transform domain analysis of des the data encryption standard des can be regarded as a nonlinear feedback shift register nlfsr with input from this point of view the tools for pseudo random sequence analysis are applied to the s boxes in des the properties of the s boxes of des under the fourier transform hadamard transform extended hadamard transform and the avalanche transform are investigated two important results about the s boxes of des are found the first result is that nearly two thirds of the total 32 functions from gf 2 sup 6 to gf 2 which are associated with the eight s boxes of des have the maximal linear span g3 and the other one third have linear span greater than or equal to 57 the second result is that for all s boxes the distances of the s boxes approximated by monomial functions has the same distribution as for the s boxes approximated by linear functions some new criteria for the design of permutation functions for use in block cipher algorithms are discussed

feedback reconstruction and implementations of pseudorandom number generators from composited de bruijn sequences a binary de bruijn sequence of order n is a sequence of zeros and ones of period 2 n that contains every binary n tuple exactly once in a period of the sequence a composited construction of a de bruijn sequence is a construction of a nonlinear feedback shift register nlfsr that generates a de bruijn sequence where the composited feedback function of the nlfsr is the sum of a feedback function with k th order composition and a sum of k 1 product of sum terms the goals of this article are to perform a profound analysis of composited de bruijn sequences for use in cryptography and find an efficient implementation of the composited feedback function we first determine the lower bound of the linear complexity of a composited de bruijn sequence and then conduct a profound analysis on the composited construction by introducing the notion of the higher order d morphic preimages of a binary sequence our analysis aims at the reconstruction of a composited de bruijn sequence from a segment known as k th order d morphic order n de bruijn preimages n k dmdps of length 2 n k and k th order d morphic order n m sequence preimages n k dmmps of length 2n k for a nonlinearly and linearly generated composited de bruijn sequence respectively we also provide the success probability of finding an n k dmmp dmdp from a composited de bruijn sequence for the reconstruction furthermore we develop a new iterative technique with its parallel extension for computing the feedback function and the new technique is faster than other known techniques for producing de bruijn sequences of long period in addition we present three instances of composited de bruijn sequences of period 2 64 together with their software implementations and performances

a spectral characterization of correlation immune combining functions it is shown that a boolean combining function f x of n variables is mth order correlation immune if and only if its walsh transform f omega vanishes for all omega with hamming weight between 1 and m inclusive this result is used to extend slightly siegenthaler s ieee trans comput vol c 34 pp 81 85 jan 1985 characterization of the algebraic normal form of correlation immune combining functions

effective hydraulic conductivity of bounded strongly heterogeneous porous media we develop analytical expressions for the effective hydraulic conductivity ke of a three dimensional heterogeneous porous medium in the presence of randomly prescribed head and flux boundaries the log hydraulic conductivity y forms a gaussian statistically homogeneous and anisotropic random field with an exponential autocovariance by effective hydraulic conductivity of a finite volume in such a field we mean the ensemble mean expected value of all random equivalent conductivities that one could associate with a similar volume under uniform mean flow we start by deriving a first order approximation of an exact expression developed in 1993 by neuman and orr we then generalize this to strongly heterogeneous media by invoking the landau lifshitz conjecture upon evaluating our expressions we find that ke decreases rapidly from the arithmetic mean ka toward an asymptotic value as distance between the prescribed head boundaries increases from zero to about eight integral scales of y the more heterogeneous is the medium the larger is ke relative to its asymptote at any given separation distance our theory compares well with published results of spatially power averaged expressions and with a first order expression developed intuitively by kitanidis in 1990

wavelet based filters for accurate computation of derivatives let f x be a smooth function whose derivative of a given order must be computed the signal f x is affected by two kinds of perturbation the perturbation caused by the presence of the machine epsilon m of the computer may be considered to be an extremely high frequency noise of very small amplitude the way to minimize its effect consists of choosing an appropriate value for the step size of the difference quotient the second perturbation caused by the presence of noise requires first the signal to be treated in some way it is the purpose of this work to construct a wavelet based band pass filter that deals with the two cited perturbations simultaneously in effect our wavelet acts like a smoothed difference quotient whose stepsize is of the same order as that of the usual difference quotient moreover the wavelet effectively removes the noise and computes the derivative with an accuracy equal to the one obtained by the corresponding difference quotient in the absence of noise

fundamental mode shape and its derivatives in structural damage localization abstract various techniques for structural damage detection involving modal parameters have widely been used over the past few decades this is because the modal parameters of a structure can easily be obtained from forced free or ambient vibration measurements in many of these techniques mode shape curvature has been used for localization of damage in this paper a mathematical basis is provided to show the correlation between a structural damage and a change in the fundamental mode shape and its derivatives this has been achieved by deriving the expression of a damaged mode shape utilizing a perturbation approach for a cantilever shear beam discretized into a large number of elements this approach demonstrates that the change in the fundamental mode shape due to any damage is an excellent indicator of damage localization as it is found to be discontinuous at the location of damage further the change in higher derivatives i e slope and curvature of the fundamental mode shape is shown to be sensitive enough in damage localization a numerical study involving a shear building and a steel moment resisting frame is conducted to show the effectiveness of the proposed approach in damage localization it has been found from this study that spline fitting of mode shapes in case of a limited number of degrees of freedom which is generally adopted for plotting mode shapes and their derivatives may lead to false detection of damage

upscaling of permeability of anisotropic heterogeneous formations 3 applications the general results of indelman and dagan this issue a b are applied to some cases of interest in applications the exact solution of upscaling of stratified formations is used in order to assess the range of applicability of the first order analysis the explicit expressions of the two first moments of the upscaled permeability are derived for media of two dimensional isotropic and three dimensional axisymmetric actual log permeability of gaussian covariance and for parallelepipedic rectangular for two dimensional averaging elements the effects of the actual log permeability covariance and of the anisotropy of partition elements on the upscaled permeability statistics are analyzed the asymptotic behavior of the upscaled permeability for large and small blocks is discussed for general heterogeneity the strong dependence of the small element limit on the properties of actual heterogeneity and on the shape of the elements is illustrated by typical examples in contrast the homogenization of the upscaled media by large blocks depends primarily on the volume area for two dimensions of the partition element relative to the one based on the correlation scale of the log permeability

three dimensional unsaturated flow modeling using cellular automata 1 the parabolic partial differential equation describing fluid flow in partially saturated porous media richards equation is highly nonlinear due to pressure head dependencies in the specific soil moisture capacity and relative hydraulic conductivity terms in order to solve richards equation several numerical techniques have been developed which starting from the discretization of the partial differential equation produced even more accurate models leading to complex and computationally expensive simulations for large scale systems a three dimensional unsaturated flow modeling developed through a simulation environment based on cellular automata ca is described in this paper the proposed model represents an extension of the original computational paradigm of cellular automata because it uses a macroscopic ca approach where local laws with a clear physical meaning govern interactions among automata this ca structure aimed at simulating a large scale system is based on functionalities capable of increasing its computational capacity both in terms of working environment and in terms of the optimal number of processors available for parallel computing the model has been validated with reference multidimensional solutions taken from benchmarks in literature showing a good agreement even in the cases where nonlinearity is very marked furthermore some analyses have been carried out considering quantization techniques aimed at transforming the ca model into an asynchronous structure the use of these techniques in a three dimensional benchmark allowed a considerable reduction in the number of local interactions among adjacent automata without changing the efficiency of the model especially when simulations are characterized by scarce mass exchanges finally from a computational point of view the higher efficiency values were achieved running the model on a parallel architecture obtaining a high speedup very close to the optimal with the maximum number of processors available

