突发性水污染事件溯源方法

为快速准确地求解突发性水污染溯源问题,在微分进化与蒙特卡罗基础上提出了一种新的溯源方法。该方法将溯源问题视为贝叶斯估计问题,推导出污染源强度、位置和排放时刻等未知参数的后验概率密度函数;结合微分进化和蒙特卡罗模拟方法对后验概率分布进行采样,进而估计出这些未知参数,确定污染源项。通过算例与贝叶斯-蒙特卡罗方法进行对比,结果表明:该方法可使迭代次数有效缩减3/4,污染源强度、位置和排放时刻的平均相对误差分别减少1.23%、2.23%和4.15%,均值误差分别降低0.39%、0.83%和1.49%,其稳定性和可靠性明显高于贝叶斯-蒙特卡罗方法,能较好地识别突发性水污染源,为解决突发水污染事件中的追踪溯源难点问题提供了新的思路和方法。     

地下水污染源反演的Hooke Jeeves吸引扩散粒子群混合算法

根据污染物质量浓度监测数据进行地下水污染源反演是一类典型的地下水逆问题,该问题可转化为决策变量为污染源位置和强度的最优化问题进行求解。基于Hooke-Jeeves粒子群混合算法,引入吸引扩散粒子群(ARPSO)算法的粒子群发散算子,保证混合算法的种群多样性,并提出HJ-ARPSO混合算法,再结合地下水污染物迁移模型MT3DMS反演地下水污染源的位置和强度信息。在已知污染源位置和未知污染源位置两种情形下,分别利用HJ-ARPSO算法、HJ-PSO算法和GA算法进行地下水污染源反演。在两种情形下,HJ-ARPSO算法均具有较高的寻优成功率(分别对应为100%和90%);与之相比,未引入粒子群发散算子的HJ-PSO算法在未知污染源位置情形下其寻优成功率迅速降为60%;GA算法寻优效率则最低。算例结果表明,HJ-ARPSO算法是一种有效的地下水污染源反演优化算法。     

地下水污染源识别的数学方法研究进展

地下水污染源识别模型可利用有限的观测资料估计污染源位置、污染物泄露强度及其泄露过程,是制定地下水污染修复方案的依据。在阐明地下水污染源识别基本问题基础上,综述了污染源识别研究的两大类数学方法,一类为直接方法,包括反向追踪法和基于正则化的方法;另一类为间接方法,包括基于优化的方法和基于概率统计的方法。同时指出了当前污染源识别数学方法应用中存在的主要问题:地下水污染源识别问题的复杂性、地下水有机污染问题和模型求解效率的低下性。对土壤-地下水的联合管理、基于物联网的地下水污染监测、对非水相流体（Non-aqueous Phase Liquid,NAPL）类污染源识别以及基于图形处理器（GPU）的异构并行计算将是未来研究的重点方向。     

先验概率分布及似然函数模型的选择对边坡可靠度评价影响的定量评估

受工程勘察成本及试验场地限制,可获得的试验数据通常有限,基于有限的试验数据难以准确估计岩土参数统计特征和边坡可靠度。贝叶斯方法可以融合有限的场地信息降低对岩土参数不确定性的估计进而提高边坡可靠度水平。但是,目前的贝叶斯更新研究大多假定参数先验概率分布为正态、对数正态和均匀分布,似然函数为多维正态分布,这种做法的合理性有待进一步验证。总结了岩土工程贝叶斯分析常用的参数先验概率分布及似然函数模型,以一个不排水黏土边坡为例,采用自适应贝叶斯更新方法系统探讨了参数先验概率分布和似然函数对空间变异边坡参数后验概率分布推断及可靠度更新的影响。计算结果表明：参数先验概率分布对空间变异边坡参数后验概率分布推断及可靠度更新均有一定的影响,选用对数正态和极值I型分布作为先验概率分布推断的参数后验概率分布离散性较小。选用Beta分布和极值I型分布获得的边坡可靠度计算结果分别偏于保守和危险,选用对数正态分布获得的边坡可靠度计算结果居中。相比之下,似然函数的影响更加显著。与其他类型似然函数相比,由多维联合正态分布构建的似然函数可在降低对岩土参数不确定性估计的同时,获得与场地信息更为吻合的计算结果。另外,构建似然函数时不同位置处测量误差之间的自相关性对边坡后验失效概率也具有一定的影响。     

