlations in the variance stability in power law tails of distributions abstract we model the time series of the s p500 index by a combined process the ar garch process where ar denotes the autoregressive process which we use to account for the short range correlations in the index changes and garch denotes the generalized autoregressive conditional heteroskedastic process which takes into account the long range correlations in the variance we study the ar garch process with an initial distribution of truncated levy form we find that this process generates a new probability distribution with a crossover from a levy stable power law to a power law with an exponent outside the levy range beyond the truncation cutoff we analyze the sum of n variables of the ar garch process and find that due to the correlations the ar garch process generates a probability distribution which exhibits stable behavior in the tails for a broad range of values n a feature which is observed in the probability distribution of the s p500 index we find that this power law stability depends on the characteristic scale in the correlations we also find that inclusion of short range correlations through the ar process is needed to obtain convergence to a limiting gaussian distribution for large n as observed in the data

distribution of natural trends in long term correlated records a scaling approach we estimate the exceedance probability w x ensuremath alpha l that in a long term correlated gaussian distributed sub record of length l characterized by a fluctuation exponent ensuremath alpha between 0 5 and 1 5 a relative increase ensuremath delta ensuremath sigma t of size larger than x occurs where ensuremath delta is the total observed increase measured by linear regression and ensuremath sigma t is the standard deviation around the regression line we consider l between 500 and 2000 which is the typical length scale of monthly local and reconstructed annual global temperature records we use scaling theory to obtain an analytical expression for w x ensuremath alpha l from this expression we can determine analytically for a given confidence probability q the boundaries ifmmode pm else textpm fi x q ensuremath alpha l of the confidence interval in the presence of an external linear trend the total observed increase is the sum of the natural and the external increase an observed relative increase ensuremath delta ensuremath sigma t is considered unnatural when it is above x q ensuremath alpha l in this case the size of the external relative increase is bounded by ensuremath delta ensuremath sigma t ifmmode pm else textpm fi x q ensuremath alpha l we apply this approach to various global and local climate data and discuss the different results for the significance of the observed trends

anisotropy lacunarity and upscaled conductivity and its autocovariance in multiscale random fields with truncated power variograms it has been shown by di federico and neuman 1997 1998a b that observed multiscale behaviors of subsurface fluid flow and transport variables can be explained within the context of a unified stochastic framework which views hydraulic conductivity as a random fractal characterized by a power variogram such a random field is statistically nonhomogeneous but possesses homogeneous spatial increments di federico and neuman 1997 have demonstrated that the power variogram and associated spectra of a statistically isotropic fractal field can be constructed as a weighted integral from zero to infinity an infinite hierarchy of exponential or gaussian variograms and spectra of mutually uncorrelated fields modes that are homogeneous and isotropic we show in this paper that the same holds true when the field and its constituent modes are statistically anisotropic provided the ratios between principal integral spatial correlation scales are the same for all modes we then analyze the effect of filtering out truncating modes of low high and intermediate spatial frequency from this infinite hierarchy in the real and spectral domains a low frequency cutoff renders the truncated hierarchy homogeneous the integral scales of the lowest and highest frequency cutoff modes are related to length scales of the sampling window domain and data support sample volume respectively taking the former to be proportional to the latter renders expressions for the integral scale and variance of the truncated field dependent on window and support scale in a manner previously shown to be consistent with observations in the isotropic case it also allows in principle bridging across scales at a specific locale as well as among locales by adopting either site specific or generalized variogram parameters the introduction of intermediate cutoffs allows us to account in a straightforward manner for lacunarity due to gaps in the multiscale hierarchy created by the absence of modes associated with discrete ranges of scales for example where textural and structural features are associated with distinct ranges of scale such as fractures having discrete ranges of trace length and density which dissect the rock into matrix blocks having corresponding ranges of sizes we explore mathematically and graphically the effects that anisotropy and lacunarity have on the integral scale variance covariance and spectra of a truncated fractal field we then develop an expression for the equivalent hydraulic conductivity of a box shaped porous block embedded within a multiscale log hydraulic conductivity field under mean uniform flow the block is larger than the support scale of the field but is smaller than a surrounding sampling window consequently its equivalent hydraulic conductivity is a random variable whose variance and spatial autocorrelation function conditioned on a known mean value of support scale conductivity across the window are given explicitly by our multiscale theory

