models for disc infiltrometers disc infiltrometers are popular devices for determining in situ hydraulic properties of unsaturated soils this paper compares alternative solutions of richards equation for both steady state and time dependent cases the steady state solutions were in general agreement using alternative hydraulic conductivity functions of the same capillary length scale small time solutions for the nonlinear cases were consistent with linear diffusion from a disc source using an average diffusivity value which is simply related to the capillary length this offers a refinement over the one dimensional solution for short times in that the geometric effect of the circular source is included simulations for two examples indicate that the approach to the steady state solution may take considerably longer than what is commonly reported in the literature for field applications

any controllable driftless system with 3 inputs and 5 states is flat abstract a very simple classification of pfaffian systems of dimension 2 in 5 variables is given it is used to show that any controllable driftless system with 3 inputs and 5 states is 0 flat and can be put into multi input chained form by dynamic feedback and coordinate change

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

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

on the possible multiplicities of the eigenvalues of a hermitian matrix whose graph is a tree we consider the general problem of determining which lists of multiplicities for the eigenvalues occur among hermitian matrices the graph of whose off diagonal entries is a given tree several restrictions are cited and a construction strategy is given together these are sufficient to characterize all lists for each tree in two infinite classes the double paths and generalized stars and to tabulate all lists for trees on fewer than nine vertices such tables should be useful for formulating and dispelling general conjectures

estimating unsaturated soil hydraulic properties from tension disc infiltrometer data by numerical inversion tension disc infiltrometers are becoming increasingly popular devices for in situ measurement of the unsaturated hydraulic properties of soil tension infiltration data are generally used to evaluate the parametersks and ain gardner s exponential model of the unsaturated hydraulic conductivity either two measurements using different disc diameters or measurements with a single disc but using multiple pressure heads are then used in this paper we describe a parameter estimation procedure which combines the levenberg marquardt nonlinear parameter optimization method involving weighted least squares with a quasi three dimensional numerical model which solves the variably saturatedflow equation by numerical inversion of richards equation the unknown parameters in van genuchten s model of the unsaturated soil hydraulic properties are estimated from observed cumulative infiltration data during transient waterflow additional measurements of the pressure head or water content as well as a penalty function for constraining the unknown parameters to remain in some feasible region bayesian estimation can be optionally included into the parameter estimation procedure the problem of optimal sampling design that is selecting the best points in space and time for making measurements is addressed by studying the sensitivity of the objective function to changes in the optimized hydraulic parameters we calculate objective functions based on available cumulative infiltration pressure head and water content measurements and also on several combinations of these data the behavior of the objective function in three dimensional parameter space is evaluated by means of a series of two dimensional response surfaces the utility of the parameter estimation procedure is demonstrated using numerically generated data the sensitivity of the procedure to different initial estimates of the model parameters is also discussed

application of the finite difference heterogeneous multiscale method to the richards equation 1 this paper extends the finite difference heterogeneous multiscale method fdhmm to simulate transient unsaturated water flow problems in random porous media the numerical method is based on the use of two different schemes for the original equation at different grid levels which allows numerical results at a lower cost than solving the original equations the main feature of fdhmm is that the necessary data for the macroscopic model are supplied by solving the microscopic model on a sparse spatial domain the generated code is verified by applying the linearization model of the richards equation considering two different constitutive relationships this method is applied to several test examples with different soil textures and boundary conditions both the dirichlet and the periodic boundary conditions are considered for solving the local microscopic model when the water flow in heterogeneous unsaturated soils is simulated by fdhmm the numerical experiments demonstrate that fdhmm can effectively simulate the transient unsaturated water flow in the specific soils the numerical experiments also demonstrate that fdhmm can achieve accurate global mass balance and is a globally convergent algorithm and the spatial correlation length of random coefficients under the specific standard deviation has relatively little influence on the accuracy of the method

inverse maximal eigenvalues problems for leslie and doubly leslie matrices abstract in this paper we deal with leslie and doubly leslie matrices of order n in particular with the companion and doubly companion matrices we study three inverse eigenvalues problems which consist of constructing these matrices from the maximal eigenvalues of its all leading principal submatrices for leslie and doubly companion matrices an eigenvector associated with the maximal eigenvalue of the matrix is additionally considered and for the doubly leslie matrix also an eigenvector associated with the maximal eigenvalue of leading principal submatrix of order n 1 is required we give necessary and sufficient conditions for the existence of a leslie matrix and a companion matrix and sufficient conditions for the existence of a doubly leslie matrix and a doubly companion matrix our results are constructive and generate an algorithmic procedure to construct these special kinds of matrices

scaling of constitutive relationships in unsaturated heterogeneous media a numerical investigation numerical models of groundwater flow in the unsaturated zone require relative conductivity and moisture retention curves at a discretization scale much larger than that at which these constitutive relationships are measured this study seeks to identify simple scaling laws linking parameters of block scale constitutive relationships with the parameters of their measurement scale counterparts represented here using gardner russo models the two parameters of these models saturated conductivity and pore size distribution parameter are viewed as cross correlated regionalized variables with isotropic or anisotropic spatial correlation structures and are simulated using the sequential gaussian method upscaled constitutive curves for steady gravity drainage through three dimensional heterogeneous domains are then constructed by repeating numerical flow experiments for a series of applied infiltration rates and plotting the resulting upscaled relative conductivities and average water saturations against average matric potentials the experiments reveal clear log linear relationships between upscaled relative conductivities and average matric potentials the slopes of which are used to define an upscaled pore size distribution parameter the reciprocal of this quantity which represents an upscaled characteristic capillary length is very well approximated by a spatial power average of its measurement scale values furthermore the averaging exponent is found to be the same as that associated with the upscaling of saturated conductivities over the same flow domains when average matric potentials are normalized using this upscaled characteristic capillary length relative conductivity and moisture saturation values for widely different flow field realizations collapse along well defined curves however for wet conditions the form of these curves differs from that of the models assumed at the measurement scale

