[1] We propose a novel direct method for estimating steady state hydrogeological model parameters and model state variables in an aquifer where boundary conditions are unknown. The method is adapted from a recently developed potential theory technique for solving general inverse/reconstruction problems. Unlike many inverse techniques used for groundwater model calibration, the new method is not based on fitting and optimizing an objective function, which usually requires forward simulation and iterative parameter updates. Instead, it directly incorporates noisy observed data (hydraulic heads and flow rates) at the measurement points in a single step, without solving a boundary value problem. The new method is computationally efficient and is robust to the presence of observation errors. It has been tested on two-dimensional groundwater flow problems with regular and irregular geometries, different heterogeneity patterns, variances of heterogeneity, and error magnitudes. In all cases, parameters (hydraulic conductivities) converge to the correct or expected values and are thus unique, based on which heads and flow fields are constructed directly via a set of analytical expressions. Accurate boundary conditions are then inferred from these fields. The accuracy of the direct method also improves with increasing amount of observed data, lower measurement errors, and grid refinement. Under natural flow (i.e., no pumping), the direct method yields an equivalent conductivity of the aquifer, suggesting that the method can be used as an inexpensive characterization tool with which both aquifer parameters and aquifer boundary conditions can be inferred.
The problem considered in this paper deals with reconstruction of heat flux from temperature measurements, where due to lack of data, numerical integration is impossible. The problem is reduced to the determination of unknown complex coefficients of a linear piecewise holomorphic function representing heat flux properties and a quadratic holomorphic function representing complex temperature distribution. This leads to an over-determined system of linear algebraic equations, which are subjected to experimental errors. The reconstruction of heat flux is unique; however, reconstruction of flux directions is non-unique. It contains one free additive parameter. The method is useful in situations where limited data on temperature are provided.
This study presents an approach for reconstruction of harmonic functions in three dimensions from the finite number of field and surface measurements. The approach, based on the Trefftz method, performs reconstruction as the best fit to the data and provides smoothness of the reconstructed function. Two particular algorithms are proposed; the first one uses specific radial basis functions and the second one is of finite element type. Either of them can be applied to analyse different data types but the latter can handle larger problems. The data types considered in this study also cover direct and inverse boundary value problems. Therefore, the proposed approach is universal and capable of dealing with both well-posed and ill-posed formulations. Examples from steady heat conduction and elastostatics are examined in order to investigate the efficiency of the approach.
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