is transmissivity a meaningful property of natural formations conceptual issues and model development 1 at regional scale it is common to model groundwater flow as 2 d in the x y horizontal plane by integrating the full 3 d equations over the vertical furthermore adopting the dupuit assumption results in the local transmissivity t as a formation property equal to the vertically integrated hydraulic conductivity k in practice the related block transmissivity tb defined for a volume of area square of side l in the horizontal plane and height d is the property of interest however most aquifers are of a heterogeneous 3 d structure and y lnk is commonly modeled as a normal and stationary random function which is characterized by the variance y2 the horizontal i and vertical iv integral scales the dupuit assumption is generally not obeyed for formations of 3 d spatially variable y and transmissivity is no more a meaningful property independent of flow conditions useful generalizations of local and block transmissivity are possible for steady mean uniform flow in the horizontal direction and formations of constant thickness in that case t and tb become random stationary variables characterized by their mean variance and integral scales these moments are determined for the first time in an analytical form or by a few quadratures by adopting a first order approximation in y2 and they depend on the ratio d iv e iv i and l i the block conductivity expected values are compared with the numerical solutions of dykaar and kitanidis 1993 and the agreement is very good for y2 1 the main conclusion of the study is that for this simple flow configuration and for common parameter values tb is practically deterministic and equal to keff e d where keff is the effective conductivity in uniform flow in an unbounded formation at regional scale tb may be regarded as a local property which changes slowly in the horizontal plane analysis of numerous field data shows that this variation is also random and characterized by integral scales ireg of the order of kilometers the separation of scales makes possible to regard the local tb as determined along the lines of the present study in a support volume of extent of a few d as a point value at the regional scale practical implications and topics for future investigations are outlined

singularity detection and processing with wavelets the mathematical characterization of singularities with lipschitz exponents is reviewed theorems that estimate local lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed it is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their lipschitz exponents the wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately the local frequency of such oscillations is measured from the wavelet transform modulus maxima it has been shown numerically that one and two dimensional signals can be reconstructed with a good approximation from the local maxima of their wavelet transform modulus as an application an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales in two dimensions the wavelet transform maxima indicate the location of edges in images

upscaling of permeability of anisotropic heterogeneous formations 1 the general framework in numerical simulations of flow through heterogeneous formations the domain is partitioned into numerical elements the solution requires assigning physical properties to each numerical block the process of transferring information from the scale of actual heterogeneity to that of the numerical elements is known as upscaling we consider steady one phase flow and the only property of interest is the permeability which is regarded as a random space function and the same is true for the upscaled conductivity in this paper we establish the necessary and sufficient conditions to be satisfied by upscaling the necessary conditions are expressed with the aid of the global response of the formation e g the total flux caused by a constant pressure head drop applied on the boundaries the requirement is that the expected value and the variance of the total flux are the same in the actual and upscaled domains these are supplemented by local conditions which stipulate that space averages of flow variables over the numerical blocks in the upscaled domain tend to those in the actual formation when the elements size tends to 0 the result of this analysis is that the upscaled permeability must be such as to ensure the equality in a statistical sense between the space averaged dissipation in the two media by assuming an unbounded domain average uniform flow and stationarity simplified relationships between the moments of the dissipation in the upscaled and actual media are derived it is shown that the effective permeability of the two media is the same the approach is illustrated for stratified formations for which exact composition rules are possible even in this simple case the upscaled permeability is a tensor characterized at second order by two mean components and three covariance functions

a stochastic approach to the problem of upscaling of conductivity in disordered media theory and unconditional numerical simulations a general problem in groundwater modeling is how to assign conductivity values to blocks of a numerical model in this study a method is proposed to upscale the conductivity measurements observed at a given scale to block conductivity values for arbitrary block size assuming a multilognormal distribution for the point conductivities the block conductivity appears as a random function whose distribution conditional to point measurements can be derived this conditional distribution allows filling the flow simulator blocks with equiprobable realizations of block conductivities locally conditioned to actual data these realizations could also be conditioned to conductivity values measured at any scale if the size of the measurement support is known the analytical results are compared to results obtained through a monte carlo analysis and a good agreement was found even for variances of ln t larger than 1

a geostatistical approach to the inverse problem in groundwater modeling steady state and one dimensional simulations the problem of estimating hydrogeologic parameters in particular permeability from input output measurements is reexamined in a geostatistical framework the field of the unknown parameters is represented as a random field and the estimation procedure consists of two main steps first the structure of the parameter field is identified i e mathematical representations of the variogram and the trend are selected and their parameters are established by using all available information including measurements of hydraulic head and permeability second linear estimation theory is applied to provide minimum variance and unbiased point estimates of hydrogeologic parameters kriging structure identification is achieved iteratively in three substeps structure selection maximum likelihood estimation and model validation and diagnostic checking the methodology was extensively tested through simulations on a simple one dimensional case the results are remarkably stable and well behaved the estimated field is smooth while small scale variability is statistically described as the quality of measurements improves the procedure reproduces more features of the original field the results are also shown to be rather insensitive to deviations from assumptions about the geostatistical structure of the field

a construction of codebooks associated with binary sequences an n k codebook is a set of n unit norm code vectors in a k dimensional vector space for its applications it is desired that the maximum magnitude of inner products between a pair of distinct code vectors should be as small as possible meeting the welch bound equality strictly or asymptotically in this paper an n k codebook is constructed from a k n partial matrix with k n where each code vector is equivalent to a column of the matrix to obtain the k n matrix k rows are selected from a j n matrix associated with a binary sequence of length j and hamming weight k where a set of the selected row indices is equivalent to the index set of nonzero entries of the binary sequence it is then discovered that the maximum magnitude of inner products between a pair of distinct code vectors is determined by the maximum magnitude of transform of the binary sequence thus constructing a codebook with small magnitude of inner products is equivalent to finding a binary sequence where the maximum magnitude of its transform is as small as possible from the discovery new classes of codebooks with nontrivial bounds on the maximum inner products are constructed from fourier and hadamard matrices associated with binary sequences

uniqueness for an elliptic inverse problem for the elliptic equation nabla cdot p x nabla v lambda q x v f x in omega subset bbr n the problem of determining when one or more of the coefficient functions p q and f are defined uniquely by a knowledge of one or more of the solution functions v v p q f lambda is considered

damage estimation on beam like structures using the multi resolution analysis abstract this work proposes a vibration based damage evaluation method that can detect locate and size damage utilizing only a few of the lower mode shapes the proposed method is particularly advantageous for beam like structures with uncertain applied axial load mass density and foundation stiffness based on a small damage assumption a linear relationship between damaged and undamaged curvatures is revealed in the context of elasticity it turns out that the resulting damage index equation inherently suffers from singularities near inflection nodes the transformation of the problem into the multi resolution wavelet domain provides a set of coupled linear equations with the aid of the singular value decomposition technique the solution to the damage index equation is achieved in the wavelet space next the desired physical solution to the damage index equation is reconstructed from the one in the wavelet space the performance of the proposed method is compared with two existing damage detection methods using a set of numerical simulations the proposed method attempts to resolve the mode selection problem the singularity problem the axial force problem and the absolute severity estimation problem all of which remained unsolved by earlier attempts

parameter identification for elliptic problems abstract the problem of computing a principal coefficient function p in the differential equation p x u f x r m m 1 on a bounded region from a knowledge of the solution function u and the right hand side f where u f are known only approximately and p may have mild discontinuities is solved by minimization of an associated functional

on stable numerical differentiation based on a regularized volterra equation two different approaches for numeri cal differentiation are considered the first approach consists of solving a regularized volterra equation while the second approach is based on solving a disretized version of the regularized volterra equation numerical experiments show that these methods are efficient and compete fa vorably with the variational regularization method for stable calculating the derivatives of noisy functions

variational second order estimates for nonlinear composites it is shown here that the approximate second order method of ponte castaneda for estimating the effective behaviour of nonlinear composite materials can be given a variational status this is accomplished by means of a slight generalization of the variational structure of talbot and willis consisting of the replacement of the infimum over the polarization field by the more general corresponding stationary operation since the relevant functions are not convex in general this replacement provides additional flexibility that can be exploited advantageously the net result is a stationary principle from which stationary variational estimates can be generated under certain hypotheses on the choice of the linear comparison composite it is shown that the resulting estimates are identical to the second order estimates thus showing that these second order estimates are indeed variational in character although only of the stationary variety the connection of these second order estimates to earlier bounds obtained from related extremum variational principles is also explored

