The Water Residence Time distribution is the equivalent of the impulse response of a linear system allowing the propagation of water through a medium, e.g. the propagation of rain water from the top of the mountain towards the aquifers. We consider the output aquifer levels as the convolution between the input rain levels and the Water Residence Time, starting with an initial aquifer base level. The estimation of Water Residence Time is important for a better understanding of hydro-bio-geochemical processes and mixing properties of wetlands used as filters in ecological applications, as well as protecting fresh water sources for wells from pollutants. Common methods of estimating the Water Residence Time focus on cross-correlation, parameter fitting and non-parametric deconvolution methods. Here we propose a 1D full-deconvolution, regularized, non-parametric inverse problem algorithm that enforces smoothness and uses constraints of causality and positivity to estimate the Water Residence Time curve. Compared to Bayesian non-parametric deconvolution approaches, it has a fast runtime per test case; compared to the popular and fast cross-correlation method, it produces a more precise Water Residence Time curve even in the case of noisy measurements. The algorithm needs only one regularization parameter to balance between smoothness of the Water Residence Time and accuracy of the reconstruction. We propose an approach on how to automatically find a suitable value of the regularization parameter from the input data only. Tests on real data illustrate the potential of this method to analyze hydrological datasets.
In the case of the @cite_0 article, the direct model is similar to ours with some differences in its formulation: where @math is the output of the system, known; @math is the input of the system, known; @math is the Toeplitz matrix of the input of the system; @math is the impulse response of the system, to estimate; @math is the hyper-parameter to estimate with Fischer's Statistic method; @math denotes the Hessian of @math
@cite_0
2058013427
Solute transport in soils can be described in terms of solute travel time probability density functions (pdfs). In the past, parameterized functional forms for the travel time pdf have been derived based on models of physical transport processes and the properties of the soil, or have been obtained from observed transport behavior. Alternatively, a nonparametric pdf for a particular soil can be measured experimentally. In general, this measurement requires deconvoluting the pdf, which is mathematically an ill-posed problem. In this paper we perform the deconvolution using a constrained regularization approach. The regularization method uses a basis function representation of the travel time pdf which eliminates numerical problems associated with the standard regularization approach. The method is tested using synthetic and experimental data. Regularized solutions are evaluated using Provencher's (Provencher, 1982a) F test statistic along with a visual inspection of their fit to the data. We analyze data from previously published field experiments and the results suggest multimodal travel time pdfs may be appropriate for modeling transport in soils.