first order variance of travel time in nonstationary formations 1 evaluating uncertainty in solute transport under nonstationary flow conditions is a computationally demanding task this is particularly true for cases with a two point covariance function of log conductivity depending on the actual positions of the points rather than their distance vector these may occur when the geological formation exhibits a trend nonstationarity can also be the result of uncertainty in the trend parameters of the mean log conductivity value or it may originate from conditioning of the log conductivity field to measurements of for example head or conductivity we present an efficient numerical method for evaluating the variance of travel time in such formations we cover cases in which the nonstationary covariance functions are constructed from stationary counterparts either by scaling functions or by summation with nonstationary functions resulting from marginalization or conditioning we apply a matrix based first order second moment fosm method for uncertainty propagation using the continuous adjoint state method for coupled systems to evaluate the sensitivity matrix the resulting matrix matrix multiplications are accelerated by fast fourier transformation fft techniques after periodic embedding of the covariance matrices referring to the stationary counterparts the combination of these methods makes it possible to compute the travel time uncertainty in domains discretized by several hundred thousand log conductivity values on standard personal computers within a reasonable time frame for demonstration we apply the method to a binary medium and a medium exhibiting a continuous trend in the covariance function in the latter application we also demonstrate the effects of marginalization and conditioning

nonstationary stochastic analysis in well capture zone design using first order taylor s series approximation 1 nonstationarity of flow fields due to pumping wells and its impact on advective transport is of particular interest in well capture zone design and wellhead protection however techniques based on monte carlo methods to characterize the associated capture zone uncertainty are time consuming and cumbersome this paper introduces an alternative approach the mean and covariance of system state variables i e head pore water velocity and particle trajectory are approximated using a first order taylor s series with sensitivity coefficients estimated from the adjoint operator for a system of discrete equations the approach allows nonstationarity due to several sources e g transmissivity pumping boundary conditions to be treated by employing numerical solution methods it is able to handle irregular geometry varying boundary conditions complicated sink source terms and different covariance functions all of which are important factors for real world applications a comparison of results for the taylor s series approximation with those from monte carlo analysis showed in general good agreement for most of the tested particles particle trajectory variance calculated using taylor s series approximation is then used to predict well capture zone probabilities under the assumption of normality of the mass transport s state variables verification of this assumption showed that not all particle trajectories depending on their starting location are normally or log normally distributed however the risk of using the first order method to delineate the confidence interval of a well capture zone is minimal since it marginally overestimates the 2 5 probability contour furthermore this should be balanced against its greater computation efficiency over the monte carlo approach

numerical solutions to statistical moment equations of groundwater flow in nonstationary bounded heterogeneous media in this paper we investigate the combined effect of nonstationarity in log hydraulic conductivity and the presence of boundaries on flow in heterogeneous aquifers we derive general equations governing the statistical moments of hydraulic head for steady state flow by perturbation expansions due to their mathematical complexity we solve the moment equations by the numerical technique of finite differences the numerical approach has flexibility in handling moderately irregular geometry different boundary conditions various trends in the mean log hydraulic conductivity spatial variabilities in the magnitude and direction of mean flow and different covariance functions all of which are important factors to consider for real world applications the effect of boundaries on the first two statistical moments involving head is strong and persistent for example in the case of stationary log hydraulic conductivity the head variance is always finite in a bounded domain while the head variance may be infinite in an unbounded domain as in many other stochastic models the statistical moment equations are derived under the assumption that the variance of log hydraulic conductivity is small accounting for a spatially varying large scale trend in the log hydraulic conductivity field reduces the variance of log hydraulic conductivity although this makes the conductivity field nonstationary and significantly increases the mathematical complexity in the problem it justifies the small variance assumption for many aquifers