基于MCMC法的非饱和土渗流参数随机反分析

基于贝叶斯理论,以马尔可夫链蒙特卡罗方法（Markov chain Monte Carlo Simulation, MCMC法）的自适应差分演化Metropolis算法为参数后验分布抽样计算方法,建立利用时变测试数据的参数随机反分析及模型预测方法。以香港东涌某天然坡地降雨入渗测试为算例,采用自适应差分演化Metropolis算法对时变降雨条件下非饱和土一维渗流模型参数进行随机反分析,研究参数后验分布的统计特性,并分别对校准期和验证期内模型预测孔压和实测值进行比较。研究结果表明,DREAM算法得到的各随机变量后验分布标准差较先验分布均显著减小;经过实测孔压数据的校准,模型计算精度很高,校准期内95%总置信区间的覆盖率达到0.964;验证期第2～4个阶段95%总置信区间的覆盖率分别为0.52、0.79和0.79,模型预测结果与实测值吻合程度较高。     

水环境逆边界逆动态混合控制精确算法

根据逆边界逆动态控制理论,将河流水污染动态控制问题提为逆边界逆动态混合控制问题。针对多个或单个污染源排放浓度和排放总量计算,提出了一维对流-扩散方程逆控制的精确算法。该方法与现行最优控制方法相比,其优点是充分考虑了河流沿程的稀释混合容量,并能充分考虑水质动态标准和社会经济变化等因素,可获得动态控制精确解的近似解。     

基于贝叶斯理论面波频散曲线随机反演

面波频散曲线反演是获得地下横波速度结构的重要地球物理方法。常规基于迭代最小二乘等线性反演方法依赖于初始模型,且存在多极值、容易陷入局部最小、反演精度低等问题。基于贝叶斯理论的随机反演方法是一种可以融合先验信息的非线性反演方法,该方法无需人为给定初始模型,仅利用先验信息对模型进行随机采样,根据概率分布筛选接受合适的后验概率密度估计结果,可达到对细节信息的准确估计。本文针对瑞利面波频散曲线,提出了基于GPR数据先验资料约束的贝叶斯马尔科夫蒙特卡洛(MCMC)随机反演方法,通过随机改变模型参数并计算其频散曲线与实际频散曲线的似然函数来选择是否接受新的模型参数,不断重复此过程,最终得到与实际频散曲线拟合效果最佳的最优解以及横波速度解的后验概率密度分布。通过理论模型以及实际数据反演测试,验证了该方法与常规无约束的随机反演相比,可以有效地提高反演速度和反演精度。     

基于朴素贝叶斯的判别频率估计算法在油水层识别中的应用

贝叶斯网络分类器,已经广泛应用于各种分类问题。对于固定的贝叶斯网络结构,可以通过生成参数和判别参数2种方法进行学习。生成参数学习效率较高但是分类精度较低,而判别参数学习与之相反。通过对数据集中参数出现频率计算来进行参数学习,并加入一个判别参数来加强实例属性与分类目标值之间的关联性,在此基础上提出了一种简单、快速、有效的判别频率估计(DFE)算法。实验结果表明:在油水层模式识别当中,这种判别频率估计方法相对于其他算法在分类精度上能够提高5%～10%。     

基于贝叶斯方法的船坞双排钢板桩围堰整体稳定性可靠度分析

双排钢板桩围堰的整体稳定性具有较大的不确定性,同样安全系数情况下对应的失稳概率可能不同。为了更准确地分析双排钢板桩围堰的整体稳定性,降低稳定性分析中的不确定性,采用贝叶斯方法对船坞双排钢板桩围堰的整体稳定性进行可靠度分析。首先通过统计数据获取土体参数的先验分布,然后基于实测数据采用贝叶斯方法更新参数以得到后验分布,最后根据参数的后验分布采用一次二阶矩计算围堰结构的可靠度。贝叶斯方法从理论的角度解决了已有工程经验和实际案例数据两方面信息有效综合的问题,能在更接近实际情况的前提下进行可靠度分析。     

基于随机加权先验的P-Ⅲ分布参数贝叶斯估计

研究随机加权先验法进行P-Ⅲ分布参数贝叶斯估计。应用随机加权法确定分布参数的先验分布,MCMC自适应采样算法(AM)进行参数的后验分布采样,并与矩法、极大似然法和概率权重矩法等传统水文频率分析方法进行比较。实例表明,AM方法估算参数下,实测样本与对应频率设计值离差平方和最小,是一种可行的水文频率分析途径。     

Bayesian Reservoir History Matching Considering Model and Parameter Uncertainties

This paper presents a consistent Bayesian solution for data integration and history matching for oil reservoirs while accounting for both model and parameter uncertainties. The developed method uses Gaussian Process Regression to build a permeability map conforming to collected data at well bores. Following that, an augmented Markov Chain Monte Carlo sampler is used to condition the permeability map to dynamic production data. The selected proposal distribution for the Markov Chain Monte Carlo conforms to the Gaussian process regression output. The augmented Markov Chain Monte Carlo sampler allows transition steps between different models of the covariance function, and hence both the parameter and model space are effectively explored. In contrast to single model Markov Chain Monte Carlo samplers, the proposed augmented Markov Chain Monte Carlo sampler eliminates the selection bias of certain covariance structures of the inferred permeability field. The proposed algorithm can be used to account for general model and parameter uncertainties.     

