Characterization of groundwater contaminant source using Bayesian method

Contaminant source identification in groundwater system is critical for remediation strategy implementation, including gathering further samples and analysis, as well as implementing and evaluating different remediation plans. Such problem is usually solved with the aid of groundwater modeling with lots of uncertainty, e.g. existing uncertainty in hydraulic conductivity, measurement variance and the model structure error. Monte Carlo simulation of flow model allows the input uncertainty onto the model predictions of concentration measurements at monitoring sites. Bayesian approach provides the advantage to update estimation. This paper presents an application of a dynamic framework coupling with a three dimensional groundwater modeling scheme in contamination source identification of groundwater. Markov Chain Monte Carlo (MCMC) is being applied to infer the possible location and magnitude of contamination source. Uncertainty existing in heterogonous hydraulic conductivity field is explicitly considered in evaluating the likelihood function. Unlike other inverse-problem approaches to provide single but maybe untrue solution, the MCMC algorithm provides probability distributions over estimated parameters. Results from this algorithm offer a probabilistic inference of the location and concentration of released contamination. The convergence analysis of MCMC reveals the effectiveness of the proposed algorithm. Further investigation to extend this study is also discussed.     

基于MCMC的叠前地震反演方法研究

马尔科夫链蒙特卡洛方法(MCMC)是一种启发式的全局寻优算法[1].它在贝叶斯框架下,利用已有资料进行约束,既可使最优解满足参数的统计特性,又通过融入的先验信息,提高解的精度;寻优过程可跳出局部最优,得到全局最优解.利用MCMC方法,可以得到大量来自于后验概率分布的样本,不仅可以得到每个未知参数的估计值,而且可以得到与...     

Identification of uncertainty in low flow frequency analysis using Bayesian MCMC method

This study employs the Bayesian Markov Chain Monte Carlo (MCMC) method with the Metropolis–Hastings algorithm and maximum likelihood estimation (MLE) using a quadratic approximation of the likelihood function for the evaluation of uncertainties in low flow frequency analysis using a two‐parameter Weibull distribution. The two types of prior distributions, a non‐data‐based distribution and a data‐based distribution using regional information collected from neighbouring stations, are used to establish a posterior distribution. Eight case studies using the synthetic data with a sample size of 100, generated from two‐parameter Weibull distribution, are performed to compare with results of analysis using MLE and Bayesian MCMC. Also, Bayesian MCMC and MLE are applied to 36 years of gauged data to validate the efficiency of the developed scheme. These examples illustrate the advantages of Bayesian MCMC and the limitations of MLE based on a quadratic approximation. From the point of view of uncertainty analysis, Bayesian MCMC is more effective than MLE using a quadratic approximation when the sample size is small. In particular, Bayesian MCMC method is more attractive than MLE based on a quadratic approximation because the sample size of low flow at the site of interest is mostly not enough to perform the low flow frequency analysis. Copyright © 2007 John Wiley & Sons, Ltd.     

Uncertainty-based multi-criteria calibration of rainfall-runoff models: a comparative study

This study compares formal Bayesian inference to the informal generalized likelihood uncertainty estimation (GLUE) approach for uncertainty-based calibration of rainfall-runoff models in a multi-criteria context. Bayesian inference is accomplished through Markov Chain Monte Carlo (MCMC) sampling based on an auto-regressive multi-criteria likelihood formulation. Non-converged MCMC sampling is also considered as an alternative method. These methods are compared along multiple comparative measures calculated over the calibration and validation periods of two case studies. Results demonstrate that there can be considerable differences in hydrograph prediction intervals generated by formal and informal strategies for uncertainty-based multi-criteria calibration. Also, the formal approach generates definitely preferable validation period results compared to GLUE (i.e., tighter prediction intervals that show higher reliability) considering identical computational budgets. Moreover, non-converged MCMC (based on the standard Gelman–Rubin metric) performance is reasonably consistent with those given by a formal and fully-converged Bayesian approach even though fully-converged results requires significantly larger number of samples (model evaluations) for the two case studies. Therefore, research to define alternative and more practical convergence criteria for MCMC applications to computationally intensive hydrologic models may be warranted.     