estimation of root water uptake parameters by inverse modeling with soil water content data 1 in this paper we have tested the feasibility of the inverse modeling approach to derive root water uptake parameters rwup from soil water content data using numerical experiments for three differently textured soils and for an optimal drying period the rwup of interest are the rooting depth and the bottom root length density in a first step a thorough sensitivity analysis was performed this showed that soil water content dynamics is relatively insensitive to rwup and that the sensitivity depends on the texture of the considered soil for medium fine textured soil the sensitivity is particularly low due to relatively high unsaturated hydraulic conductivity values these ones allow a compensating effect to occur i e vertical unsaturated water fluxes overshadowing in some way the root water uptake in a second step we analyzed the well posedness of the solution stability and nonuniqueness when only rwup are optimized for this case the inverse problem is clearly ill posed except for the estimation of the rooting depth parameter for coarse and the very fine textured soils in a third step we addressed the case where rwup are estimated simultaneously with additional parameters of the system i e with soil hydraulic parameters for this case our study showed that the inverse problem is well posed for the coarse and very fine textured soils allowing for the estimation of both rwup of interest provided that a powerful global optimization algorithm is used on the contrary the estimation of rwup is unfeasible for medium fine textured soil due to the compensating effect of the vertical unsaturated water flows in conclusion we can state that the inverse modeling approach can be applied to derive rwup for some soils coarse and very fine textured and that the feasibility is strongly improved if the rwup are simultaneously optimized with additional parameters nevertheless more detailed research is needed to apply the inverse modeling approach to real cases for which additional issues are likely to be encountered such as soil heterogeneity and root dynamics

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

constructive controllability algorithms for motion planning and optimization presents algorithms for planning feasible and minimum energy paths the algorithms rely on series expansions to characterize planning problems for polynomial control systems the resulting inversion problem is solved through an iterative contraction or a power series inversion while the design methodology is general our focus is on a class of polynomial control systems for which we provide explicit convergence guarantees we demonstrate performance and numerical characteristics using one dimensional and six dimensional systems

parameter equivalence for the brooks corey and van genuchten soil characteristics preserving the effective capillary drive this paper provides a simple way to convert brooks corey bc parameters to van genuchten vg parameters and vice versa for use primarily in situations where saturated conditions are likely to be encountered essential in this conversion is the preservation of the maximum value of a physical characteristic the effective capillary drive h morel seytoux and khanji 1974 defined with a good approximation for a soil water and air system as h jg k dh where k is relative permeability or conductivity to water and ii is capillary pressure head a positive quantity with this conversion infiltration calculations are essentially insensitive to the model used to represent the soil hydraulic properties it is strictly a matter of convenience for the user which expression is used on the other hand the paper shows that other equivalences may lead to great variations in predictions of infiltration capacity consequently the choice of the proper equivalence to use in calculations for rainfall runoff modeling or for low level radioactive waste disposal design is a serious matter

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

stochastic modeling of large scale transient unsaturated flow systems a new framework for modeling large scale transient unsaturated flow systems in spatially variable soils is proposed in order to overcome the problem of limited information about the local details of spatial soil variability a stochastic approach which assumes that local soil properties are realizations of three dimensional random fields is followed for derivation of a large scale model representation structure the three dimensionality of the local flow and the nonlinear dependence of the local flow output on the local soil properties are considered the large scale model structure is derived by averaging the local governing flow equation over the ensemble of realizations of the underlying soil property random fields the resulting mean model representation is in the form of a partial differential equation in which averaged or effective model parameters occur these effective model parameters i e effective hydraulic conductivity and effective specific moisture capacity are evaluated using a quasi linearized fluctuation equation and a spectral representation of stationary processes the large scale model representation considers the large scale effects of soil property variability and has relatively few parameters the general conclusions of this study are that soil property variability produces large scale hysteresis and anisotropy of the effective parameters the potential theoretical and practical ramifications of these results in the area of unsaturated flow modeling need to be investigated the general stochastic modeling framework developed here is applicable not only to unsaturated flow but also to other distributed parameter systems e g saturated flow and transport geothermal and oil reservoir modeling

stochastic analysis of one dimensional steady state unsaturated flow a comparison of monte carlo and perturbation methods the stochastic nature of moisture content in a soil profile under steady state unsaturated infiltration is examined where the saturated hydraulic conductivity is taken as a stationary stochastic process two different techniques are employed in determining the stochastic output these are monte carlo simulations and an analytic derivation from a first order perturbation solution in comparing the two techniques the liability of the perturbation expressions is investigated if the monte carlo results are assumed to exactly represent the stochastic nature of the moisture content in four test examples a relatively good comparison is obtained between the results of the two techniques the analytic nature of the perturbation solution then readily provides information concerning the general stochastic properties of the moisture content and in this way additional monte carlo runs are avoided from this analysis it is shown that the moisture content is a stationary stochastic process at distances far from boundary regions the distance from the boundary to regions of stationarity is dependent upon soil properties and the boundary conditions imposed

nonholonomic motion planning steering using sinusoids methods for steering systems with nonholonomic c onstraints between arbitrary configurations are investigated suboptimal trajectories are derived for systems that are not in canonical form systems in which it takes more than one level of bracketing to achieve controllability are considered the trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level a class of systems that can be steered using sinusoids claimed systems is defined conditions under which a class of two input systems can be converted into this form are given

