【摘要】：由于具有水量穩(wěn)定,水質(zhì)好等優(yōu)點(diǎn),地下水是人類最為重要的飲用水源之一。然而人類活動往往導(dǎo)致地下水系統(tǒng)受到各類污染物的污染。為了更好地管理地下水以及評價地下水污染的環(huán)境風(fēng)險,我們需要借助于數(shù)值模擬對污染物的去向進(jìn)行準(zhǔn)確預(yù)測。而地下水溶質(zhì)運(yùn)移模型的關(guān)鍵參數(shù),例如污染源位置、污染源釋放強(qiáng)度、含水層的滲透系數(shù)等,往往難以直接獲得,需要基于監(jiān)測井獲取的觀測數(shù)據(jù),通過求解反問題來獲得對它們的估計。如何進(jìn)行監(jiān)測井網(wǎng)的最優(yōu)設(shè)計,為反問題提供最有價值的觀測值,從而準(zhǔn)確地獲得對模型參數(shù)的估計,是地下水水文學(xué)的一個研究熱點(diǎn)。此外,監(jiān)測設(shè)計和參數(shù)反演往往需要進(jìn)行數(shù)以萬計的模型調(diào)用,這在大尺度問題中會造成極高的計算代價。針對這些問題,本文以地下水污染物運(yùn)移中的源識別為研究目標(biāo),發(fā)展了基于替代系統(tǒng)的貝葉斯不確定性分析方法,進(jìn)行高效、準(zhǔn)確的監(jiān)測設(shè)計和參數(shù)反演,具體工作如下:(1)為了使觀測數(shù)據(jù)的價值最大化,我們以參數(shù)先驗到后驗相對熵的期望為目標(biāo)函數(shù)實施了監(jiān)測井網(wǎng)的最優(yōu)設(shè)計,其中使目標(biāo)函數(shù)值最大的采樣位置即為最優(yōu)采樣方案。在利用最優(yōu)采樣方案獲得濃度觀測值之后,我們采用了馬爾科夫鏈蒙特卡洛(Markov chain Monte Carlo,MCMC)法來反演未知模型參數(shù)。為了提高計算效率,我們使用自適應(yīng)稀疏格子插值(adaptivesparsegridinterpolation)法在參數(shù)的先驗空間上構(gòu)造了多項式替代系統(tǒng),并將它應(yīng)用到了監(jiān)測設(shè)計和參數(shù)反演中,避免了反復(fù)求解原始模型,即地下水流與溶質(zhì)運(yùn)移控制方程。為了消除替代系統(tǒng)帶來的誤差,我們采用了一種兩階段MCMC模擬來反演未知參數(shù),即先采用替代系統(tǒng)對參數(shù)的后驗分布進(jìn)行充分探索,再使用原始模型對參數(shù)后驗進(jìn)行準(zhǔn)確采樣。數(shù)值算例的結(jié)果表明,在滲透系數(shù)非均質(zhì)性條件下,我們提出的方法可以有效而準(zhǔn)確地識別出污染源參數(shù)和滲透系數(shù)參數(shù)。(2)在兩階段MCMC模擬的第二階段里,我們?nèi)耘f需要多次求解原始模型,因而兩階段MCMC模擬所需的計算量依然較高。為了進(jìn)一步降低計算代價,我們提出了在參數(shù)后驗空間上自適應(yīng)地構(gòu)造替代系統(tǒng)的思想。這里,我們使用了高斯過程(Gaussianprocess,GP)來構(gòu)造替代系統(tǒng),并在反演參數(shù)時將MCMC模擬與替代系統(tǒng)構(gòu)造耦合起來,通過自適應(yīng)地增加接近后驗的基點(diǎn),來不斷提高替代系統(tǒng)在參數(shù)后驗空間上的精度。此外,由于GP的優(yōu)良特性,我們得以量化替代系統(tǒng)的誤差并將之反映到參數(shù)的后驗分布中。數(shù)值模擬的結(jié)果表明,基于后驗替代系統(tǒng)的過程要比基于先驗替代系統(tǒng)的過程更加高效和準(zhǔn)確。(3)在高維問題里,替代系統(tǒng)構(gòu)造和MCMC反演的效果都欠佳。為了解決高維問題中的最優(yōu)監(jiān)測設(shè)計和參數(shù)反演問題,我們提出了一種基于集合(ensemble)的方法。