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Dan Lu

Publications and source records attributed to Dan Lu.

5 recordsLinked to original sources
USGS2015

Maximum likelihood Bayesian model averaging and its predictive analysis for groundwater reactive transport models

While Bayesian model averaging (BMA) has been widely used in groundwater modeling, it is infrequently applied to groundwater reactive transport modeling because of multiple sources of uncertainty in the coupled hydrogeochemical processes and because of the long execution time of each model run. To resolve these problems, this study analyzed different levels of uncertainty in a hierarchical way, and used the maximum likelihood version of BMA, i.e., MLBMA, to improve the computational efficiency. This study demonstrates the applicability of MLBMA to groundwater reactive transport modeling in a synthetic case in which twenty-seven reactive transport models were designed to predict the reactive transport of hexavalent uranium (U(VI)) based on observations at a former uranium mill site near Naturita, CO. These reactive transport models contain three uncertain model components, i.e., parameterization of hydraulic conductivity, configuration of model boundary, and surface complexation reactions that simulate U(VI) adsorption. These uncertain model components were aggregated into the alternative models by integrating a hierarchical structure into MLBMA. The modeling results of the individual models and MLBMA were analyzed to investigate their predictive performance. The predictive logscore results show that MLBMA generally outperforms the best model, suggesting that using MLBMA is a sound strategy to achieve more robust model predictions relative to a single model. MLBMA works best when the alternative models are structurally distinct and have diverse model predictions. When correlation in model structure exists, two strategies were used to improve predictive performance by retaining structurally distinct models or assigning smaller prior model probabilities to correlated models. Since the synthetic models were designed using data from the Naturita site, the results of this study are expected to provide guidance for real-world modeling. Limitations of applying MLBMA to the synthetic study and future real-world modeling are discussed.

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USGS2014

A computer program for uncertainty analysis integrating regression and Bayesian methods

This work develops a new functionality in UCODE_2014 to evaluate Bayesian credible intervals using the Markov Chain Monte Carlo (MCMC) method. The MCMC capability in UCODE_2014 is based on the FORTRAN version of the differential evolution adaptive Metropolis (DREAM) algorithm of Vrugt et al. (2009), which estimates the posterior probability density function of model parameters in high-dimensional and multimodal sampling problems. The UCODE MCMC capability provides eleven prior probability distributions and three ways to initialize the sampling process. It evaluates parametric and predictive uncertainties and it has parallel computing capability based on multiple chains to accelerate the sampling process. This paper tests and demonstrates the MCMC capability using a 10-dimensional multimodal mathematical function, a 100-dimensional Gaussian function, and a groundwater reactive transport model. The use of the MCMC capability is made straightforward and flexible by adopting the JUPITER API protocol. With the new MCMC capability, UCODE_2014 can be used to calculate three types of uncertainty intervals, which all can account for prior information: (1) linear confidence intervals which require linearity and Gaussian error assumptions and typically 10s–100s of highly parallelizable model runs after optimization, (2) nonlinear confidence intervals which require a smooth objective function surface and Gaussian observation error assumptions and typically 100s–1,000s of partially parallelizable model runs after optimization, and (3) MCMC Bayesian credible intervals which require few assumptions and commonly 10,000s–100,000s or more partially parallelizable model runs. Ready access allows users to select methods best suited to their work, and to compare methods in many circumstances.

Environmental Modelling and Software↗
USGS2013

Effects of error covariance structure on estimation of model averaging weights and predictive performance