Probabilistic risk assessment of major accidents: application to offshore blowouts in the Gulf of Mexico

Major accidents are low-frequency, high-consequence accidents which are not well supported by conventional statistical methods due to the scarcity of directly relevant data. Modeling and decomposition techniques such as event tree have been proved as robust alternatives as they facilitate incorporation of partially relevant near accident data–accident precursor data—in probability estimation and risk analysis of major accidents. In this study, we developed a methodology based on event tree and hierarchical Bayesian analysis to establish informative distributions for offshore blowouts using data of near accidents, such as kicks, leaks, and failure of blowout preventers collected from a variety of offshore drilling rigs. These informative distributions can be used as predictive tools to estimate relevant failure probabilities in the future. Further, having a set of near accident data of a drilling rig of interest, the informative distributions can be updated to render case-specific posterior distributions which are of great importance in quantitative risk analysis. To cope with uncertainties, we implemented the methodology in a Markov Chain Monte Carlo framework and applied it to risk assessment of offshore blowouts in the Gulf of Mexico.     

Characterising geotechnical model uncertainty by hybrid Markov Chain Monte Carlo simulation

Geotechnical models are usually associated with considerable amounts of model uncertainty. In this study, the model uncertainty of a geotechnical model is characterised through a systematic comparison between model predictions and past performance data. During such a comparison, model input parameters (such as soil properties) may also be uncertain, and the observed performance may be subjected to measurement errors. To consider these uncertainties, the model uncertainty parameters, uncertain model input parameters and actual performance variables are modelled as random variables, and their distributions are updated simultaneously using Bayes’ theorem. When the number of variables to update is large, solving the Bayesian updating problem is computationally challenging. A hybrid Markov Chain Monte Carlo simulation is employed in this paper to decompose the high-dimensional Bayesian updating problem into a series of updating problems in lower dimensions. To increase the efficiency of the Markov chain, the model uncertainty is first characterised with a first order second moment method approximately, and the knowledge learned from the approximate solution is then used to design key parameters in the Markov chain. Two examples are used to illustrate the proposed methodology for model uncertainty characterisation, with insights, discussions, and comparison with previous methods.     

Probabilistic parameter estimation and predictive uncertainty based on field measurements for unsaturated soil slope

A key issue in assessment of rainfall-induced slope failure is a reliable evaluation of pore water pressure distribution and its variations during rainstorm, which in turn requires accurate estimation of soil hydraulic parameters. In this study, the uncertainties of soil hydraulic parameters and their effects on slope stability prediction are evaluated, within the Bayesian framework, using the field measured temporal pore-water pressure data. The probabilistic back analysis and parameter uncertainty estimation is conducted using the Markov Chain Monte Carlo simulation. A case study of a natural terrain site is presented to illustrate the proposed method. The 95% total uncertainty bounds for the calibration period are relatively narrow, indicating an overall good performance of the infiltration model for the calibration period. The posterior uncertainty bounds of slope safety factors are much narrower than the prior ones, implying that the reduction of uncertainty in soil hydraulic parameters significantly reduces the uncertainty of slope stability.     

基于随机理论的地下水环境风险评价

针对确定性模型难以描述含水层非均质空间分布的问题,提出基于随机理论的地下水环境风险评价方法。以矩形场地地下水污染风险评价为例,采用蒙特卡罗法生成大量渗透系数随机场,模拟含水层参数各种可能的非均质空间分布,在此基础上建立场地地下水流模型与溶质运移模型,分别计算污染物在地下水中的迁移转化情况。统计大量随机模拟中污染事故发生的频率,当模拟次数足够多时,污染频率收敛于污染概率,污染风险即通过污染概率体现出来。该方法将模型参数设为满足一定分布特征的随机变量,避免了确定性方法得出的武断的评价结果,可为工厂的选址、水源地的选址等工作提供科学指导。     

A review of Markov Chain Monte Carlo and information theory tools for inverse problems in subsurface flow