impact of sampling volume on the probability density function of steady state concentration 1 in recent years statistical theory has been used to compute the ensemble mean and variance of solute concentration in aquifer formations with second order stationary velocity fields the merit of accurately estimating the mean and variance of concentration however remains unclear without knowing the shape of the probability density function pdf in a setup where a conservative solute is continuously injected into a domain the concentration is bounded between zero and the concentration value in the injected solution at small travel distances close to the fringe of the plume an observation point may fall into the plume or outside so that the statistical concentration distribution clusters at the two limiting values obviously this results in non gaussian pdf s of concentration with increasing travel distance the lateral plume boundaries are smoothed resulting in increased probability of intermediate concentrations likewise averaging the concentration in a larger sampling volume as typically done in field measurements leads to higher probabilities of intermediate concentrations we present semianalytical results of concentration pdf s for measurements with point like or larger support volumes based on stochastic theory applied to stationary media to this end we employ a reversed auxiliary transport problem in which we use analytical expressions for first and second central spatial lateral moments with an assumed gaussian pdf for the uncertainty of the first lateral moment and gauss like shapes in individual cross sections the resulting concentration pdf can be reasonably fitted by beta distributions the results are compared to monte carlo simulations of flow and steady state transport in 3 d heterogeneous domains in both methods the shape of the concentration pdf changes with distance to the contaminant source near the source the distribution is multimodal whereas it becomes a unimodal beta distribution far away from the contaminant source the semianalytical and empirical pdf s differ slightly which we contribute to the numerical artifacts in the monte carlo simulations but also to hard assumptions made in the semianalytical approach our results imply that geostatistical techniques for interpolation and other statistical inferences based on gaussian distributions such as kriging and cokriging may be feasible only far away from the contaminant source for calculations near the source the beta like distribution of concentration should be accounted for

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

identification of large scale spatial trends in hydrologic data it is often useful to distinguish different scales of variability in hydrologic properties such as hydraulic conductivity in the simplest two scale case large scale variations can be viewed as a trend while small scale fluctuations about this trend can be viewed as a random residual this paper describes a method for estimating spatial trends from scattered field measurements the basic concept is to treat both the trend and the residual as stationary random functions these functions are distinguished by their spatial spectral or covariance properties which may be estimated from available data or simply hypothesized we present two versions of a general algorithm for estimating spatial trends 1 a discrete version which is useful in practical applications where data are limited and irregularly spaced and 2 a continuous version which can be used to study the effects of using incorrect spectral parameters applications of the discrete algorithm to both synthetically generated data and field measurements yield satisfactory trend estimates an analysis based on the continuous algorithm shows that the estimation error lower bound for these applications depends on two dimensionless ratios that is the scale disparity ratio of the trend and residual correlation scales and the signal to noise ratio ratio of the trend and residual variances these ratios may be used to evaluate the feasibility of trend estimation before field samples are actually collected

simple closed form formulas for predicting groundwater flow model uncertainty in complex heterogeneous trending media 1 this note presents approximate closed form formulas for predicting groundwater velocity variances caused by unmodeled small scale heterogeneity in hydraulic conductivity these formulas rely on a linearization of the groundwater flow equation but do not require the common statistical stationarity assumptions the formulas are illustrated with a two dimensional analysis of steady state flows in complex multidimensional trending media and compared with a first order numerical analysis and monte carlo simulation this comparison indicates that despite the simplifications the closed form formulas capture the strongly nonstationary distributions of the velocity variances and match well with the first order numerical model and the monte carlo simulation except in the immediate proximity of prescribed head boundaries and mean conductivity discontinuities the complex trending examples illustrate that the closed form formulas have many of the capabilities of a full stochastic numerical model while retaining the convenience of analytical results

decoherence functionals for von neumann quantum histories boundedness and countable additivity gell mann and hartle have proposed a significant generalisation of quantum theory in which decoherence functionals perform a key role verifying a conjecture of isham linden schreckenberg the author analysed the structure of bounded finitely additive decoherence functionals for a general von neumann algebra a where a has no type i2 direct summand isham et al had already given a penetrating analysis for the situation where a is finite dimensional the assumption of countable additivity for a decoherence functional may seem more plausible physically than that of boundedness the results of this note are obtained much more generally but when specialised to l h the algebra of all bounded linear operators on a separable hilbert space h give