determining soil hydraulic functions from evaporation experiments by a parameter estimation approach experimental verifications and numerical studies a parameter estimation method is developed for the determination of unsaturated soil hydraulic properties from evaporation experiments under laboratory conditions variables used in the inversion procedure are soil water pressure heads at various positions and total soil weights both measured as a function of time during the experiment unknown parameters of different analytical expressions used to describe the soil hydraulic properties are estimated by coupling a finite difference solution of the richards equation with a nonlinear optimization problem this problem is formulated by minimizing the deviations between the numerical solution of the transient flow process and the real system response measured during the experiment minimization of the objective function is performed using a version of the levenberg marquardt s method the procedure also provides information about the uncertainty in parameter estimates the performance of the selected parametric relationships is evaluated and the applicability and accuracy of the proposed inverse method are shown by comparing estimated water retention and hydraulic conductivity functions with experimental results obtained via the instantaneous profile method parameter sensitivity analyses and stability of the inverse solutions are discussed with reference to the originally designed evaporation experiment and to an evaporation method that is developed with a view to reducing experimental efforts further insights into the properties of existence and uniqueness of inverse solutions are gained by examining contour plots of the objective functions under varying experimental conditions the results confirm the reliability and flexibility of the proposed method and suggest that the evaporation flux imposed at the upper soil surface may determine the well posedness of the optimization problem

an efficient method for simulating steady unsaturated flow in random porous media using an analytical perturbation solution as initial guess to a numerical model numerical simulation of flow through multidimensional heterogeneous soils under unsaturated conditions is a computationally intensive task the governing unsaturated flow equation is nonlinear the degree of nonlinearity depends on the unsaturated hydraulic properties of the soil and the degree of heterogeneity in this paper we present a highly efficient method to simulate unsaturated steady state flow through random porous media our method utilizes the analytical approximate solution derived by a perturbation spectral method as an initial guess solution for a numerical model to simulate two dimensional vertical infiltration problems it is found that this approach which we call asigning reduces the required cpu time by one to two orders of magnitude asigning is demonstrated to operate successfully under a wide variety of boundary conditions which may substantially deviate from those imposed on the initial guess solution a large range of mean and variances in the independent variables ln ks and or alternatively in has been tested and it is shown that the method works well for variances of the unsaturated hydraulic conductivity ln2k 5 and average 0 1 cm 1

the gs algorithm for exact linearization to brunovsky normal form an algorithm utilizing the minimal number of integrations for the exact linearization of nonlinear systems to brunovsky normal form under nonlinear feedback is presented the tools which are involved are based on classical constructions appearing in the theory of exterior differential systems

parameter estimation in flow through partially saturated porous materials a class of numerical simulators were developed and critically evaluated to be incorporated as the solver of a forward problem in the framework of an inverse modeling strategy the strategy couples a mass lumped galerkin linear finite element solution of the mixed form richards equation with an experimental time space series and the osborne more revised form of the levenberg marquardt algorithm to retrieve hydraulic parameters of a partially saturated porous medium the numerical simulator shows excellent agreement with a reference solution obtained on a dense grid and infinitesimal time step in terms of fluid pressure head fluid content and fluid volumetric flux density and perfectly conserves the global mass an adaptive algorithm was implemented to estimate sensitivity matrix in the inverse algorithm a multi criterion stopping rule was developed and successfully implemented to end the inverse code at the solution the result of the optimization was compared with a large scale in situ soil moisture space time series measured during the course of a drainage experiment and excellent agreements were found analysis of the parameter response surfaces and hyper space plots closeness of the gradient of the penalty function at minimum to zero and positive definiteness of the approximation for the hessian at the solution eigs h 0 indicate that the obtained solution is a strong local minimum a state of the art sensitivity analysis carried out to quantify sensitivity of the state variable with respect to uncertainty and changes in different model parameters

questions conjectures and data about multiplicity lists for trees abstract we review and discuss a number of questions and conjectures about multiplicity lists occurring among real symmetric matrices whose graph is a tree our investigation is aided by a new electronic database containing all multiplicity lists for trees on fewer than 12 vertices some questions and conjectures are familiar and some are new and new information is given about several

nonnegative realization of spectra having negative real parts abstract the nonnegative inverse eigenvalue problem niep is the problem of determining necessary and sufficient conditions for a list of complex numbers to be the spectrum of a nonnegative matrix in this paper the problem is completely solved in the case when all numbers in the given list except for one the perron eigenvalue have real parts smaller than or equal to zero

homogenization of richards equation in permeability fields with different connectivities 1 large scale modeling of transient flow in the unsaturated zone is important for the estimation of the water budget and solute transport in the vadose zone upscaled flow models need to capture the impact of small scale heterogeneities which are not resolved by the model on large scale flow we perform upscaling of the richards equation in heterogeneous porous media with continuous distributions of the soil hydraulic parameters using homogenization theory and stochastic averaging techniques we restrict the analysis to flow regimes in which the capillary equilibrium assumption holds on the small scale in order to account for effects of capillary entry pressure we apply the brooks corey model for the soil retention and relative permeability curves and consider leverett scaling for the coupling of intrinsic permeability and entry pressure for this model we derive and analyze the ensemble averaged parameter functions for the macroscopic flow equations the effects of a definite entry pressure vanish with increasing variance of the log intrinsic permeability we compare the statistically averaged parameter functions to numerically calculated effective functions for parameter fields with different connectivity properties these results illustrate that soils with well connected coarse materials differ in the relative permeability from those with well connected fine materials or those without particular connectedness

stochastic analysis of unsaturated flow in heterogeneous soils 1 statistically isotropic media steady unsaturated flow with vertical mean infiltration through unbounded heterogeneous porous media is analyzed using a perturbation approximation of the stochastic flow equation which is solved by spectral representation techniques the hydraulic conductivity k is related to the capillary pressure head by k ks exp where ks is the saturated conductivity and is a soil parameter a general formulation is presented for the case with ks and represented as statistically homogeneous spatial random fields in part 1 solutions are developed assuming is constant and representing ks variability by one dimensional and three dimensional isotropic random fields results are obtained for head variances and covariance functions effective hydraulic conductivities variances of the unsaturated hydraulic conductivity flux variances and variance of pressure gradient when the parameter is relatively large corresponding to coarse textured soils the head variance decreases and all of the results demonstrate a trend toward gravitationally dominated one dimensional vertical flow the effective conductivity is dependent on the correlation scale of ln ks and the mean hydraulic gradient

