sensitivity analysis for steady state groundwater flow using adjoint operators adjoint sensitivity theory is currently being considered as a potential method for calculating the sensitivity of nuclear waste repository performance measures to the parameters of the system for groundwater flow systems performance measures of interest include piezometric heads in the vicinity of a waste site velocities or travel time in aquifers and mass discharge to biosphere points the parameters include recharge discharge rates prescribed boundary heads or fluxes formation thicknesses and hydraulic conductivities the derivative of a performance measure with respect to the system parameters is usually taken as a measure of sensitivity to calculate sensitivities adjoint sensitivity equations are formulated from the equations describing the primary problem the solution of the primary problem and the adjoint sensitivity problem enables the determination of all of the required derivatives and hence related sensitivity coefficients in this study adjoint sensitivity theory is developed for equations of two dimensional steady state flow in a confined aquifer both the primary flow equation and the adjoint sensitivity equation are solved using the galerkin finite element method the developed computer code is used to investigate the regional flow parameters of the leadville formation of the paradox basin in utah the results illustrate the sensitivity of calculated local heads to the boundary conditions alternatively local velocity related performance measures are more sensitive to hydraulic conductivities

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

efficient geostatistical inverse methods for structured and unstructured grids 1 the geostatistical inverse approach of kitanidis 1995 requires the covariance matrix of a discretized spatial random variable and matrix multiplications of the covariance matrix the explicit computation of the full covariance matrix is prohibitive for large scale problems discrete spectral methods which help to reduce computational costs dramatically are restricted to regular grids to make the geostatistical inverse approach applicable to unstructured grids which may be of great interest for practical applications we use the functional parameterization of the spatial random field by the karhunen loeve kl expansion in which the spatial variable is parameterized by weighted base functions that are derived from the covariance function in the inverse approach the estimation of the spatial variable becomes estimating the weights of base functions the prior covariance matrix of the weights is the identity matrix the base functions are continuous in space and can be discretized in any fashion for the derivation of the base functions we embed the domain into a larger unit cell of a periodic domain then the base functions are sinusoidal and can be derived by spectral methods which are independent of the discretization used for forward and inverse modeling for illustration we present case studies using synthetic data to estimate hydraulic conductivities on regular and unstructured grids the results show that we can efficiently obtain a good estimate using the inverse method in conjunction with the kl expansion we also demonstrate how to estimate structural parameters and to generate conditional realizations in the framework of our method

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

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

sensitivity methods for time continuous spatially discrete groundwater contaminant transport models existing sensitivity methods for spatially discrete groundwater contaminant transport models have been developed for time stepping numerical algorithms and cannot be readily used with time continuous approaches to transport simulation such as the laplace transform galerkin technique we develop direct and adjoint sensitivity methods in which sensitivity coefficients are computed in the laplace domain and inverted numerically to the time domain the methods are computationally efficient when used in conjunction with time continuous transport equations the relative efficiency of the two methods depends on the number of model parameters number of performance measures and number of spatial discretization nodes the adjoint method is favored when the number of performance measures is much smaller than the number of model parameters the adjoint method is limited in that performance measures are restricted to being linear functions of state variables a two dimensional transport example is developed in detail and sensitivities with respect to nodal hydraulic conductivities are computed in the problems analyzed the direct and adjoint methods are 9 to 156 times faster than the perturbation method with the computational savings increasing as the size of the problem is increased

a statistical methodology for estimating transport parameters theory and applications to one dimensional advectivec dispersive systems a simulation nonlinear multiple regression methodology for estimating parameters that characterize the transport of contaminants is developed and demonstrated finite difference contaminant transport simulation is combined with a nonlinear weighted least squares multiple regression procedure the technique provides optimal parameter estimates and gives statistics for assessing the reliability of these estimates under certain general assumptions about the distributions of the random measurement errors monte carlo analysis is used to estimate parameter reliability for a hypothetical homogeneous soil column for which concentration data contain large random measurement errors the value of data collected spatially versus data collected temporally was investigated for estimation of velocity dispersion coefficient effective porosity first order decay rate and zero order production the use of spatial data gave estimates that were 2 3 times more reliable than estimates based on temporal data for all parameters except velocity comparison of estimated linear and nonlinear confidence intervals based upon monte carlo analysis showed that the linear approximation is poor for dispersion coefficient and zero order production coefficient when data are collected over time in addition examples demonstrate transport parameter estimation for two real one dimensional systems first the longitudinal dispersivity and effective porosity of an unsaturated soil are estimated using laboratory column data we compare the reliability of estimates based upon data from individual laboratory experiments versus estimates based upon pooled data from several experiments second the simulation nonlinear regression procedure is extended to include an additional governing equation that describes delayed storage during contaminant transport the model is applied to analyze the trends variability and interrelationship of parameters in a mourtain stream in northern california