stochastic modeling of groundwater flow by unconditional and conditional probabilities the inverse problem the inverse problem is defined here as follows determine the transmissivity at varius points given the shape and boundary of the aquifer and recharge intensity and given a set of measured log transmissivity y and head h values at a few points the log transmissivity distribution is regarded as a realization of a random function of normal and stationary unconditional probability density function pdf the solution of the inverse problem is the conditional normal pdf of y conditioned on measured h and y which is expressed in terms of the unconditional joint pdf of y and h the problem is reduced to determining the unconditional head log transmissivity covariance and head variogram for a selected y covariance which depends on a few unknown parameters this is achieved by solving a first order approximation of the flow equations the method is illustrated for an exponential y covariance and the effect of head and transmissivity measurements upon the reduction of uncertainty of y is investigated systematically it is shown that measurement of h has a lesser impact than those of y but a judicious combination may lead to significant reduction of the predicted variance of y possible applications to real aquifers are outlined

upscaling of permeability of anisotropic heterogeneous formations 2 general structure and small perturbation analysis the general methodology developed in part 1 indelman and dagan this issue of this study is applied to the detailed analysis of upscaled permeability first the structure of the dissipation function for the general conductivity a tensor of stationary random components is examined with the aid of dimensional analysis it is shown that for arbitrary shapes of the numerical elements the upscaled permeability has also this general structure and the numbers of unknown parameters and equations match this result suggests that the upscaling problem has an unique solution in the particular case of scalar permeability of isotropic covariance of the actual formation it is shown that a similar upscaled permeability is possible only for spherical circular numerical elements otherwise the upscaled permeability has to be a tensor of anisotropic covariance if the actual formation has a scalar permeability of axisymmetric covariance upscaling preserves the last property only for axisymmetric partition elements i e for a sphere cylinder and ellipsoid explicit expressions for the first moments of the upscaled permeability mean covariance are derived at first order in the log permeability variance the detailed computations for a scalar permeability of axisymmetric covariance and for axisymmetric numerical elements lead to simple results the upscaled permeability expected values are components of an axisymmetric tensor whereas the fluctuations are determined by a scalar random space function of anisotropic covariance the general case of upscaling by a first order approximation is examined in appendix b these results will be applied to a few particular cases in part 3 indelman this issue

general explicit difference formulas for numerical differentiation through introducing the generalized vandermonde determinant the linear algebraic system of a kind of vandermonde equations is solved analytically by use of the basic properties of this determinant and then we present general explicit finite difference formulas with arbitrary order accuracy for approximating first and higher derivatives which are applicable to unequally or equally spaced data comparing with other finite difference formulas the new explicit difference formulas have some important advantages basic computer algorithms for the new formulas are given and numerical results show that the new explicit difference formulas are quite effective for estimating first and higher derivatives of equally and unequally spaced data

wg a family of stream ciphers with designed randomness properties in this paper we present a family of stream ciphers which generate a keystream with ideal two level autocorrelation the design also guarantees other randomness properties i e balance long period ideal tuple distribution and high and exact linear complexity we discuss how these properties are achieved by the proposed design and show how to select various parameters to obtain an efficient stream cipher for the desired security level we also show that the proposed generators are secure against time memory data tradeoff attacks algebraic attacks and correlation attacks finally we present wg 128 as a concrete example of a wg stream cipher with a key size of 128bits

aquifer parameter identification with optimum dimension in parameterization this paper presents a systematic procedure whereby the functional coefficients imbedded in a two dimensional partial differential equation which governs unsteady groundwater flow are optimally identified the coefficients to be identified are transmissivities which vary spatially finite elements are used to represent the unknown transmissivity function parametrically in terms of nodal values over a suitable discretization of a flow region a modified gauss newton algorithm is used for parameter optimization covariance analysis is used to estimate the reliability of the estimated parameters as the dimension of the unknown parameter increases the modeling error represented by a least squares criterion will generally decrease but errors in data would be propagated to a greater degree into the estimated parameters thus reducing the reliability of estimation the reliability of the estimated parameters is characterized by a norm of the covariance matrix this information is used for the determination of the optimum dimension in parameterization

binary pseudorandom sequences of period 2 sup n 1 with ideal autocorrelation in this correspondence we present five new classes of binary sequences of period 2 sup n 1 with ideal autocorrelation these sequences which correspond to new cyclic hadamard difference sets were found by extensive computer search conjectures on the general construction of these sequences are formulated

damage identification for beams in noisy conditions based on teager energy operator wavelet transform modal curvature abstract modal curvatures have been widely used in the detection of structural damage attractive features of modal curvature include great sensitivity to damage and instant determination of damage location however an intrinsic deficiency in a modal curvature is its susceptibility to the measurement noise present in the displacement mode shape that produces the modal curvature likely obscuring the features of damage to address this deficiency the teager energy operator together with wavelet transform is tactically utilized to treat modal curvature producing a new modal curvature termed the teager energy operator wavelet transform modal curvature this new modal curvature features distinct capabilities of suppressing noise canceling global trends and intensifying the singular feature caused by damage for a measured mode shape involving noise these features maximize the sensitivity to damage and accuracy of damage localization the proposed modal curvature is demonstrated in several analytical cases of cracked pinned pinned clamped free and clamped clamped beams with emphasis on characterizing damage in noisy conditions and it is further validated by an experimental program using a scanning laser vibrometer to acquire mode shapes of a cracked aluminum beam the teager energy operator wavelet transform modal curvature essentially overcomes the deficiency of conventional modal curvature providing a new dynamic feature well suited for damage characterization in noisy environments the matlab code for implementing teager energy operator wavelet transform modal curvature can be provided by the corresponding author on request

new hardware implementationsof wg bf 29 11 and wg bf 16 streamciphers using polynomial basis the wg stream ciphers are based on the wg welch gong transformation and possess proved randomness properties in this paper we propose nine new hardware designs for the two classes of wg 29 11 and wg 16 for each class we design and implement three versions of standard pipelined and serial for the first time we use the polynomial basis pb representation to design and implement the wg 29 11 and wg 16 we consider traditional pb multiplier for the wg 29 11 and the traditional and karatsuba multipliers for the wg 16 for efficient field operations we propose an irreducible trinomial for the wg 29 11 for the wg 16 a new formulation of its permutation which requires only 8 multipliers is introduced in these designs the multipliers in the transforms arefurther reduced by utilizing a novel computation for the trace of the multiplication of two field elements we have implemented theproposed designs in asic using cmos 65 nm technology the results show that the proposed standard wg 29 11 consumes less area and slightly enhances the normalized throughput compared to the existing counterparts for the wg 16 throughput of theproposed pipelined instance outperforms the previous designs moreover the speed of the proposed wg 16 designs meet the peak bit rates for the 4 g specifications

damage detection from changes in curvature mode shapes abstract a damage in a structure alters its dynamic characteristics the change is characterized by changes in the eigenparameters i e natural frequency damping values and the mode shapes associated with each natural frequency considerable effort has been spent in obtaining a relationship between the changes in eigenparameters the damage location and the damage size most of the emphasis has been on using the changes in the natural frequencies and the damping values to determine the location and the size of the damage in this paper a new parameter called curvature mode shape is investigated as a possible candidate for identifying and locating damage in a structure by using a cantilever and a simply supported analytical beam model it is shown here that the absolute changes in the curvature mode shapes are localized in the region of damage and hence can be used to detect damage in a structure the changes in the curvature mode shapes increase with increasing size of damage this information can be used to obtain the amount of damage in the structure finite element analysis was used to obtain the displacement mode shapes of the two models by using a central difference approximation curvature mode shapes were then calculated from the displacement mode shapes

convergence rates for total variation regularization of coefficient identification problems in elliptic equations ii abstract we investigate the convergence rates for total variation regularization of the problem of identifying i the coefficient q in the neumann problem for the elliptic equation div q u f in q u n g on ii the coefficient a in the neumann problem for the elliptic equation u a u f in u n g on r d d 1 when u is imprecisely given by z h 1 u z h 1 0 we regularize these problems by correspondingly minimizing the strictly convex functionals 1 2 q u q z 2 d x 1 2 q l 2 2 q and 1 2 u a z 2 d x 1 2 a u a z 2 d x 1 2 a l 2 2 a over admissible sets where u q u a is the solution of the first second neumann boundary value problem 0 is the regularization parameter taking the solutions of these optimization problems as the regularized solutions to the corresponding identification problems we obtain the convergence rates of them to the solution of the inverse problem in the sense of the bregman distance and in the l 2 norm under relatively simple source conditions without the smallness requirement on the source functions