a mass conservative switching algorithm for modeling fluid flow in variably saturated porous media numerical solutions of flow equation in fluid content based form or in fluid pressure head based form are often tradeoffs between speed accuracy and convenience the fluid content based form can be solved quite rapidly with low cpu time and perfect mass balance however it cannot be used in saturated regions as diffusivity function becomes infinite and strictly becomes invalid in composite layered and real heterogeneous porous materials due to singularity and discontinuity in fluid content profile this formulation also gives misleading impression that gradient in fluid content causes the flow of fluid in porous materials where in reality gravity and fluid pressure potential gradient produce the motion the pressure head based form on the other hand is more flexible but due to its highly nonlinear nature is much more time consuming and produces poor global mass balance for dry initial conditions very fine spatial and temporal discretizations are needed to maintain mass balance property for these scenarios the mixed form of the flow equation partially solves these issues as it maintains acceptable mass balance and is applicable to layered heterogeneous and composite fractured foundations however it is only applicable in unsaturated zones in this study a switching algorithm was proposed and implemented in which the mass conservative mixed form and the pressure head based form were respectively used in the unsaturated and saturated zones of an initial boundary value flow problem involving a variably saturated porous medium the algorithm showed excellent agreement with a reference solution obtained on a very fine spatiotemporal mesh the simulator was then calibrated with several real world large scale experimental datasets in all cases the proposed algorithm exhibited close agreements with the experimental time space series the algorithm poses excellent mass balance property and can easily be used in both saturated and unsaturated regions without special treatment of fluid content discontinuities in heterogeneous and layered porous media the proposed algorithm can also be extended to simulate multiphase and multidimensional flow problems

nonnegative matrices with prescribed extremal singular values we consider the problem of constructing nonnegative matrices with prescribed extremal singular values in particular given 2n 1 real numbers s 1 j and s j j j 1 n we construct an nxn nonnegative bidiagonal matrix b and an nxn nonnegative semi bordered diagonal matrix c such that s 1 j and s j j are respectively the minimal and the maximal singular values of certain submatrices b j and c j of b and c respectively by using a singular value perturbation result we also construct an nxn nonnegative matrix with prescribed singular values s 1 s n

the inverse eigenvalue problem for symmetric quasi anti bidiagonal matrices in this paper we construct the symmetric quasi anti bidiagonal matrix that its eigenvalues are given and show that the problem is also equivalent to the inverse eigenvalue problem for a certain symmetric tridiagonal matrix which has the same eigenvalues not only elements of the tridiagonal matrix come from quasi anti bidiagonal matrix but also the places of elements exchange based on some conditions

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

an iterative geostatistical inverse method for steady flow in the vadose zone an iterative geostatistical inverse approach is developed to estimate conditional effective unsaturated hydraulic conductivity parameters soil water pressure head and degree of saturation in heterogeneous vadose zones this approach is similar to the classical cokriging technique and it uses a linear estimator that depends on covariances and cross covariances of unsaturated hydraulic parameters soil water pressure head and degree of saturation the linear estimator is however improved successively by solving the governing flow equation and by updating the residual covariance and cross covariance functions in an iterative manner as a result the nonlinear relationship between unsaturated hydraulic conductivity parameters and head is incorporated in the estimation and the estimated fields are approximate conditional means the ability of the iterative approach is demonstrated through some numerical examples

on the identification of macroscopic root water uptake parameters from soil water content observations 1 in this paper we analyze the identification problem of macroscopic root water uptake parameters from soil water content observations for this study the macroscopic root water uptake is considered to be linearly decreasing with depth with a and b parameters conditioning respectively the maximum root water uptake at the surface and the decreasing rate with depth for identification of parameter a and b two different identification approaches are tested using a detailed soil moisture data set consisting of vertical profiles measured with time domain reflectometry tdr and neutron probe on 28 locations within a small maize cropped field during a dry period of 26 days the first approach is based on a simplified water balance while the second one uses an integrated soil vegetation atmosphere transfer svat model in an inverse mode results of the simplified water balance show first that the root water uptake measured within the field is quite variable with cv ranging from 22 to 34 for the root uptake parameters a and b respectively furthermore positive correlation between a and b suggests that low superficial root water uptake could be compensated by high deep root water uptake second numerical simulations used to test the validity of this simplified approach show that the method is quite robust at least for a certain range of a parameter values 0 008 a 0 013 for b and for all investigated differently textured soils except the coarse sand to investigate the feasibility and the robustness of the second approach we address the problems of insensitivity instability and nonuniqueness by means of numerical simulations results show for soils with different textures that the soil water content and to a lesser extent the time derivative of soil water content are quite insensitive to root water uptake parameters at least compared to soil hydraulic parameters furthermore instability analysis clearly illustrates that root water uptake parameters are quite and even very instable with respect to small uncertainties on the apparently known parameters e g soil parameters finally results show that if root water uptake plus additional other parameters e g soil parameters are simultaneously optimized nonuniqueness of parameter sets must be expected therefore a robust identification of root water uptake parameters by means of a full svat inversion is unlikely to be achieved during a dry period with soil water content observations at least when some other parameters driving the system like soil hydraulic properties are not error free in this context further in depth research is needed to investigate if the use of longer periods characterized by both drying and redistribution events may increase the success of a full svat inversion finally to extend the results of this study we recommend applying a similar analysis to other macroscopic conceptual root water uptake models

on dynamic feedback linearization abstract we address the problem of transforming a nonlinear multi input system into a linear controllable one via nonsingular dynamic feedback and extended state space diffeomorphism we show that in the single input case if a system is dynamic feedback linearizable it is also static feedback linearizable we then give sufficient conditions for a class of multi input systems to be dynamic feedback linearizable all three dimensional systems with two controls and a controllable linear approximation about the origin are shown to belong to such a class