periodic solutions for damped differential equations with a weak repulsive singularity abstract this paper deals with the existence of positive t periodic solutions for the following damped differential equation x p t x q t x f t x c t where p q c l 1 r are t periodic functions and f car r r r is t periodic in the first variable we will prove that a weak repulsive singularity enables the achievement of new existence criteria through a basic application of schauder s fixed point theorem

a proposed upstream weight numerical method for simulating pollutant transport in groundwater an alternative upstream weighting numerical method is proposed to simulate the transport process in groundwater under a variety of conditions the discretized system of the solute transport equations is derived from the integral form that expresses the balance of mass in each element an upstream weight that depends on the local peclet number is directly added to the basis functions when convective transport dominates the dispersive transport this technique can eliminate the oscillation of solutions efficiently with only a nominal increase of computation effort over that of the general linear triangular galerkin finite element method the accuracy of the proposed numerical model is tested against two classical problems for which analytical solutions exist the agreement is nearly exact an additional numerical example is used to illustrate the applicability of this model

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

a proposed stepwise regression method for model structure identification this paper proposes a new methodology for constructing groundwater models the proposed methodology which determines simultaneously both model structure and model parameters is based on the following ideas 1 when solving the inverse problem different model structures always produce different model parameters 2 since the number of possible model structures of an aquifer is infinite the number of possible representative parameters is also infinite 3 to obtain a set of appropriate representative model parameters we must have an appropriate model structure and 4 an appropriate model structure should be determined not only by observation data and prior information but also by the accuracy requirements of model applications in this proposed methodology we start with a homogeneous model structure and step by step gradually increase the complexity of the model structure at each level of complexity we calculate not only the fitting residual of parameter identification but also the error of model structure to determine if a more complex model structure is needed the model structure error of using one model structure to replace another model structure is defined by a maximum minimum max min problem that is based on the distance between the two models and is measured in parameter observation and prediction or decision spaces this proposed methodology is used to solve a hypothetical remediation design problem in which the true hydraulic conductivity is a random field with a certain trend we have found that for the example problem virtually identical pumping policy is obtained when a five zone model with an optimized zonation pattern is used to represent the nonstationary random field we have also found that observation errors have minimum impact on management solution in comparison with structure errors to calculate the model structure error for this example the inverse solution is coupled with a management problem we have also developed an effective iteration method to handle nonlinear water quality constraints

coupled inverse problems in groundwater modeling 1 sensitivity analysis and parameter identification some basic concepts and methods for solving the coupled inverse problem in groundwater modeling are presented in a series of two papers because there are crossover effects between state variables and parmeters coupled inverse problems are different from single inverse problems in several aspects especially in the selection of objective functions identifiability and experimental design this paper discusses the general definition of coupled inverse problems posed in the framework of multiobjective optimization in order to obtain gradient vectors of objective functions as well as all sensitivity coefficients a general procedure for deriving the adjoint state equations and their associated conditions is described using the rules of adjoint operation presented in this paper the derivation of adjoint problems becomes very easy even for nonlinear and transient coupled problems as examples the adjoint problem of flow mass transport or flow heat transport the adjoint problem of saltwater intrusion and the adjoint problem of two phase flow are derived numerical experiments on a two dimensional flow mass transport problem are given to explain the procedure of sensitivity analysis and parameter identification for coupled problems the hydraulic conductivity can be identified by using observations of head and or solute concentration

bayesian calibration of a large scale geothermal reservoir model by a new adaptive delayed acceptance metropolis hastings algorithm 1 the aim of this research is to estimate the parameters of a large scale numerical model of a geothermal reservoir using markov chain monte carlo mcmc sampling within the framework of bayesian inference all feasible parameters that are consistent with the measured data are summarized by the posterior distribution and hence parameter estimation and uncertainty quantification are both given by calculating expected values of statistics of interest over the posterior distribution it appears to be computationally infeasible to use the standard metropolis hastings algorithm mh to sample the high dimensional computationally expensive posterior distribution to improve the sampling efficiency a new adaptive delayed acceptance mh algorithm adamh is implemented to adaptively build a stochastic model of the error introduced by the use of a reduced order model this use of adaptivity differs from existing adaptive mcmc algorithms that tune proposal distributions of the metropolis hastings algorithm mh though adamh also implements that technique for the 3 d geothermal reservoir model we present here adamh shows a great improvement in the computational efficiency of the mcmc sampling and promising results for parameter estimation and uncertainty quantification are obtained this algorithm could offer significant improvement in computational efficiency when implementing sample based inference in other large scale inverse problems

