on unifying the concepts of scale instrumentation and stochastics in the development of multiphase transport theory a generalized theory of multiphase transport is presented which combines the concepts of scale instrumentation stochastics and time series with the development of transport equations by defining the filtering process as a convolution of a measure p with a field property we are able to exploit the fourier transform to place plausible restrictions on an instrument in frequency space so as to make its measurement relevant to its physical environment an ideal instrument is defined which filters out high frequency noise corresponding to short distances and yet does not alter the structure of low frequencies using ideal instruments we successively filter out lower frequency noise in a multiscale multiphase environment formulas are developed to relate the autocorrelation of a field property on one scale of motion to that on any other scale while taking into account the types of instruments used in the measuring process an equation relating the integral scale on one scale of motion to the integral scale on any other scale of motion is developed power spectra are developed which relate spectra on different scales to the measuring instrument used by successively applying filtering theorems a hierarchy of multiscale transport equations is developed filtered properties in the transport equations are mass averages different properties are allowed to be measured by different instruments and different instruments are allowed on different scales of motion and for different phases the concept of a wide sense stationary ergodic process is introduced to develop mass average autocorrelations and spectra over scales of motion as a function of measuring devices

numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media a numerical procedure for the determination of equivalent grid block permeability tensors for heterogeneous porous media is presented the method entails solution of the fine scale pressure equation subject to periodic boundary conditions to yield upon appropriate averaging of the fine scale velocity field the coarse scale or equivalent grid block permeability when the region over which this coarse scale permeability is computed constitutes a representative elementary volume rev the resulting equivalent permeability may be interpreted as the effective permeability of the region solution of the pressure equation on the fine scale is accomplished through the application of an accurate triangle based finite element numerical procedure which allows for the modeling of geometrically complex features the specification of periodic boundary conditions is shown to yield symmetric positive definite equivalent permeability tensors in all cases the method is verified through application to a periodic model problem and is then applied to the scale up of areal and cross sections with fractally generated permeability fields the applicability and limitations of the method for these more general heterogeneity fields are discussed

ensembles of fuzzy linear model trees for the identification of multioutput systems we address the task of discrete time modeling of nonlinear dynamic systems with multiple outputs using measured data in the area of control engineering this task is typically converted into a set of classical regression problems one for each output which can then be solved with any nonlinear regression approach fuzzy models in the takagi sugeno form are popular in this context we use lolimot which is a tree learning method to build fuzzy linear model trees in this paper we propose implement and empirically evaluate three extensions of fuzzy linear model trees first we consider and evaluate multioutput models second we propose to use ensembles of such models third we investigate the use of a search heuristic based on simulation error as opposed to one step ahead prediction error specific to the context of modeling dynamic systems finally we perform an empirical evaluation and compare these approaches on six multioutput case studies using both measured and simulated data with noise the case studies include modeling of the inverse dynamics of a robot arm as well as five additional process industry systems ensembles improve the performance of both single and multioutput trees while the heuristic specific to modeling dynamic systems only improves performance very slightly multioutput model trees exhibit comparable or worse predictive performance to a set of single output models while providing a more compact model overall we can recommend the use of bagging of single output lolimot models learned by using the simulation error as a search heuristic

multi scale modelling of flow in periodic solid structures through spatial averaging this paper presents spatially averaged navier stokes equations for modelling macro scale flow in devices with periodic solid structures such as fin and tube arrays the properties of steady and unsteady periodically developed flow are investigated to assess different strategies for determining the closure terms in the macro scale flow equations it is shown that the spatial averaging technique requires an appropriate weighting function to ensure that the closure terms are spatially constant for periodically developed flow moreover through an appropriate choice of the weighting function the closure terms can be obtained by solving a local closure problem on a unit cell of the periodic structures the theoretical framework of this paper is applied as multi scale modelling technique for flow through a cylindrical tube array this case study illustrates the advantages of the weighted spatial averaging technique over the volume averaging technique

on measurement scale and scaling the relationships between measurement scale and scaling are discussed and analyzed in detail it is pointed out that only certain fundamental properties can be measured and that these properties do not include constitutive variables the interrelationships between scale and measurement are shown to manifest themselves in two types of heterogeneity which we label as explicit and implicit an important issue that is addressed for the first time is the necessity of scaling instruments when scaling laboratory experiments up to field scale or vice versa explicit constraints that scaled instruments must satisfy are derived for the frequency response of the instrument it is shown that linear interpolators can be viewed as filters instruments but that these interpolators don t preserve the scale constraints of the instrument used in making the field measurements the relationship between signal conditioning and the instrument s window is isolated and shown to be a multiple convolution over the spatial coordinates corresponding to the window followed by a temporal convolution for the signal conditioning

an evaluation of mathematical models of the transport of biologically reacting solutes in saturated soils and aquifers three different conceptual frameworks have been adopted in the past for the development of mathematical models of bacterial growth and biologically reacting solute transport in saturated porous media two schools of thought are based upon assuming that the pore scale geometrical configuration of the attached bacteria consists of biofilms or microcolonies the third school of thought represents the traditional approach where pore scale processes are neglected and the bacteria are assumed to respond to the macroscopic bulk fluid substrate concentration on the basis of a schematic block diagram representation of a saturated porous medium hosting a microbial population it is shown that these frameworks share a common theoretical foundation and that they differ only by the choice of particular constitutive equations for several transfer parameters using one possible option in this respect we derive a mathematical model that involves no unwarranted assumption about the distribution of the microorganisms in the pore space the governing equations of this latter model are shown to be formally identical to those obtained by f j molz et al 1986 using the concept of microcolony and to those that would result from adopting a simple form of biofilm model to describe bacterial growth in the pore space some of the consequences of this formal similarity between macroscopic transport equations obtained in different conceptual frameworks are discussed from an operational standpoint and in terms of model validation

volume averaging for determining the effective dispersion tensor closure using periodic unit cells and comparison with ensemble averaging in this work we use the method of volume averaging to determine the effective dispersion tensor for a heterogeneous porous medium closure for the averaged equation is obtained by solution of a concentration deviation equation over a periodic unit cell our purpose is to show how the method of volume averaging with closure can be rectified with the results obtained by other upscaling methods under particular conditions although this rectification is something that is generally believed to be true there has been very little research that explores this issue explicitly we show that under certain limiting but mild assumptions the closure problem provides a fourier series solution for the effective dispersion tensor when second order spatial stationarity is imposed on the velocity field the method yields a simple fourier series that converges to an integral form in the limit as the period of the unit cell approaches infinity this limiting result is identical to the quasi fickian forms that have been developed previously via ensemble averaging by deng et al 1993 and recently by fiori and dagan 2000 except in the definition of the averaging operation as a second objective we have conducted a numerical study to evaluate the influence of the size of the averaging volume on the effective dispersion tensor and its volume averaged statistics this second objective is complimentary in many ways to recent research reported by rubin et al 1999 via ensemble averaging and by wang and kitanidis 1999 via volume averaging on the block averaged effective dispersion tensor the variability of the effective dispersion tensor from realization to realization is assessed by computing the volume averaged effective dispersion tensor for an ensemble of finite fields with the same ensemble statistics ensembles were generated using three different sizes of unit cells all three unit cell sizes yield similar results for the value of the mean effective dispersion tensor however the coefficient of variation depends strongly upon the size of the unit cell and our results are consistent with those developed by fiori 1998 from the ensemble averaging perspective this implies that in applications the actual value of the effective dispersion tensor may be significantly different than expected on the basis of unconditioned hydraulic conductivity statistics and this variation should be considered when applying macrodispersion to real world systems

