地下水污染监测井优化设计及污染源识别
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摘要
在地下水污染源识别过程中,针对监测井监测值信息量不充分或监测值与模型参数关联性较弱的问题,提出一种基于贝叶斯公式与信息熵的监测井优化方法.构建二维地下水溶质运移模型,并运用GMS软件进行数值求解.为减少监测井优化设计及污染源识别过程中反复调用数值模型的计算负荷,采用克里金法建立数值模型的替代模型.以信息熵作为优化指标,筛选出不同监测类型的最优监测方案,并以监测成本和反演精度为参考因素,选定相应监测方案,最后运用差分进化自适应Metropolis算法进行污染源识别.算例研究表明:7口监测井的克里金替代模型的决定系数均大于0.98,可较好地替代原数值模型.基于监测成本最小的方案1(3号单井),其信息熵为12.772;兼顾监测成本和反演精度的方案2(井(2,3)组合),其信息熵为9.723;基于反演精度较高的方案3(3井(2,3,5)组合),其信息熵为9.377.方案1到方案3参数后验分布范围及标准差均逐渐减小,验证了信息熵是参数后验分布不确定性的有效量度.
In the process of identifying groundwater pollution sources,a monitoring well optimization method based on Bayesian formula and information entropy is proposed for the problem that the monitoring value of monitoring wells is insufficient or the correlation between monitoring values and model parameters is weak. The two-dimensional groundwater contaminant transport model was numerically solved by GMS software. To reduce the computational load of the numerical model repeatedly in the optimization design of the monitoring wells and the identification process of the pollution source, the Kriging method was used to establish the surrogate model of the numerical model.As an optimization index, the optimal monitoring schemes of different monitoring types were selected, and the monitoring cost and inversion accuracy were taken as reference factors for the corresponding monitoring schemes. Then,the differential evolution adaptive Metropolis algorithm was used to identify the pollution source. The case study results show that: The determination coefficient of the Kriging surrogate models of the 7 monitoring wells was greater than 0.98, which indicated that the Kriging surrogate models can well replace the original numerical model. The scheme 1(single well No. 3)based on the lowest monitoring cost has an information entropy of 12.772;The scheme 2(the combination of well No.2 and No.3)taking the monitoring cost and inversion accuracy into account has an information entropy of 9.723;The scheme 3(the combination of well No.2,3 and 5)with higher inversion precision has an information entropy of 9.377. Both the posterior distribution ranges and the standard deviation of model parameters from scheme 1 to scheme 3 were gradually reduced, which verifies that the information entropy is an effective measure of the uncertainty of the posterior distribution of the parameters.
In the process of identifying groundwater pollution sources,a monitoring well optimization method based on Bayesian formula and information entropy is proposed for the problem that the monitoring value of monitoring wells is insufficient or the correlation between monitoring values and model parameters is weak. The two-dimensional groundwater contaminant transport model was numerically solved by GMS software. To reduce the computational load of the numerical model repeatedly in the optimization design of the monitoring wells and the identification process of the pollution source, the Kriging method was used to establish the surrogate model of the numerical model.As an optimization index, the optimal monitoring schemes of different monitoring types were selected, and the monitoring cost and inversion accuracy were taken as reference factors for the corresponding monitoring schemes. Then,the differential evolution adaptive Metropolis algorithm was used to identify the pollution source. The case study results show that: The determination coefficient of the Kriging surrogate models of the 7 monitoring wells was greater than 0.98, which indicated that the Kriging surrogate models can well replace the original numerical model. The scheme 1(single well No. 3)based on the lowest monitoring cost has an information entropy of 12.772;The scheme 2(the combination of well No.2 and No.3)taking the monitoring cost and inversion accuracy into account has an information entropy of 9.723;The scheme 3(the combination of well No.2,3 and 5)with higher inversion precision has an information entropy of 9.377. Both the posterior distribution ranges and the standard deviation of model parameters from scheme 1 to scheme 3 were gradually reduced, which verifies that the information entropy is an effective measure of the uncertainty of the posterior distribution of the parameters.
引文
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[32]周圣武,李金玉,周长新.概率论与数理统计[M].2版.北京:煤炭工业出版社,2007:208-215.ZHOU S W,LI J Y,ZHOU C X.Probability theory and mathematical statistics[M].2nd ed.Beijin:China Coal Industry Publishing House,2007:208-215.(In Chinese)
[2]HUAN X,MARZOUK Y M.Simulation-based optimal Bayesian experimental design for nonlinear systems[J].Journal of Computational Physics,2013,232(1):288-317.
[3]LINDLEY D V.On a measure of the information provided by an experiment[J].The Annals of Mathematical Statistics,1956,27(4):986-1005.
[4]SHANNON C E.A mathematical theory of communication[J].The Bell System Technical Journal,1948,27(3):379-423.
[5]SOHN M D,SMALL M J,PANTAZIDOU M.Reducing uncertainty in site characterization using bayes monte carlo methods[J].Journal of Environmental Engineering-asce,2000,126(10):893-902.
[6]CHEN M,IZADY A,ABDALLA O A,et al.A surrogate-based sensitivity quantification and Bayesian inversion of a regional groundwater flow model[J].Journal of Hydrology,2018,557:826-837.
