基于贝叶斯公式的地下水污染源及含水层参数同步反演
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摘要
针对非均质地下含水层污染源识别及含水层参数反演过程中监测方案优化问题,提出一种基于贝叶斯公式及信息熵最小的累进加井的多井监测方案优化方法.首先,构建假想案例下的二维非均质各向同性潜水含水层水流及溶质运移模型,运用GMS软件进行数值模拟求解.采用最优拉丁超立方抽样方法和Kriging法建立数值模拟模型的替代模型.然后以参数后验分布的信息熵最小为目标函数,采用累进加井的方式进行多井监测方案优化设计.最后根据优化后的监测方案,采用差分进化自适应Metropolis算法进行污染源及含水层参数的同步反演.算例研究表明:在兼顾反演精度及监测成本,并保证每个参数分区内至少有1眼监测井的条件下,5眼井组合监测方案(6,5,1,2,8)为最优监测方案.与信息熵最小的10眼井组合监测方案(1,2,3,4,5,6,7,8,9,10)的参数反演结果相比,5眼井组合监测方案对11个参数α=(XS,YS,T1,T2,QS,K1,K2,K3,DL1,DL2,DL3)的后验均值偏离率的虽增大1.2%,但监测成本却是10眼井组合监测方案的50%.
Aiming at the optimization of monitoring schemes in the process of the identification of pollution source and the inversion of aquifer parameters in the heterogeneous underground aquifer, this paper proposes an optimization method for the multi-well monitoring schemes based on Bayesian formula and progressive addition of wells with minimum information entropy. Firstly, the two-dimensional heterogeneous isotropic subsurface groundwater flow and solute transport models under hypothetical case were constructed, and the numerical simulation models were solved by GMS software. The surrogate model of the numerical simulation model was established by the optimal Latin hypercube sampling method and Kriging method. Then Taking the minimum information entropy of the parameter posterior distribution as the objective function, the optimization design of multi-well monitoring schemes was carried out by means of progressive addition of wells. Finally, the differential evolution adaptive Metropolis algorithm was used to inverse the pollution source and aquifer parameters synchronously according to the optimized monitoring scheme. The case study results showed that: The 5 combination monitoring scheme(6, 5, 1, 2, 8) under the condition of taking into account the inversion accuracy and monitoring cost and ensuring that there was at least one monitoring well in each parameter section was the optimal monitoring scheme. Compared with the 10 combined monitoring scheme(1, 2, 3, 4, 5, 6, 7, 8, 9, 10) with the smallest information entropy, the 11 parameters posterior mean deviation rate increased by 1.2%, but the monitoring cost was reduced by 50%.
Aiming at the optimization of monitoring schemes in the process of the identification of pollution source and the inversion of aquifer parameters in the heterogeneous underground aquifer, this paper proposes an optimization method for the multi-well monitoring schemes based on Bayesian formula and progressive addition of wells with minimum information entropy. Firstly, the two-dimensional heterogeneous isotropic subsurface groundwater flow and solute transport models under hypothetical case were constructed, and the numerical simulation models were solved by GMS software. The surrogate model of the numerical simulation model was established by the optimal Latin hypercube sampling method and Kriging method. Then Taking the minimum information entropy of the parameter posterior distribution as the objective function, the optimization design of multi-well monitoring schemes was carried out by means of progressive addition of wells. Finally, the differential evolution adaptive Metropolis algorithm was used to inverse the pollution source and aquifer parameters synchronously according to the optimized monitoring scheme. The case study results showed that: The 5 combination monitoring scheme(6, 5, 1, 2, 8) under the condition of taking into account the inversion accuracy and monitoring cost and ensuring that there was at least one monitoring well in each parameter section was the optimal monitoring scheme. Compared with the 10 combined monitoring scheme(1, 2, 3, 4, 5, 6, 7, 8, 9, 10) with the smallest information entropy, the 11 parameters posterior mean deviation rate increased by 1.2%, but the monitoring cost was reduced by 50%.
引文
[1]Dougherty D E,Marryott R A.Optimal groundwater management:1.Simulated annealing[J].Water Resources Research,1991,27(10):2493-2508.
[2]Mahinthakumar G,Sayeed M.Hybrid genetic algorithm-local search methods for solving groundwater source identification inverse problems[J].Journal of Water Resources Planning and Management,2005,131(1):45-57.
[3]Glover F,Laguna M.Tabu Search[M].Boston,MA:Kluwer Academic Publisher,1997:125-151.