evolution towards ergodic behavior of stationary fractal random processes with memory application to the study of long range correlations of nucleotide sequences in dna abstract the possible occurrence of ergodic behavior for large times is investigated in the case of stationary random processes with memory it is shown that for finite times the time average of a state function is generally a random variable and thus two types of cumulants can be introduced for the time average and for the statistical ensemble respectively in the limit of infinite time a transition from the random to the deterministic behavior of the time average may occur resulting in an ergodic behavior the conditions of occurrence of this transition are investigated by analyzing the scaling behavior of the cumulants of the time average a general approach for the computation of these cumulants is developed explicit computations are presented both for short and long memory in the particular case of separable stationary processes for which the cumulants of a statistical ensemble can be factorized into products of functions depending on binary time differences in both cases the ergodic behavior emerges for large times provided that the cumulants of a statistical ensemble decrease to zero as the time differences increase to infinity the analysis leads to the surprising conclusion that the scaling behavior of the cumulants of the time average is relatively insensitive to the type of memory considered both for short and long memory the cumulants of the time average obey inverse different from zero for large time differences then the time averaage is random even as the length of the total time interval tends to infinity and the ergodic behavior no longer holds the theory is applied to the study of long range correlations of nucleotide sequences in dna in this case the length t of a sequence of nucleotides plays the role of the time variable a proportionality relationship is established between the cumulants of the pyrimidine excess in a sequence of length t and the cumulants of the time length average of the probability of occurrence of a pyrimidine it is shown that the statistical analysis of the dna data presented in the literature is consistent with the occurrence of the ergodic behavior for large lengths the implications of the approach to the analysis of the large time behavior of stochastic cellular automata and of fractional brownian motion are also investigated

multifractal analyses of hydraulic conductivity distributions the concept of a universal multifractal a generalization of a monofractal is a recently developed scaling model for natural phenomena characterized by irregularity presented herein is an effort to use universal multifractal concepts to deal with spatial variations of hydraulic conductivity k which have a significant effect on contaminant transport in the subsurface structure function analyses of four k data sets show that some vertical variations of k display multifractal structures while others are consistent with monofractal behavior in order to make multifractal concepts more useful multifractal noise is introduced and defined as the increments of a multifractal it is concluded that the multifractal formalism of schertzer and lovejoy 1987 has provided a rather general approach for modeling in k variations in the vertical with the exception of horizontal variations of the borden data all results fell within the domain of universal multifractal behavior which includes the monofractal case parameters were well defined in an empirical sense and easy to calculate indicating a robust formalism results were consistent with the recent finding of liu and molz 1997 also non gaussian and scale variant behavior in hydraulic conductivity distributions submitted to water resources research 1997 hereinafter referred to as submitted paper that k variations display increasing heterogeneity at decreasing scales

irreversible processes the generalized affinities within tsallis statistics abstract within the generalized statistical mechanics introduced recently by tsallis the generalized form of the affinities the quantities which drive a process in the theory of irreversible thermodynamics was derived the affinities were obtained by considering changes in the generalized entropy s q aub of a system composed by two subsystems a and b the non extensive character of the tsallis entropy is taken into account at the end the equilibrium condition is discussed and the zeroth law in the framework of the generalized thermodynamics is consistently recovered as in the usual case two systems which are in equilibrium with a third one are necessarily in equilibrium among them and share the same temperature

non power law scale properties of rainfall in space and time 1 the identification of general relationships linking statistical properties of rainfall aggregated at different temporal and spatial scales possesses clear theoretical and practical relevance among other properties it is important to characterize the scale dependence of rainfall variability which may for many purposes be summarized by its variance this paper presents theoretical results and observational analyses connecting the variance of temporal and spatial rainfall to the scale of observation under an assumption of second order stationarity previous results regarding the theoretical form of the temporal variance as a function of aggregation are refined and extended to the spatial case it is shown that the variance of aggregated rainfall exhibits an overall nonscaling form within the range of scales of usual hydrological interest it is further shown that the theoretically derived form of the variance as a function of scale is incompatible with power law relations usually assumed for the second order statistical moment within multiscaling approaches to rainfall modeling predictions of the theoretical derivations are validated by use of observations representing a wide variety of resolutions from 2 s to 1 hour in time and 4 km in space climates and measurement instruments the validation shows a good agreement between observations and the theoretically predicted forms of the variance as a function of aggregation the proposed theoretical framework thus provides a useful context for rainfall analysis modeling and downscaling in space and time and suggests careful reexaminations of usual multiscaling assumptions

scaling features of texts images and time series in the given paper we consider the scaling features of long letter sequences like human writings discretized images and discretized 3nancial data using several approaches we show that the symbolic strings and time series being analyzed have a complex multiscale structure and demonstrate di5erent scalings for large and small 6uctuations we discuss complex phenomena in the scaling behavior of partition functions in the case of high frequency dax future data c 2001 elsevier science b v all rights reserved

scaling limits and self similarity in precipitation fields it has been recognized in the literature on rainfall models in hydrology that the exact calculations of probabilities for models which incorporate cluster structure is typically not possible consequently one is led to considerations of limiting approximations however because of the presence of dependencies and or fat tails in several of the current models the nature of the approximations involves some important issues for both theoretical as well as experimental consideration in precipitation research the objective of the present paper is to systematically describe these issues as they pertain to rainfall models some specific large scale approximations are given for models of the type currently in the literature

