A method for solving the inverse problem in hydrogeology is presented. This method is suitable for computing the interblock transmissivities (harmonic mean or others) referred to the sides of the network blocks of a nonhomogeneous, anisotropic aquifer in steady state flow. The interblock transmissivities computing procedure is based on the comparison between real gradients and the ones generated by a 'comparison model' whose initial transmissivity value is arbitrarily chosen and constant through out the surveyed area. Alternative solutions of interblock transmissivities, one for each constant initial transmissivity value utilized in the comparison model, are obtained. Generally, these solutions differ from each other, but all of them present the characteristics of being able to reproduce, with the precision of an arbitrarily small e, the real piezometric heads, respecting the geometry and the boundary conditions of the real aquifer. The selection of the value, or set of values, of the initial transmissivity to put into the comparison model in order to obtain one solution of computed interblock transmissivities close to the real ones, both as trends and as absolute values, is rather critical. The minimum head anomaly criterion and the bottleneck criterion enable one to select a value or a set of values of initial transmissivity. These criteria exploit only the data needed for the solution of the inverse problem (real piezometric heads, geometry, and boundary conditions), and they lead to a suitable initial transmissivity that, put into the comparison model, allow one to obtain a good and meaningful solution of computed interblock transmissivities. The comparison model method and the two aforementioned criteria have been applied to a numerical case study bearing good results. The method has been tested, in order to evaluate its effectiveness, even in the case of noisy piezometric heads (piezometric heads with random errors). Even in this case the method bore satisfactory and useful results. Finally, a promising approach to the inverse problem solution, when one has at his disposal at least four sets of data coming from as many hydraulic situations, has been briefly described. objective function composed of some measure of the difference between observed and computed heads. Among the authors who affronted the identification problem, we will mention Emsellem and DeMarsily [1971], Neuman [1973], Neuman et al. [1980], Frind and Pinder [1973], Hefez et al. [1975], Sagar et al. [1975], Neuman and Yakowitz [1979], and Irmay [1980]. Scarascia and Ponzini [1972] presented a method for the solution of the inverse problem that minimizes the anomalies between the measured piezometric heads and the computed ones. The present paper, which from some aspects is a continuation of the Scarascia and Ponzini [!972] paper, describes a method suitable for the identification of the transmissivities of a single-layer aquifer in steady state flow, in which the boundary conditions (piezometric heads at the boundaries or known flow rates, fl...

Modern and effective water management in large alluvial plains that have intensive agricultural activity requires the integrated modeling of soil and groundwater. The models should be complex enough to properly simulate several, often non-linear, processes, but simple enough to be effectively calibrated with the available data. An operative, practical approach to calibration is proposed, based on three main aspects. First, the coupling of two models built on wellvalidated algorithms, to simulate (1) the irrigation system and the soil water balance in the unsaturated zone and (2) the groundwater flow. Second, the solution of the inverse problem of groundwater hydrology with the comparison model method to calibrate the model. Third, the use of appropriate criteria and cross-checks (comparison of the calibration results and of the model outputs with hydraulic and hydrogeological data) to choose the final parameter sets that warrant the physical coherence of the model. The approach has been tested by application to a large and intensively irrigated alluvial basin in northern Italy.