effective permeability and dispersion in locally heterogeneous aquifers the global effective permeability and hydrodynamic dispersion tensors due to steady flow in locally heterogeneous and statistically anisitropic aquifers are analysed using the self consistent approximation for small perturbations the results for the statistically anisotropic case are shown to be consistent with those derived using stochastic continuum theory however for larger perturbations the stochastic continuum theory has not yet produced rigorous results that can be compared with those in this paper

steady state groundwater flow simulation with imprecise parameters a methodology based on fuzzy set theory is developed to incorporate imprecise parameters into steady state groundwater flow models in this case fuzzy numbers are used to represent parameter imprecision as such they are also used as a measure for the uncertainty associated with the hydraulic heads due to the imprecision in the input parameters the imprecise input parameters may come from indirect measurements subjective interpretation and expert judgment of available information in the methodology a finite difference method is combined with level set operations to formulate the fuzzy groundwater flow model this fuzzy modeling technique can handle imprecise parameters in a direct way without generating a large number of realizations two numerical solution methods are used to solve the fuzzy groundwater flow model the groundwater model operator method proposed in this methodology and the iterative algorithm based on conventional interval arithmetics the iterative method is simple but may overestimate the uncertainty of hydraulic heads the groundwater model operator method not only provides the hull of the solution set for the hydraulic heads but also considers the dependence of hydraulic head coefficients which are functions of imprecise parameters sensitivity analysis shows that the dependence of hydraulic head coefficients has a critical impact on the model results and neglecting this dependence may result in significant overestimation of the uncertainty of hydraulic heads a numerical model based on the methodology is tested by comparing it with the analytical solution for a homogeneous radial flow problem it is also applied to a simplified two dimensional heterogeneous flow case to demonstrate the methodology

estimation of spatial covariance structures by adjoint state maximum likelihood cross validation 1 theory this series of three papers describes a cross validation method to estimate the spatial covariance structure of intrinsic or nonintrinsic random functions from point or spatially averaged data that may be corrupted by noise any number of relevant parameters including nugget effect can be estimated the theory described in this paper is based on a maximum likelihood approach which treats the cross validation errors as gaussian various a posteriori statistical tests are used to verify this hypothesis and to show that in many cases correlation between these errors is weak the log likelihood criterion is optimized through a combination of conjugate gradient algorithms an adjoint state theory is used to efficiently calculate the gradient of the estimation criterion optimize the step size downgradient and compute a lower bound for the covariance matrix of the estimation errors issues related to the identifiability stability and uniqueness of the resulting adjoint state maximum likelihood cross validation asmlcv method are discussed this paper also describes the manner in which asmlcv allows one to use model structure identification criteria to select the best covariance model among a given set of alternatives practical aspects of asmlcv and its application to synthetic data are presented in paper 2 samper and neuman this issue a applications to real hydrogeological data transmissivities and groundwater levels have been presented elsewhere while hydrochemical and isotopic data are analyzed by asmlcv in paper 3 samper and neuman this issue b

nonlocal reactive transport with physical and chemical heterogeneity localization errors the origin of nonlocality in macroscale models for subsurface chemical transport is illustrated it is argued that media that are either nonperiodic e g media with evolving heterogeneity or periodic viewed on a scale wherein a unit cell is discernible must display some nonlocality in the mean a metaphysical argument suggests that owing to the scarcity of information on natural scales of heterogeneity and on scales of observation associated with an instrument window constitutive theories for the mean concentration should at the outset of any modeling effort always be considered nonlocal the intuitive appeal to nonlocality is reinforced with an analytical derivation of the constitutive theory for a conservative tracer without appeal to any mathematical approximations deng et al 1993 present a first order nonlocal eulerian theory for transport of a conservative solute in an infinite nondeforming domain under steady flow conditions hu et al this issue extended these results to account for nonequilibrium linear sorption with random partition coefficient kd but deterministic constant reaction rate kr these theories are localized herein and comparisons are made between the fully nonlocal fnl nonlocal in time nlt and fully localized fl theories for conservative transport there is little difference between the first order fl and fnl models for spatial moments up to and including the third however for conservative transport the first order nlt model differs significantly from the fnl model in the third spatial moments for reactive transport all spatial moments differ between the fnl and fl models the second transverse horizontal and third longitudinal horizontal moments for the nlt model differ from the fnl model these results suggest that localized first order transport models for conservative tracers are reasonable if only lower order moments are desired however when the chemical reacts with its environment the localization approximation can lead to significant error in all moments and a fnl model will in general be required for accurate simulation

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

an evaluation of boundary conditions for one dimensional solute transport 1 mathematical development the laplace transform method is employed to obtain the solution to several boundary value problems in which mixing occurs in reservoirs attached to a porous medium flow is assumed to be uniformly one dimensional throughout the porous medium solutions are obtained for both continuous and discontinuous concentrations at the reservoir medium boundary and for resident and flux averaged concentrations a mass balance conducted on each solution shows that neither continuity condition can be proved generally superior an evaluation of the parametric sensitivity of the solutions is also conducted and shows that the distinction between the continuity conditions and the flux and resident concentrations is only important at small peclet numbers the only exception to this is the distinction between flux averaged versus resident continuous concentrations in a finite solution domain in this case the flux concentrations were found to be insensitive to the volume of the downstream reservoir and consequently the flux and resident concentrations can differ substantially for larger reservoirs in consideration of the general behavior of the solutions three conceptual problems were identified which must be investigated by physical modeling 1 determination of the appropriate solution at small peclet number 2 the possibility that dispersive mass flux occurs from the upstream reservoir under conditions of continuous concentration and 3 the use of the flux transformation for continuous concentration in a finite solution domain the physical modeling is discussed in a companion paper

modeling the large scale dynamics of saturated groundwater flow using spatial filtering theory 1 theoretical development we develop a model and a framework for understanding the explicitly resolved large scale dynamics of groundwater flow in fully saturated heterogeneous porous media the approach is based upon the large eddy simulation les methodology from geophysical fluid dynamics the idea behind the les approach is to explicitly resolve the larger scales of flow and provide a closure model for the effect of subgrid scales upon resolved scales the technical foundation of the method rests on a spatial filtering operation which allows one to precisely define the scale of the model we spatially filter darcy s law and the continuity equation to produce a model for the resolved scale dynamics we develop les approximations for the resulting flow equations and a closure model for unresolved subgrid scale terms we model the subgrid terms using resolved scale quantities only and thus do not require subgrid scale information the accuracy of these closure models depends upon the scale at which hydraulic conductivity is measured by field tests we characterize the approximations used to model subgrid scale effects asymptotically with a small parameter the dimensionless filter width defined as the ratio of the filter width to the dominant length scale of the explicitly resolved variables

mechanics of heterogeneous porous media with several spatial scales by extending the theory of homogenization we consider a heterogeneous porous medium whose material structure is characterized by multiple periodicity over several disparate length scales because for geological and engineering applications the driving force e g the pressure gradient and the global motion of primary interest is usually at a scale much larger than the largest of structural periodicity we derive the phenomenological equations of motion by using the perturbation technique of multiple scales all the effective coefficients are defined by boundary value problems on unit cells in smaller scales and no constitutive assumptions are added aside from the basic equations governing the mechanics of the pore fluid and the solid matrix in part i the porous matrix is assumed to be rigid the case of three scales is treated first symmetry and positiveness of the effective darcy permeability tensor are proven extensions to four and more scales are then discussed in part ii we allow the solid matrix to be deformable and deduce the equations of consolidation of a two phase medium the coupling of fluid flow and the quasi static elastic deformation of the matrix is considered when there are three disparate length scales effective coefficients of various kinds are deduced in terms of cell problems on the scale of the pores several general symmetry relations as well as specific properties of a medium composed of layers of porous matrix are discussed

stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers 2 impervious boundary the spatial variability of the head h in a two dimensional flow in the vicinity of an impervious boundary is investigated this study is a continuation of a previous one rubin and dagan 1988 which dealt with the influence of a given head boundary by representing h and the log transmissivity y as space random function and by solving the flow equation first order closed form analytical expressions for the head variogram and the head log transmissivity y covariance are developed these expressions offer the means to evaluate the influence of the boundary on the joint y h random field the assumption of unbounded domain is examined and it is found to be a good approximation at distances larger than three log transmissivity integral scales from the boundary by using conditional probabilities a simple method to estimate the statistical moments close to the boundary is presented it requires the knowledge of the moments of y and h in the unbounded domain only the method is compared with the exact analytical solution showing favorable results

versal deformations of elements of classical jordan algebras every involution defining by x x the elements of one of the classical lie algebras o n c and sp 2n c over c and o p q sp 2n r u p q o 2n and sp 2p 2q over r also defines symmetric matrices y y spanning the classical jordan algebras the paper contains an explicit description of all perturbations of symmetric matrices considered up to equivalence under the action of the corresponding classical lie group such exhaustive perturbations are called versal deformations

on the optimization of fuzzy decision trees the induction of fuzzy decision trees is an important way of acquiring imprecise knowledge automatically fuzzy id3 and its variants are popular and efficient methods of making fuzzy decision trees from a group of training examples this paper points out the inherent defect of the likes of fuzzy id3 presents two optimization principles of fuzzy decision trees proves that the algorithm complexity of constructing a kind of minimum fuzzy decision tree is np hard and gives a new algorithm which is applied to three practical problems the experimental results show that with regard to the size of trees and the classification accuracy for unknown cases the new algorithm is superior to the likes of fuzzy id3

a cloned linguistic decision tree controller for real time path planning in hostile environments abstract the idea of a cloned controller to approximate optimised control algorithms in a real time environment is introduced a cloned controller is demonstrated using linguistic decision trees ldts to clone a model predictive controller mpc based on mixed integer linear programming milp for unmanned aerial vehicle uav path planning through a hostile environment modifications to the ldt algorithm are proposed to account for attributes with circular domains such as bearings and discontinuous output functions the cloned controller is shown to produce near optimal paths whilst significantly reducing the decision period further investigation shows that the cloned controller generalises to the multi obstacle case although this can lead to situations far outside of the training dataset and consequently result in decisions with a high level of uncertainty a modification to the algorithm to improve the performance in regions of high uncertainty is proposed and shown to further enhance generalisation the resulting controller combines the high performance of mpc milp with the rapid response of an ldt while providing a degree of transparency interpretability of the decision making

boundary element solution for stochastic groundwater flow random boundary condition and recharge this paper presents an integral equation technique for solving stochastic boundary value problems in groundwater flow the aquifer considered has deterministic hydraulic conductivity but is subject to random boundary condition and domain recharge using the distribution of fictitious sources and dipoles stochastic integral equations for the mean and covariance of head and flux are derived an iterative boundary element technique is applied for numerical solution two one dimensional examples are examined and compared with exact solution a two dimensional problem is then presented

large scale averaging analysis of single phase flow in fractured reservoirs the method of large scale averaging is introduced to derive and analyze the most general form of the dual porosity model of single phase flow in naturally fractured reservoirs the dual porosity model contains the usual equations based on darcy s law and the coupling terms representing the fluid transfer between the matrix and the fractures a transient closure problem is developed to obtain and analyze the fracture and matrix permeability tensors and the fluid transfer terms the spatial averaging theorem is presented from a mathematical point of view and proved rigorously by the distribution theory the problem of wellposedness of the dual porosity model is also considered the techniques developed here are not restricted to either regular geometric fractures or spatially periodic reservoirs

stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers 1 constant head boundary the spatial variability of log transmissivity y and head h in aquifers assumes a major role in determining the solutions of the direct flow problem as well as of the inverse problem investigation of the joint y h variability has been cast in the past in numerical and analytic frames analytic solutions for the y h statistical moments in two dimensions assumed the flow to occur in an unbounded domain in this study exact analytical expressions for the y h moments are derived which account for the presence of a given head boundary in a half plane this result is achieved by a first order approximation and for an exponential y covariance the result offers the opportunity to examine the effects of boundaries on the semivariogram of h and on the y h cross covariance the assumption of unbounded domain is examined and shown to be accurate at distances larger than two y integral scales from the boundary by using conditional probabilities a general and simple method to estimate the moments close to the boundary is presented it requires the knowledge of the moments of y and h in the unbounded domain only the method is compared with the exact analytic solution showing excellent results

simplest miniversal deformations of matrices matrix pencils and contragredient matrix pencils abstract for a family of linear operators a u u over c that smoothly depend on parameters 1 k v i arnold obtained the simplest normal form of their matrices relative to a smoothly depending on change of a basis in u we solve the same problem for a family of linear operators a u u over r for a family of pairs of linear mappings a u v b u v over c and r and for a family of pairs of counter linear mappings a u v b v u over c and r

stochastic analysis of three dimensional flow in a bounded domain a commonly accepted first order approximation of the equation for steady state flow in a fully saturated spatially random medium has the form of poisson s equation this form allows for the advantageous use of green s functions to solve for the random output hydraulic heads in terms of a convolution over the random input the logarithm of hydraulic conductivity a solution for steady state three dimensional flow in an aquifer bounded above and below is presented consideration of these boundaries is made possible by use of green s functions to solve poisson s equation within the bounded domain the medium hydraulic conductivity is assumed to be a second order stationary random process as represented by a simple three dimensional covariance function upper and lower boundaries are taken to be no flow boundaries the mean flow vector lies entirely in the horizontal dimensions the resulting hydraulic head covariance function exhibits nonstationary effects resulting from the imposition of boundary conditions comparisons are made with existing infinite domain solutions

disjunctive normal random forests we develop a novel supervised learning classification method called disjunctive normal random forest dnrf a dnrf is an ensemble of randomly trained disjunctive normal decision trees dndt to construct a dndt we formulate each decision tree in the random forest as a disjunction of rules which are conjunctions of boolean functions we then approximate this disjunction of conjunctions with a differentiable function and approach the learning process as a risk minimization problem that incorporates the classification error into a single global objective function the minimization problem is solved using gradient descent dnrfs are able to learn complex decision boundaries and achieve low generalization error we present experimental results demonstrating the improved performance of dndts and dnrfs over conventional decision trees and random forests we also show the superior performance of dnrfs over state of the art classification methods on benchmark datasets