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

an inverse eigenvalue problem for symmetrical tridiagonal matrices we consider the following inverse eigenvalue problem to construct a symmetrical tridiagonal matrix from the minimal and maximal eigenvalues of all its leading principal submatrices we give a necessary and sufficient condition for the existence of such a matrix and for the existence of a nonnegative symmetrical tridiagonal matrix our results are constructive in the sense that they generate an algorithmic procedure to construct the matrix

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

a hybrid global optimization method for inverse estimation of hydraulic parameters annealing simplex method inverse estimation of unsaturated hydraulic parameters is often a highly nonlinear optimization problem with multiple parameters the objective functions involved are often topographically complex and may contain many local minima because of these reasons the inverse solutions are commonly very sensitive to the initial guess of the parameters when conventional optimizers are used this paper presents an annealing simplex method that incorporates simulated annealing strategies into a classical downhill simplex method an upward infiltration experiment was used as an example of inverse estimation to test the method numerical experiments of both minimizing an algebraic function and inversion of upward infiltration data showed that the new method successfully converged to the global minimum in all cases irrespective of the initial hydraulic parameter estimates while the classical downhill method often converged to unfavorable local minima the cpu times needed for the annealing simplex method to estimate 5 and 7 hydraulic parameters simultaneously are about a half hour and 1 hour on a pc respectively additionally no special requirements are imposed on the objective function and the method is independent of the details of the simulation submodel therefore the proposed method should be applicable to other optimization problems in water resources when it is important to have a robust global search capability

inverse spectral problems for collections of leading principal submatrices of tridiagonal matrices abstract which assignments from 2 n 1 arbitrary distinct real numbers as eigenvalues of designated leading principal submatrices permit a real symmetric tridiagonal matrix we raise this question motivated both by known results and recent work on multiplicities and interlacing equalities in symmetric matrices whose graph is a given tree known results are reviewed a general conjecture is given and several new partial results are proved

stochastic analysis of three phase flow in heterogeneous porous media 1 spectral perturbation approach stochastic analysis of steady state three phase water oil and air flow in heterogeneous porous media was performed using the perturbation theory and spectral representation technique the governing equations describing flow of water oil and air are coupled and nonlinear the key stochastic input variables are intrinsic permeability k and the soil grain size distribution index three different stochastic combinations of these two input parameters were considered in the analysis the spectral perturbation approach was used to develop closed form expressions to describe stochastic variability of key output processes such as capillary and individual phase pressures the second paper of this two part series abdin and kaluarachchi this issue will discuss the applicability and the validity of the spectral perturbation technique to study the stochastic behavior of three phase flow using numerical simulations

the implicit construction of multiplicity lists for classes of trees and verification of some conjectures abstract for the problem of understanding what multiplicities are possible for eigenvalues among real symmetric matrices with a given graph constructing matrices with conjectured multiplicities is generally more difficult than finding constraining conditions here the implicit function theorem method for constructing matrices with a given graph and given multiplicity list is refined and extended in particular the breadth of known circumstances in which the jacobian is nonsingular is increased this allows characterization of all multiplicity lists for binary diametric depth one trees in addition the degree conjecture and a conjecture about the minimum number of multiplicities equal to 1 is proven for diametric trees finally an intriguing conjecture about the eigenvalues of a matrix whose graph is a path and its submatrices is given along with a discussion of some ides that would support a proof of the degree conjecture and the minimum number of 1 s conjecture in general

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

the quasilinear approximation for unsaturated porous media flow this paper gives a review of the quasilinear approximation an exponential relationship between hydraulic conductivity and moisture potential and the uses this has been put to since the time of its creation one of the major attractions of this approximation is that under steady regimes the highly nonlinear richards equation governing unsaturated flow can be linearized this simplification has enabled analytic and semianalytic solutions to be found for many unsaturated flow problems these are reviewed and the current research areas in which the quasilinear approximation is being used are discussed

the inverse eigenvalue problem for entanglement witnesses abstract we consider the inverse eigenvalue problem for entanglement witnesses which asks for a characterization of their possible spectra or equivalently of the possible spectra resulting from positive linear maps of matrices we completely solve this problem in the two qubit case and we derive a large family of new necessary conditions on the spectra in arbitrary dimensions we also establish a natural duality relationship with the set of absolutely separable states and we completely characterize witnesses i e separating hyperplanes of that set when one of the local dimensions is 2

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

the maximum of the minimal multiplicity of eigenvalues of symmetric matrices whose pattern is constrained by a graph abstract in this paper we introduce a parameter mm g defined as the maximum over the minimal multiplicities of eigenvalues among all symmetric matrices corresponding to a graph g we develop basic properties of mm g and compute it for several families of graphs

determining soil hydraulic properties by parameter estimation on the selection of a model for the hydraulic properties parameter estimation procedures involving solution of the inverse flow problem pertinent to a transient experiment constitute a powerful method of determining the soil hydraulic properties one of the problems associated with these procedures is the selection of an appropriate model to describe the soil hydraulic properties three different models were considered those of van genuchten vg and brooks and corey bc model and the exponential model of gardner for the hydraulic conductivity soil water pressure relationships coupled with a new expression for the soil water content soil water pressure relationship gr model for a given model the soil hydraulic properties were determined from simulated and measured outflow experiments supplemented with water content at a soil water pressure head of 15 000 cm h2o using the parameter estimation procedure of kool et al 1985a and data from two soils a hypothetical sandy loam soil assumed to be described by the vg model and a silt loam soil model validation tests were performed and the most appropriate model was selected from the candidate models by discrimination tests using the akaike information criterion aic in the case of the hypothetical sandy loam the vg model was found to be the most accurate and most consistent with the data the performance of both the bc and the gr models in terms of the aic was less good but very similar to each other in the case of the silt loam soil the vg model with additional parameter m which accounts for the correlation between pores and for the flow path tortuosity m 2 02 rather than a constant m 0 5 was found to be most accurate and most consistent with the data the performance of the bc and the gr models was similar when m was considered as an unknown parameter instead of a constant