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

convergence rates for tikhonov regularization of a two coefficient identification problem in an elliptic boundary value problem we investigate the convergence rates for tikhonov regularization of the problem of simultaneously estimating the coefficients q and a in the neumann problem for the elliptic equation rm div q nabla u au f rm in omega q partial u partial n g on the boundary partial omega omega subset mathbb r d d geq 1 when u is imprecisely given by z delta in h 1 omega u z delta h 1 omega le delta delta 0 we regularize this problem by minimizing the strictly convex functional of q a begin array lll int limits omega left q nabla u q a z delta 2 a u q a z delta 2 right dx quad rho left q q 2 l 2 omega a a 2 l 2 omega right end array over the admissible set k where 0 is the regularization parameter and q a is an a priori estimate of the true pair q a which is identified and consider the unique solution of these minimization problem as the regularized one to that of the inverse problem we obtain the convergence rate mathcal o sqrt delta as 0 and for the regularized solutions under the simple and weak source condition rm there is a function w in v rm such that u prime q dagger a dagger w q dagger q a dagger a with q dagger a dagger being the q a minimum norm solution of the coefficient identification problem u the frechet derivative of u v the sobolev space on which the boundary value problem is considered our source condition is without the smallness requirement on the source function which is popularized in the theory of regularization of nonlinear ill posed problems furthermore some concrete cases of our source condition are proved to be simply the requirement that the sought coefficients belong to certain smooth function spaces

on the condition number of lagrangian numerical differentiation given distinct nodes u u 0 u 1 u n in 1 1 consider the differentiation rule k 0 h email protected n j 0w email protected hu j where w j 1 k j 0 u 1 j t u p n and 1 j u i u d i j the condition number l k n u s n j 0 w j is a measure of the error due to perturbations in the function values hu j t j rivlin siam j numer anal 1975 theoretically determined sets u which minimize the condition number for fixed k n results of rivlin are used to exhibit an explicit formula for the optimal value of l k n u a description is given of a simple and efficient algorithm and a corresponding fortran subroutine for computing derivatives of interpolating polynomials condition numbers obtained from the subroutine are tabulated for optimal equidistant and geometric nodes and for 1 k n 15 andk n of the same parity the computed values whose relative errors are theoretically bounded by machine precision are remarkably free from roundoff amplification the subroutine can be used as the basis of a simple program that returns approximate derivatives and corresponding error estimates

optimal spatial sampling interval for damage detection by curvature or strain energy mode shapes vibration based methods of non destructive damage detection utilizing curvature or strain energy mode shapes have been applied in a variety of applications attractive features of these methods include high sensitivity to damage and instant determination of the damage location however the quality of damage detection achieved in practice depends upon determining a proper sampling interval for discretization of the displacement mode shapes experimental observations show that both undersampling and oversampling of the displacement mode shapes may have adverse effects on the quality of damage detection this paper presents an analysis aimed at determining the optimal sampling interval that would minimize the effects of measurement noise and truncation errors on the calculation of the curvature and strain energy mode shapes thus maximizing sensitivity to damage and accuracy of damage localization derivation of the formulas for the optimal sampling interval is based on the most commonly used numerical methods for the computation of the curvature and the strain energy mode shapes numerical verification has shown very good performance of the suggested formulas in predicting the optimal sampling interval

spatial variability of soil moisture and the scale issue a geostatistical approach we study the standard deviation of water saturation sds as function of the mean saturation s by a stochastic model of unsaturated flow which is based on the first order solution of the three dimensional richards equation the model assumes spatially variable soil properties following a given geostatistical description and it explicitly accounts for the different scales involved in the determination of the spatial properties of saturation the extent l i e the domain size the spacing among measurements and the dimension l associated to the sampling measurement it is found that the interplay between those scales and the correlation scale i of the hydraulic properties rules the spatial variability of saturation a scale effect manifests for small to intermediate l i for which sds increase with the extent l this nonergodic effect depends on the structural and hydraulic parameters as well as the scales of the problem and it is consistent with a similar effect found in field experiments in turn the influence of the scale l is to decrease the saturation variability and increase its spatial correlation although the solution focuses on the medium heterogeneity as the main driver for the spatial variability of saturation neglecting other important components it explicitly links the spatial variation of saturation to the hydraulic properties of the soil their spatial variability and the sampling schemes it can provide a useful tool to assess the impact of scales on the saturation variability also in view of the several applications that involve the saturation variability

computationally efficient algorithms for parameter estimation and uncertainty propagation in numerical models of groundwater flow finite difference and finite element methods are frequently used to study aquifer flow however additional analysis is required when model parameters and hence predicted heads are uncertain computational algorithms are presented for steady and transient models in which aquifer storage coefficients transmissivities distributed inputs and boundary values may all be simultaneously uncertain innovative aspects of these algorithms include a new form of generalized boundary condition a concise discrete derivation of the adjoint problem for transient models with variable time steps an efficient technique for calculating the approximate second derivative during line searches in weighted least squares estimation and a new efficient first order second moment algorithm for calculating the covariance of predicted heads due to a large number of uncertain parameter values the techniques are presented in matrix form and their efficiency depends on the structure of sparse matrices which occur repeatedly throughout the calculations details of matrix structures are provided for a two dimensional linear triangular finite element model

a variational method for numerical differentiation a new method is given for effecting numerical differentiation by means of an optimization procedure the method is shown to be effective for the differentiation of noisy functions and stability and convergence results for a steepest descent implementation are proved

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

the behavior of higher order mode shape derivatives in damaged beam like structures this work considers the effect that damage has upon the higher order derivatives of mode shapes of structures having primarily beam like vibration via numerical investigations the sensitivity of various damage related parameters in inducing changes in these higher order modal derivatives is determined leading to a more complete understanding of what factors make the most contribution to significant changes in these derivatives it is concluded that higher order mode shape derivatives e g modal curvature third derivative and fourth derivative are better indicators of damage than the mode shapes three distinct types of response for the damage induced higher order derivative discontinuities are identified as three key parameters the mass loss stiffness loss and damage radius scale vary from this formal approximations are obtained for the expected forms of the higher order derivative discontinuities based upon the underlying behavior predicted by a simple relation among these three parameters these approximations are checked with numerical simulations and an excellent level of agreement is observed under appropriate conditions finally the potential of these higher order derivative changes for indicating the presence and location of damage in a global setting is examined

am fm energy detection and separation in noise using multiband energy operators this paper develops a multiband or wavelet approach for capturing the am fm components of modulated signals immersed in noise the technique utilizes the recently popularized nonlinear energy operator psi s s sup 2 ss to isolate the am fm energy and an energy separation algorithm esa to extract the instantaneous amplitudes and frequencies it is demonstrated that the performance of the energy operator esa approach is vastly improved if the signal is first filtered through a bank of bandpass filters and at each instant analyzed via psi and the esa using the dominant local channel response moreover it is found that uniform worst case performance across the frequency spectrum is attained by using a constant q or multiscale wavelet like filter bank the elementary stochastic properties of psi and of the esa are developed first the performance of psi and the esa when applied to bandpass filtered versions of an am fm signal plus noise combination is then analyzed the predicted performance is greatly improved by filtering if the local signal frequencies occur in band these observations motivate the multiband energy operator and esa approach ensuring the in band analysis of local am pm energy in particular the multi bands must have the constant q or wavelet scaling property to ensure uniform performance across bands the theoretical predictions and the simulation results indicate that improved practical strategies are feasible for tracking and identifying am fm components in signals possessing pattern coherencies manifested as local concentrations of frequencies

a stable computation of log derivatives from noisy drawdown data pumping tests interpretation is an art that involves dealing with noise coming from multiple sources and conceptual model uncertainty interpretation is greatly helped by diagnostic plots which include drawdown data and their derivative with respect to log time called log derivative log derivatives are especially useful to complement geological understanding in helping to identify the underlying model of fluid flow because they are sensitive to subtle variations in the response to pumping of aquifers and oil reservoirs the main problem with their use lies in the calculation of the log derivatives themselves which may display fluctuations when data are noisy to overcome this difficulty we propose a variational regularization approach based on the minimization of a functional consisting of two terms one ensuring that the computed log derivatives honor measurements and one that penalizes fluctuations the minimization leads to a diffusion like differential equation in the log derivatives and boundary conditions that are appropriate for well hydraulics i e radial flow wellbore storage fractal behavior etc we have solved this equation by finite differences we tested the methodology on two synthetic examples showing that a robust solution is obtained we also report the resulting log derivative for a real case