Numerical simulation and inversion of MT fields in the 3D electric conductivity model of the Vesuvius volcano

Possibilities for three-dimensional (3D) magnetotelluric (MT) sounding of local objects contained in the Earth’s crust are estimated in a case study of the magma chamber of the Vesuvius volcano. Stochastic inversion of the model MT data by the Markov Chain Monte Carlo (MCMC) method has shown that the most efficient approach is not simultaneous but successive estimation of the geometry and the depth of the anomaly and the assessment of the conductivity distribution within the anomalous region. A zone of equivalence is revealed between the a priori estimate of the depth of the anomalous zone and the a posteriori distribution of electric conductivity within it. Based on the present estimation and previous results, an algorithm for determination of the parameters of local crustal anomaly is proposed.     

Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling?

In recent years, a strong debate has emerged in the hydrologic literature regarding what constitutes an appropriate framework for uncertainty estimation. Particularly, there is strong disagreement whether an uncertainty framework should have its roots within a proper statistical (Bayesian) context, or whether such a framework should be based on a different philosophy and implement informal measures and weaker inference to summarize parameter and predictive distributions. In this paper, we compare a formal Bayesian approach using Markov Chain Monte Carlo (MCMC) with generalized likelihood uncertainty estimation (GLUE) for assessing uncertainty in conceptual watershed modeling. Our formal Bayesian approach is implemented using the recently developed differential evolution adaptive metropolis (DREAM) MCMC scheme with a likelihood function that explicitly considers model structural, input and parameter uncertainty. Our results demonstrate that DREAM and GLUE can generate very similar estimates of total streamflow uncertainty. This suggests that formal and informal Bayesian approaches have more common ground than the hydrologic literature and ongoing debate might suggest. The main advantage of formal approaches is, however, that they attempt to disentangle the effect of forcing, parameter and model structural error on total predictive uncertainty. This is key to improving hydrologic theory and to better understand and predict the flow of water through catchments.     

基于局部采样MCMC方法的时移探地雷达反演

马尔科夫链蒙特卡洛方法(MCMC)是一种启发式的全局寻优算法,可以用来解决概率反演的问题.基于MCMC方法的反演不依赖于准确的初始模型,可以引入任意复杂的先验信息,通过对先验概率密度函数的采样来获得大量的后验概率分布样本,在寻找最优解的过程中可以跳出局部最优得到全局最优解.MCMC方法由于计算量巨大,应用难度较高,在地...     

Response to comment by Keith Beven on “Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling?”

In recent years, a strong debate has emerged in the hydrologic literature regarding what constitutes an appropriate framework for uncertainty estimation. Particularly, there is strong disagreement whether an uncertainty framework should have its roots within a proper statistical (Bayesian) context, or whether such a framework should be based on a different philosophy and implement informal measures and weaker inference to summarize parameter and predictive distributions. In this paper, we compare a formal Bayesian approach using Markov Chain Monte Carlo (MCMC) with generalized likelihood uncertainty estimation (GLUE) for assessing uncertainty in conceptual watershed modeling. Our formal Bayesian approach is implemented using the recently developed differential evolution adaptive metropolis (DREAM) MCMC scheme with a likelihood function that explicitly considers model structural, input and parameter uncertainty. Our results demonstrate that DREAM and GLUE can generate very similar estimates of total streamflow uncertainty. This suggests that formal and informal Bayesian approaches have more common ground than the hydrologic literature and ongoing debate might suggest. The main advantage of formal approaches is, however, that they attempt to disentangle the effect of forcing, parameter and model structural error on total predictive uncertainty. This is key to improving hydrologic theory and to better understand and predict the flow of water through catchments.     

Stratigraphic Columns Modeling and Cyclicity Analysis of the Misoa Formation,Maracaibo Lake,Venezuela, using Markov Chains

A stochastic characterization of a hydrocarbon reservoir, constituted by a sedimentary sequence of sandstones interbbeded with siltstones and shales, has been performed. The stratigraphic unit studied here mainly comprises the C4 sands of the Misoa Formation, located in the Lama Field, Maracaibo Lake (Venezuela). A Markov Chain algorithm, based on the definition of genetic lithofacies relationships along stratigraphic columns, was developed. The application of the Monte Carlo stochastic method using this algorithm, to log data from 11 wells, allowed the generation of pseudo sequences at 20 new locations. This algorithm was able to properly model pseudo stratigraphic sequences and to quantify the relative facies percentage, showing a 82% confidence level related to the proportional content of sediments at a test well. The net sand map obtained integrating the stratigraphic columns, derived from the well information, and the Markov pseudo-columns, suggests the presence of sand bodies with a northeast-southwest orientation that agree with previous geological studies in the area. This map could help in the definition of prospective zones in the field. The existence of stratigraphic memory along the evaluated columns was recognized after applying the algorithm. The embedded Markov method used in the cyclicity analysis of the whole area indicates cyclic transitions just from sandstones to siltstones and from shales to siltstones. Hence for the study area, on average, fining upward and coarsening upward processes can be identified with the Markovian approach, as was expected for the tide-dominated deltaic system associated to the analyzed reservoir.     

