地下水污染源解析的贝叶斯监测设计与参数反演方法

发布时间：2018-07-31 13:25

【摘要】：由于具有水量稳定,水质好等优点,地下水是人类最为重要的饮用水源之一。然而人类活动往往导致地下水系统受到各类污染物的污染。为了更好地管理地下水以及评价地下水污染的环境风险,我们需要借助于数值模拟对污染物的去向进行准确预测。而地下水溶质运移模型的关键参数,例如污染源位置、污染源释放强度、含水层的渗透系数等,往往难以直接获得,需要基于监测井获取的观测数据,通过求解反问题来获得对它们的估计。如何进行监测井网的最优设计,为反问题提供最有价值的观测值,从而准确地获得对模型参数的估计,是地下水水文学的一个研究热点。此外,监测设计和参数反演往往需要进行数以万计的模型调用,这在大尺度问题中会造成极高的计算代价。针对这些问题,本文以地下水污染物运移中的源识别为研究目标,发展了基于替代系统的贝叶斯不确定性分析方法,进行高效、准确的监测设计和参数反演,具体工作如下:(1)为了使观测数据的价值最大化,我们以参数先验到后验相对熵的期望为目标函数实施了监测井网的最优设计,其中使目标函数值最大的采样位置即为最优采样方案。在利用最优采样方案获得浓度观测值之后,我们采用了马尔科夫链蒙特卡洛(Markov chain Monte Carlo,MCMC)法来反演未知模型参数。为了提高计算效率,我们使用自适应稀疏格子插值(adaptivesparsegridinterpolation)法在参数的先验空间上构造了多项式替代系统,并将它应用到了监测设计和参数反演中,避免了反复求解原始模型,即地下水流与溶质运移控制方程。为了消除替代系统带来的误差,我们采用了一种两阶段MCMC模拟来反演未知参数,即先采用替代系统对参数的后验分布进行充分探索,再使用原始模型对参数后验进行准确采样。数值算例的结果表明,在渗透系数非均质性条件下,我们提出的方法可以有效而准确地识别出污染源参数和渗透系数参数。(2)在两阶段MCMC模拟的第二阶段里,我们仍旧需要多次求解原始模型,因而两阶段MCMC模拟所需的计算量依然较高。为了进一步降低计算代价,我们提出了在参数后验空间上自适应地构造替代系统的思想。这里,我们使用了高斯过程(Gaussianprocess,GP)来构造替代系统,并在反演参数时将MCMC模拟与替代系统构造耦合起来,通过自适应地增加接近后验的基点,来不断提高替代系统在参数后验空间上的精度。此外,由于GP的优良特性,我们得以量化替代系统的误差并将之反映到参数的后验分布中。数值模拟的结果表明,基于后验替代系统的过程要比基于先验替代系统的过程更加高效和准确。(3)在高维问题里,替代系统构造和MCMC反演的效果都欠佳。为了解决高维问题中的最优监测设计和参数反演问题,我们提出了一种基于集合(ensemble)的方法。我们采用了数据价值分析(data-worth analysis)来寻找信息量最大的采样方案,然后使用集合平滑器(ensemble smoother,ES)来对模型参数进行反演。为了验证方法的效果,我们测试了一个高维的数值算例。在这个算例里,我们考虑了 8个未知污染源参数和3321个未知渗透系数参数。经过12个监测时间步的设计,我们获得了 24个最优采样位置。利用这24个最优采样位置上获得的浓度和水头观测值,我们可以将3329个未知参数准确地反演出来。(4)虽然ES算法适用于高维情况,但是它基于线性估计理论,无法解决参数分布为多峰的反演问题。为了解决高维非高斯情况下的参数反演问题,我们提出了一种名为迭代局部更新集合平滑(iterative local updating ensemble smoother,ILUES)的算法。在实施该算法的过程中,我们没有直接对集合中的每个样本进行更新,而是对每个样本的局部样本集合进行更新,来对可能的多峰分布进行充分探索。另外,在非线性问题里,为了提高反演效果,我们在ILUES算法中采用了一种简单的迭代过程。ILUES算法无需聚类分析,就可以准确地将参数的多峰分布识别出来。为了验证ILUES算法的效果,我们测试了五个数值算例,分别考虑了参数先验多峰,参数后验多峰和参数高维等不同的场景。这些算例都很好地展示了ILUES算法在复杂模型参数反演中的效果。与常用的MCMC算法相比,ILUES算法具有计算量上的显著优势。(5)由于替代系统构造在高维问题里效率很低,这极大地限制了替代系统的应用范围。为了解决这个问题,我们提出了将降维和替代系统构造结合的思想,并将它应用到地下水污染风险评估和分析中。在估计失效概率(即超过风险值的概率)的时候,采用直接的蒙特卡洛(MonteCarlo,MC)模拟通常需要大量调用系统模型。为了减小失效概率分析的计算代价,人们往往会在MC模拟中使用替代系统。然而,直接对高维地下水模型构造替代系统非常困难。而且,替代系统的使用会不可避免地引入误差。为了解决上述问题,我们提出了一种两阶段MC模拟方法,来准确而有效地开展失效概率分析。在第一阶段,我们结合Karhunen-Loeve展开和分段逆回归(sliced inverseregression)法对空间非均质的渗透系数参数进行充分降维,并在此基础上利用混沌多项式展开(polynomial chaos expansion)构造出较为准确的替代系统。利用该替代系统,我们可以有效地计算出大量样本的关注量(quantity of interest,QoI);在第二阶段,为了消除替代系统引入的误差,我们用原始模型重新计算了失效边界附近样本的QoI值。这样,我们可以消除替代系统引入的误差,并得到对失效概率的准确估计。为了验证上述方法的效果,我们将它应用到了一个高维地下水污染物运移模拟的例子里,并将流向下游的总污染物的量作为QoI。结果证明,两阶段MC模拟可以以不足1%的计算代价,得到和基于原始模型的模拟完全一致的结果。
[Abstract]:Groundwater is one of the most important drinking water sources for human beings. However, human activities often cause the groundwater system to be polluted by various pollutants. In order to better manage the groundwater and evaluate the environmental risk of groundwater pollution, we need to use numerical simulation to determine the direction of the pollutant. The key parameters of the groundwater solute transport model, such as the location of the pollution source, the intensity of the pollution source, the permeability coefficient of the aquifer and so on, are often difficult to be obtained directly. It is necessary to obtain the observation data based on the monitoring well to obtain the estimation of it by solving the inverse problem. How to do the optimal design of the monitoring well network It is a hot spot for groundwater hydrology to provide the most valuable observation values, and thus accurately estimate the parameters of the model. In addition, the monitoring design and parameter inversion often require tens of thousands of model calls, which will cause extremely high calculation costs in large scale problems. The source identification of