  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.