Combining discrete cosine transform and convolutional neural networks to speed up the Hamiltonian Monte Carlo inversion of pre-stack seismic data

Markov chain Monte Carlo algorithms are commonly employed for accurate uncertainty appraisals in non-linear inverse problems. The downside of these algorithms is the considerable number of samples needed to achieve reliable posterior estimations, especially in high-dimensional model spaces. To overcome this issue, the Hamiltonian Monte Carlo algorithm has recently been introduced to solve geophysical inversions. Different from classical Markov chain Monte Carlo algorithms, this approach exploits the derivative information of the target posterior probability density to guide the sampling of the model space. However, its main downside is the computational cost for the derivative computation (i.e. the computation of the Jacobian matrix around each sampled model). Possible strategies to mitigate this issue are the reduction of the dimensionality of the model space and/or the use of efficient methods to compute the gradient of the target density. Here we focus the attention to the estimation of elastic properties (P-, S-wave velocities and density) from pre-stack data through a non-linear amplitude versus angle inversion in which the Hamiltonian Monte Carlo algorithm is used to sample the posterior probability. To decrease the computational cost of the inversion procedure, we employ the discrete cosine transform to reparametrize the model space, and we train a convolutional neural network to predict the Jacobian matrix around each sampled model. The training data set for the network is also parametrized in the discrete cosine transform space, thus allowing for a reduction of the number of parameters to be optimized during the learning phase. Once trained the network can be used to compute the Jacobian matrix associated with each sampled model in real time. The outcomes of the proposed approach are compared and validated with the predictions of Hamiltonian Monte Carlo inversions in which a quite computationally expensive, but accurate finite-difference scheme is used to compute the Jacobian matrix and with those obtained by replacing the Jacobian with a matrix operator derived from a linear approximation of the Zoeppritz equations. Synthetic and field inversion experiments demonstrate that the proposed approach dramatically reduces the cost of the Hamiltonian Monte Carlo inversion while preserving an accurate and efficient sampling of the posterior probability.     

Exposure estimation using repeated blood concentration measurements

Physiologically based toxicokinetic (PBTK) modeling has been well established to study the distributions of chemicals in target tissues. In addition, the hierarchical Bayesian statistical approach using Markov Chain Monte Carlo (MCMC) simulations has been applied successfully for parameter estimation. The aim was to estimate the constant inhalation exposure concentration (assumed) using a PBTK model based on repeated measurements in venous blood, so that exposures could be estimated. By treating the constant exterior exposure as an unknown parameter of a four-compartment PBTK model, we applied MCMC simulations to estimate the exposure based on a hierarchical Bayesian approach. The dataset on 16 volunteers exposed to 100 ppm (≅0.538 mg/L) trichloroethylene vapors for 4 h was reanalyzed as an illustration. Cases of time-dependent exposures with a constant mean were also studied via 100 simulated datasets. The posterior geometric mean of 0.571, with narrow 95% posterior confidence interval (CI) (0.506, 0.645), estimated the true trichloroethylene inhalation concentration (0.538 mg/L) with very high precision. Also, the proposed method estimated the overall constant mean of the simulated time-dependent exposure scenarios well with slightly wider 95% CIs. The proposed method justifies the accuracy of exposure estimation from biomonitoring data using PBTK model and MCMC simulations from a real dataset and simulation studies numerically, which provides a starting point for future applications in occupational exposure assessment.     

Seismic Tomography by Monte Carlo Sampling

The paper discusses the performance and robustness of the Bayesian (probabilistic) approach to seismic tomography enhanced by the numerical Monte Carlo sampling technique. The approach is compared with two other popular techniques, namely the damped least-squares (LSQR) method and the general optimization approach. The theoretical considerations are illustrated by an analysis of seismic data from the Rudna (Poland) copper mine. Contrary to the LSQR and optimization techniques the Bayesian approach allows for construction of not only the “best-fitting” model of the sought velocity distribution but also other estimators, for example the average model which is often expected to be a more robust estimator than the maximum likelihood solution. We demonstrate that using the Markov Chain Monte Carlo sampling technique within the Bayesian approach opens up the possibility of analyzing tomography imaging uncertainties with minimal additional computational effort compared to the robust optimization approach. On the basis of the considered example it is concluded that the Monte Carlo based Bayesian approach offers new possibilities of robust and reliable tomography imaging.     

