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

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

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

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

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

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

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

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

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

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

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

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

error estimation and adaptivity in elastodynamics abstract the paper discusses adaptive procedures for elastodynamic problems using semi discrete finite elements the adaptivity in space may be done by mesh refinement h version by increase of order of the approximation polynomials p version or a combination of these hp version in time the integration is either made by mode superposition of characteristic functions or by direct integration the adaptation in time may either be made by change of the global time step h version or increase of the order of approximation polynomials p version crucial parts of the analysis are the error estimator and the error indicator which are based either on interpolation theory or on total energy obtained from postprocessed stresses for the mesh generation and regeneration an isoline technique is utilized by which the mesh can be smoothed by interpolation theory the isolines are created for some characteristic scalar function for time discretization a new simple a posteriori local error estimator for the newmark scheme is described it is derived by a post processing technique without solving new equations so the additional computational cost is small and the implementation is convenient the paper also describes a sequence of nested time integration methods for dynamic problems formulated as two first order equations in time these nested integration methods in time based on a hierarchical formulation in time are a stable schemes of order 2k and l stable schemes of order 2k 1 a considerable improvement in terms of accuracy as well as effectiveness is obtained compared to currently available methods

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

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

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

continuation of eigenvalues and invariant pairs for parameterized nonlinear eigenvalue problems invariant pairs have been proposed as a numerically robust means to represent and compute several eigenvalues along with the corresponding generalized eigenvectors for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter in this work we consider nonlinear eigenvalue problems that depend on an additional parameter and our interest is to track several eigenvalues as this parameter varies based on the concept of invariant pairs a theoretically sound and reliable numerical continuation procedure is developed particular attention is paid to the situation when the procedure approaches a singularity that is when eigenvalues included in the invariant pair collide with other eigenvalues for the real generic case it is proven that such a singularity only occurs when two eigenvalues collide on the real axis it is shown how this situation can be handled numerically by an appropriate expansion of the invariant pair the viability of our continuation procedure is illustrated by a numerical example

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 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

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