effective groundwater model parameter values influence of spatial variability of hydraulic conductivity leakance and recharge a stochastic simulation approach was used to inspect the influence of spatial variability in aquifer and recharge properties on effective averaged groundwater model parameters a two dimensional unconfined aquifer model was set up for an area similar to that near livermore california monte carlo simulations were generated for different sets of spatial correlation structures assuming stationarity and an exponential semivariogram for each monte carlo realization groundwater flow was simulated and estimates of the means and standard deviations of the hydraulic head were obtained nine different cases were studied involving different correlation lengths of hydraulic conductivity conditional simulations parameter zonation and spatial recharge distributions results indicate that there is no single uniform value of the hydraulic conductivity that reproduces the expected head values spatial averaging to obtain effective values was based on the p norm power norm which ranges from 1 for the arithmetic mean to 1 for the harmonic mean with 0 representing the geometric mean for the unconditional case the 0 4 p norm seems to best reproduce the expected values the 0 2 p norm gave the best result for the conditional simulation case if the geometric mean were used instead the heads would deviate from the expected heads by about 2 m near the wells some value between the arithmetic and the geometric means of riverbed leakance will give results close to the expected head values the arithmetic mean of aerially distributed recharge produced results with small deviations from the expected head values these particular results may only apply to this modeled system as the pumping pattern and magnitudes exhibited strong influences on zones of maximum head deviation the implication of this work is that if an effective hydraulic conductivity is used in a simulation model it must be selected for a particular set of wells and pumping rates in an aquifer with significant pumping centers the best effective value for a two dimensional unconfined flow model is most likely between the geometric and harmonic mean if the wells are turned off the effective hydraulic conductivity for that flow model reverts to the geometric mean

nonlocal properties of nonuniform averaged flows in heterogeneous media the properties of nonuniform average potential flows in media of stationary random conductivity are studied the mathematical model of average flow is derived as a system of governing equations to be satisfied by mean velocity and mean head the averaged darcy s law determines the effective conductivity as an integral operator of the convolution type relating the mean velocity to the mean head gradient in a nonlocal way in fourier domain the mean velocity is proportional to the mean head gradient the coefficient of proportionality is referred to as the effective conductivity tensor and is derived by perturbation methods in terms of functionals of the conductivity spatial moments it is shown that the effective conductivity cannot be defined uniquely for potential flows however this nonuniqueness does not affect the spatial distribution of the mean head and velocity analytical expressions of the effective conductivity tensor are derived for two and three dimensional flows and for exponential and gaussian correlations of isotropic conductivity the fundamental solution of the governing equations in effective media mean green function is calculated for the same cases two new asymptotic models of the averaged darcy s law are developed to be applicable to large and small scales of heterogeneity several asymptotic expansions of tue two dimensional and three dimensional mean green functions are derived for exponential and gaussian correlations

implementation of the three dimensional turning bands random field generator numerical techniques to generate replicates of spatially correlated random fields are often used to synthesize sets of highly variable physical quantities in stochastic models of naturally heterogeneous systems within the realm of hydrologic research for example such tools are widely used to develop hypothetical rainfall distributions hydraulic conductivity fields fracture set properties and other surface or subsurface flow parameters the turning bands method is one such algorithm which generates two and three dimensional fields by combining values found from a series of one dimensional simulations along lines radiating outward from a coordinate origin previous work with two dimensional algorithms indicates that radial lines evenly spaced about the unit circle lead to enhanced convergence properties the same can be said for the three dimensional models but it is more difficult to choose an arbitrary number of evenly spaced lines about the unit sphere the current investigation shows that the use of larger numbers of randomly oriented lines 100 can enhance the performance of the three dimensional algorithm this improved performance is needed to effectively simulate problems characterized by full three dimensionality and or anisotropy in either monte carlo or single realization applications use of a large number of lines will also reduce the presence of a distortion effect manifested as linelike patterns in the field increased computational costs can be reduced by employing a fast fourier transform technique to generate the line processes

complex orthogonal and symplectic matrices depending on parameters versal deformations of elements of complex orthogonal and symplectic lie algebras are studied for a general element m of o n c or sp 2n c a normal form ma is found which unlike the jordan normal form mj depends holomorphically on m and on the similarity transformation ma gmg 1 from the corresponding group orthogonal and symplectic cases are treated simultaneously in order to underline their close relation bundles of matrices of low codimension are listed and bifurcation diagrams of two parameter families of orthogonal matrices are shown finally versal deformations of all elements of o 6 c are explicitly shown

stochastic diagrammatic analysis of groundwater flow in heterogeneous porous media the diagrammatic approach is an alternative to standard analytical methods for solving stochastic differential equations governing groundwater flow with spatially variable hydraulic conductivity this approach uses diagrams instead of abstract symbols to visualize complex multifold integrals that appear in the perturbative expansion of the stochastic flow solution and reduces the original flow problem to a closed set of equations for the mean and the covariance functions diagrammatic analysis provides an improved formulation of the flow problem over conventional first order series approximations which are based on assumptions such as constant mean hydraulic gradient infinite flow domain and neglect of cross correlation terms this formulation includes simple schemes like finite order diagrammatic perturbations that account for mean gradient trends and boundary condition effects as well as more advanced schemes like diagrammatic porous media description operators which contain infinite order correlations in other words diagrammatic analysis covers not only the cases where low order diagrams lead to good approximations of flow but also those situations where low order perturbation is insufficient and a more sophisticated analysis is needed diagrams lead to a nonlocal equation for the mean hydraulic gradient in terms of which necessary conditions are formulated for the existence of an effective hydraulic conductivity three dimensional flow in an isotropic bounded domain with dirichlet boundary conditions is considered and an integral equation for the mean hydraulic head is derived by means of diagrams this formulation provides an explicit expression for the boundary effects within the three dimensional flow domain in addition to these theoretical results the numerical performance of the diagrammatic approach is tested and useful insight is obtained by means of one dimensional flow examples where the exact stochastic solutions are available

on constructing a fuzzy inference framework using crisp decision trees this paper proposes a framework which consists of a novel fuzzy inference algorithm to generate fuzzy decision trees from induced crisp decision trees fuzzy theoretical techniques are used to fuzzify crisp decision trees in order to soften sharp decision boundaries at decision nodes inherent in this type of trees a framework for the investigation of various types of membership functions and fuzzy inference techniques is proposed once the decision tree has been fuzzified all branches throughout the tree will fire resulting in a membership grade being generated at each branch five different fuzzy inference mechanisms are used to investigate the degree of interaction between membership grades on each path in the decision tree which ultimately leads to a final crisp classification a genetic algorithm is used to optimize and automatically determine the set of fuzzy regions for all branches and simultaneously the degree in which the inference parameters will be applied comparisons between crisp trees and the fuzzified trees suggest that the later fuzzy tree is significantly more robust and produces a more balanced classification in addition the results obtained from five real world data sets show that there is a significant improvement in the accuracy of the fuzzy trees when compared with crisp trees

a higher order approximation to effective conductivity in media of anisotropic random structure properties of the effective conductivity tensor keff are studied by deriving the second order terms in its expansion in the variance 2 of normally distributed log conductivity it is shown that for media of anisotropic structure the components of the effective conductivity tensor are expressed by a functional of the log conductivity covariance that is it depends on the shape of the correlation function and not only on anisotropy ratios variance 2 and space dimensions however the trace of keff is independent of the log conductivity autocovariance and for a given mean conductivity depends only on 2 the dependence of the effective conductivity on the correlation structure is illustrated for gaussian and exponential autocovariances of log conductivity and for two and three dimensional flows

geno fuzzy classification trees making the non terminal nodes of a binary tree classifier fuzzy can mitigate tree brittleness using a genetic algorithm two optimization techniques are explored in one case each generation minimizes classification error by optimizing a common fuzzy percent pt used to determine parameters at every node in the other case each generation yields a sequence of minimized node specific parameters the output value is determined through defuzzification after input vectors in general take both paths at each node with a weighting factor determined by the node membership functions experiments conducted using this geno fuzzy approach yield an improvement compared with other classical algorithms