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

an evaluation of jacob s method for the interpretation of pumping tests in heterogeneous formations most pumping tests are interpreted using the classical theis assumption of large scale homogeneity with various corrections to account for early time behavior of drawdown curves when drawdowns are plotted versus log time late time data often delineate a straight line which is consistent with jacob s approximation of theis solution but may seem surprising in view of the heterogeneity of natural media the aim of our work is to show that jacob s method leads to a good approximation of the effective transmissivity of heterogeneous media when constrained to late time data a review of several multiwell pumping tests demonstrates that when drawdown curves from each observation well are interpreted separately they produce very similar transmissivity t estimates however the corresponding estimates for storativity span a broad range this behavior is verified numerically for several models of formation heterogeneity a very significant finding of the numerical investigation is that t values estimated using simulated drawdown from individual observation wells are all very close to the effective t value for parallel flow this was observed even in nonmultigaussian t fields where high t zones are well connected and where the effective t is larger than the geometric average of point values this implies that jacob s method can be used for estimating effective t values in many if not most formations

an identification technique for non linear dynamical systems under stochastic excitations an identification technique is devised for sdof dynamical mechanical systems under random excitations the system is assumed to be governed by a non linear equation of motion in general form in which the restoring force and the dissipative terms are given by arbitrary power functions algebraic equations are obtained for the expectations of some suitable excitation and response quantities it is shown that these equations are valid for any stationary random excitations if the system attains the steady state based on these equations an identification technique has been devised and verified experimentally for white noise and coloured pink noise random excitations

an application of the geostatistical approach to the inverse problem in two dimensional groundwater modeling the geostatistical approach to the estimation of transmissivity from head and transmissivity measurements is developed for two dimensional steady flow the field of the logarithm of transmissivity log transmissivity is represented as a zero order intrinsic random field its spatial structure is described in this application through a two term covariance function that is linear in the parameters 1 and 2 linearization of the discretized flow equations allows the construction of the joint covariance matrix of the head and log transmissivity measurements as a linear function of 1 and 2 in this particular application the coefficient matrices are calculated numerically in a noniterative fashion maximum likelihood estimation is employed to estimate 1 and 2 as well as additional parameters from measurements linear estimation theory cokriging then yields point or block averaged estimates of transmissivity the approach is first applied to a test case with favorable results it is shown that the application of the methodology gives good estimates of transmissivities it is also shown that when the transmissivities are used in a numerical model they reproduce the head measurements quite well results from the application of the methodology to the jordan aquifer in iowa are also presented

estimation of aquifer parameters under transient and steady state conditions 1 maximum likelihood method incorporating prior information in this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions the parameters include values and directions of principal hydraulic conductivities or transmissivities in anisotropic media specific storage or storativity interior and boundary recharge or leakage rates coefficients of head dependent interior and boundary sources and boundary heads in transient situations the initial head distribution can also be estimated if the system is originally at a steady state paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters the inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters since not all the factors which contribute to the prior errors can be quantified statistically at the outset the covariance matrices of these errors are expressed in terms of several parameters which if unknown can be estimated jointly with the hydraulic parameters by a stagewise optimization process when transient head data are separated by a fixed time interval the temporal structure of these data is approximated by a lag one autoregressive model with a correlation coefficient that can be treated as another unknown parameter estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature all of which are based on the maximum likelihood concept might be useful for selecting the best groundwater model or the best method of parameterizing a particular model among a number of given alternatives

transmissivity upscaling in numerical aquifer models of steady well flow unconditional statistics 1 numerical solution of regional scale aquifer flow requires discretizing the transmissivity t typically the numerical element scale r and the transmissivity integral scale i are both of the order of hundreds of meters then the upscaled block transmissivity is random and its statistical moments depend on those of t on flow conditions and on r i modeling y ln t as a two dimensional normally distributed stationary random field we derive unconditional statistics of the upscaled ln accurate to the first order in y2 for strongly nonuniform flow in a circular block of radius r centered on a well of radius rw in an unbounded domain after a proper definition it was found that the ensemble mean is approximately but not exactly equal to ln tg the geometric mean and that the variance ratio 2 y2 which depends on rw i and the shape of the correlation drops slowly from unity for r i 0 to approximately 0 6 for r i 10 hence the variance of the upscaled transmissivity in radial flow is larger than that determined previously for uniform flow additionally defining the equivalent teq as a deterministic value for which the solution of the flow problem renders directly the mean upscaled head drop and specific discharge we find that in radial flow teq th the harmonic mean and grows slowly with increasing r i whereas for mean uniform flow teq tg application of the procedure is illustrated for an example of aquifer with selected values of parameters

aquifer parameter identification with kriging and optimum parameterization a new parameter identification method is developed for a two dimensional unsteady state groundwater flow model the method is a direct approach as it requires no iterative searching of parameters the unknown functional coefficients parameters are optimally determined by a generalized matrix inversion scheme kriging an unbiased minimum variance estimator is used as a presampling filter to reconstruct the head distribution for the entire flow domain the ill posedness of the inverse problem as characterized by instability and nonuniqueness is controlled by reparameterization utilizing the technique of finite elements a set of statistics are developed to quantify explicitly the tradeoff between the system modeling error and the error associated with the parameter uncertainty these two types of errors are shown to be a function of the parameter dimension residual mean square and trace of the covariance matrix of the estimated parameters serve as good indicators for the determination of the optimum parameter dimension a set of numerical experiments are conducted to illustrate the methodology

numerical aspects for approximating governing equations using data abstract we present effective numerical algorithms for approximating unknown governing differential equations from measurement data we employ a set of standard basis functions e g polynomials to approximate the governing equation with high accuracy upon recasting the problem into a function approximation problem we discuss several important aspects for accurate approximation most notably we discuss the importance of using a large number of short bursts of trajectory data rather than using data from a single long trajectory several options for the numerical algorithms to perform accurate approximation are then presented along with an error estimate of the final equation approximation we then present an extensive set of numerical examples of both linear and nonlinear systems to demonstrate the properties and effectiveness of our equation approximation algorithms

binary pseudorandom sequences of period 2 sup m 1 with ideal autocorrelation generated by the polynomial z sup d z 1 sup d we present a construction for binary pseudorandom sequences of period 2 sup m 1 with ideal autocorrelation property using the polynomial z sup d z 1 sup d we show that the sequence obtained from the polynomial becomes an m sequence for certain values of d we also find a few values of d which yield new binary sequences with ideal autocorrelation property when m is 3k spl plusmn 1 where k is a positive integer these new sequences are represented using a trace function and the results are tabulated