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

a comparison of several methods of solving nonlinear regression groundwater flow problems computational efficiency and computer memory requirements for four methods of minimizing functions were compared for four test nonlinear regression steady state groundwater flow problems the fastest methods were the marquardt and quasi linearization methods which required almost identical computer times and numbers of iterations the next fastest was the quasi newton method and last was the fletcher reeves method which did not converge in 100 iterations for two of the problems the fastest method per iteration was the fletcher reeves method and this was followed closely by the quasi newton method the marquardt and quasi linearization methods were slower for all four methods the speed per iteration was directly related to the number of parameters in the model however this effect was much more pronounced for the marquardt and quasi linearization methods than for the other two hence the quasi newton and perhaps fletcher reeves method might be more efficient than either the marquardt or quasi linearization methods if the number of parameters in a particular model were large although this remains to be proven the marquardt method required somewhat less central memory than the quasi linearization metilod for three of the four problems for all four problems the quasi newton method required roughly two thirds to three quarters of the memory required by the marquardt method and the fletcher reeves method required slightly less memory than the quasi newton method memory requirements were not excessive for any of the four methods

parameter estimation for coupled unsaturated flow and transport this paper deals with the estimation of soil hydraulic and transport parameters from transient unsaturated flow and tracer experiments using a combined simulations optimization approach hydraulic properties are defined by a modified form of van genuchten s 1980 parametric model for two phase permeability saturation pressure relations and transport properties are defined by an empirical parametric dispersion model a nonlinear weighted least squares algorithm is used to estimate unknown model parameters by minimizing deviations between concentrations water contents and pressure heads obtained from hypothetical infiltration redistribution evaporation experiments and those predicted by solving a numerical model of coupled unsaturated flow and transport simultaneous estimation of hydraulic and transport properties is found to yield smaller estimation errors for model parameters than sequential inversion of hydraulic properties from water content and pressure head data followed by inversion for transport properties from concentration data effects of random noise in data measurements soil layering and choice of improper parametric model on the parameter estimation process are discussed

simultaneous inversion of hydrogeologic and thermal data 1 theory and application using hydraulic head data one goal in solving the inverse problem in groundwater hydrology is to obtain parameter estimates for use in a simulation model a philosophy about the inverse is adopted here that takes the form of constructing a variety of models under different norms large differences in parameter estimates obtained using different norms suggests the possibility of an incorrect interpretational model an optimization procedure is used that is based on the constrained simplex method this algorithm easily handles constraints does not require derivatives of the functional with respect to the parameters and can be employed to minimize complicated functionals the algorithm produces a cluster of search points in parameter space about the global minimum of the objective function each of these points generates data that matches observed values within a prescribed tolerance the centroid of the search points is taken as the best estimate of the parameters the covariances of the estimated parameters and estimated data can also be computed this inverse technique is applied to a sparse inaccurate hydraulic head data set observed at downie slide british columbia canada for this cross sectional problem with noisy data and a strong drift component interpolation of hydraulic head values using kriging techniques was not possible the domain is represented as a homogeneous isotropic medium four parameters are estimated the free surface position of the water table and three boundary conditions further simulations using a synthetic data set with assumed properties similar to that of downie slide show that with noise free data and an adequate spacing between measurement points it is possible to interpolate an unbiased estimate of hydraulic head data at all nodes in the equivalent discretized domain when the inverse technique is applied the hydraulic conductivity structure is correctly identified the implications for downie slide are that in order to resolve more than a simple hydraulic conductivity structure accurate field measurements of hydraulic head a better spacing between measurements and more detailed knowledge of the boundary conditions are required

variational sensitivity analysis data requirements and parameter identification in a leaky aquifer system the adjoint state sensitivity method is developed to aid the analysis of a pumping test design for parameter identification in a leaky aquifer system the system considered consists of an unconfined aquifer overlaying an aquitard which overspreads a confined aquifer the adjoint equations are derived in the continuous form without discretizing the governing equations and are expressed in terms of convection diffusion equations with time dependent coefficients the multiple cell balance mcb method is used to solve the nonlinear governing equations as well as their adjoint equations algorithms are developed for calculating sensitivity coefficients of heads in each layer of an aquifer with respect to various parameters a concept called contribution of an observation is introduced which can be used to measure the usefulness of an observation in connection with data requirements for parameter identification it links experimental design with observation errors and the requirement of model application the sufficiency of an observation system for parameter identification can be evaluated at the design stage before extensive resources are committed to data collection a numerical example is given to verify the proposed method and to explain its applicability to the design of a pumping test