[7]SNODGRASS M F,KITANIDIS P K.A geostatistical approach to contaminant source identification[J].Water Resources Research,1997,33(4):537-546.
[8]RUZEK B,KVASNICKA M.Differential evolution algorithm in the earthquake hypocenter location[J].Pure and Applied Geophysics,2001,158:667-693.
[9]GIACOBBO F,MARSEGUERRA M,ZIO E.Solving the inverse problem of parameter estimation by genetic algorithms:the case of a groundwater contaminant transport model[J].Annals of Nuclear Energy,2002,29(8):967-981.
[10]DOUGHERTY D E,MARRYOTT R A.Optimal groundwater management:simulated annealing[J].Water Resources Research,1991,27(10):2493-2508.
[11]TANNER M A.Tools for statistical inference:methods for the expectation of posterior distribution and likelihood functions[M].Berlin:Springer,1996:14-39.
[12]ROBERTS C P,CASELLA G.Monte carlo statistical methods[M].2nd ed.Berlin:Springer,2004:79-122.
[13]METROPOLIS N,ROSENBLUTH A W,ROSENBLUTH M N,et al.Equation of state calculations by fast computing machines[J].The Journal of Chemical Physics,1953,21(6):1087-1092.
[14]HASTINGS W K.Monte Carlo sampling methods using Markov chains and their applications[J].Biometrika,1970,57(1):97-109.
[15]TIERNEY L,MIRA A.Some adaptive Monte Carlo methods for bayesian inference[J].Statistics in Medicine,1999,18:2507-2515.
[16]MIRA A.Ordering and improving the performance of Monte Carlo Markov Chains[J].Statistical Science,2002,16:340-350.
[17]HAARIO H,SAKSMAN E,TAMMINEN J.An adaptive metropolis algorithm[J].Bernoulli,2001,7(2):223-242.
[18]HAARIO H,LAINE M,MIRA A.DRAM:efficient adaptive MCMC[J].Statistics and Computing,2006,16(4):339-354.
[19]张江江.地下水污染源解析的贝叶斯监测设计与参数反演方法[D].杭州:浙江大学环境与资源学院,2017.ZHANG J J.Bayesian monitoring design and parameter inversion for groundwater contaminant source identification[D].Hangzhou:College of Environmental and Resource Sciences,Zhejiang University,2017.(In Chinese)
[20]TER BRAAK C J F.A Markov Chain Monte Carlo version of the genetic algorithm differential evolution:easy Bayesian computing for real parameter spaces[J].Statistics and Computing,2006,16(3):239-249.
[21]VRUGT J A,TER BRAAK C J F,DIKS C G H,et al.Accelerating Markov Chain Monte Carlo simulation by differential evolution with self-adaptive randomized subspace sampling[J].International Journal of Nonlinear Sciences and Numerical Simulation,2009,10(3):273-290.
[22]KNILL D L,GIUNTA A A,BAKER C A,et al.Response surface models combining linear and Euler aerodynamics for supersonic transport design[J].Journal of Aircraft,1999,36(1):75-86.
[23]LI J,CHEN Y,PEPPER D.Radial basis function method for 1-Dand 2-D groundwater contaminant transport modeling[J].Computational Mechanics,2003,32(1):10-15.
[24]肖传宁,卢文喜,赵莹,等.基于径向基函数模型的优化方法在地下水污染源识别中的应用[J].中国环境科学,2016,36(7):2067-2072.XIAO C N,LU W X,ZHAO Y,et al.Optimization method of identification of groundwater pollution sources based on radial basis function model[J].China Environmental Science,2016,36(7):2067-2072.(In Chinese)
[25]CHRISTIE M,DEMYANOV V,ERBAS D.Uncertainty quantification for porous media flows[J].Journal of Computational Physics,2006,217(1):143-158.
[26]MATHERTON G.Principles of geostatistics[J].Economic Geology,1963,58(8):1246-1266.
[27]SACKS J,WELCH W J,MITCHELL T J,et al.Design and analysis of computer experiments[J].Statistical Science,1989,4(4):409-435.
[28]高月华.基于Kriging代理模型的优化设计方法及其在注塑成型中的作用[D].大连:大连理工大学运载工程与力学学部,2008.GAO Y H.Optimization methods based on Kriging surrogate model and their application in injection molding[D].Dalian:Faculty of Vehicle Engineering and Mechanics,Dalian University of Technology,2008.(In Chinese)
[29]LOPHAVEN S N,NIELSEN H B,SONDERGAARD J.Dace:AMATLAB Kriging toolbox[R].Kongens Lyngby:Technical University of Denmark,Technical Report No.IMM-TR-2002-12.
[30]GELMAN A G,RUBIN D B.Inference from iterative simulation using multiple sequences[J].Statistical Science,1992,7:457-472.[31 HICKERNELL F J.A generalized discrepancy and quadrature error bound[J].Mathematics of Computation of the American Mathematical Society,1998,67(221):299-322.
[32]周圣武,李金玉,周长新.概率论与数理统计[M].2版.北京:煤炭工业出版社,2007:208-215.ZHOU S W,LI J Y,ZHOU C X.Probability theory and mathematical statistics[M].2nd ed.Beijin:China Coal Industry Publishing House,2007:208-215.(In Chinese)
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