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

strategic placement of localization devices such as pilot points and anchors in inverse modeling schemes 1 localization schemes using devices such as pilot points or anchors localization devices are used in inversion schemes as a tool for gleaning information from measurements that can be used to constrain inversion schemes with the added benefit in the form of a potential for reduction in the number of parameters employed in the inversion scheme this paper proposes and demonstrates a method for strategic placement of the localization devices the method combines stochastic singular value decomposition with parameter rejection methods to produce a map of intensity scores which allows the inverse modeler to place the localization devices at locations defined by their probability for producing informative probabilistic constraints on the statistical distributions of the target variables we also show that the method can be used for selecting measurement locations when designing data acquisition campaigns the method is demonstrated for the case of steady state flow in a heterogeneous conductivity field using mad method of anchored distributions in summary the proposed localization scheme intends to increase the information gleaned from measurements while reducing the associated computational costs associated with stochastic inversion

variational method for multiple parameter identification in elliptic pdes abstract in the present paper we investigate the inverse problem of identifying simultaneously the diffusion matrix source term and boundary condition in the neumann boundary value problem for an elliptic partial differential equation pde from a measurement data which is weaker than required of the exact state a variational method based on energy functions with tikhonov regularization is here proposed to treat the identification problem we discretize the pde with the finite element method and prove the convergence as well as analyze error bounds of this approach

a reassessment of the groundwater inverse problem this paper presents a functional formulation of the groundwaterflow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a possibly infinite basis function expansion with random coefficients the unknown parameter function is related to the measurements used for estimation by a forward operator which describes the measurement process in the particular case considered here the parameter of interest is the large scale log hydraulic conductivity the measurements are point values of log conductivity and piezometric head and the forward operator is derived from an upscaled groundwaterflow equation the inverse algorithm seeks the most probable or maximum a posteriori estimate of the unknown parameter function when the measurement errors and parameter function are gaussian and independent the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness of fit and prior terms when the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate this constraint tends to make the problem well posed by limiting the range of admissible solutions the gaussian maximum a posteriori problem may be solved with variational methods using functional generalizations of gauss newton or gradient based search techniques several popular groundwater inverse algorithms are either special cases of or variants on the functional maximum a posteriori algorithm these algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use linear versus nonlinear the accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a bayesian extension of the cramer rao lower bound on the estimation error covariance this bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements

a continuous time inverse operator for groundwater and contaminant transport modeling model identifiability general identifiability conditions are derived for the model coefficients of the spatially discretized continuous time diffusion equation these conditions in turn determine some sufficient and necessary conditions for the existence and uniqueness of the continuous time inverse operator described in a previous article the identifiability conditions are in terms of the initial conditions and the flux vector containing boundary condition information and the eigenspace of the spatially discretized system the derivation of these conditions gives some insight into the location of the solution of the inverse problem in terms of the direct operator under investigation in the space of the laplace transform and limited insight into the stability of the solution near that location

stochastic subsurface hydrology from theory to applications research on stochastic analysis of subsurface flow has developed rapidly in the last decade but applications of this approach have been very limited the purpose of this paper is to illustrate how currently available techniques and results can be used to answer important questions about the large scale behavior of naturally heterogeneous aquifers perturbation based spectral theory which presumes local statistical homogeneity provides generic theoretical results for the head variance effective conductivity tensor and macrodispersivity tensor in a field these results emphasize the key role of the variance and spatial correlation scales of the log hydraulic conductivity field field information of variances and correlation scales of natural materials is summarized the validity of some of the generic stochastic results is evaluated through comparisons with monte carlo simulations and field observations a specific field application example is developed to illustrate how the stochastic results are used to estimate large scale parameters and determine the reliability of three dimensional numerical simulations using typical log conductivity covariance parameters the effective hydraulic conductivity tensor and the macrodispersivity tensor are estimated the calculated head variance based on the simulated mean hydraulic gradient is used as a measure of the adequacy of the calculation of the steady state flow model discussion emphasizes limitations and extensions of this approach and ongoing field evaluations of the results

descent methods for inverse problems of fundamental importance in modelling is the problem of recovering coe cient functions in a difierential equation from appropriate measurements on the solution thisisinvestigatedbymeansofsteepestdescenttechniquesusingcertainnew banach space gradients

estimation of aquifer parameters under transient and steady state conditions 2 uniqueness stability and solution algorithms paper 2 of this three part series starts with a discussion of the question under what conditions is the aquifer inverse problem well posed after defining the terms uniqueness identifiability and stability theoretical considerations and synthetic examples are used to demonstrate that ill posedness can be mitigated by including prior information about the parameters in the estimation criterion to be minimized at the same time the inclusion of such information is shown to be insufficient to guarantee uniqueness and stability in all cases several test problems in the recent literature which have resulted in pessimistic conclusions about the solvability of the aquifer inverse problem are shown to be ill posed a question is thus raised about the validity of these conclusions in the general case various conjugate gradient algorithms coupled with the adjoint state finite element method for computing the gradient of the estimation criterion and with newton s method for determining the optimum step size downgradient are compared a marked improvement in the rate at which these algorithms converge is shown to be achieved by switching from one method to another when the former slows down or fails to converge

field statistics in nonlinear composites i theory this work presents a means for extracting the statistics of the local fields in nonlinear composites from the effective potential of suitably perturbed composites the idea is to introduce a parameter in the local potentials generally a tensor such that differentiation of the corresponding effective potential with respect to the parameter yields the volume average of the desired quantity in particular this provides a generalization to the nonlinear case of well known formulas in the context of linear composites which express phase averages and second moments of the local fields in terms of derivatives of the effective potential such expressions are useful since they allow the generation of estimates for the field statistics in nonlinear composites directly from homogenization estimates for appropriately defined effective potentials here use is made of these expressions in the context of the variational tangent second order and second order homogenization methods to obtain rigorous estimates for the first and second moments of the fields in nonlinear composites while the variational estimates for these quantities are found to be identical to those proposed in previous works the tangent second order and second order estimates are found be different in particular the new estimates for the first moments given in this work are found to be entirely consistent with the corresponding estimates for the macroscopic behaviour sample results for two phase power law composites are provided in part ii of this work

a synthesis of approaches to upscaling of hydraulic conductivities simulation of flow through heterogeneous media often requires discretizing the flow domain into blocks and assigning an equivalent block conductivity value to each one of them the process of defining block conductivities from point values is termed upscaling a number of approaches to upscaling are available most of which consider the uncertainty associated with any natural property so that they cast the problem in a stochastic frame recently indelman and dagan 1993a b provided a general stochastic methodology to upscaling in heterogeneous anisotropic formations by means of the dissipation energy function unfortunately they did not provide any practical method to compute block values from point ones the objective of this work is twofold first we analyze different practical approaches to compute block conductivities and find that all of them provide very similar results in terms of actual computed values second we check that all approaches verify approximately a number of conditions stated by indelman and dagan 1993a specifically we show analytically that for regular blocks the methodologies of both rubin and gomez hernandez 1990 and desbarats 1992 which we call practical methodologies satisfy the condition that the effective conductivity obtained from a field where the elementary conductivities are defined over a certain support we call this the actual formation is identical to that obtained from the same field with conductivities defined at a larger support upscaled formation the analysis is carried out by working with the logarithm of block conductivities and using a small perturbation expansion and thus is strictly valid for small variances on the other hand we show numerically that the two methodologies satisfy approximately an important condition stated in terms of the dissipation energy that block averaged dissipation values computed are indeed very close to the true dissipation values in each block the agreement is even better if we consider statistical moments instead of point values as an important conclusion we should note that all practical methodologies considered in this work perform equally well and more important constitute a simple way to treat an otherwise very complex problem

a simple and efficient space domain implementation of the turning bands method given a correlation function c x with x in rn the turning bands method tbm for n 2 and 3 constructs realizations of an n dimensional and stationary process y n o c x from appropriately summed line processes therefore an implementation of tbm calls naturally for fast and accurate generations of line realizations these have generally been generated by a spectral approach since the fourier transform of the line covariances is linked in a very simple fashion to the n dimensional fourier transform of c x however we show that in many instances a space domain implementation of tbm with line realizations obtained from a circulant embedding method has advantages over a spectral implementation in that without increase in computational costs 1 the line realizations display exactly the required covariance structure 2 knowledge of the spectral density of the process y is not required and 3 fine tuning of line process parameters is not needed