eulerian kinematics of flow through spatially periodic models of porous media for spatially periodic models of porous media and incompressible fluids we consider here in detail thephysical interpretation of averaged velocity fields in terms of a purely continuum level eulerian seepage flux relation involving the mass flow across an arbitrarily oriented macroscale surface element we prove that the appropriately averaged microscale velocities do indeed satisfy a macroscale eulerian flux relation albeit to some approximation provided that the surface element is large compared with the pore lengthscale microscale a quantitative discussion of an intermediate scale lying between the pore scale and a characteristic macroscale is achieved through the use of locally periodic velocity fields within the context of seepage velocity kinematics our developments provide a conceptual foundation that renders continuum theories of flow of incompressible fluids through periodic porous media entirely selfcontained on the macroscale free from any details of the discrete microscale structure inaccessible to macroscopic observers from which this continuum theory sprang the error bounds and scaling laws derived are of special interest when the macroscale microscale disparity is not very wide with reference to current rational mechanical theories of immiscible mixtures our analysis provides aconstructive existence proof at least for periodic models of porous media for phase specific velocities heretofore such velocities were usually accepted in a lagrangian sense on an axiomatic basis thus clarification and quantification of thekinematical aspects of the continuum hypothesis for certain classes of multiphase systems can be effecteda priori by our arguments independently of any subsequent macroscaledynamical modelling our analysis whereby we systematically proceed from a discrete picture of the underlying geometry to a purely continuum picture may be likened to that achieved in classical statistical mechanics as in that case we hope that the general results obtained transcend the underlying discrete i e spatially periodic model of the subcontinuum structure

decision tree based fuzzy classifier of h1 magnetic resonance spectra from cerebrospinal fluid samples this paper presents a method for classifying cerebrospinal fluid csf samples studied by proton magnetic resonance spectroscopy h1 mrs into clinical subgroups by means of a fuzzy classifier the method focuses on the analysis of a low signal to noise region of the spectra and is designed to use a small number of samples because sampling can only be done through an invasive technique the proposed method involves the fusion of classifiers based on decision trees designed using fuzzy techniques the fusion step was carried out by ordered weighted averaging owa operators the quality of the proposed classifier was evaluated by efficiency and robustness quality indexes using a method based on a cross validation technique results show excellent classification levels and satisfactory robustness in both training and test sets

generic families of matrix pencils and their bifurcation diagrams abstract v i arnold russian math surveys 26 2 1971 29 constructed smooth generic families of matrices with respect to similarity transformations depending smoothly on the entries of matrices and got bifurcation diagrams of such families with a small number of parameters we extend these results to pencils of matrices

globally optimal fuzzy decision trees for classification and regression a fuzzy decision tree is constructed by allowing the possibility of partial membership of a point in the nodes that make up the tree structure this extension of its expressive capabilities transforms the decision tree into a powerful functional approximant that incorporates features of connectionist methods while remaining easily interpretable fuzzification is achieved by superimposing a fuzzy structure over the skeleton of a cart decision tree a training rule for fuzzy trees similar to backpropagation in neural networks is designed this rule corresponds to a global optimization algorithm that fixes the parameters of the fuzzy splits the method developed for the automatic generation of fuzzy decision trees is applied to both classification and regression problems in regression problems it is seen that the continuity constraint imposed by the function representation of the fuzzy tree leads to substantial improvements in the quality of the regression and limits the tendency to overfitting in classification fuzzification provides a means of uncovering the structure of the probability distribution for the classification errors in attribute space this allows the identification of regions for which the error rate of the tree is significantly lower than the average error rate sometimes even below the bayes misclassification rate

top down induction of fuzzy pattern trees fuzzy pattern tree induction was recently introduced as a novel machine learning method for classification roughly speaking a pattern tree is a hierarchical tree like structure whose inner nodes are marked with generalized fuzzy logical operators and whose leaf nodes are associated with fuzzy predicates on input attributes a pattern tree classifier is composed of an ensemble of such pattern trees one for each class label this type of classifier is interesting for several reasons for example since a single pattern tree can be considered as a kind of logical description of a class it is quite appealing from an interpretation point of view moreover in terms of classification accuracy the method has shown promising performance in first experimental studies yet as will be argued in this paper the algorithm that has originally been proposed for learning fuzzy pattern trees from data offers scope for improvement here we propose a new method that modifies the original proposal in several ways notably our learning algorithm reverses the direction of pattern tree construction from bottom up to top down additionally a different termination criterion is proposed that is more adapted to the learning problem at hand experimentally it will be shown that our approach is indeed able to outperform the original learning method in terms of predictive accuracy

identification and simplification of t s fuzzy neural networks based on incremental structure learning and similarity analysis abstract this paper proposes a novel identification method and a simplification scheme for t s fuzzy neural networks which consists of two steps the first step refers to the structure design based on an incremental clustering approach whose basic ideas are that the structure identification of fuzzy neural networks is guided by the attenuation of output approximation error in each cluster and processed by a recursive refined clustering iteration with the input space clustering and sub clustering as the main steps once the structure of a t s fuzzy neural network is identified by the incremental clustering approach its parameters are further learned and refined by the levenberg marquardt optimization algorithm the second step refers to the structure simplification including removing redundant fuzzy rules and merging highly similar fuzzy rules furthermore the performances of several similarity calculating methods are analyzed and discussed which provides a basis for the selection of the appropriate similarity analysis and effective calculating method for the merging of fuzzy rules and system simplification that is the given performance analysis provides the methodology basis and design guide for structure simplification based on similarity analysis and merger of fuzzy rules several experiments are implemented to illustrate the feasibility and effectiveness of the proposed approach

macroscopic balance equations in soils and aquifers the case of space and time dependent instrumental response macroscopic balance equations for mass and linear momentum in soils and aquifers are derived in the relativist concept under the condition that all physical variables are associated with a single instrumental weighting function that depends on both position and time the resulting balance equations differ in concept from expressions obtained recently in the representative elementary volume rev approach by exhibiting additional terms that depend on the time and space derivatives of the instrumental weighting function these terms indicate that mass and linear momentum balance at the macroscopic scale are not preserved unless the weighting function has space time invariance properties over the period and spatial domain of measurement

effects of kriging and inverse modeling on conditional simulation of the avra valley aquifer in southern arizona the avra valley aquifer in southern arizona is modeled stochastically at three levels of uncertainty the highest level of uncertainty occurs when log transmissivity estimates are based on measured values of this parameter but without regard to the geographic location of each measurement point the resulting steady state hydraulic heads in the aquifer computed by unconditional simulation with the aid of a multivariate normal random number generator coupled with a finite element model have a relatively large variance this variance can be reduced by conditioning the log transmissivity estimates on the spatial arrangement of the data by means of kriging when conditional simulation of the aquifer is performed at this intermediate level of uncertainty with the aid of the same technique as in the unconditional case the variance of the predicted head values is reduced by a factor of 3 2 the lowest level of uncertainty is achieved when the log transmissivity estimates are further conditioned on data relating to the flow regime such as flow rates and water levels in observation wells by means of a statistical inverse procedure conditional simulation at this level of uncertainty is a novel concept it results in a hydraulic head prediction variance that is 14 3 times lower than the corresponding variance based on kriged log transmissivities the net effect of conditioning by means of kriging and inverse modeling is to reduce the prediction variance by a factor of 46 0 a similar study performed by binsarti 1980 on the cortaro aquifer in southern arizona showed insignificant variance reduction due to kriging but a factor of four reduction after inverse modeling these results indicate that the conditioning effect of the inverse method may be much greater than that of kriging the fact that conditioning can reduce the hydraulic head prediction variance by as much as a factor of 46 implies that one should be cautious when dealing with the results of stochastic aquifer models in which conditioning effects are disregarded

