Generalisation of the solution of the inverse Richards' problem

In inverse problems defined by models that include partial differential equations, a part of the boundary conditions are unknown and are to be estimated from experimental measurements. We have shown in a previous contribution that the solution of the inverse Richards' problem can allow estimating percolation rates at the bottom of landfills through the use of measurements at the surface only. This can be a useful complement of the information furnished by the vadose measurement system, pointing to the possible presence of biases of in-situ equipment, and making it possible to use inexpensive mobile equipment to carry out surface measurements. In this article, we consider a generalisation which makes it possible to consider the presence of unknown nonlinear parameters, such as the effective hydraulic conductivity and the root uptake coefficients. This is accomplished using the method of separation of variables in the resulting estimation problem. Thanks to the linearity of the model, all these conditions can be expressed as linear functions of the unknown lower boundary condition. Otherwise, the relevant non-linear parameters are to be estimated from the data as well. Obviously, the correlation between the linear parameters contained in the unknown lower boundary conditions and the non-linear parameters can reduce the reliability of the monitoring procedure and hence the necessity of limiting the number of the latter

A general method for the solution of inverse problems in transport phenomena

The typical inverse problems in transport phenomena are given by partial differential equations with unknown boundary conditions, which are to be estimated from measurements corresponding to solutions of the PDEs or of their gradients. The resulting problem is an ill-posed problem, which can't be solved unless it is adequately regularized, because arbitrarily small errors on the data can give rise to very large deviations in the reconstruction of the unknown boundary conditions. In other words the estimated solution does not depend continuously on the data. The method proposed in this paper is the generalization of methods already applied to a number of problems (diffusion, heat transfer, percolation). The unknown boundary condition is replaced by a piecewise constant (or linear) functions with unknown coefficients. This approximation makes it possible to solve the resulting equation analytically and to estimate the coefficients by comparing theoretical and experimental values by means of linear least squares methods. The ill-posedness of this class of problems (resulting in singular matrices of the least squares problems) can be tackled using the well-known method proposed by Tikhonov. The value of the parameter contained in Tikhonov's method is estimated from the statistical noise on data by means of the Morozov's discrepancy principle. If additional information on the structure of the solution is available, specialised algorithms can be developed on a case-by-case basis. The presence of additional unknown non-linear parameters in the partial differential equations (such as diffusion coefficients, conductivities or adsorption coefficients of plant roots) leads to a particular model known as separable least squares. This procedure makes use of the conditional optimality principle for carrying out the overall identification problem in terms of the non-linear parameters only

Approximate solution of the inverse Richards' problem

We propose a method for the estimation of time dependent distributions of pressure head, water content, and fluid flow in homogeneous unsaturated soils with unknown lower boundary conditions using surface measurements only. The unknown boundary condition is replaced by a piecewise constant temporal function and the resulting discontinuity is alleviated by the introduction of a mass balance condition on the solution at discontinuity points. This approach makes it possible to express the analytical solution of Richards' one-dimensional equation as a linear function of a finite number of variables corresponding to the unknown coefficients of the piecewise constant function. While the estimation of unknown boundary belongs to a class of typically ill-posed inverse problems, the simplifications introduced in the algorithm provide for the regularization of this particular problem without the use of traditional smoothing techniques, such as Tikhonov's method and Morozov's discrepancy principle. A Bayesian estimation method and a unimodal regression algorithm have been employed to test the overall algorithm using simulated data

Reconciliation of flow rate measurements in the presence of solid particles

The reconciliation of flow rates of fluids entraining and reacting with solid particles is considered in this article. It is shown that, in case the amount of solid particles is not high, the reconciliation problem can be addressed formally as a rectification that includes additional components and solved using the techniques proposed by earlier research [Crowe, C. M., Garcia Campos, Y. A, Hrymak, A. AIChE J. 1983, 29, 881-888]. Furthermore, it is shown that the introduction of the interaction law between the fluid and the solid phase can improve the overall reliability of the reconciliation procedure

Improved remediation processes through cost-effective estimation of soil properties from surface measurements

A wide range of technologies is presently available for the remediation of contaminated soils. The optimal selection depends on a number of soil characteristics. However, if the depth of the contaminated layer is considerable, the direct measurement of these properties can be costly and sometimes outright infeasible. In this paper, a method originally developed for the early detection of leaks in landfill liners has been properly modified to accommodate the estimation of soil characteristics. In particular, while the soil properties were considered known parameters in the previous model, they are now present as non-linear parameters and their estimation constitutes the main goal of the article. The resulting algorithm consists in the optimization of a suitable objective function with respect to both linear and non-linear variables and makes it possible to estimate soil characteristics from surface measurements. In particular, it is shown that partitioning linear and non-linear variables into two different sets and regularizing the inverse problems resulting from the discretization of the Richards\ue2\u80\u99 problem with unknown boundary conditions provides a robust numerical procedure. As an example, the method has been applied to the estimation of soil porosity, demonstrating its robustness and reliability potential. Including an inexpensive estimation procedure for proper soil parameters in the preliminary analysis can greatly improve the performances of remediation technologies whose convenience depends critically on the parameters selected case by case