Dimensionality reduction for efficient Bayesian estimation of groundwater flow in strongly heterogeneous aquifers

We focus on the Bayesian estimation of strongly heterogeneous transmissivity fields conditional on data sampled at a set of locations in an aquifer. Log-transmissivity, Y, is modeled as a stochastic Gaussian process, parameterized through a truncated Karhunen–Loève (KL) expansion. We consider Y fields characterized by a short correlation scale as compared to the size of the observed domain. These systems are associated with a KL decomposition which still requires a high number of parameters, thus hampering the efficiency of the Bayesian estimation of the underlying stochastic field. The distinctive aim of this work is to present an efficient approach for the stochastic inverse modeling of fully saturated groundwater flow in these types of strongly heterogeneous domains. The methodology is grounded on the construction of an optimal sparse KL decomposition which is achieved by retaining only a limited set of modes in the expansion. Mode selection is driven by model selection criteria and is conditional on available data of hydraulic heads and (optionally) Y. Bayesian inversion of the optimal sparse KLE is then inferred using Markov Chain Monte Carlo (MCMC) samplers. As a test bed, we illustrate our approach by way of a suite of computational examples where noisy head and Y values are sampled from a given randomly generated system. Our findings suggest that the proposed methodology yields a globally satisfactory inversion of the stochastic head and Y fields. Comparison of reference values against the corresponding MCMC predictive distributions suggests that observed values are well reproduced in a probabilistic sense. In a few cases, reference values at some unsampled locations (typically far from measurements) are not captured by the posterior probability distributions. In these cases, the quality of the estimation could be improved, e.g., by increasing the number of measurements and/or the threshold for the selection of KL modes.     

Multiscale data integration using coarse-scale models

In this paper we combine a multiscale data integration technique introduced in [Lee SH, Malallah A, Datta-Gupta A, Hidgon D. Multiscale data integration using Markov Random Fields. SPE Reservoir Evaluat Eng 2002;5(1):68–78] with upscaling techniques for spatial modeling of permeability. The main goal of this paper is to find fine-scale permeability fields based on coarse-scale permeability measurements. The approach introduced in the paper is hierarchical and the conditional information from different length scales is incorporated into the posterior distribution using a Bayesian framework. Because of a complicated structure of the posterior distribution Markov chain Monte Carlo (MCMC) based approaches are used to draw samples of the fine-scale permeability field.     

A nonhomogeneous Poisson process geostatistical model

This paper introduces a new geostatistical model for counting data under a space-time approach using nonhomogeneous Poisson processes, where the random intensity process has an additive formulation with two components: a Gaussian spatial component and a component accounting for the temporal effect. Inferences of interest for the proposed model are obtained under the Bayesian paradigm. To illustrate the usefulness of the proposed model, we first develop a simulation study to test the efficacy of the Markov Chain Monte Carlo (MCMC) method to generate samples for the joint posterior distribution of the model’s parameters. This study shows that the convergence of the MCMC algorithm used to simulate samples for the joint posterior distribution of interest is easily obtained for different scenarios. As a second illustration, the proposed model is applied to a real data set related to ozone air pollution collected in 22 monitoring stations in Mexico City in the 2010 year. The proposed geostatistical model has good performance in the data analysis, in terms of fit to the data and in the identification of the regions with the highest pollution levels, that is, the southwest, the central and the northwest regions of Mexico City.     

Markov chain Monte Carlo algorithms for target-oriented and interval-oriented amplitude versus angle inversions with non-parametric priors and non-linear forward modellings