continuous time and consistent histories we discuss the use of histories labelled by a continuous time in the approach to consistent histories quantum theory in which propositions about the history of the system are represented by projection operators on a hilbert space this extends earlier work by two of us c j isham and n linden j math phys 36 5392 5408 1995 where we showed how a continuous time parameter leads to a history algebra that is isomorphic to the canonical algebra of a quantum field theory we describe how the appropriate representation of the history algebra may be chosen by requiring the existence of projection operators that represent propositions about the time average of the energy we also show that the history description of quantum mechanics contains an operator corresponding to velocity that is quite distinct from the momentum operator finally the discussion is extended to give a preliminary account of quantum field theory in this approach to the consistent histories formalism

the structure of decoherence functionals for von neumann quantum histories gell mann and hartle have proposed a significant generalization of quantum theory in which decoherence functionals perform a key role isham linden schreckenberg have given a penetrating analysis of decoherence functionals for l h where h is finite dimensional dimension greater than 2 in this note their conjecture on the significance of boundedness is verified in particular it is shown that when d is a bounded decoherence functional associated with a von neumann algebra a then provided a has no direct summand of type i2 d can be represented as the difference between semi innerproducts on a

comparison of gaussian conditional mean and kriging estimation in the geostatistical solution of the inverse problem two separate applications of the geostatistical solution to the inverse problem in groundwater modeling are presented both applications estimate the transmissivity field for a two dimensional model of a confined aquifer under steady flow conditions the estimates are based on point observations of transmissivity and hydraulic head and also on a model of the aquifer which includes prescribed head boundaries leakage and steady state pumping the model used to describe the spatial variability of the log transmissivity describes large scale fluctuations through a linear mean or drift intermediate and small scale fluctuations through a two parameter covariance function the first application presented estimates the log transmissivities using gaussian conditional mean estimation the second application uses an extended form of cokriging the two methods are compared and their relative merits discussed the extended cokriging application is applied to the jordan aquifer of iowa a comparison is also made between the conditional mean application and an analytical approach

symmetry and history quantum theory an analog of wigner s theorem the basic ingredients of the consistent histories approach to quantum theory are a space up of history propositions and a space d of decoherence functionals in this article we consider such history quantum theories in the case where up is given by the set of projectors p v on some hilbert space v we define the notion of a physical symmetry of a history quantum theory pshqt and specify such objects exhaustively with the aid of an analog of wigner s theorem in order to prove this theorem we investigate the structure of d define the notion of an elementary decoherence functional and show that each decoherence functional can be expanded as a certain combination of these functionals we call two history quantum theories that are related by a pshqt physically equivalent and show explicitly in the case of history quantum mechanics how this notion is compatible with one that has appeared previously

on tracial operator representations of quantum decoherence functionals a general quantum history theory can be characterized by the space of histories and by the space of decoherence functionals in this note we consider the situation where the space of histories is given by the lattice of projection operators on an infinite dimensional hilbert space h we study operator representations for decoherence functionals on this space of histories we first give necessary and sufficient conditions for a decoherence functional being representable by a trace class operator on h h an infinite dimensional analogue of the isham linden schreckenberg representation for finite dimensions since this excludes many decoherence functionals of physical interest we then identify the large and physically important class of decoherence functionals which can be represented canonically by bounded operators on h h

completeness of decoherence functionals the basic ingredients of the consistent histories approach to a generalized quantum theory are histories and decoherence functionals the main aim of this program is to find and to study the behavior of consistent sets associated with a particular decoherence functional d in its recent formulation by isham quantum logic and the histories approach to quantum theory j math phys 35 2157 2185 1994 it is natural to identify the space up of propositions about histories with an orthoalgebra or lattice when up is given by the lattice of projectors p v in some hilbert space v consistent sets correspond to certain partitions of the unit operator in v into mutually orthogonal projectors 1 2 such that the function d is a probability distribution on the boolean algebra generated by 1 2 using the classification theorem for decoherence functionals proven in isham et al the classification of decoherence functionals an analog of gleason s theorem j math phys 35 6

