Lingzao Zeng

An adaptive Gaussian process‐based method for efficient Bayesian experimental design in groundwater contaminant source identification problems

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.

A Probabilistic Collocation-Based Kalman Filter for History Matching

Original SPE manuscript received for review 17 January 2010. Revised manuscript received for review 9 May 2010. Paper (SPE 140737) peer approved 13 July 2010. Summary The ensemble Kalman filter (EnKF) has been used widely for data assimilation. Because the EnKF is a Monte Carlo-based method, a large ensemble size is required to reduce the sampling errors. In this study, a probabilistic collocation-based Kalman filter (PCKF) is developed to adjust the reservoir parameters to honor the production data. It combines the advantages of the EnKF for dynamic data assimilation and the polynomial chaos expansion (PCE) for efficient uncertainty quantification. In this approach, all the system parameters and states and the production data are approximated by the PCE. The PCE coefficients are solved with the probabilistic collocation method (PCM). Collocation realizations are constructed by choosing collocation point sets in the random space. The simulation for each collocation realization is solved forward in time independently by means of an existing deterministic solver, as in the EnKF method. In the analysis step, the needed covariance is approximated by the PCE coefficients. In this study, a square-root filter is employed to update the PCE coefficients. After the analysis, new collocation realizations are constructed. With the parameter collocation realizations as the inputs and the state collocation realizations as initial conditions, respectively, the simulations are forwarded to the next analysis step. Synthetic 2D water/oil examples are used to demonstrate the applicability of the PCKF in history matching. The results are compared with those from the EnKF on the basis of the same analysis. It is shown that the estimations provided by the PCKF are comparable to those obtained from the EnKF. The biggest improvement of the PCKF comes from the leading PCE approximation, with which the computational burden of the PCKF can be greatly reduced by means of a smaller number of simulation runs, and the PCKF outperforms the EnKF for a similar computational effort. When the correlation ratio is much smaller, the PCKF still provides estimations with a better accuracy for a small computational effort.

Uncertainty quantification of contaminant transport and risk assessment with conditional stochastic collocation method

Solute transport prediction is always subject to uncertainty due to the scarcity of observation data. The data worth of limited measurements can be explored by conditional simulation. This paper presents an efficient approach for the conditional simulation of solute transport in a randomly heterogeneous aquifer. The conditioning conductivity field is parameterized by the Karhunen–Loève (KL) expansion, and the concentration field is represented by Lagrange polynomials of random variables in the KL expansion. After employing the stochastic collocation method (SCM), stochastic governing advection–dispersion equations are reduced to a series of uncoupled deterministic equations. The concentration realizations can be obtained by sampling the established Lagrange polynomials instead of solving governing equations repeatedly. We assess the accuracy and computational efficiency of this method in comparison to the conditional Monte Carlo simulation. The influence of conditioning to hydraulic conductivity measurements on transport is analyzed. Numerical results demonstrate that the SCM can efficiently derive the conditional statistics of concentration as well as the probability of the aquifer to be contaminated. It is shown that the contamination risk is significantly influenced by measurements conditioning.

An Iterative Local Updating Ensemble Smoother for Estimation and Uncertainty Assessment of Hydrologic Model Parameters With Multimodal Distributions

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.

Inverse Modeling of Hydrologic Systems with Adaptive Multifidelity Markov Chain Monte Carlo Simulations

Markov chain Monte Carlo (MCMC) simulation methods are widely used to assess parametric uncertainties of hydrologic models conditioned on measurements of observable state variables. However, when the model is CPU-intensive and high-dimensional, the computational cost of MCMC simulation will be prohibitive. In this situation, a CPU-efficient while less accurate low-fidelity model (e.g., a numerical model with a coarser discretization, or a data-driven surrogate) is usually adopted. Nowadays, multi-fidelity simulation methods that can take advantage of both the efficiency of the low-fidelity model and the accuracy of the high-fidelity model are gaining popularity. In the MCMC simulation, as the posterior distribution of the unknown model parameters is the region of interest, it is wise to distribute most of the computational budget (i.e., the high-fidelity model evaluations) therein. Based on this idea, in this paper we propose an adaptive multi-fidelity MCMC algorithm for efficient inverse modeling of hydrologic systems. In this method, we evaluate the high-fidelity model mainly in the posterior region through iteratively running MCMC based on a Gaussian process (GP) system that is adaptively constructed with multi-fidelity simulation. The error of the GP system is rigorously considered in the MCMC simulation and gradually reduced to a negligible level in the posterior region. Thus, the proposed method can obtain an accurate estimate of the posterior distribution with a small number of the high-fidelity model evaluations. The performance of the proposed method is demonstrated by three numerical case studies in inverse modeling of hydrologic systems.

Bayesian Monitoring Design for Streambed Heat Tracing: Numerical Simulation and Sandbox Experiments: Bayesian Monitoring Design for Streambed Heat Tracing: Numerical Simulation and Sandbox Experiments

Heat tracing methods have been widely employed for subsurface characterization. Nevertheless, there were very few studies regarding the optimal monitoring design for heat tracing in heterogeneous streambeds. In this study, we addressed this issue by proposing an efficient optimal design framework to collect the most informative diurnal temperature signal for Bayesian estimation of streambed hydraulic conductivities. The data worth (DW) was measured by the expected relative entropy between the prior and posterior distributions of the conductivity field. An adaptively refined Gaussian process surrogate was employed to alleviate the computational burden, resulting in at least three orders of magnitude of speed-up. The applicability of the optimal experimental design framework was evaluated by both numerical and sandbox experimental cases. Results showed that the most informative locations centered in the transition zones among the main patterns of the hydraulic conductivity field, while the most informative times centered in a short period after the minimum/maximum temperature appeared. With the fixed number of measurements, extending the calibration period was more beneficial than increasing the monitoring frequency in improving the estimation results. To our best knowledge, this work is the first study on Bayesian monitoring design for streambed characterization with the heat tracing method. The method and results can provide guidance on selecting monitoring strategies under budget-limited conditions.