correlation immunity of nonlinear combining functions for cryptographic applications corresp pseudonoise generators for cryptographic applications consisting of several linear feedback shift registers with a nonlinear combining function have been proposed as running key generators in stream ciphers these running key generators eau sometimes be broken by ciphertext only correlation attacks on individual subsequences a new class of combining functions is presented which provides better security against such attacks the security is quantified by the smallest number m 1 of subsequences that must be simultaneously considered in a correlation attack a necessary condition for such m th order correlation immunity is proved a recursive construction is given that permits the construction of an m th order immune combining function for n subsequences for any m and n with 1 leq m finally the trade off between the length of the linear equivalent of the nonlinear generator and the order m of its immunity against correlation attacks is considered

parameter identification of groundwater aquifer models a generalized least squares approach this paper concerns the methods of estimating aquifer transmissivities on the basis of unsteady state hydraulic head data traditionally the criterion of minimizing the sum of the squares of errors has been used to match the observed data with the model response the data used for optimization usually contain noise that is not necessarily uncorrelated it is well understood that the results of identification methods are very sensitive to measurement errors in data in this study the ordinary least squares ols technique is carried out along with a generalized least squares gls technique specifically designed to reduce the effect of correlated errors the trace of the covariance matrix is used as a measure of overall accuracy and reliability of the estimated parameters the effectiveness of the ols and gls techniques in dealing with noisy data is demonstrated by using a hypothetical example the results of numerical experiments suggest that gls offers a promising approach in efficiently improving the reliability of the estimated parameters

incorporation of prior information on parameters into nonlinear regression groundwater flow models 1 theory prior information on the parameters of a groundwater flow model can be used to improve parameter estimates obtained from nonlinear regression solution of a modeling problem two scales of prior information can be available 1 prior information having known reliability that is bias and random error structure and 2 prior information consisting of best available estimates of unknown reliability a regression method that incorporates the second scale of prior information assumes the prior information to be fixed for any particular analysis to produce improved although biased parameter estimates approximate optimization of two auxiliary parameters of the formulation is used to help minimize the bias which is almost always much smaller than that resulting from standard ridge regression it is shown that if both scales of prior information are available then a combined regression analysis may be made

numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials the homogenization of nonlinear heterogeneous materials is much more difficult than the homogenization of linear ones this is mainly due to the fact that the general form of the homogenized behavior of nonlinear heterogeneous materials is unknown at the same time the prevailing numerical methods such as concurrent methods require extensive computational efforts a simple numerical approach is proposed to compute the effective behavior of nonlinearly elastic heterogeneous ma terials at small strains the proposed numerical approach comprises three steps at the rst step a representative volume element rve for a given nonlinear heterogeneous material is de ned and a loading space consisting of all the boundary conditions to be imposed on the rve is discretized into a sufficiently large number of points called nodes at the second step the boundary condition corresponding each node is prescribed on the surface of the rve and the resulting nonlinear boundary value problem is solved by the nite element method fem so as to determine the effective response of the heterogeneous material to the loading as sociated to each node of the loading space at the third step the nodal effective responses are interpolated via appropriate interpolation functions so that the effective strain energy stress strain relation and tangent stiffness tensor of the nonlinear heterogeneous material are provided in a numerically explicit way this leads to a non concurrent nonlinear multiscale approach to the computation of structures made of nonlinearly heterogeneous materials the rst version of the proposed approach uses mutidimensional cubic splines to interpolate effective nodal responses while the second version of the proposed approach takes advantage of an outer product decomposition of multidimensional data into rank one tensors to interpolate effective nodal responses and avoid high rank data these two versions of the proposed approach are applied to a few examples where nonlinear composites whose phases are characterized by the power law model are involved the numerical results given by our approach are compared with available analytical estimates exact results and full fem or concurrent multilevel fem solutions

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

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

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 1 theory and computational approach we consider the effect of measuring randomly varying hydraulic conductivitiesk x on one s ability to predict numerically without resorting to either monte carlo simulation or upscaling steady state flow in bounded domains driven by random source and boundary terms our aim is to allow optimum unbiased prediction of hydraulic heads h x and fluxes q x by means of their ensemble moments h x c and q x c respectively conditioned on measurements of k x these predictors have been shown by neuman and orr 1993a to satisfy exactly an integrodifferential conditional mean flow equation in which q x c is nonlocal and non darcian here we develop complementary integrodifferential equations for second conditional moments of head and flux which serve as measures of predictive uncertainty obtain recursive closure approximations for both the first and second conditional moment equations through expansion in powers of a small parameter y which represents the standard estimation error of ln k x and show how to solve these equations to first order in y2 by finite elements on a rectangular grid in two dimensions in the special case where one treats k x as if it was locally homogeneous and mean flow as if it was locally uniform one obtains a localized darcian approximation q x c kc x h x c in which kc x is a space dependent conditional hydraulic conductivity tensor this leads to the traditional deterministic darcian steady state flow equation which however acquires a nontraditional meaning in that its parameters and state variables are data dependent and therefore inherently nonunique it further explains why parameter estimates obtained by traditional inverse methods tend to vary as one modifies the database localized equations yield no information about predictive uncertainty our stochastic derivation of these otherwise standard deterministic flow equations makes clear that uncertainty measures associated with estimates of head and flux obtained by traditional inverse methods are generally smaller often considerably so than measures of corresponding predictive uncertainty which can be assessed only by means of stochastic models such as ours we present a detailed comparison between finite element solutions of nonlocal and localized moment equations and monte carlo simulations under superimposed mean uniform and convergent flow regimes in two dimensions paper 1 presents the theory and computational approach and paper 2 guadagnini and neuman this issue describes unconditional and conditional computational results

a differential homogenization method for estimating the macroscopic response and field statistics of particulate viscoelastic composites abstract this work provides an analytical method for determining the time dependent effective constitutive response of linear viscoelastic composites with particulate microstructures by means of a differential algebraic system das of equations by making use of differential equations instead of difference equations and utilizing as primary variables the variances of the field fluctuations instead of the properties of fictitious linear comparison composites the new formulation is found to be more robust and to alleviate some of the issues associated with earlier incremental approaches remarkably the new formulation can be shown to recover the exact results of the correspondence principle for isotropic incompressible composites subjected to proportional loading as well as for isotropic compressible composites subjected to purely hydrostatic loading more generally the proposed formulation is not exact but still delivers reasonably accurate estimates in most cases lastly a generalization of the das formulation is provided for nonlinear viscoelastic composites with incompressible phases and also found to be more robust than earlier approaches

bayesian inversion for facies detection an extensible level set framework 1 in many cases it has been assumed that the variability in hydrologic parameters can be adequately described through a simple geostatistical model with a given variogram in other cases variability may be best described as a series of jumps in parameter behavior e g those that occur at geologic facies contacts when using indirect measurements such as pump tests to try to map such heterogeneity during inverse modeling the resulting images of the subsurface are always affected by the assumptions invoked in this paper we discuss inversion for parameter fields where prior information has suggested that major variability can be described by boundaries between geologic units or facies in order to identify such parameter fields we propose a bayesian level set inversion protocol framework which allows for flexible zones of any shape size and number we review formulas for defining facies locations using the level set method and for moving the boundaries between zones using a gradient based technique that improves fit through iterative deformation of the boundaries we describe the optimization algorithm employed when multiple level set functions are used to represent a field with more than two facies we extend these formulas to the investigation of the inverse problem in a bayesian context in which prior information is taken into account and through which measures of uncertainty can be derived we also demonstrate that the level set method is well suited for joint inversion problems and present a strategy for integrating different data types such as hydrologic and geophysical without assuming strict petrophysical relations our framework for joint inversion also contrasts with many previous methods in that all data sources e g both hydrologic and geophysical contribute to boundary delineation at once

improved methodology for parameter inference in nonlinear hydrologic regression models a new method is developed for the construction of reliable marginal confidence intervals and joint confidence regions for the parameters of nonlinear hydrologic regression models a parameter power transformation is combined with measures of the asymptotic bias and asymptotic skewness of maximum likelihood estimators to determine the transformation constants which cause the bias or skewness to vanish these optimized constants are used to construct confidence intervals and regions for the transformed model parameters using linear regression theory the resulting confidence intervals and regions can be easily mapped into the original parameter space to give close approximations to likelihood method confidence intervals and regions for the model parameters unlike many other approaches to parameter transformation the procedure does not use a grid search to find the optimal transformation constants an example involving the fitting of the michaelis menten model to velocity discharge data from an australian gauging station is used to illustrate the usefulness of the methodology