evidence for non gaussian scaling behavior in heterogeneous sedimentary formations vertical and horizontal fluctuations in permeability and porosity in sedimentary formations are analyzed and are found to be consistent with scaling models based on levy stable probability distributions the approach avoids the assumption of gaussian behavior and is supported by evidence from horizontal and vertical well logs and from permeability measurements on a sandstone outcrop and segments of core from a heterogeneous formation the incremental values in these measurement sequences are accurately modeled as having levy stable distributions the width of the distribution of increments depends on the spatial scale in a manner consistent with scaling behavior the width of the distribution is smaller for horizontal increments than for vertical increments reflecting the reduced variability in the horizontal direction the scaling parameters are in the range associated with antipersistence and are roughly the same magnitude in the vertical and horizontal directions the relationships between different physical properties are briefly studied and it is suggested that they be quantified through off diagonal terms in a multivariate levy width matrix simulations designed to reproduce the observed statistical features are also described these results have some fundamental implications as levy stable distributions require a different set of statistical tools and theoretical methods compared to finite variance distributions

complex fractal dimension of the bronchial tree the architecture of the mammalian lung has been shown to be correctly described using a fractal model with a complex dimension related to a cantor set with random errors for four different species here we provide an interpretation of that model that has implications for biological evolution we argue that fractals are more error tolerant than other structures and therefore have an evolutionary advantage

effects of problem size and arithmetic operation on brain activation during calculation in children with varying levels of arithmetical fluency abstract most studies on mathematics learning in the field of educational neuroscience have focused on the neural correlates of very elementary numerical processing skills in children little is known about more complex mathematical skills that are formally taught in school such as arithmetic using functional magnetic resonance imaging the present study investigated how brain activation during single digit addition and subtraction is modulated by problem size and arithmetic operation in 28 children aged 10 12 years with different levels of arithmetical fluency commensurate with adult data large problems and subtractions activated a fronto parietal network including the intraparietal sulci the latter of which indicates the influence of quantity based processes during procedural strategy execution different from adults the present findings revealed that particularly the left hippocampus was active during the solution of those problems that are expected to be solved by means of fact retrieval i e small problems and addition suggesting a specific role of the hippocampus in the early stages of learning arithmetic facts children with low levels of arithmetical fluency showed higher activation in the right intraparietal sulcus during the solution of problems with a relatively small problem size indicating that they continued to rely to a greater extent on quantity based strategies on those problems that the children with relatively higher arithmetical fluency already retrieved from memory this might represent a neural correlate of fact retrieval impairments in children with mathematical difficulties

flow through porous media with multifractal hydraulic conductivity 1 we make a nonlinear analysis of flow through saturated porous media when the hydraulic conductivity k is an isotropic lognormal field with multifractal scale invariance in this case logk is isotropic gaussian with spectral density where d is the space dimension our main result is that the hydraulic gradient h and specific flow q are also multifractal fields whose renormalization under space contraction involves random rotation of the field and random scaling of its amplitude the scaling properties and marginal distributions of h and q are obtained analytically as functions of the space dimension d and a multifractal parameter of k the codimension ck the fields h and q are anisotropic at large scales but approach isotropy at very small scales using scaling arguments we obtain the effective conductivity of the medium keff as an explicit function of d ck and the scaling range of k

a general fractal model of flow and solute transport in randomly heterogeneous porous media 1 one of the challenges of hydrology is modeling flow and solute transport through media that are not uniformly porous this paper proposes a general fractal model of flow and solute transport in randomly heterogeneous porous media we describe the random field using fractional levy motion flm which more effectively represents a field with a high degree of variability understanding flow and solute transport in the flm field expands the applicability of stochastic theory following the work of v di federico and s p neuman 1997 and v di federico et al 1999 we derive truncated isotropic and anisotropic power variograms for the flm fields the results are analogous to but more general than those of fractional brownian motion fbm v di federico and s p neuman 1997 and v di federico et al 1999 for a random field when the levy index in flm is replaced by a constant of 2 fbm is recovered this study employs first order perturbation analyses of flow and solute transport in two dimensional flm fields for the stationary fbm and flm fields macrodispersion coefficients are compared the plume is found to grow slowest in the flm field but will show highest concentration the proposed fractal model is general in the sense that when the levy index of flm equals 2 the results of flow and solute transport for the commonly used fbm model are recovered when the upper cutoff of the truncated power variogram is close to the lower cutoff the results for the stationary exponential or gaussian model can be well approximated since the proposed general fractal model has broader applications than the stationary and fbm models it simulates flow and solute transport in more different scenarios to yield more accurate modeling results