why fuzzy decision trees are good rankers several fuzzy extensions of decision tree induction which is an established machine learning method have already been proposed in the literature so far however fuzzy decision trees have almost exclusively been used for the performance task of classification in this paper we show that a fuzzy extension of decision trees is arguably more useful for another performance task namely ranking roughly the goal of ranking is to order a set of instances from most likely positive to most likely negative the motivation for applying fuzzy decision trees to this problem originates from recent investigations of the ranking performance of conventional decision trees these investigations will be continued and complemented in this paper our results reveal some properties that seem to be crucial for a good ranking performance properties that are better and more naturally offered by fuzzy than by conventional decision trees most notably a fuzzy decision tree produces scores in terms of membership degrees on a fine granular scale using these membership degrees as a ranking criterion a key problem of conventional decision trees is solved in an elegant way namely the question of how to break ties between instances in the same leaf or more generally between equally scored instances

the use of field theoretic methods for the study of flow in a heterogeneous porous medium the effects of heterogeneities on the steady state flow of a single fluid in a porous medium are examined it is argued that incomplete knowledge of the permeability requires the use of a stochastic model of the system it is shown that the problem may be written as a field theory which allows a perturbation series to be expressed by diagrammatic means this allows the calculation of effective permeability the mean pressure and the pressure variance the method as well as recovering familiar results gives a formal means of improving the approximation and approaching more complex systems

arcing classifier with discussion and a rejoinder by the author recent work has shown that combining multiple versions of unstable classifiers such as trees or neural nets results in reduced test set error one of the more effective is bagging here modified training sets are formed by resampling from the original training set classifiers constructed using these training sets and then combined by voting freund and schapire propose an algorithm the basis of which is to adaptively resample and combine hence the acronym arcing so that the weights in the resampling are increased for those cases most often misclassified and the combining is done by weighted voting arcing is more successful than bagging in test set error reduction we explore two arcing algorithms compare them to each other and to bagging and try to understand how arcing works we introduce the definitions of bias and variance for a classifier as components of the test set error unstable classifiers can have low bias on a large range of data sets their problem is high variance combining multiple versions either through bagging or arcing reduces variance significantly

posterior probability based ensemble strategy using optimizing decision directed acyclic graph for multi class classification ensemble strategy is important to develop a decomposition and ensemble method for multi class classification problems most existing ensemble strategies use predetermined and heuristic decision rules in this work we build up the decision rules by optimizing decision directed acyclic graph oddag with classical and fuzzy decision trees to ensemble the posterior probabilities of binary classifiers from one vs one ovo or one vs all ova decomposition strategies for multi class classification problems four widely used extensible algorithms and ten decomposition and ensemble methods incorporating four binary classifiers bcs have been tested on 25 data sets the empirical results show that the methods based on ensemble strategies using oddag are among the top methods that achieve the best performance in terms of two different measures

induction of fuzzy decision trees abstract most decision tree induction methods used for extracting knowledge in classification problems do not deal with cognitive uncertainties such as vagueness and ambiguity associated with human thinking and perception in this paper cognitive uncertainties involved in classification problems are explicitly represented measured and incorporated into the knowledge induction process a fuzzy decision tree induction method which is based on the reduction of classification ambiguity with fuzzy evidence is developed fuzzy decision trees represent classification knowledge more naturally to the way of human thinking and are more robust in tolerating imprecise conflict and missing information

a comparison of methods to determine measurement support volumes we examine methods that can be used to identify the spatial filter function associated with the measurement of quantities on two scales we divide the methods into two general classes deconvolution methods and sensitivity based methods we briefly describe the two classes of methods provide examples of each from the literature and introduce a new sensitivity based method inspired by backus gilbert model resolution ideas we compare the approaches by examining the filter function that relates smaller scale grid block parameters to parameters determined by a theis model interpretation of drawdown data in general we find that sensitivity approaches are easier to implement numerically and are less subject to the problems of identifiability stability and uniqueness associated with deconvolution methods

versal deformations of elements of real classical lie algebras versal deformations of normal forms of elements of real classical lie algebras o p q sp 2n r o 2n sp 2p 2q and u p q are computed examples involving the lie algebras o p q p q or 6 are considered

what does an instrument measure empirical spatial weighting functions calculated from permeability data sets measured on multiple sample supports with the aid of linear filter theory we analyze 13 824 permeability measurements to empirically address the question what does an instrument measure by measure we mean the sample support or sample volume associated with an instrument as well as how the instrument spatially weights the heterogeneities comprising that sample support although the theoretical aspects of linear filter analysis are well documented physical data for testing the filtering behavior of an instrument particularly in the context of porous media flow are rare to nonexistent our exploration makes use of permeability data measured with a minipermeameter on a block of berea sandstone data were collected according to a uniform grid that was resampled with tip seals of increasing size i e increasing sample support spatial weighting filter functions characterizing the minipermeameter measurements were then calculated directly from the permeability data sets in this paper we limit our presentation to one of the six rock faces consisting of 2304 measurements as the general results for each rock face are similar we found that the empirical weighting functions are consistent with the basic physics of the minipermeameter measurement they decay as a nonlinear function of radial distance from the center of the tip seal consistent with the divergent flow geometry imposed by the minipermeameter the magnitude of the weighting function decreases while its breadth increases with increasing tip seal size reflecting the increasing sample support we further demonstrate both empirically and theoretically that nonadditive properties like permeability are amenable to linear filter analysis under certain limiting conditions i e small variances specifically the weighting function is independent of the power average employed in its calculation e g arithmetic versus harmonic average finally we examine the implications of these results for other instruments commonly employed in hydraulic testing e g slug and pump tests

type curve interpretation of late time pumping test data in randomly heterogeneous aquifers 1 the properties of heterogeneous media vary spatially in a manner that can seldom be described with certainty it may however be possible to describe the spatial variability of these properties in terms of geostatistical parameters such as mean integral spatial correlation scale and variance neuman et al 2004 proposed a graphical method to estimate the geostatistical parameters of natural log transmissivity on the basis of quasi steady state head data when a randomly heterogeneous confined aquifer is pumped at a constant rate from a fully penetrating well they conjectured that a quasi steady state during which heads vary in space time while gradients vary only in space develops in a statistically homogeneous and horizontally isotropic aquifer as it does in a uniform aquifer we confirm their conjecture numerically for gaussian log transmissivities show that time drawdown data from randomly heterogeneous aquifers are difficult to interpret graphically and demonstrate that quasi steady state distance drawdown data are amenable to such interpretation by the type curve method of neuman et al the method yields acceptable estimates of statistical log transmissivity parameters for fields having either an exponential or a gaussian spatial correlation function these estimates are more robust than those obtained using the graphical time drawdown method of copty and findikakis 2003 2004a we apply the method of neuman et al 2004 simultaneously to data from a sequence of pumping tests conducted in four wells in an aquifer near tubingen germany and compare our transmissivity estimate with estimates obtained from 312 flowmeter measurements of hydraulic conductivity in these and eight additional wells at the site we find that 1 four wells are enough to provide reasonable estimates of lead log transmissivity statistics for the tubingen site using this method and 2 the time drawdown method of cooper and jacob 1946 underestimates the geometric mean transmissivity at the site by 30 40