large time behavior of macrodispersion in heterogeneous trending aquifers 1 this paper presents a stochastic analysis of the large time behavior of macrodispersion in a three dimensional heterogeneous aquifer with a linear trend in the mean log hydraulic conductivity to solve the problem analytically focus is placed on the particular case where the linear trend is aligned in the direction of mean hydraulic head gradient a spectral approach based on fourier stieltjes representations for the perturbed quantities is used to develop closed form expressions that describe variability of flow velocity the second order mean flow and asymptotic macrodispersion the impact of the mean log hydraulic conductivity gradient on these results is examined it is found that the asymptotic longitudinal and transverse macrodispersion coefficients decrease with the increasing trend gradient of mean log hydraulic conductivity in the case of finite peclet numbers this feature is a consequence of the reduction in variability of flow velocity with the trend gradient

effects of first order approximations on head and specific discharge covariances in high contrast log conductivity the hydraulic head and the specific dischargefluctuations depend nonlinearly on the hydraulic conductivity however the methods most commonly used in the stochastic analysis of groundwaterflow are based upon the linearization of these relations in this paper we apply a numerical spectral approach to investigate the range of validity of the small perturbation approximation for head and specific discharge moments in two dimensionalfinite domains wefind that the small perturbation approximation tends to underestimate the variance of large scale head and specific dischargefluctuations and error increases with increasing log conductivity variance and increasing domain size the headfluctuations do not appear ergodic even when the small perturbation approximation predicts they will be ergodic the specific dischargefluctuations on the other hand do appear ergodic the small perturbation approximation performs well in estimating specific discharge variance in the longitudinal direction but significantly underestimates transverse specific discharge variance

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

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

a nonstationary spectral method for solving stochastic groundwater problems unconditional analysis stochastic analyses of groundwater flow and transport are frequently based on partial differential equations which have random coefficients or forcing terms analytical methods for solving these equations rely on restrictive assumptions which may not hold in some practical applications numerically oriented alternatives are computationally demanding and generally not able to deal with large three dimensional problems in this paper we describe a hybrid solution approach which combines classical fourier transform concepts with numerical solution techniques our approach is based on a nonstationary generalization of the spectral representation theorem commonly used in time series analysis the generalized spectral representation is expressed in terms of an unknown transfer function which depends on space time and wave number the transfer function is found by solving a linearized deterministic partial differential equation which has the same form as the original stochastic flow or transport equation this approach can accomodate boundary conditions spatially variable mean gradients measurement conditioning and other sources of nonstationarity which cannot be included in classical spectral methods here we introduce the nonstationary spectral method and show how it can be used to derive unconditional statistics of interest in groundwater flow and transport applications

physical pictures of transport in heterogeneous media advection dispersion random walk and fractional derivative formulations 1 the basic conceptual picture and theoretical basis for development of transport equations in porous media are examined the general form of the governing equations is derived for conservative chemical transport in heterogeneous geological formations for single realizations and for ensemble averages of the domain the application of these transport equations is focused on accounting for the appearance of non fickian anomalous transport behavior the general ensemble averaged transport equation is shown to be equivalent to a continuous time random walk ctrw and reduces to the conventional forms of the advection dispersion equation ade under highly restrictive conditions fractional derivative formulations of the transport equations both temporal and spatial emerge as special cases of the ctrw in particular the use in this context of levy flights is critically examined in order to determine chemical transport in field scale situations the ctrw approach is generalized to nonstationary systems we outline a practical numerical scheme similar to those used with extended geological models to account for the often important effects of unresolved heterogeneities