In geophysical inverse problems, the posterior model can be analytically assessed only in case of linear forward operators, Gaussian, Gaussian mixture, or generalized Gaussian prior models, continuous model properties, and Gaussian-distributed noise contaminating the observed data. For this reason, one of the major challenges of seismic inversion is to derive reliable uncertainty appraisals in cases of complex prior models, non-linear forward operators and mixed discrete-continuous model parameters. We present two amplitude versus angle inversion strategies for the joint estimation of elastic properties and litho-fluid facies from pre-stack seismic data in case of non-parametric mixture prior distributions and non-linear forward modellings. The first strategy is a two-dimensional target-oriented inversion that inverts the amplitude versus angle responses of the target reflections by adopting the single-interface full Zoeppritz equations. The second is an interval-oriented approach that inverts the pre-stack seismic responses along a given time interval using a one-dimensional convolutional forward modelling still based on the Zoeppritz equations. In both approaches, the model vector includes the facies sequence and the elastic properties of P-wave velocity, S-wave velocity and density. The distribution of the elastic properties at each common-mid-point location (for the target-oriented approach) or at each time-sample position (for the time-interval approach) is assumed to be multimodal with as many modes as the number of litho-fluid facies considered. In this context, an analytical expression of the posterior model is no more available. For this reason, we adopt a Markov chain Monte Carlo algorithm to numerically evaluate the posterior uncertainties. With the aim of speeding up the convergence of the probabilistic sampling, we adopt a specific recipe that includes multiple chains, a parallel tempering strategy, a delayed rejection updating scheme and hybridizes the standard Metropolis–Hasting algorithm with the more advanced differential evolution Markov chain method. For the lack of available field seismic data, we validate the two implemented algorithms by inverting synthetic seismic data derived on the basis of realistic subsurface models and actual well log data. The two approaches are also benchmarked against two analytical inversion approaches that assume Gaussian-mixture-distributed elastic parameters. The final predictions and the convergence analysis of the two implemented methods proved that our approaches retrieve reliable estimations and accurate uncertainties quantifications with a reasonable computational effort.     

ANN测井相分析方法研究

本文研究了用于测井相分析识别岩性的人工神经网络（ＡＮＮ）模型设计并在ＳＵＮ工作站上用基于距离Ｄ－ＫｏｈｏｎｅｎＮＮ、Ｄ－ＢＰＮＮ两个网络建立了ＡＮＮ自动测井相分析系统。在实际应用中对比了ＡＷ岩相识别和传统多元统计岩相识别的效果，证明了ＡＮＮ模式识别技术用于测井相分析的可行性和优越性。     

Multi-point source identification of sudden water pollution accidents in surface waters based on differential evolution and Metropolis–Hastings–Markov Chain Monte Carlo

Sudden water pollution accidents in surface waters occur with increasing frequency. These accidents significantly threaten people’s health and lives. To prevent the diffusion of pollutants, identifying these pollution sources is necessary. The identification problem of pollution source, especially for multi-point source, is one of the difficulties in the inverse problem area. This study examines this issue. A new method is designed by combining differential evolution algorithm (DEA) and Metropolis–Hastings–Markov Chain Monte Carlo (MH–MCMC) based on Bayesian inference to identify multi-point sudden water pollution sources. The effectiveness and accuracy of this proposed method is verified through outdoor experiments and comparison between DEA and MH–MCMC. The average absolute error of the sources’ position and intensity, the relative error and the average standard deviations obtained using the proposed method are less than those of DEA and MH–MCMC. Moreover, the relative error and the sampling relative error under four different standard deviations of measurement error (σ = 0.01, 0.05, 0.1, 0.15) are less than 2 and 0.11 %, respectively. The proposed method (i.e., DEMH–MCMC) is effective even when the standard deviation of the measurement error increases to 0.15. Therefore, the proposed method can identify sources of multi-point sudden water pollution accidents efficiently and accurately.     

随机反演在储层预测中的应用

针对隐蔽油气藏储层预测的需要,开展了地震反演研究,根据目前的实际应用将储层预测中的基于模型的地震反演分为三个实施阶段:即构造反演、声波波阻抗或弹性波阻抗反演以及岩性反演,并对每个阶段的目的、关键技术及其原理进行了详细描述,尤其是详细描述了基于马尔科夫链的蒙特卡罗随机模拟技术.最后给出了一个综合应用测井、地质、地震资料进行反演,从而进行储层预测的实例.     

Bayesian uncertainty quantification for flows in heterogeneous porous media using reversible jump Markov chain Monte Carlo methods

In this paper, we study the uncertainty quantification in inverse problems for flows in heterogeneous porous media. Reversible jump Markov chain Monte Carlo algorithms (MCMC) are used for hierarchical modeling of channelized permeability fields. Within each channel, the permeability is assumed to have a log-normal distribution. Uncertainty quantification in history matching is carried out hierarchically by constructing geologic facies boundaries as well as permeability fields within each facies using dynamic data such as production data. The search with Metropolis–Hastings algorithm results in very low acceptance rate, and consequently, the computations are CPU demanding. To speed-up the computations, we use a two-stage MCMC that utilizes upscaled models to screen the proposals. In our numerical results, we assume that the channels intersect the wells and the intersection locations are known. Our results show that the proposed algorithms are capable of capturing the channel boundaries and describe the permeability variations within the channels using dynamic production history at the wells.