[4]Hassan A E,Bekhit H M,Chapman J B.Uncertainty assessment of a stochastic groundwater flow model using GLUE analysis[J].Journal of Hydrology,2008,362(1/2):89-109.
[5]Snodgrass M F,Kitanidis P K.A geostatistical approach to contaminant source identification[J].Water Resources Research,1997,33(4):537-546.
[6]Rasmussen J,Madsen H,Jensen K H,et al.Data assimilation in integrated hydrological modeling using ensemble Kalman filtering:evaluating the effect of ensemble size and localization on filter performance[J].Hydrology and Earth System Sciences,2015,12(2):2267-2304.
[7]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.
[8]Roberts C P,Casella G.Monte Carlo statistical methods(Second edition)[M].Berlin:Springer,2004:79-122.
[9]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.
[10]Hastings W K.Monte Carlo sampling methods using Markov chains and their applications[J].Biometrika,1970,57(1):97-109.
[11]Haario H,Saksman E,Tamminen J.An adaptive Metropolis algorithm[J].Bernoulli,2001,7(2):223-242.
[12]Tierney L,Mira A.Some adaptive Monte Carlo methods for bayesian inference[J].Statistics in Medicine,1999,18:2507-2515.
[13]Mira A.Ordering and improving the performance of Monte CarloMarkov Chains[J].Statistical Science,2002,16:340-350.
[14]Haario H,Laine M,Mira A.DRAM:Efficient adaptive MCMC[J].Statistics and Computing,2006,16(4):339-354.
[15]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.
[16]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.
[17]Vrugt J A.Markov chain Monte Carlo simulation using the DREAMsoftware package:Theory,consepts,and Matlab implementation[J].Enviromental Modeling&Software,2016,75:273-316.
[18]Czanner G,Sarma S V,Eden U T,et al.A Signal-to-Noise Ratio Estimator for Generalized Linear Model Systems[J].Lecture Notes in Engineering&Computer Science,2008,2171:1063-1069.
[19]Lindley D V.On a measure of the information provided by an experiment[J].The Annals of Mathematical Statistics,1956,27(4):986-1005.
[20]Huan X,Marzouk,Y M.Simulation-based optimal Bayesian experimental design for nonlinear systems[J].Journal of Computational Physics,2013,232(1):288-317.
[21]Shannon C E.A mathematical theory of communication[J].The Bell System Technical Journal,1948,27(3):379-423.
[22]张双圣,强静,刘汉湖,等.基于贝叶斯公式的地下水污染源识别[J].中国环境科学,2019,39(4):1568-1578.Zhang S,Qiang J,Liu H,et al.Identification of groundwater pollution sources based on Bayes’theorem[J].China Environmental Science,2019,39(3):1568-1578.
[23]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.
[24]Li J,Chen Y,Pepper,D.Radial basis function method for 1-D and2-D groundwater contaminant transport modeling[J].Computational Mechanics,2003,32(1):10-15.
[25]安永凯,卢文喜,董海彪,等.基于克里格法的地下水流数值模拟模型的替代模型研究[J].中国环境科学,2014,34(4):1073-1079.An Y,Lu W,Dong H,et al.Surrogate model of numerical simulation model of groundwater based on Kriging[J].China Environmental Science,2014,34(4):1073-1079.
[26]Lophaven S N,Nielsen H B,Sondergaard J.Dace:A MATLABKriging toolbox[R].Kongens Lyngby:Technical University of Denmark,Technical Report No.IMM-TR-2002-12.
[27]Grlman A G,Rubin D B.Inference from iterative simulation using multiplesequences[J].Statistical Science,1992,7:457-472.
[28]Hickernell,F.A.A generalized discrepancy and quadrature error bound[J].Mathematics of Computation of the American Mathematical Society,1998,67(221):299-322.
[29]初海波.基于自适应替代模型的DNAPLs污染地下水修复优化设计及其不确定性分析[D].长春:吉林大学,2015.Chu H.The optimization design and uncertainty analysis of DNAPLscontaminated groundwater remediation based on the adaptive surrogate model[D].Changchun:Jilin University,2015.
[30]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,2017.Zhang J.Bayesian monitoring design and parameter inversion for groundwater contaminant source identification[D].Hangzhou:Zhejiang University,2017.
[31]Lenhart L,Eckhardt K,Fohrer N,et al.Comparison of two different approaches of sensitivity analysis[J].Physics and Chemistry of the Earth,2002,27:645-654.