symmetries of history quantum theories and decoherence functionals recently schreckenberg investigated symmetries in the context of history quantum theories in the case that the space of histories is given by the set of projectors on some finite dimensional hilbert space he obtained first a complete characterization of all physical symmetries of history quantum theories an analog of wigner s theorem and second a complete mathematical characterization of symmetries of single decoherence functionals in this paper we extend schreckenberg s results to the case where the underlying space of histories is given by the set of projectors on some infinite dimensional hilbert space

temporal scaling of hydraulic head fluctuations nonstationary spectral analyses and numerical simulations 1 temporal scaling of the hydraulic head time series h t was found in a previous analysis of hourly measured head data this issue is further investigated in this paper with nonstationary spectral analyses and numerical simulations the results show that temporal scaling may indeed exist in h t which fluctuates like a fractional brownian motion in most aquifers on the basis of a linear reservoir model with a white noise recharge input we show that the variance and covariance of h t are functions of time the head variance increases with time and approaches a constant limit as time progresses while the covariance decreases with the separation time interval for a fixed time and approaches the typical exponential covariance as time increases the spectra of the simulated h t using a one dimensional transient groundwater flow model with a white noise recharge in both homogeneous and heterogeneous aquifers are shown to be proportional to f where f is frequency and 1 84 or h 0 42 heterogeneity in the hydraulic conductivity may affect the fractal dimension of h t in highly permeable aquifers but not in the low permeable aquifer simulated in this study

a distributed parameter approach for evaluating the accuracy of groundwater model predictions 2 application to groundwater flow the distributed parameter theory presented by mclaughlin and wood this issue is used to evaluate the accuracy of a groundwater flow model the special case investigated assumes that the model s prediction errors are due primarily to data limitations in this case approximate prediction error moments mean and covariance are obtained by solving two sets of coupled partial differential equations which have the same basic structure as the original flow equation solutions can be obtained with spectral green s function or numerical techniques depending on the assumptions made two examples are solved using a finite element approach the first two dimensional example confirms steady state infinite domain results obtained with spectral methods the second three dimensional example investigates the influence of spatial variability sampling strategy and suboptimal estimation on model prediction accuracy

stochastic analysis of spatial variability in two dimensional steady groundwater flow assuming stationary and nonstationary heads two dimensional steady groundwater flow in a confined aquifer with spatially variable transmissivity t is analyzed stochastically using spectral analysis and the theory of intrinsic random functions conditions that ensure a stationary statistically homogeneous head process are derived and using two convenient forms for the covariance function of the ln t process the head covariance function is studied in addition the head variogram is obtained for a particular nonstationary case and the asymptotic head variogram is derived under very general conditions results are compared to those obtained by gelhar 1976 for one and two dimensional phreatic flow and bakr et al 1978 for one and three dimensional confined flow multidimensional flow analysis results in a significantly reduced head variance the head correlation remains high over much greater distances than the ln t correlation the variogram obtained when stationary heads are assumed is identical to that obtained for nonstationary heads for dimensionless lag distances up to 2 times the correlation scale of the log transmissivity the variogram for nonstationary heads continues to grow logarithmically as lag distance increases independent of the form of the input covariance in the nonstationary case the conditions for stationarity are contrasted with the corresponding results obtained for the one and three dimensional cases of gutjahr and gelhar 1981 the head variance calculated from the stationary theory is found to agree with that of previous monte carlo simulations

characterization of permeability anisotropy using wavelet analysis 1 wavelet analysis is an image analysis technique that can extract local information at multiple scales because of this capability wavelet analysis can be used to identify dominant scales in statistically heterogeneous and anisotropic random fields we develop and test a wavelet analysis method for identifying dominant scales and orientations in permeability fields and for identifying boundaries between regions with different dominant orientations we evaluate three different wavelets fully anisotropic morlet wavelet mexican hat wavelet and cauchy wavelet and show that the morlet wavelet is the most effective of these three wavelets in identifying dominant orientations we also investigate the use of several different quantitative wavelet measures in identifying dominant scales and orientations in permeability fields the technique is demonstrated using both a synthetic data set with known characteristics and a laboratory collected permeability data set from massillon sandstone