Parameter identification is one of the key elements in the construction of models in geosciences. However, inherent difficulties such as the instability of ill-posed problems or the presence of multiple local optima may impede the execution of this task. Regularization methods and Bayesian formulations, such as the maximum a posteriori estimation approach, have been used to overcome those complications. Nevertheless, in some instances, a more in-depth analysis of the inverse problem is advisable before obtaining estimates of the optimal parameters. The Markov Chain Monte Carlo (MCMC) methods used in Bayesian inference have been applied in the last 10 years in several fields of geosciences such as hydrology, geophysics or reservoir engineering. In the present paper, a compilation of basic tools for inference and a case study illustrating the practical application of them are given. Firstly, an introduction to the Bayesian approach to the inverse problem is provided together with the most common sampling algorithms with MCMC chains. Secondly, a series of estimators for quantities of interest, such as the marginal densities or the normalization constant of the posterior distribution of the parameters, are reviewed. Those reduce the computational cost significantly, using only the time needed to obtain a sample of the posterior probability density function. The use of the information theory principles for the experimental design and for the ill-posedness diagnosis is also introduced. Finally, a case study based on a highly instrumented well test found in the literature is presented. The results obtained are compared with the ones computed by the maximum likelihood estimation approach.     

Unraveling reservoir compaction parameters through the inversion of surface subsidence observations

In an attempt to derive more information on the parameters driving compaction, this paper explores the feasibility of a method utilizing data on compaction-induced subsidence. We commence by using a Bayesian inversion scheme to infer the reservoir compaction from subsidence observations. The method’s strength is that it incorporates all the spatial and temporal correlations imposed by the geology and reservoir data. Subsequently, the contributions of the driving parameters are unravelled. We apply the approach to a synthetic model of an upscaled gas field in the northern Netherlands. We find that the inversion procedure leads to coupling between the driving parameters, as it does not discriminate between the individual contributions to the compaction. The provisional assessment of the parameter values shows that, in order to identify adequate estimate ranges for the driving parameters, a proper parameter estimation procedure (Markov Chain Monte Carlo, data assimilation) is necessary.     

Stochastic Fracture Propagation Modelling for Enhanced Geothermal Systems

Fractures and fracture networks are the fundamental components of enhanced geothermal systems and determine their technical and economic viability. A realistic fracture model that can adequately describe a fracture-stimulated reservoir is critical for subsequent flow and heat transfer analyses of the system. Fractures in these systems are essentially the product of hydraulic stimulations of the reservoir that, together with ground conditions and the local stress regime, determine how fractures are formed and propagated. This paper describes three methods for generating realistic fracture models for enhanced geothermal systems; two of them incorporate the fracture propagation process in the modelling and hence provide a stochastic fracture propagation model. The methods are: a Bayesian framework in the form of Markov Chain Monte Carlo simulation, an extended Random Sampling Consensus model and a Point and Surface Association Consensus model. The conditioning data used in these methods are seismic events recorded during fracture stimulation. Geodynamics’ Habanero reservoir in the Cooper Basin of South Australia is used as a case study to test these methods.     

基于不同邻域系统的马尔可夫链模型的储层岩相随机模拟

针对在油气储层随机模拟中马尔可夫链模型的不同方向的转移概率矩阵求取困难的问题,提出一种二维剖面中不同方向的转移概率矩阵求取方法,这种方法的提出使得不同阶次的各向同性和各向异性的邻域系统的转移概率矩阵的求取变得容易可行。随后,对不同邻域系统的马尔可夫链模型采用蒙特卡罗抽样方法进行了储层岩相随机模拟试验。最后比较了不同邻域系统岩相模拟的结果并探讨了在储层研究中的适用性。     

Comparing the Gradual Deformation with the Probability Perturbation Method for Solving Inverse Problems

Inverse problems are ubiquitous in the Earth Sciences. Many such problems are ill-posed in the sense that multiple solutions can be found that match the data to be inverted. To impose restrictions on these solutions, a prior distribution of the model parameters is required. In a spatial context this prior model can be as simple as a Multi-Gaussian law with prior covariance matrix, or could come in the form of a complex training image describing the prior statistics of the model parameters. In this paper, two methods for generating inverse solutions constrained to such prior model are compared. The gradual deformation method treats the problem of finding inverse solution as an optimization problem. Using a perturbation mechanism, the gradual deformation method searches (optimizes) in the prior model space for those solutions that match the data to be inverted. The perturbation mechanism guarantees that the prior model statistics are honored. However, it is shown with a simple example that this perturbation method does not necessarily draw accurately samples from a given posterior distribution when the inverse problem is framed within a Bayesian context. On the other hand, the probability perturbation method approaches the inverse problem as a data integration problem. This method explicitly deals with the problem of combining prior probabilities with pre-posterior probabilities derived from the data. It is shown that the sampling properties of the probability perturbation method approach the accuracy of well-known Markov chain Monte Carlo samplers such as the rejection sampler. The paper uses simple examples to illustrate the clear differences between these two methods