measurement scale network sampling scale and groundwater model parameters the scales at which model parameters are measured with local field tests distributed on a sampling network are examined two scales are defined to characterize the problem 1 the measurement scale associated with the resolution of a single field test and 2 the network scale associated with the separation between samples on a network using a spatial filtering approach it is shown that a network of measurements can only resolve a larger scale component of a parameter field the smaller scale component of the parameter field not seen by measurements here called the subgrid component can only vary on scales larger than the measurement scale and smaller than the network scale these unobserved subgrid scales give rise to the so called closure problem and consequent modeling errors when a significant proportion of the parameter variability is contained in the subgrid scales not only will the closure problem be significant but aliasing errors will also pollute the estimate of the large scale component of the parameter field these concepts are illustrated by performing simple analytical and numerical calculations

a complete fuzzy decision tree technique in this paper a new method of fuzzy decision trees called soft decision trees sdt is presented this method combines tree growing and pruning to determine the structure of the soft decision tree with refitting and backfitting to improve its generalization capabilities the method is explained and motivated and its behavior is first analyzed empirically on 3 large databases in terms of classification error rate model complexity and cpu time a comparative study on 11 standard uci repository databases then shows that the soft decision trees produced by this method are significantly more accurate than standard decision trees moreover a global model variance study shows a much lower variance for soft decision trees than for standard trees as a direct cause of the improved accuracy

miniversal deformations of matrices of bilinear forms arnold v i arnold on matrices depending on parameters russian math surveys 26 2 1971 29 43 constructed miniversal deformations of square complex matrices under similarity that is a simp

pattern trees induction a new machine learning method fuzzy classification is one of the most important applications in fuzzy set and fuzzy logic related research its goal is to find a set of fuzzy rules that form a classification model most of the existing fuzzy rule induction methods e g the fuzzy decision trees fdts induction method focus on searching rules consisting of triangular norms t norms i e and only but not triangular conorms t conorms or explicitly this may lead to the omission of generating important rules that involve t conorms explicitly this paper proposes a type of tree termed pattern trees pts that makes use of different aggregations including both t norms and t conorms like decision trees pts are an effective tool for classification applications this paper discusses the difference between decision trees and pts and also shows that the subsethood based method sbm and the weighted subsethood based method wsbm are two specific cases of pt induction a novel pt induction method is proposed using similarity measure and fuzzy aggregations the comparison to other classification methods including sbm wsbm c4 5 nearest neighbor support vector machine and fdt induction shows that 1 pts can obtain high accuracy rates in classifications 2 pts are robust to overfltting and 3 pts especially simple pattern trees spts maintain compact tree structures

learning pairwise image similarities for multi classification using kernel regression trees we are often faced with the problem of distinguishing between visually similar objects that share the same general appearance characteristics as opposed to object categorization this task is focused on capturing fine image differences in a pose dependent fashion our research addresses this particular family of problems and is centered around the concept of learning from example pairs formally we construct a parameterized visual similarity function optimally separating the pairs of images that depict the objects of the same class or identity from the pairs representing different object classes identities it combines various image distances that are quantified by comparing local descriptor responses at the corresponding locations in both paired images to find the best combinations we train ensembles of so called kernel regression trees which model the desired similarity function as a hierarchy of fuzzy decision stumps the obtained function is then used within a k nn like framework to address complex multi classification problems through the experiments with several image datasets we demonstrate the numerous advantages of the proposed approach high classification accuracy scalability ease of interpretation and the independence of the feature representation

spatial averaging of transmissivity in heterogeneous fields with flow toward a well steady state nonuniform flow in a heterogeneous medium is investigated using a combined numerical empirical approach the objective of the study is to determine the effective transmissivity of a square field with a constant head external boundary and a constant rate well at its center transmissivity at the point scale is modeled as an isotropic and multivariate lognormal spatial random function block transmissivities are defined using an empirical scaling up of point support values within the field the scaling process is a weighted spatial geometric averaging where log transmissivities are weighted by the inverse square of their distance from the well block transmissivities thus obtained and true effective values calculated using a numerical flow model are compared for realizations a finely discretized transmissivity field the principal finding of this study is that effective transmissivities obtained from the numerical flow model and block values obtained by spatial averaging are in excellent agreement for low to moderate variances of log transmissivity and moderately eccentric field geometries the geostatistical model for point scale transmissivity and the deterministic spatial averaging law are combined in order to create a geostatistical model for transmissivity at the block scale exact expressions are derived for the ensemble moments of block averaged transmissivities and well drawdowns in both the nonconditional case and the conditional case when transmissivity at the well bore is known the ensemble mean of block averaged transmissivity is found to decrease from the ensemble arithmetic mean toward the ensemble geometric mean as the field size becomes large compared to the integral range of spatial correlation this result contradicts earlier published work numerical results presented in this study also disagree with the analytical results of a recent stochastic analysis of specific discharge in radial systems

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

the universal structure of the groundwater flow equations most groundwater flow models are not sufficiently detailed to allow an explicit representation of all dynamical scales instead most models are constructed at such a coarse scale of resolution that unresolved subgrid variability often exists it is therefore important to understand the interaction between unresolved dynamics and explicitly resolved dynamics the concept of universality is central to this interaction and is here related to the problem of rescaling models of groundwater flow we examine when it is possible to construct an accurate model for the explicitly resolved large scales without an explicit description of the subgrid scale dynamics we demonstrate how unresolved subgrid scale dynamics interact with resolved scale dynamics we show that a universally valid resolved scale model can be constructed if the resolved dynamics are sufficiently independent of the details of the subgrid scale dynamics in that event a resolved scale model is composed of a universal structure and accompanying model parameters the model parameters represent the effect of unresolved dynamics upon resolved dynamics we examine two possible model structures for groundwater flow we show theoretically and numerically that a local darcy law is a universally valid resolved scale model if the resolved and subgrid scales of the hydraulic conductivity are separated in scale by a so called spectral gap if however the hydraulic conductivity possesses many scales of variability then a more general nonlocal darcy law is a more appropriate model structure when the nonlocal darcy s law is more appropriate numerical experiments suggest that errors using a local darcy s law with effective parameters are most significant at the smallest resolving scale of the model and are minimal at scales between 8 and 16 times the resolving scale

a monte carlo assessment of eulerian flow and transport perturbation models monte carlo studies of flow and transport in two dimensional synthetic conductivity fields are employed to evaluate first order flow and eulerian transport theories hydraulic conductivity is assumed to obey fractional brownian motion fbm statistics with infinite integral scale or to have an exponential covariance structure with finite integral scale the flow problem is solved via a block centered finite difference scheme and a random walk approach is employed to solve the transport equation for a conservative tracer the model is tested for mass conservation and convergence of computed statistics and found to yield accurate results it is then used to address several issues in the context of flow and transport the validity of the first order relation between the fluctuating velocity covariance and the fluctuating log conductivity is examined the simulations show that for exponential covariance this approximation is justified in the mean flow direction for log conductivity variance f2 of the order of unity however as f2 increases the relation for the transverse velocity component deviates from the fully nonlinear monte carlo results eulerian transport models neglect triplet correlation functions that appear in the nonlocal macroscopic flux the relative importance of the triplet correlation term for conservative chemicals is examined this term appears to be small relative to the convolution flux term in mildly heterogeneous media as f2 increases or the integral scale grows the triplet correlation becomes significant in purely convective transport the triplet correlation term is significant if the heterogeneity is evolving the exact nonlocal macroscale flux for the purely convective case significantly differs from that of the convective dispersive transport this is in agreement with recent theoretical analysis and numerical studies and it suggests that neglecting local scale dispersion may lead to large errors localization errors in the flux term are evaluated using monte carlo simulations the nonlocal in time model significantly differs from the fully nonlocal model for small variance and integral scale there is a slight difference between the fully localized flux and the fully nonlocal convolution flux this is also in agreement with recent theories that suggest that moments through the second for the two models are nearly identical for conservative tracers the fully localized model does not perform well in the purely convective cases