numerical solutions of moment equations for flow in heterogeneous composite aquifers 1 we analyze flow in heterogeneous media composed of multiple materials whose hydraulic properties and geometries are uncertain our analysis relies on the composite media theory of winter and tartakovksy 2000 2002 which allows one to derive and solve moment equations even when the medium is highly heterogeneous we use numerical solutions of darcy flows through a representative composite medium to investigate the robustness of perturbation approximations in porous medium with total log conductivity variances as high as 20 we also investigate the relative importance of the two sources of uncertainty in composite media material properties and geometry in our examples the uncertain geometry by itself captures the main features of the mean head estimated by the full composite model even when the within material conductivities are deterministic however neglecting randomness within materials leads to head variance estimates that are qualitatively and quantitatively wrong we compare the composite media approach to approximations that replace statistically inhomogeneous conductivity fields with pseudohomogeneous random fields with deterministic trends we demonstrate that models with a deterministic trend can be expected to give a poor estimate of the statistics of head

analysis of the spatial structure of properties of selected aquifers data from 31 regional aquifers are analyzed to identify the horizontal spatial correlation structure of transmissivity hydraulic conductivity and storage coefficient three parameters are estimated the variance of small scale variability or nugget and the variance and integral scale range of an exponential covariance function the assumption of normality of the aforementioned geohydrologic parameters and their logarithmic transformations is also examined results are obtained which indicate that the logarithms of these properties generally pass normality tests

asymptotic properties of the eulerian truncation approximation analysis of the perfectly stratified transport problem it is well known that stochastic groundwater problems are difficult to solve without making approximations of one kind or another a popular approximation used in both subsurface flow and transport applications is the so called eulerian truncation this approximation which relies on a perturbation expansion of the governing equation neglects certain terms involving products of small fluctuations several recent publications e g dagan and neuman 1991 neuman and orr 1993 neuman 1993 question the validity of eulerian truncation arguing that it is inconsistent because neglected terms are of the same order as those retained in this paper we analyze the asymptotic properties of the eulerian truncation solution using a taylor series approach which clearly reveals the properties of different order approximations we show that the eulerian truncation does in fact generate an asymptotic ensemble mean expansion for the perfectly stratified example considered by dagan and neuman 1991 moreover we show that the eulerian truncation solution to this problem is identical to the well known cumulant expansion approximation which is also based on an asymptotic expansion

using the nonstationary spectral method to analyze flow through heterogeneous trending media this paper describes a nonstationary spectral theory for analyzing flow in a heterogeneous porous medium with a systematic trend in log hydraulic conductivity this theory relies on a linearization of the groundwater flow equation but does not require the stationarity assumptions used in classical spectral theories the nonstationary theory is illustrated with a two dimensional analysis of a linear trend aligned with the mean flow direction in this case closed form solutions can be obtained for the effective hydraulic conductivity head covariance and log conductivity head cross covariance the effective hydraulic conductivity decreases from the geometrical mean as the mean slope of the log conductivity increases trending leads to a reduction of head variance and a structural change in the head covariance and the log conductivity head cross covariance such changes have important implications for measurement conditioning or cokriging methods which rely on the head covariance and log conductivity head covariance the nonstationary spectral analysis is also compared with classical spectral analysis this comparison indicates that the classical spectral method correctly predicts the normalized head covariance in a linear trending media the stationary spectral method fails to capture the qualitative influence of trends on the effective hydraulic conductivity and the log conductivity head cross covariance although the magnitude of the error is relatively small for realistic values of the mean log conductivity slope the stationary and nonstationary results are the same when there is no trend in log conductivity the trending conductivity example illustrates that the nonstationary spectral method has all the capabilities of the classical spectral approach while not requiring as many restrictive assumptions