water pollutant migration is the research goal, and the Bayesian uncertainty analysis method based on the alternative system is developed to carry out efficient, accurate monitoring design and parameter inversion. (1) in order to maximize the value of the observed data, we carry out the expectation of the prior to the posterior relative entropy to the objective function. The optimal design of the monitoring well network, in which the maximum sampling position of the target function is the optimal sampling scheme, we use the Markoff Montecalo (Markov chain Monte Carlo, MCMC) method to inverse the unknown model parameters after using the optimal sampling scheme. In order to improve the computational efficiency, we use the self We adapt to the sparse lattice interpolation (adaptivesparsegridinterpolation) method to construct a polynomial substitution system in the prior space of parameters, and apply it to the monitoring design and parameter inversion. It avoids the repeated solution of the original model, that is the control square of the groundwater flow and solute transport. In order to eliminate the error caused by the alternative system, we adopt the method. A two stage MCMC simulation is used to invert the unknown parameters, that is, an alternative system is used to fully explore the posterior distribution of the parameters, and then the original model is used to accurately sample the parameter posterior. The numerical example shows that the method proposed by us can identify the pollution effectively and accurately under the condition of permeability coefficient heterogeneity. Source parameters and osmotic coefficient parameters. (2) in the second stage of the two phase MCMC simulation, we still need to solve the original model many times, so the amount of computation required for the two phase MCMC simulation is still high. In order to further reduce the computational cost, we propose the idea of constructing the alternative system adaptively in the parameter posterior space. Here, I We use the Gauss process (Gaussianprocess, GP) to construct an alternative system, and combine the MCMC simulation with the substitutes in the inversion of the parameters, and increase the accuracy of the alternative system in the parametric posterior space by self adaptively increasing the base point near the posterior. In addition, we are able to quantify the substitution line due to the excellent properties of the GP. The error of the system is reflected in the posterior distribution of the parameters. The results of the numerical simulation show that the process based on the posterior substitution system is more efficient and accurate than the process based on the prior alternative system. (3) in the high dimensional problem, the construction of the alternative system and the effect of the MCMC inversion are not good. And the problem of parameter inversion, we propose a method based on the set (ensemble). We use the data value analysis (data-worth analysis) to find the maximum information sampling scheme, and then use the set smoother (ensemble smoother, ES) to retrieve the model parameters. In order to verify the effect of the method, we tested one of the methods. In this case, we consider 8 unknown pollution source parameters and 3321 unknown osmotic coefficient parameters in this example. Through the design of 12 monitoring time steps, we obtain 24 optimal sampling positions. We can use the 3329 unknown parameters by using the concentration and water head observed on the 24 optimal sampling locations. (4) (4) although the ES algorithm is suitable for the high dimensional case, it is based on