a global multilevel coordinate search procedure for estimating the unsaturated soil hydraulic properties we present a new inverse modeling procedure to characterize the hydraulic properties of partially saturated soils from soil moisture measurements during a natural transient flow experiment the inversion of the governing one dimensional richards equation is carried out using the global multilevel coordinate search optimization algorithm in sequential combination with the local nelder mead simplex algorithm gmcs nms we introduce this optimization method in the area of unsaturated zone hydrology since it is adapted for solving accurately and efficiently complex nonlinear problems several numerical experiments have been conducted to evaluate the proposed inversion method using synthetic error free and error contaminated data for different textured soils inversion of the simulated error free data and examination of the related response surfaces demonstrated the uniqueness of the inverse solution and the suitability of the gmcs nms strategy when identifying four key parameters of the hydraulic functions described by the mualem van genuchten model inversion of the error contaminated data proved further the good stability of the inverse solution that is consistent with the needs required by real experiments

conditional well posedness for an elliptic inverse problem for the elliptic equation nabla cdot p x nabla v lambda q x v f x in omega subset mathbb r n the problem of determining when one or more of the coefficient functions p q and f depends continuously on the solution functions v v p q f lambda is considered

the use of modal curvatures for damage localization in beam type structures abstract the localization of stiffness variation in damaged beams through modal curvatures i e second derivative of mode shapes is studied by exploiting a perturbative solution of the euler bernoulli equation it is shown that for low order modes the difference between undamaged and damaged modal curvatures has only one distinct peak if the damage is localized in a narrow region this phenomenon is independent of the presence of experimental noise and of the technique used to reconstruct the curvature mode shapes from the displacement mode shapes broader damages cause the modal curvature difference to have several peaks outside the damage region that could result in a false damage localization the same effect is present at higher modes for both narrow and broad damages as a result modal curvatures can be effectively used to localize structural damages only once they have been properly filtered here the perturbative solution is used to introduce an effective damage measure able to localize correctly narrow and broad damages and also single and multiple damages cases

multimodel bayesian analysis of groundwater data worth we explore the way in which uncertain descriptions of aquifer heterogeneity and groundwater flow impact one s ability to assess the worth of collecting additional data we do so on the basis of maximum likelihood bayesian model averaging mlbma by accounting jointly for uncertainties in geostatistical and flow model structures and parameter hydraulic conductivity as well as system state hydraulic head estimates given uncertain measurements of one or both variables previous description of our approach was limited to geostatistical models based solely on hydraulic conductivity data here we implement the approach on a synthetic example of steady state flow in a two dimensional random log hydraulic conductivity field with and without recharge by embedding an inverse stochastic moment solution of groundwater flow in mlbma a moment equations based geostatistical inversion method is utilized to circumvent the need for computationally expensive numerical monte carlo simulations the approach is compatible with either deterministic or stochastic flow models and consistent with modern statistical methods of parameter estimation admitting but not requiring prior information about the parameters it allows but does not require approximating lead predictive statistical moments of system states by linearization while updating model posterior probabilities and parameter estimates on the basis of potential new data both before and after such data are actually collected

conditional stochastic averaging of steady state unsaturated flow by means of kirchhoff transformation we consider the effect of measuring randomly varying soil hydraulic properties on one s ability to predict steady state unsaturated flow subject to random sources and or initial and boundary conditions our aim is to allow optimum unbiased prediction of system states pressure head water content and fluxes deterministically without upscaling and without linearizing the constitutive characteristics of the soil it has been shown by neuman et al 1999 that such prediction is possible by means of first ensemble moments of system states and fluxes conditioned on measured values of soil properties when the latter scale in a linearly separable fashion as proposed by vogel et al 1991 the uncertainty associated with such predictions can be quantified by means of the corresponding conditional second moments the derivation of moment equations for soils whose properties do not scale in the above manner requires linearizing the corresponding constitutive relations which may lead to major inaccuracies when these relations are highly nonlinear as is often the case in nature when the scaling parameter of pressure head is a random variable independent of location the steady state unsaturated flow equation can be linearized by means of the kirchhoff transformation for gravity free flow linearization is also possible in the presence of gravity when hydraulic conductivity varies exponentially with pressure head for the latter case we develop exact conditional first and second moment equations which are nonlocal and therefore non darcian we solve these equations analytically by perturbation for unconditional vertical infiltration and compare our solution with the results of numerical monte carlo simulations our analytical solution demonstrates in a rigorous manner that the concept of effective hydraulic conductivity does not apply to ensemble averaged unsaturated flow except when gravity is the sole driving force

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

incorporation of prior information on parameters into nonlinear regression groundwater flow models 2 applications this paper investigates factors influencing the degree of improvement in estimates of parameters of a nonlinear regression groundwater flow model by incorporating prior information of unknown reliability consideration of expected behavior of the regression solutions and results of a hypothetical modeling problem lead to several general conclusions first if the parameters are properly scaled linearized expressions for the mean square error mse in parameter estimates of a nonlinear model will often behave very nearly as if the model were linear second by using prior information the mse in properly scaled parameters can be reduced greatly over the mse of ordinary least squares estimates of parameters third plots of estimated mse and the estimated standard deviation of mse versus an auxiliary parameter the ridge parameter specifying the degree of influence of the prior information on regression results can help determine the potential for improvement of parameter estimates fourth proposed criteria can be used to make appropriate choices for the ridge parameter and another parameter expressing degree of overall bias in the prior information results of a case study of truckee meadows reno sparks area washoe county nevada conform closely to the results of the hypothetical problem in the truckee meadows case incorporation of prior information did not greatly change the parameter estimates from those obtained by ordinary least squares however the analysis showed that both sets of estimates are more reliable than suggested by the standard errors from ordinary least squares

bounds for the effective constitutive relation of a nonlinear composite for a nonlinear composite a bound on its effective energy density does not induce a corresponding bound on its constitutive relation because differentiating a bound on a function does not automatically bound its derivative in this work a method introduced by milton and serkov for bounding directly the constitutive relation is refined by employing a linear comparison material in a way similar to that employed by the present authors to obtain bounds of hashin shtrikma type for the effective energy of a nonlinear composite the original milton serkov approach produces bounds with a close relationship to the classical energy bounds of voigt and reuss type the bounds produced in the present implementation are closely related to bounds of hashin shtrikman type for the composite indeed in the case of a linear composite the present method delivers the hashin shtrikman bounds in their generalized form valid for any two point statistics it is demonstrated for nonlinear examples that the approximate constitutive relation that is obtained by differentiating the energy bound can be on the boundary of the bounding set for the exact constitutive relation but a simple counterexample is presented to show that this is not always the case

cryptographic properties of the welch gong transformation sequence generators welch gong wg transformation sequences are binary sequences of period 2 sup n 1 with two level autocorrelation these sequences were discovered by golomb gong and gaal 1998 and they verified the validity of their construction for 5 spl les n spl les 20 later no chung and yun 1998 found another way to construct the wg sequences and verified their result for 5 spl les n spl les 20 dillon 1998 first proved this result for odd n and finally dobbertin and dillon 1999 proved it for even n in this paper we investigate a two faced property of the wg transformation sequences for application in stream ciphers and pseudorandom number generators one is to present the randomness or unpredictability of the wg transformation sequences the other is to exhibit the security properties of the wg transformations regarded as boolean functions in particular we prove that the wg transformation sequences in addition to the known two level autocorrelation and three level cross correlation with m sequences have the ideal 2 tuple distribution and large linear span increasing exponentially with n moreover it can be implemented efficiently this is the first type of pseudorandom sequences with good correlation statistic properties large linear span and efficient implementation when wg transformations are regarded as boolean functions they have high nonlinearity we derive a criterion for the boolean representation of wg transformations to be r resilient and show that they are at least 1 resilient under some basis of the finite field gf 2 sup n an algorithm to find such bases is given the degree and linear span of wg transformations are presented as well