In this study, an efficient full Bayesian approach is developed for the optimal sampling well location design and source parameters identification of groundwater contaminants. An information measure, i.e., the relative entropy, is employed to quantify the information gain from concentration measurements in identifying unknown parameters. In this approach, the sampling locations that give the maximum expected relative entropy are selected as the optimal design. After the sampling locations are determined, a Bayesian approach based on Markov Chain Monte Carlo (MCMC) is used to estimate unknown parameters. In both the design and estimation, the contaminant transport equation is required to be solved many times to evaluate the likelihood. To reduce the computational burden, an interpolation method based on the adaptive sparse grid is utilized to construct a surrogate for the contaminant transport equation. The approximated likelihood can be evaluated directly from the surrogate, which greatly accelerates the design and estimation process. The accuracy and efficiency of our approach are demonstrated through numerical case studies. It is shown that the methods can be used to assist in both single sampling location and monitoring network design for contaminant source identifications in groundwater.

Ensemble smoother (ES) has been widely used in inverse modeling of hydrologic systems. However, for problems where the distribution of model parameters is multimodal, using ES directly would be problematic. One popular solution is to use a clustering algorithm to identify each mode and update the clusters with ES separately. However, this strategy may not be very efficient when the dimension of parameter space is high or the number of modes is large. Alternatively, we propose in this paper a very simple and efficient algorithm, i.e., the iterative local updating ensemble smoother (ILUES), to explore multimodal distributions of model parameters in nonlinear hydrologic systems. The ILUES algorithm works by updating local ensembles of each sample with ES to explore possible multimodal distributions. To achieve satisfactory data matches in nonlinear problems, we adopt an iterative form of ES to assimilate the measurements multiple times. Numerical cases involving nonlinearity and multimodality are tested to illustrate the performance of the proposed method. It is shown that overall the ILUES algorithm can well quantify the parametric uncertainties of complex hydrologic models, no matter whether the multimodal distribution exists.

A greenhouse pot experiment was conducted to study the effects of conventional nitrogen fertilization on soil acidity and salinity. Three N rates (urea; N0, 0 kg N ha(-1); N1, 600 kg N ha(-1); and N2, 1,200 kg N ha(-1)) were applied in five soils with different greenhouse cultivation years to evaluate soil acidification and salinization rate induced by nitrogen fertilizer in lettuce production. Both soil acidity and salinity increased significantly as N input increased after one season, with pH decrease ranging from 0.45 to 1.06 units and electrolytic conductivity increase from 0.24 to 0.68 mS cm(-1). An estimated 0.92 mol H(+) was produced for 1 mol (NO2 (-) + NO3 (-))-N accumulation in soil. The proton loading from nitrification was 14.3-27.3 and 12.1-58.2 kmol H(+) ha(-1) in the center of Shandong Province under N1 and N2 rate, respectively. However, the proton loading from the uptake of excess bases by lettuces was only 0.3-4.5 % of that from nitrification. Moreover, the release of protons induced the direct release of base cations and accelerated soil salinization. The increase of soil acidity and salinity was attributed to the nitrification of excess N fertilizer. Compared to the proton loading by lettuce, nitrification contributed more to soil acidification in greenhouse soils.

Surrogate models are commonly used in Bayesian approaches such as Markov Chain Monte Carlo (MCMC) to avoid repetitive CPU‐demanding model evaluations. However, the approximation error of a surrogate may lead to biased estimation of the posterior distribution. This bias can be corrected by constructing a very accurate surrogate or implementing MCMC in a two‐stage manner. Since the two‐stage MCMC requires extra original model evaluations after surrogate evaluations, the computational cost is still high. If the information of measurement is incorporated, a locally accurate surrogate can be adaptively constructed with low computational cost. Based on this idea, we integrate Gaussian process (GP) and MCMC to adaptively construct locally accurate surrogates for Bayesian experimental design in groundwater contaminant source identification problems. Moreover, the uncertainty estimate of GP approximation error is incorporated in the Bayesian formula to avoid over‐confident estimation of the posterior distribution. The proposed approach is tested with a numerical case study. Without sacrificing the estimation accuracy, the new approach achieves about 200 times of speed‐up compared to our previous work which implemented MCMC in a two‐stage manner.

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