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.