lognormal distribution model for unsaturated soil hydraulic properties the soil water retention model developed by kosugi was modified to be compatible with mualem s model in order to derive an analytical expression for the relative hydraulic conductivity kr the modified water retention model is to be derived by applying a lognormal distribution law to the soil pore radius distribution function parameters of this retention model have physical significance on the water content capillary pressure curve and are related directly to the statistics of the pore radius distribution the accuracy of the resulting combined water retention hydraulic conductivity model is verified for observed data sets for six soils results showed that the proposed model produces acceptable matches with observed water retention curves and adequate predictions of hydraulic conductivities in five out of six cases the and kr or kr curves generated by this model are generally similar to those generated by van genuchten s model

analyses of infiltration events in relation to determining soil hydraulic properties by inverse problem methodology at the field scale because of the relatively large number of observation points required to estimate the spatial distributions of hydraulic properties application of inversion procedures must be based on a characteristic of the flow that for a given point in the field can be measured with relative ease and can be used as input for the inversion procedure suggesting the use of infiltration data for this purpose and recognizing the fact that a basic requirement for the design of a transient flow experiment is that the resulting inverse problem be sufficiently well posed to allow solution this paper addresses the question under which circumstances the resulting inverse problem is well posed utilizing the concepts of identifiability uniqueness and stability for this purpose three different models of the soil hydraulic properties were analyzed the main conclusion of this study is that when infiltration data measured at the darcy scale are used as input for the inversion procedure the inclusion of prior information on a single measurable parameter the saturated conductivity ks in the estimation criterion will enhance the likelihood of uniqueness and stability of the inverse solution provided that the structure of the hydraulic model is sufficiently simple however since for a particular situation it is impossible to determine a priori whether the resultant inverse problem is well posed or not this must be carried out only a posteriori by solving the problem several times with different initial parameter estimates accompanied by an analysis of the associated estimation errors

statistical analysis of spatial variability in unsaturated flow parameters core scale estimates of soil parameters of the gardner russo and van gemachten models of the hydraulic conductivity and water retention functions were obtained for 417 undisturbed soil cores taken from a wall of a trench 20 m long 2 5 m deep using a procedure based on inverse problem methodology these estimates were used to evaluate the first two statistical moments of the underlying random space functions rsfs using the restricted maximum likelihood estimation procedure coupled with the weighted least squares procedure to estimate parameters of models of the covariance and the drift functions of the pertinent rsfs the fitted models were used to evaluate the mean and covariance functions of the hydraulic conductivity and water rentention functions for given water saturations covariance functions of log saturated conductivity log ks and the shape parameters of the gardner russo and van genuchten models exhibited statistical anisotropy characterized by aspect ratios that vary between 3 to 4 correlation scales of log ks were larger than those of the shape parameters consequently the product of the variance of log unsaturated conductivity and its correlation scale remained essentially invariant for a considerable range of water saturation the implications of these results regarding stochastic modeling of transport in heterogeneous porous formations and possible applications of the results of this study are discussed briefly

complementary basic matrices abstract we show that an n n matrix which has both subdiagonal and superdiagonal rank at most one even if we distribute the diagonal positions except the first and last completely between the subdiagonal and superdiagonal part then this matrix can be factorized into a product of n 1 matrices each consisting of a 2 2 principal submatrix in two consecutive rows and columns in all possible of the n 1 positions and completed by ones along the diagonal the converse is also true it is shown that the spectrum does not depend on the order of the factors

extremal inverse eigenvalue problem for bordered diagonal matrices abstract the following inverse eigenvalue problem was introduced and discussed in j peng x y hu l zhang two inverse eigenvalue problems for a special kind of matrices linear algebra appl 416 2006 336 347 to construct a real symmetric bordered diagonal matrix a from the minimal and maximal eigenvalues of all its leading principal submatrices however the given formulae in 4 theorem 1 to compute the matrix a may lead us to a matrix which does not satisfy the requirements of the problem in this paper we rediscuss the problem to give a sufficient condition for the existence of such a matrix and necessary and sufficient conditions for the existence of a nonnegative such a matrix results are constructive and generate an algorithmic procedure to construct the matrices

stochastic analysis of unsaturated flow one dimensional monte carlo simulations and comparisons with spectral perturbation analysis and field observations a numerical experiment was designed to study the stochastic behavior of one dimensional transient unsaturated flow in a monte carlo setting soil hydraulic properties log saturated hydraulic conductivity ln ks pore size distribution parameter and the specific water capacity c are assumed to be statistically homogeneous random fields described by exponential correlation functions with identical correlation lengths fifty realizations of each soil hydraulic property with statistical properties obtained from a field experiment are generated by using a nearest neighbor model numerical solutions of the one dimensional flow equation are used repeatedly to obtain realizations of soil water pressure head and flux q corresponding to the realizations of in ks and c the dependence of and q on the statistical properties of ln ks and c and the magnitude of the hydraulic head gradient g prevailing at the lower boundary are investigated in addition results of monte carlo analysis and spectral perturbation analyses are compared with field observations the greatest variability of and q are observed when soil hydraulic properties are uncorrelated and have large variances and integral scales and when g at the lower boundary is unity the monte carlo and spectral perturbation analyses tend to agree reasonably well for the flow domains in which ergodicity of local soil hydraulic properties is assured results of the monte carlo and spectral perturbation analyses are also supported by field observations

on the construction of a jacobi matrix from its mixed type eigenpairs in this paper an inverse eigenvalue problem of constructing a jacobi matrix from its mixedtype eigenpairs is considered the necessary and sufficient conditions for the existence and uniqueness of the solutions are derived one numerical algorithm and two numerical examples are given