an upstream weight multiple cell balance finite element method for solving three dimensional convection dispersion equations an upstream weight multiple cell balance finite element method is developed to simulate the three dimensional contaminant transport in aquifers the method combines the concepts of local mass balance and finite element approximations when convective transport dominates the dispersive transport the proposed technique effectively eliminates the oscillation of numerical solutions with only a nominal increase of formulation effort over that of the general galerkin finite element method two numerical examples for which analytical solutions exist are used to verify the proposed numerical solutions an additional example is presented to demonstrate the practical applicability of the proposed method

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

the distribution of bidegrees of smooth surfaces in gr 1 p3 the study of algebraic surfaces in gr 1 p3 the grassmann variety parametrising lines in projective three space was a popular one for algebraic geometers of the late nineteenth and early twentieth centuries calling them line congruences researchers such as kummer fano roth and many others published many papers on the topic classifying congruences and studying their invariants since that time the field has lain dormant until very recently the classical geometers identified two numbers associated with a given congruence the order and the class thinking of a congruence as a two dimensional family of lines the order is the number of lines in the family passing through a general point in p3 and the class the number of lines in the family contained in a general plane together these two numbers make up the bidegree of the congruence in modern terms the bidegree gives the class of the congruence in the chow ring of gr 1 p3 in this paper we consider the question for what values of a and b does there exist or not exist a smooth congruence of bidegree a b in particular we try to find restrictions on the bidegree using an approach suggested by dolgachev and reider in 8 this approach is to study the restriction of the universal bundle g of gr 1 p3 which appears in the exact sequence

illustration and verification of adjoint sensitivity theory for steady state groundwater flow the application of adjoint sensitivity theory to steady state ground water flow is illustrated with three one dimensional flow problems adjoint states are analytically derived for four performance measures of these test problems hydraulic head at a point spatially average hydraulic head darcy velocity at a point and flux from a prescribed head boundary sensitivity coefficients are analytically calculated for average head the adjoint states are interpreted and their usefulness is discussed the implementation of a numerical adjoint sensitivity flow code to solve for these adjoint states is described and the computed adjoint states are used in the code to evaluate the sensitivities of model results to model input parameters the one dimensional flow problems provide a set of verification tests for the numerical code the numerical code successfully reproduces both the analytically derived adjoint states including those involving jump conditions and the sensitivity coefficients for model output values that are nonlinear with respect to model parameters

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

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

coupled inverse problems in groundwater modeling 2 identifiability and experimental design parameter identifiability and experimental design in the context of solving the coupled inverse problem in groundwater modeling are considered in part 2 of the two paper series the following three new definitions of extended identifiability for a distributed parameter system are presented interval identifiability ini prediction equivalence identifiability pei and management equivalence identifiability mei the uniqueness requirement of the inverse solution is relaxed by these definitions an identified parameter is said to be extendedly identifiable if it is able to satisfy an accuracy requirement in a specific application of the model when a generalized least squares norm is used the proposed identifiabilities are applicable to any coupled problems in which the state variables and parameters may have different order of magnitudes a salient feature of the new definitions is that the extended identifiabilities are closely related to experimental design consequently the sufficiency of an experimental design for assuring a given level of an extended identifiability can be estimated prior to field experiments the methodology developed can be used to determine the reliability of an experimental design even for nonlinear models a numerical example on the management of a wastewater disposal is given to explain how the concepts and methods presented in this paper can be used for solving practical problems

identification of parameter structure in groundwater inverse problem this paper develops an optimization method that can be used to identify parameter structure as well as its values in connection with the inverse problem in groundwater modeling the complexity of the parameter structure is determined by the quantity and quality of observation data output least square error is used as the criterion of identification sensitivity coefficients at finite element nodes are evaluated by the variational method the problem of identifying the parameter structure is formulated as a problem of combinatorial optimization a new parameterization technique is presented for describing the parameter structure it can represent various distributions of parameters from piecewise constant to continuous variability based on this representation the combinatorial optimization problem for determining the parameter structure is then solved by a simple one at a time searching procedure numerical experimentations are used to demonstrate the efficiency and practicality of the proposed method