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.
Another example is @cite_6 with an algorithm based on ridge regression, where the direct model is similar to ours but has two hyper-parameters to be set. @cite_11 is another article where Bayesian Monte-Carlo deconvolution is done through an inverse problem setup. Here positivity and causality are implicitly enforced by the method of images applied to reflected Brownian motion that gives "" @cite_11 . The MCMC is here implemented with the Gibbs sampling algorithm. Similar to @cite_31 the result is also a pdf with zones of interest for the residence time curve. Even if the computational time for Bayesian MCMC deconvolution methods is deemed "" @cite_11 , probably even more so with current hardware, the need for a fast method seems necessary for the community, and we expand on this in the next paragraph.
@cite_31 @cite_6 @cite_11
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Reactive transport modeling is often used in support of bioremediation and chemicaltreatment planning and design. There remains a pressing need for practical andefficient models that do not require (or assume attainable) the high level ofcharacterization needed by complex numerical models. We focus on a linear systems ortransfer function approach to the problem of reactive tracer transport in a heterogeneoussaprolite aquifer. Transfer functions are obtained through the Bayesian geostatisticalinverse method applied to tracer injection histories and breakthrough curves. We employnonparametric transfer functions, which require minimal assumptions about shape andstructure. The resulting flexibility empowers the data to determine the nature of thetransfer function with minimal prior assumptions. Nonnegativity is enforced through areflected Brownian motion stochastic model. The inverse method enables us to quantifyuncertainty and to generate conditional realizations of the transfer function. Complexinformation about a hydrogeologic system is distilled into a relatively simple butrigorously obtained function that describes the transport behavior of the system betweentwo wells. The resulting transfer functions are valuable in reactive transport models basedon traveltime and streamline methods. The information contained in the data,particularly in the case of strong heterogeneity, is not overextended but is fully used. Thisis the first application of Bayesian geostatistical inversion to transfer functions inhydrogeology but the methodology can be extended to any linear system. For a rainfall-runoff event, quadratic programming is particularly suited to the estimation of unit graph as linear and positivity constraints on unit graph ordinates can be naturally implemented. In this paper, the quadratic programming framework is invoked in a novel way to stabilize the unit graph estimation procedure via the use of difference norms. The advantage of the latter over standard ridge regression is that penalties are placed on oscillations of the unit graph rather than on the size of its ordinates, Application of the methodology to real rainfall-runoff data is provided and comparisons with existing approaches are made. The latter indicate that our approach generally yields unit graph estimates of more realistic shape and smaller variance resulting in a better fit to the runoff data. Received 28 May 2002; revised 18 September 2002; accepted 24 September 2002; published 14 February 2003. [1] When an inverse problem is solved to estimate an unknown function such as the hydraulic conductivity in an aquifer or the contamination history at a site, one constraint is that the unknown function is known to be everywhere nonnegative. In this work, we develop a statistically rigorous method for enforcing function nonnegativity in Bayesian inverse problems. The proposed method behaves similarly to a Gaussian process with a linear variogram (i.e., unrestricted Brownian motion) for parameter values significantly greater than zero. The method uses the method of images to define a prior probability density function based on reflected Brownian motion that implicitly enforces nonnegativity. This work focuses on problems where the unknown is a function of a single variable (e.g., time). A Markov chain Monte Carlo (MCMC) method, specifically, a highly efficient Gibbs sampler, is implemented to generate conditional realizations of the unknown function. The new method is applied to the estimation of the trichloroethylene (TCE) and perchloroethylene (PCE) contamination history in an aquifer at Dover Air Force Base, Delaware, based on concentration profiles obtained from an underlying aquitard. INDEX TERMS: 1831 Hydrology: Groundwater quality; 1869 Hydrology: Stochastic processes; 3260 Mathematical Geophysics: Inverse theory; KEYWORDS: stochastic inverse modeling, contaminant source identification, inference under constraints, Markov chain Monte Carlo (MCMC), Gibbs sampling, Bayesian inference