parameter estimation of hydraulic properties from one step outflow data the one step outflow method was presented by kool et al 1985a b as a fast procedure to estimate unsaturated soil hydraulic properties many investigators however encountered difficulties obtaining a unique solution to the problem the purpose of this paper is to evaluate some of the problems related to this method by looking at response surfaces and to make recommendations for improvements in the methodology the one step outflow procedure was investigated using synthetic data sets generated for 7 6 cm diameter by 7 6 cm high soil cores a finite difference solution of richards equation was used to generate the synthetic data sets response surfaces were calculated for the van genuchten parameters n and ks for various combinations of matric potential water content and outflow data from initially saturated soil cores subjected to a step change in pressure at one of its boundaries the results clearly show that with outflow data alone the parameter estimation procedure will have difficulty converging however if matric potential data are available for a point in the core at some distance from the outflow boundary parameter estimation sensitivity is greatly improved the procedure used was found to be most helpful for determining which auxiliary variables should be measured for parameter estimation from outflow experiments

effective soil water characteristics and ensemble soil water profiles in heterogeneous soils one dimensional vertical flows through spatially heterogeneous areas are studied with numerical solutions for a variety of soil types to ascertain the behavior of the ensemble soil heterogeneity is treated using the distributions of parameters describing the characteristic relations of conductivity water content and pressure head values of the critical parameters taken from a modified brooks corey or van genuchten model have previously 7lsqb smith and diekkruger 1992 been shown in two extensive data sets to exhibit little interdependence the static ensemble characteristics except for saturated and residual water content parameters are shown to have regular bias compared to a soil with the mean characteristic parameters the measured data also are shown not to support the popular similar media heterogeneity model using both analytic approximate flow models and an ensemble solution of richards equation the effective dynamic ensemble characteristics in three example soils are studied as suggested in previous studies such effective characteristics are conditional to flow conditions and are here shown to be quite different from ensemble static characteristics this indicates that irrespective of the sample size effective soil dynamic characteristics are not measurable for samples possessing significant heterogeneity

graphs that allow all the eigenvalue multiplicities to be even abstract let g be an undirected graph on n vertices and let s g be the set of all n n real symmetric matrices whose nonzero off diagonal entries occur in exactly the positions corresponding to the edges of g the inverse eigenvalue problem for a graph g is a problem of determining all possible lists that can occur as the lists of eigenvalues of matrices in s g this question is in general hard to answer and several variations were studied most notably the minimum rank problem in this paper we introduce the problem of determining for which graphs g there exists a matrix in s g whose characteristic polynomial is a square i e the multiplicities of all its eigenvalues are even we solve this question for several families of graphs

the reconstruction of a specially structured jacobi matrix with an application to damage detection in rods a finite element discretization of the differential equation for the axial vibration of a rod with varying cross section leads to a specially structured n x n matrix pencil the reconstruction of this pencil from its spectrum can be achieved by the reconstruction of a unique jacobi matrix using half of its spectrum and half of the spectrum of its principal submatrix of order n 1 the technique is used in an optimization problem formulated for damage detection in rods defined in terms of changes in the effective cross sectional area

a triangular canonical form for a class of 0 flat nonlinear systems this article proposes a triangular canonical form for a class of 0 flat nonlinear systems necessary and sufficient geometrical conditions are given in order to guarantee the existence of a local diffeomorphism to transform the studied nonlinear systems into the proposed 0 flat canonical form which enables us to compute the flat output as well

two inverse eigenvalue problems for a special kind of matrices in this paper a special kind of matrices which are symmetric all elements are equal to zero except for the first row the first column and the diagonal elements and the elements of the first row are positive except for the first one are considered two inverse problems are discussed one is to construct one of this kind of matrices by the minimal and maximal eigenvalues of its all leading principal submatrices the other is to construct one of this kind of matrix by one of its eigenpair and eigenvalues of its all leading principal submatrices the necessary and sufficient conditions for the solvability of the two problems are derived furthermore corresponding numerical algorithms and some examples are given

the niep for four dimensional leslie and doubly stochastic matrices with zero trace from the coefficients of the characteristic polynomial abstract the constraints on the coefficients of the characteristic polynomials of four dimensional doubly stochastic matrices with zero trace and leslie stochastic matrices with zero trace are determined then on the basis of these constraints the regions of the complex plane consisting of those points which can serve as characteristic roots of four dimensional doubly stochastic matrices with zero trace and leslie stochastic matrices with zero trace are characterized

unsaturated flow in spatially variable fields 3 solute transport models and their application to two fields an approximate model of salt transport in a spatially variable field during infiltration and redistribution is presented the water flow is assumed to be vertical and both pore water velocity and depth of wetting front change in the horizontal plane due to the variability of soil hydraulic properties these variables are computed with the aid of the model developed in part 1 dagan and bresler this issue and part 2 bresler and dagan this issue of the present study the salt transport in the vertical profile is computed by using an approximate closed form solution of the convection dispersion equation with the value of the dispersivity increasing from zero to its maximum 3 cm as the front propagates downward the concentration profiles vary in the horizontal plane because of the aforementioned variation of water flow variables the expectation value and the variance of the concentration are computed as function of depth and time for the two variable soils considered in part 2 panoche with large variability and bet dagan with lesser variability the results are compared with those based on a numerical simulation of the water flow and salt transport as well as with those pertaining to an equivalent deterministic uniform soil for the expected value it is shown that the approximate model yields quite accurate results when compared with the numerical simulations for the field of large variability while some differences are present for the more uniform one in any case the results for the expected value based on the traditional approach of replacing the variable field by an equivalent uniform one include a much larger error it is suggested that the approximate simplified model presented here may serve as a basis for simulation of additional effects not considered here e g adsorption root extraction and change of soil properties due to salt content the main conclusion is that in spite of the apparent complexity the statistical moments of salt concentration in a spatially variable field can be determined by using simple flow models therefore further refinement of models of transport in homogeneous columns does not seem to be warranted and the research effort should be concentrated on the study of field variability