the linear estimation theory, and can not solve the inverse problem of multi peak in the parameter distribution. In order to solve the problem of parameter inversion in the case of high dimensional non Gauss, we propose a kind of iterative local updating ensemble smoother called iterative local update set smoothing. In the process of implementing this algorithm, we do not update each sample in the set directly, instead, we update the local sample set of each sample to fully explore the possible multi peak distribution. In addition, in the nonlinear problem, in order to improve the inversion effect, we use one kind in the ILUES algorithm. In simple iterative process,.ILUES algorithm can identify the multi peak distribution of parameters without clustering analysis. In order to verify the effect of the ILUES algorithm, we tested five numerical examples, taking into account the different peaks of the parameters, the posterior multi peak and the high dimension of the parameters respectively. These examples all show the ILUES well. The effect of the algorithm in the parameter inversion of the complex model. Compared with the common MCMC algorithm, the ILUES algorithm has the significant advantage of the computational complexity. (5) because the structure of the replacement system is very inefficient in the high dimensional problem, it greatly limits the application scope of the alternative system. In order to solve this problem, we propose a construction of the substitutes for the substitutes and the substitutes. The idea of combining it and applying it to the assessment and analysis of the risk of groundwater pollution. When estimating the failure probability (i.e. the probability of exceeding the risk value), the direct Monte Carlo (MonteCarlo, MC) simulation usually requires a large number of system models. In order to reduce the cost of the failure probability analysis, people often make the MC simulation Using alternative systems. However, it is very difficult to construct a substituting system directly for the high dimensional groundwater model. Moreover, the use of the alternative system will inevitably introduce errors. In order to solve the above problems, we have proposed a two stage MC simulation method to accurately and effectively carry out the failure probability analysis. In the first stage, we combine the Karhunen-Loe The ve expansion and the piecewise inverse regression (sliced inverseregression) method can fully reduce the parameters of the permeability coefficient of spatial heterogeneity, and on this basis, a more accurate replacement system is constructed by using the chaotic polynomial expansion (polynomial chaos expansion). By using this alternative system, we can effectively calculate the attention of a large number of samples. (quantity of interest, QoI); in the second stage, in order to eliminate the error introduced by the alternative system, we recalculated the QoI value of the samples near the failure boundary with the original model. In this way, we can eliminate the error introduced by the alternative system and get the accurate estimation of the failure probability. In order to verify the effect of the above method, we apply it In the example of a high dimensional groundwater pollutant migration simulation, and the amount of total pollutants flowing downstream as a result of QoI., the two stage MC simulation can be calculated at a cost of less than 1%, and the result is completely consistent with the simulation based on the original model.
【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：X523;X832

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