我們采用了數(shù)據(jù)價值分析(data-worth analysis)來尋找信息量最大的采樣方案,然后使用集合平滑器(ensemble smoother,ES)來對模型參數(shù)進(jìn)行反演。為了驗證方法的效果,我們測試了一個高維的數(shù)值算例。在這個算例里,我們考慮了 8個未知污染源參數(shù)和3321個未知滲透系數(shù)參數(shù)。經(jīng)過12個監(jiān)測時間步的設(shè)計,我們獲得了 24個最優(yōu)采樣位置。利用這24個最優(yōu)采樣位置上獲得的濃度和水頭觀測值,我們可以將3329個未知參數(shù)準(zhǔn)確地反演出來。(4)雖然ES算法適用于高維情況,但是它基于線性估計理論,無法解決參數(shù)分布為多峰的反演問題。為了解決高維非高斯情況下的參數(shù)反演問題,我們提出了一種名為迭代局部更新集合平滑(iterative local updating ensemble smoother,ILUES)的算法。在實施該算法的過程中,我們沒有直接對集合中的每個樣本進(jìn)行更新,而是對每個樣本的局部樣本集合進(jìn)行更新,來對可能的多峰分布進(jìn)行充分探索。另外,在非線性問題里,為了提高反演效果,我們在ILUES算法中采用了一種簡單的迭代過程。ILUES算法無需聚類分析,就可以準(zhǔn)確地將參數(shù)的多峰分布識別出來。為了驗證ILUES算法的效果,我們測試了五個數(shù)值算例,分別考慮了參數(shù)先驗多峰,參數(shù)后驗多峰和參數(shù)高維等不同的場景。這些算例都很好地展示了ILUES算法在復(fù)雜模型參數(shù)反演中的效果。與常用的MCMC算法相比,ILUES算法具有計算量上的顯著優(yōu)勢。(5)由于替代系統(tǒng)構(gòu)造在高維問題里效率很低,這極大地限制了替代系統(tǒng)的應(yīng)用范圍。為了解決這個問題,我們提出了將降維和替代系統(tǒng)構(gòu)造結(jié)合的思想,并將它應(yīng)用到地下水污染風(fēng)險評估和分析中。在估計失效概率(即超過風(fēng)險值的概率)的時候,采用直接的蒙特卡洛(MonteCarlo,MC)模擬通常需要大量調(diào)用系統(tǒng)模型。為了減小失效概率分析的計算代價,人們往往會在MC模擬中使用替代系統(tǒng)。然而,直接對高維地下水模型構(gòu)造替代系統(tǒng)非常困難。而且,替代系統(tǒng)的使用會不可避免地引入誤差。為了解決上述問題,我們提出了一種兩階段MC模擬方法,來準(zhǔn)確而有效地開展失效概率分析。在第一階段,我們結(jié)合Karhunen-Loeve展開和分段逆回歸(sliced inverseregression)法對空間非均質(zhì)的滲透系數(shù)參數(shù)進(jìn)行充分降維,并在此基礎(chǔ)上利用混沌多項式展開(polynomial chaos expansion)構(gòu)造出較為準(zhǔn)確的替代系統(tǒng)。利用該替代系統(tǒng),我們可以有效地計算出大量樣本的關(guān)注量(quantity of interest,QoI);在第二階段,為了消除替代系統(tǒng)引入的誤差,我們用原始模型重新計算了失效邊界附近樣本的QoI值。這樣,我們可以消除替代系統(tǒng)引入的誤差,并得到對失效概率的準(zhǔn)確估計。為了驗證上述方法的效果,我們將它應(yīng)用到了一個高維地下水污染物運(yùn)移模擬的例子里,并將流向下游的總污染物的量作為QoI。結(jié)果證明,兩階段MC模擬可以以不足1%的計算代價,得到和基于原始模型的模擬完全一致的結(jié)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.
【學(xué)位授予單位】：浙江大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：X523;X832

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