stochastic analysis of unsaturated flow with probabilistic collocation method 1 in this study we present an efficient approach called the probabilistic collocation method pcm for uncertainty analysis of flow in unsaturated zones in which the constitutive relationship between the pressure head and the unsaturated conductivity is assumed to follow the van genuchten mualem model spatial variability of soil parameters leads to uncertainty in predicting flow behaviors the aim is to quantify the uncertainty associated with flow quantities such as the pressure head and the effective saturation in the proposed approach input random fields i e the soil parameters are represented via the karhunen loeve expansion and the flow quantities are expressed by polynomial chaos expansions pces the coefficients in the pces are determined by solving the equations for a set of carefully selected collocation points in the probability space to illustrate this approach we use two dimensional examples with different input variances and correlation scales and under steady state and transient conditions we also demonstrate how to deal with multiple input random parameters to validate the pcm we compare the resulting mean and variance of the flow quantities with those from monte carlo mc simulations the comparison reveals that the pcm can accurately estimate the flow statistics with a much smaller computational effort than the mc

estimation of brooks corey parameters from water retention data the parameters of the brooks corey soil moisture characteristic may be determined by either graphical or automatic numerical procedures a popular log log procedure for automatic estimation leads to a linear least squares optimization problem apparently it has been applied incorrectly in the past when all measurements were used for parameter estimation and this realization explains the inconsistencies in some published data its proper application generally requires the solution of more than one linear problem for each data set alternatively a nonlinear procedure will yield parameter estimates directly in either case there may exist multiple local minima in the sum of squared deviations between model and measurements and the parameter estimates so derived are demonstrably unreliable this appears to be a result of the distinct air entry suction implicit in the brooks corey model and usually absent in field measured characteristic curves a new procedure based on an integral objective function overcomes the problems cited

upscaling for unsaturated flow for non gaussian heterogeneous porous media 1 large scale models of transient flow processes in the unsaturated zone require in general upscaling of the flow problem in order to capture the impact of heterogeneities on a small scale which cannot be resolved by the model effective parameters for the upscaled models are often derived from second order stochastic properties of the parameter fields such properties are good quantifications for parameter fields which are multi gaussian however the structure of soil does rarely resemble these kinds of fields the non multi gaussian field properties can lead to strong discrepancies between predictions of upscaled models and the averaged real flow process in particular the connected paths of parameter ranges of the medium are important features which are usually not taken into account in stochastic approaches they are determined here by the euler number of one cut indicator fields methods to predict effective parameters are needed that incorporate this type of information we discuss different simple and fast approaches for estimating the effective parameter for upscaled models of slow transient flow processes in the unsaturated zone where connected paths of the material may be taken into account upscaled models are derived with the assumption of capillary equilibrium the effective parameters are calculated using effective media approaches we also discuss the limits of the applicability of these methods

a lie backlund approach to equivalence and flatness of nonlinear systems a new system equivalence relation using the framework of differential geometry of jets and prolongations of infinite order is studied in this setting two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives this is a lie backlund isomorphism the authors prove that although the state dimension is not preserved the number of input channels is kept fixed they also prove that a lie backlund isomorphism can be realized by an endogenous feedback the differentially flat nonlinear systems introduced by the authors 1992 via differential algebraic techniques are generalized and the new notion of orbitally flat systems is defined they correspond to systems which are equivalent to a trivial one with time preservation or not the endogenous linearizing feedback is explicitly computed in the case of the vtol aircraft to track given reference trajectories with stability

exploring dynamic effects in capillary pressure in multistep outflow experiments 1 traditional steady state experiments to measure constitutive relations governing two phase organic aqueous flow in the subsurface often extend over periods of weeks or months alternatively one step or multistep outflow mso experiments can be combined with application of a multiphase flow simulator and an optimization algorithm to achieve a more rapid technique for parameter estimation in this work mso experiments were conducted to produce a data set for the estimation of two phase constitutive parameters using this inverse modeling approach examination of experimental results reveals significant discrepancies between observed and simulated outflow data with simulated curves tending to approach equilibrium at a faster rate than the experimental observations similar behavior has been documented by other investigators application of alternative equilibrium constitutive models in the multiphase flow optimization simulator failed to improve model fits to observed data however when model governing equations were modified to incorporate a dynamic capillary pressure term there was significant improvement in the agreement between measured and simulated cumulative water outflow and outflow rates comparisons of simulated and measured data further suggest that the dynamic capillary pressure constitutive coefficient depends on saturation attribution of the observed experimental deviations to dynamic effects in capillarity is also supported by the consistency of the fit equilibrium retention function with an independently measured static retention relation

stochastic analysis of steady state unsaturated flow in heterogeneous media comparison of the brooks corey and gardner russo models existing stochastic models of unsaturated flow and transport are usually developed using the simple gardner russo constitutive relationship though it is generally accepted that the more complex van genuchten and brooks corey relationships may perform better in describing experimental data in this paper we develop first order stochastic models for gravity dominated flow in second order stationary media with both the brooks corey and the gardner russo constitutive relationships these models also account for the spatial variability in effective water content while the spatial variability is generally neglected in most existing stochastic models analytical solutions are obtained for the case of one dimensional gravity dominated flow on the basis of the solutions we illustrate the differences between results from these two constitutive models through some one dimensional examples it is found that the impacts of the constitutive models on the statistical moments of suction head effective water content unsaturated hydraulic conductivity and velocity depend on the saturation ranges for example the mean head and the mean effective water content for the brooks corey model differ in a great manner with their counterparts for the gardner russo model near the dry and wet limits while the differences are small at the intermediate range of saturation this finding is confirmed with some two dimensional examples it is also found that the brook corey model has certain advantages over the gardner russo model in analyzing unsaturated flow in heterogeneous media for example the stochastic model developed based on the brooks corey function requires the coefficient of variation of head and soil parameter bc to be small 1 whereas that based on the gardner russo function assumes the one point cross covariance of head and olgr to be small 1 illustrative examples reveal that the latter condition may be violated because the one point covariance is found to increase rapidly to beyond unity as the soil becomes dry whereas the former may be readily satisfied