a methodology for automated fuzzy model generation in this paper we propose a generic methodology for the automated generation of fuzzy models the methodology is realized in three stages initially a crisp model is created and in the second stage it is transformed to a fuzzy one in the third stage all parameters entering the fuzzy model are optimized the proposed methodology is novel and generic since it can integrate alternative techniques in each of its stages a specific realization of this methodology is implemented using decision trees for the creation of the crisp model the sigmoid function the min max operators and the maximum defuzzifier for the transformation of the crisp model into a fuzzy one and four different optimization strategies including global and local optimization techniques as well as hybrid approaches the proposed methodology presents several advantages and novelties the transformation of the crisp model to the respective fuzzy one is straightforward ensuring its full automated nature and it introduces a set of parameters expressing the importance of each fuzzy rule the above realization is extensively evaluated using several benchmark data sets from the uci machine learning repository and the obtained results indicate its high efficiency

the operational significance of the continuum hypothesis in the theory of water movement through soils and aquifers the operational meaning of the representative elementary volume rev concept on which current foundational theories of water movement through porous media are based is analyzed critically it is concluded that the rev concept as applied to real porous media is both unnecessarily restrictive and experimentally unverifiable in its place a relativist concept is proposed in which macroscopic physical variables are defined as convolution products of microscopic properties of a porous medium with weighting functions that represent the appropriate measuring instruments the effects of resolution and precision differences among experimental measuring devices now recognized as critical in the assessment of spatial variability in the properties of soils and aquifers can be incorporated naturally within the relativist concept but are excluded a priori in the rev concept the relativist point of view unlike the rev concept has a clear operational meaning and in principle extends theoretical validity to all local measurements made on porous media it is sufficient mathematically to derive conventional macroscopic balance equations for mass and linear momentum as well as a differential equation for the isothermal transport of water through an unsaturated anisotropic deformable soil or aquifer

conditional moment analysis of steady state unsaturated flow in bounded randomly heterogeneous soils 1 we consider steady state unsaturated flow in bounded randomly heterogeneous soils under the influence of random boundary and source terms our aim is to predict pressure heads and fluxes without resorting to monte carlo simulation upscaling or linearization of the constitutive relationship between unsaturated hydraulic conductivity and pressure head we represent this relationship through gardner s exponential model while treating its exponent as a random constant and saturated hydraulic conductivity ks as a spatially correlated random field we linearize the steady state unsaturated flow equations by means of the kirchhoff transformation and integrate them in probability space to obtain exact integro differential equations for the conditional mean and variance covariance of transformed pressure head and flux after approximating these equations recursively to second order in the standard deviation y of y ln ks we solve them by finite elements for superimposed mean uniform and divergent flows in the vertical plane with and without conditioning on measured y values comparison with monte carlo solutions demonstrates that whereas our nonlocal solution is nominally restricted to mildly nonuniform media with y2 1 it yields remarkably accurate results for strongly nonuniform media with y2 at least as large as 2 this accords well with a previous theoretical analysis which shows that the solution may remain asymptotic for values of y2 as large as 2

fuzzy classification boundaries against adversarial network attacks abstract adversarial machine learning copes with the development of methods to prevent machine learning algorithms from being misled by malicious users this field is especially relevant for applications where machine learning lies in the core of security systems in the field of network security adversarial samples are actually novel network attacks or old attacks with tuned properties this paper proposes to blur classification boundaries in order to enhance machine learning robustness and improve the detection of adversarial samples that exploit learning weaknesses we test this concept by an experimental setup with network traffic in which linear decision trees are wrapped by a one class membership scoring algorithm we benchmark our proposal with plain linear decision trees and fuzzy decision trees results show that evasive attacks i e false negatives tend to be ranked with low class membership levels meaning that they are located in zones close to classification thresholds in addition classification performances improve when membership scores are added as new features using fuzzy class boundaries is highly consistent with the interpretation of many network traffic features used for malware detection moreover it prevents network attackers from exploiting classification boundaries as attack objectives

analysis of flow through heterogeneous random aquifers 2 unsteady flow in confined formations the direct problem of determining the head field in a formation of given stationary isotropic random structure is solved approximately for unsteady flow conditions under restrictive assumptions of sufficiently small conductivity variance and average head gradient slowly varying in time and space closed form expressions are derived for the effective conductivity or storativity and for the head variogram two types of unsteady flow are considered transients and periodic flows in the case of transients it is assumed that initially the head is constant no flow and that ultimately the flow becomes uniform it is found that the effective conductivity and transmissivity are time dependent and drop from the arithmetic mean initially to the steady state value during a relaxation time the latter is much larger for one dimensional flows than for two or three dimensional ones the head variogram also relaxes from the one corresponding to the initial lack of correlation to the steady state variogram during a relaxation time which grows with the lag while in three dimensional structure the relaxation time scales are relatively short and steady state analysis can be used accurately even under transient conditions the tendency to steady state may be very slow in the case of transmissivity because of its large integral scale hence steady state analysis for transients might not be warranted for regional transient flows in the case of periodic flows of seasonal nature the head gradient is assumed to be made up from a steady component and a periodic one if the amplitude of the latter is sufficiently small compared to the steady one steady state value of effective transmissivity and storativity can be adopted the head variogram has also its steady state form for large lags and quasisteady for small lags hence the variogram lies between the steady one steady component of head gradient and the quasisteady one instantaneous head gradient

neuro fuzzy id3 a method of inducing fuzzy decision trees with linear programming for maximizing entropy and an algebraic method for incremental learning abstract id3 is a popular and efficient method of making a decision tree for classification from symbolic data without much computation fuzzy reasoning rules in the form of a decision tree which can be viewed as a fuzzy partition are obtained by fuzzy id3 the aims of this paper are 1 not only the learning from examples but also the interview with domain specialists are needed for knowledge acquisition in expert systems in order to avoid dangerous simplification of the tree by discarding partial knowledge of the experts a measure of uncertainty with maximizing entropy is applied to fuzzy id3 2 basically the tree based learning is nonincremental or single step an algebraic method to facilitate incremental learning like the neural nets is adapted and the fuzzy decision tree which consists of b spline membership function is regarded as three layered neural network 3 prototypes of expert system for estimation of wheel wear characteristics and surface roughness in abrasive cut off are developed

a physical basis for calculating instrument spatial weighting functions in homogeneous systems 1 starting with a streamline based argument the theoretical basis for a spatial weighting function in a steady homogenous and isotropic flow system is developed for both compressible and incompressible flow a physical interpretation of spatial weighting functions in terms of the ratio of steady state energy dissipation rate per unit volume of porous medium to total energy dissipation rate over the entire flow domain is formulated finally applications of the spatial weighting functions are presented based on calculated gas minipermeameter mass flux fields for the conventional surface probe and a new small drill hole probe configuration spatial weighting function distributions indicate that with diverging flow field instruments such as the gas minipermeameter porous medium volumes in the inlet vicinity are heavily weighted with volumes near the seal boundaries shown to be extremely important the technique described allows one to quantify the size and shape of the subvolume of a domain contributing to a hydraulic conductivity or intrinsic permeability measurement for heterogeneous systems it is suggested that the spatial weighting function for a given instrument will be heterogeneity dependent and therefore nonunique however extending the weighting function concept to heterogeneous and or transient systems is an important research priority