consistent histories and the interpretation of quantum mechanics the usual formula for transition probabilities in nonrelativistic quantum mechanics is generalized to yield conditional probabilities for selected sequences of events at several different times called consistent histories through a criterion which ensures that within limits which are explicitly defined within the formalism classical rules for probabilities are satisfied the interpretive scheme which results is applicable to closed isolated quantum systems is explicitly independent of the sense of time i e past and future can be interchanged has no need for wave function collapse makes no reference to processes of measurement though it can be used to analyze such processes and can be applied to sequences of microscopic or macroscopic events or both as long as the mathematical condition of consistency is satisfied when applied to appropriate macroscopic events it appears to yield the same answers as other interpretative schemes for standard quantum mechanics though from a different point of view which avoids the conceptual difficulties which are sometimes thought to require reference to conscious observers or classical apparatus

stochastic modeling of groundwater flow by unconditional and conditional probabilities 1 conditional simulation and the direct problem mathematical modeling of groundwater flow in heterogeneous porous formations of large extent is investigated the formation properties hydraulic conductivity transmissivity as well as flow variables head specific discharge solute concentration are regarded as random variables subjected to uncertainty the main aim of the study is to analyze the influence of conditional probability of input variables upon the statistical structure of the dependent variables the unconditional probability density functions are supposed to be stationary multivariate normal while conditioning accounts for the measured values at a few points of the formation two problems of groundwater flow are discussed in part 1 conditional simulation and the direct problem for steady flow analytical results are obtained by using perturbation approximations average uniform head gradient flows as well as recharge and flow to wells are discussed in unconditional modeling the input variable the conductivity or transmissivity log is regarded as stationary and is represented by its constant mean and variogram the ensemble of formations on which statistical calculations are carried out represents all aquifers with same probability density distributions in conditional modeling the measured values at a few points are kept fixed and uncertainty prevails only at other points consequently statistical computations are performed for the subensemble of aquifers which preserve the measured values and as a result both input and output variables are nonstationary the main effect of conditioning is to reduce the variance i e the uncertainty of these variables this effect is particularly important in the case of flow toward wells application of the modeling method to field problems is outlined

the classification of decoherence functionals an analog of gleason s theorem gell mann and hartle have proposed a significant generalization of quantum theory with a scheme whose basic ingredients are histories and decoherence functionals within this scheme it is natural to identify the space up of propositions about histories with an orthoalgebra or lattice this raises the important problem of classifying the decoherence functionals in the case where up is the lattice of projectors p v in some hilbert space v in effect one seeks the history analog of gleason s famous theorem in standard quantum theory in the present article the solution to this problem for the case where v is finite dimensional is presented in particular it is shown that every decoherence functional d p v can be written in the form d trv v x for some operator x on the tensor product space v v

stochastic analysis of solute transport in heterogeneous aquifers subject to spatiotemporal random recharge this paper extends the work of li and graham 1998 to deal with the unconditional moments of head velocity and concentration under transient flow conditions which are assumed to be caused by a spatiotemporally random recharge semianalytical solutions are derived for the unconditional covariances for transient velocity with a constant mean recharge using a fourier transform approach results demonstrate that the velocity covariance derived for the steady state random recharge field is a limiting case of the spatiotemporally variable velocity covariance with an infinite temporal correlation scale another limiting case indicates that introduction of temporally random but spatially uniform recharge has no effect on the velocity covariances over that induced by the mean recharge on the mean head gradient thus for this limiting case there is no increased effect on the ensemble mean concentration plume spreading or the concentration prediction uncertainty following deng et al 1993 the equations for mean concentration and macrodispersive flux under zero mean transient recharge are decoupled in the laplace fourier domain and solved using a fast fourier transform algorithm which significantly reduces the computational demand the first order concentration variance is solved using three different approximate techniques an approximate fast fourier transform technique a finite element method and a direct numerical integration the simulation results show that introduction of a spatiotemporally random recharge enhances both longitudinal and lateral mean concentration plume spreading compared to the no recharge case however transient recharge produces less spreading and less concentration prediction uncertainty than the steady state spatially random recharge case