For two-dimensional groundwater flow in an isotropic confined aquifer, it has been shown elsewhere that two independent steady state sets of data, i.e., piezometric heads and source terms corresponding to different steady state flow conditions, and the value of transmissivity at one point sutfice to determine transmissivity uniquely in a connected domain. The data are independent if the hydraulic gradients are not parallel anywhere over the domain. Here transmissivity is numerically determined by integration of suitably approximated functions of the data along polygonal lines connecting the nodes of a lattice; integration starts from the node where transmissivity is given. The choice of the integration path is based on the results of the stability analysis and allows us to minimize the effects of the approximations on the data. Since the approximated solution is computed along internode segments, the internode transmissivities are immediately calculated without introducing arbitrary averages of the node transmissivities. The internode transmissivities are the quantities necessary to set up a management model within a conservative finite differences scheme. The applicability of this technique to real cases is tested with two synthetic examples. The first one was set up by ourselves, whereas the second one has been taken from the literature. The internode transmissivities identified with our procedure are compared with the synthetic reference ones. The ultimate check is performed of evaluating new head fields on the basis of the identified and reference internode transmissivities. The fit is good. The relative error for the identified internode transmissivities is very low when error-free data are used, and it varies by an amount approximately constant over the entire aquifer when an error on the initial value of transmissivity is introduced. The errors on the piezometric heads bear more relevance, but nonetheless, the affected results are still good. GIUDICI ET AL.: IDENTIFICATION OF DISTRIBUTED TRANSMISSIVITIESIn management modeling, we have to solve the forward problem, which consists of finding a solution of (2) for h, when the transmissivity and the source term are given together with appropriate boundary conditions on h along the boundary of the aquifer.Measuring the transmissivity is a formidable task, and even if the transmissivity is obtained with experimental field procedures, for instance, by means of pumping tests, it is still a debatable question how these values should be introduced in a mathematical model [Dykaar and Kitanidis, 1993]. A better approach to finding the transmissivity to be used in a forward problem is that of solving an inverse problem.Mathematically, the inverse problem can be stated as follows: given h (x, y) and f(x, y), find r(x, y) such that (2) holds. In this case, (2) is considered as a first-order linear partial differential equation, whose solution is not unique in general. The lack of uniqueness is a very important point. In fact, the solution of the inverse problem is used t...

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The comparison model method (CMM) is applied to the identification of spatially varying thermal conductivity in a one‐dimensional domain. This method deals with the discretized steady‐state heat equation written at the nodes of a lattice, a lattice which models a stack of plane parallel layers. The required data are temperature gradient and heat source (or sink) values. The unknowns of this inverse problem are not nodal values but internode thermal conductivities, which appear in the node heat balance equation. The conductivities, e.g., the solutions to the inverse problem obtained by the CMM, are a one‐parameter family. In order to achieve uniqueness (which coincides with identifiability in this case), a suitable value of this parameter must be found. To this end we consider (1) parameterization, i.e., introducing equality constraints between the unknown coefficients, (2) the use of two data sets at least in a subdomain, and (3) self‐identifiability. Each of these items formally translates the available a priori information about the system, e.g., geophysical properties of the layers. Under the assumption that an admissible solution exists, we evaluate the effects on the solution of two types of data noise: additive noise affecting temperature and a kind of multiplicative noise affecting heat‐source terms. In the numerical examples we provide, parameterization is combined with the CMM in order to obtain the unique solution to a test inverse problem, the geophysical data of which come from a well drilled across Tertiary layers in central Italy. Finally, we consider data perturbed by pseudorandom noise. More precisely, we add noise to temperature gradients and obtain stability estimates which correctly predict the numerical results. In particular, if the layers where the parameterization constraint applies contain a strong source term (e.g., due to flowing water), the solution is relatively insensitive to noise. Also, for multiplicative noise affecting heat‐source terms, theoretical stability predictions are confirmed numerically. The main advantage of the CMM is its simple algebraic formulation. Its implementation in the field by means of a pocket calculator allows both a consistency check on the data being collected and estimates of the unknown values of conductivity.

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We identify the coefficients of the transport equation in N dimensions grad c.grad h+c Delta h=d delta h/ delta t+f by solving a differential system of the form grad c+ca=b. The assignment of c at one point only yields a unique solution, found by integration along arbitrary paths. This arbitrariness guarantees a good control of the error, notwithstanding the ill-posedness of the problem. For N=2, the hypotheses allowing for this identification are satisfied when one knows two stationary potentials with non-overlapping equipotential lines and a third non-stationary one-this last needed only for determining d. The theory is applied to a numerical synthetic example, for various grid sizes or for noisy data. Notwithstanding the minimal a priori information required for the coefficients, we are able to compute these at a large number of nodes with good precision. For the sake of completeness, we give other results on identification.