A process-based model for pentachlorophenol dissipation in a flooded paddy soil

Process-based models have been widely used for predicting environmental fate of contaminants. Nevertheless, accurate modeling of pentachlorophenol (PCP) dissipation in soils at the millimeter-scale remains a challenge due to the scarcity of observation data and uncertainty associated with model assumptions and estimation of the model parameters. To provide quantitative analysis of PCP-dissipation at the anaerobic/aerobic interface of a rhizobox experiment, this study implemented Bayesian parameter estimation for a process-based reactive chemical transport model. The model considered the main transport and transformation processes of chemicals including diffusion, sorption and degradation. The contributions of the processes to PCP dissipation were apportioned both in space and time. Using the maximum-a-posteriori (MAP) estimation of parameters, our model fitted the experimental data better compared with the previous work. Our results indicated that the most reactive zone for PCP dissipation occurred in the layer of 0-2.4 mm where degradation in solid phase dominated the PCP dissipation, while upward diffusion was the main mechanism for the reduction of PCP concentration in deeper layer (2.4-4.8 mm). By considering the coupled reactive transport of PCP and Cl-, the average degrees of PCP dechlorination in each layer were estimated from corresponding total concentrations of PCP and Cl-. The degrees of PCP dechlorination in the ponding water and the top layer of soil profile were highest, while 2,3,4,5- TeCP and 3,4,5- TCP were identified as the main dechlorination products in the soil. This study demonstrated that combining Bayesian estimation with process-based reactive chemical transport model can provide more insights of PCP dissipation at the millimeter-scale. This approach can help to understand complex dissipation mechanisms for other contaminants.

A multi-medium chain modeling approach to estimate the cumulative effects of cadmium pollution on human health

Cadmium is a highly persistent and toxic heavy metal that poses severe health risks to humans. Diet is the primary source of human exposure to cadmium, especially in China. Soil, as the main medium that transfers cadmium to rice, can be used as a helpful indicator to predict human exposure to cadmium in soils. There is, however, very little work that links a soil-rice transfer model with a biokinetic model to assess health risks. In this work, we introduce a multi-medium chain model based upon a soil-rice-human continuum to address this issue. The model consists of three basic steps: (i) development and validation of a soil-rice transfer model for cadmium based on 189 pairs of measured data in Wenling of Zhejiang province in Southeast China; (ii) calculation of weekly exposure based on the nationwide monitoring and survey results; (iii) linking the exposure model with a modified biokinetic model proposed with a classic biokinetic model to predict urinary cadmium, which is a biomarker to assess the health risks. Results indicated that the developed soil-rice-human transfer model predicted well the urinary cadmium levels in humans subjected to age and exposure uncertainties. We observed a maximum of 0.71 μg g-1 creatinine in males and 1.53 μg g-1 creatinine in females at 70 years old under median cadmium exposure, which was consistent with previous studies. Sensitive analysis was also conducted to detect the sensitive parameters that have the most significant influences on the output of the model. The new risk assessment strategy proposed in this work is beneficial for predicting the cumulative cadmium levels in various exposed populations so that we can quickly identify the critical areas from basic soil properties.

Bayesian inference for kinetic models of biotransformation using a generalized rate equation

Selecting proper rate equations for the kinetic models is essential to quantify biotransformation processes in the environment. Bayesian model selection method can be used to evaluate the candidate models. However, comparisons of all plausible models can result in high computational cost, while limiting the number of candidate models may lead to biased results. In this work, we developed an integrated Bayesian method to simultaneously perform model selection and parameter estimation by using a generalized rate equation. In the approach, the model hypotheses were represented by discrete parameters and the rate constants were represented by continuous parameters. Then Bayesian inference of the kinetic models was solved by implementing Markov Chain Monte Carlo simulation for parameter estimation with the mixed (i.e., discrete and continuous) priors. The validity of this approach was illustrated through a synthetic case and a nitrogen transformation experimental study. It showed that our method can successfully identify the plausible models and parameters, as well as uncertainties therein. Thus this method can provide a powerful tool to reveal more insightful information for the complex biotransformation processes.

Application of multiscale finite element method in the uncertainty qualification of large-scale groundwater flow

In this article, we discuss the application of multiscale finite element method (MsFEM) to groundwater flow in heterogeneous porous media. We investigate the ability of MsFEM in qualifying the flow uncertainty. Monte Carlo simulation is employed to implement the stochastic analysis, and MsFEM is used to avoid a full resolution to the spatial variable conductivity field. Large-scale flow with high variability is investigated by inspecting the single realization as well as the probability distribution functions of head and velocity. The numerical results show that the performance of MsFEM depends on the ratio between the correlation length and the coarse element size. An accurate prediction to the velocity requires a much lower ratio than the head. The MsFEM has different convergence rates for the head and the velocity, while the convergence rates do not deteriorate as the variance grows.