When conducting model averaging for assessing groundwater conceptual model uncertainty, the averaging weights are often evaluated using model selection criteria such as AIC, AICc, BIC, and KIC (Akaike Information Criterion, Corrected Akaike Information Criterion, Bayesian Information Criterion, and Kashyap Information Criterion, respectively). However, this method often leads to an unrealistic situation in which the best model receives overwhelmingly large averaging weight (close to 100%), which cannot be justified by available data and knowledge. It was found in this study that this problem was caused by using the covariance matrix, C E , of measurement errors for estimating the negative log likelihood function common to all the model selection criteria. This problem can be resolved by using the covariance matrix, C ek , of total errors (including model errors and measurement errors) to account for the correlation between the total errors. An iterative two-stage method was developed in the context of maximum likelihood inverse modeling to iteratively infer the unknown C ek from the residuals during model calibration. The inferred C ek was then used in the evaluation of model selection criteria and model averaging weights. While this method was limited to serial data using time series techniques in this study, it can be extended to spatial data using geostatistical techniques. The method was first evaluated in a synthetic study and then applied to an experimental study, in which alternative surface complexation models were developed to simulate column experiments of uranium reactive transport. It was found that the total errors of the alternative models were temporally correlated due to the model errors. The iterative two-stage method using C ek resolved the problem that the best model receives 100% model averaging weight, and the resulting model averaging weights were supported by the calibration results and physical understanding of the alternative models. Using C ek obtained from the iterative two-stage method also improved predictive performance of the individual models and model averaging in both synthetic and experimental studies.

Water Resources Research↗
USGS2012

Uncertainty quantification for environmental models

Environmental models are used to evaluate the fate of fertilizers in agricultural settings (including soil denitrification), the degradation of hydrocarbons at spill sites, and water supply for people and ecosystems in small to large basins and cities—to mention but a few applications of these models. They also play a role in understanding and diagnosing potential environmental impacts of global climate change. The models are typically mildly to extremely nonlinear. The persistent demand for enhanced dynamics and resolution to improve model realism [17] means that lengthy individual model execution times will remain common, notwithstanding continued enhancements in computer power. In addition, high-dimensional parameter spaces are often defined, which increases the number of model runs required to quantify uncertainty [2]. Some environmental modeling projects have access to extensive funding and computational resources; many do not. The many recent studies of uncertainty quantification in environmental model predictions have focused on uncertainties related to data error and sparsity of data, expert judgment expressed mathematically through prior information, poorly known parameter values, and model structure (see, for example, [1,7,9,10,13,18]). Approaches for quantifying uncertainty include frequentist (potentially with prior information [7,9]), Bayesian [13,18,19], and likelihood-based. A few of the numerous methods, including some sensitivity and inverse methods with consequences for understanding and quantifying uncertainty, are as follows: Bayesian hierarchical modeling and Bayesian model averaging; single-objective optimization with error-based weighting [7] and multi-objective optimization [3]; methods based on local derivatives [2,7,10]; screening methods like OAT (one at a time) and the method of Morris [14]; FAST (Fourier amplitude sensitivity testing) [14]; the Sobol' method [14]; randomized maximum likelihood [10]; Markov chain Monte Carlo (MCMC) [10]. There are also bootstrapping and cross-validation approaches.Sometimes analyses are conducted using surrogate models [12]. The availability of so many options can be confusing. Categorizing methods based on fundamental questions assists in communicating the essential results of uncertainty analyses to stakeholders. Such questions can focus on model adequacy (e.g., How well does the model reproduce observed system characteristics and dynamics?) and sensitivity analysis (e.g., What parameters can be estimated with available data? What observations are important to parameters and predictions? What parameters are important to predictions?), as well as on the uncertainty quantification (e.g., How accurate and precise are the predictions?). The methods can also be classified by the number of model runs required: few (10s to 1000s) or many (10,000s to 1,000,000s). Of the methods listed above, the most computationally frugal are generally those based on local derivatives; MCMC methods tend to be among the most computationally demanding. Surrogate models (emulators)do not necessarily produce computational frugality because many runs of the full model are generally needed to create a meaningful surrogate model. With this categorization, we can, in general, address all the fundamental questions mentioned above using either computationally frugal or demanding methods. Model development and analysis can thus be conducted consistently using either computation-ally frugal or demanding methods; alternatively, different fundamental questions can be addressed using methods that require different levels of effort. Based on this perspective, we pose the question: Can computationally frugal methods be useful companions to computationally demanding meth-ods? The reliability of computationally frugal methods generally depends on the model being reasonably linear, which usually means smooth nonlin-earities and the assumption of Gaussian errors; both tend to be more valid with more linear

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