[2]Mahinthakumar G,Sayeed M.Hybrid genetic algorithm-local search methods for solving groundwater source identification inverse problems[J].Journal of Water Resources Planning and Management,2005,131(1):45-57.
[3]Glover F,Laguna M.Tabu Search[M].Boston,MA:Kluwer Academic Publisher,1997:125-151.
[4]Hassan A E,Bekhit H M,Chapman J B.Uncertainty assessment of a stochastic groundwater flow model using GLUE analysis[J].Journal of Hydrology,2008,362(1/2):89-109.
[5]Snodgrass M F,Kitanidis P K.A geostatistical approach to contaminant source identification[J].Water Resources Research,1997,33(4):537-546.
[6]Rasmussen J,Madsen H,Jensen K H,et al.Data assimilation in integrated hydrological modeling using ensemble Kalman filtering:evaluating the effect of ensemble size and localization on filter performance[J].Hydrology and Earth System Sciences,2015,12(2):2267-2304.
[7]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.
[8]Roberts C P,Casella G.Monte Carlo statistical methods(Second edition)[M].Berlin:Springer,2004:79-122.
[9]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.
[10]Hastings W K.Monte Carlo sampling methods using Markov chains and their applications[J].Biometrika,1970,57(1):97-109.
[11]Haario H,Saksman E,Tamminen J.An adaptive Metropolis algorithm[J].Bernoulli,2001,7(2):223-242.
[12]Tierney L,Mira A.Some adaptive Monte Carlo methods for bayesian inference[J].Statistics in Medicine,1999,18:2507-2515.
[13]Mira A.Ordering and improving the performance of Monte CarloMarkov Chains[J].Statistical Science,2002,16:340-350.
[14]Haario H,Laine M,Mira A.DRAM:Efficient adaptive MCMC[J].Statistics and Computing,2006,16(4):339-354.
[15]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.
[16]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.
[17]Vrugt J A.Markov chain Monte Carlo simulation using the DREAMsoftware package:Theory,consepts,and Matlab implementation[J].Enviromental Modeling&Software,2016,75:273-316.
[18]Czanner G,Sarma S V,Eden U T,et al.A Signal-to-Noise Ratio Estimator for Generalized Linear Model Systems[J].Lecture Notes in Engineering&Computer Science,2008,2171:1063-1069.
[19]Lindley D V.On a measure of the information provided by an experiment[J].The Annals of Mathematical Statistics,1956,27(4):986-1005.
[20]Huan X,Marzouk,Y M.Simulation-based optimal Bayesian experimental design for nonlinear systems[J].Journal of Computational Physics,2013,232(1):288-317.
[21]Shannon C E.A mathematical theory of communication[J].The Bell System Technical Journal,1948,27(3):379-423.
[22]张双圣,强静,刘汉湖,等.基于贝叶斯公式的地下水污染源识别[J].中国环境科学,2019,39(4):1568-1578.Zhang S,Qiang J,Liu H,et al.Identification of groundwater pollution sources based on Bayes’theorem[J].China Environmental Science,2019,39(3):1568-1578.
[23]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.
[24]Li J,Chen Y,Pepper,D.Radial basis function method for 1-D and2-D groundwater contaminant transport modeling[J].Computational Mechanics,2003,32(1):10-15.
[25]安永凯,卢文喜,董海彪,等.基于克里格法的地下水流数值模拟模型的替代模型研究[J].中国环境科学,2014,34(4):1073-1079.An Y,Lu W,Dong H,et al.Surrogate model of numerical simulation model of groundwater based on Kriging[J].China Environmental Science,2014,34(4):1073-1079.
[26]Lophaven S N,Nielsen H B,Sondergaard J.Dace:A MATLABKriging toolbox[R].Kongens Lyngby:Technical University of Denmark,Technical Report No.IMM-TR-2002-12.
[27]Grlman A G,Rubin D B.Inference from iterative simulation using multiplesequences[J].Statistical Science,1992,7:457-472.
[28]Hickernell,F.A.A generalized discrepancy and quadrature error bound[J].Mathematics of Computation of the American Mathematical Society,1998,67(221):299-322.
[29]初海波.基于自适应替代模型的DNAPLs污染地下水修复优化设计及其不确定性分析[D].长春:吉林大学,2015.Chu H.The optimization design and uncertainty analysis of DNAPLscontaminated groundwater remediation based on the adaptive surrogate model[D].Changchun:Jilin University,2015.
[30]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,2017.Zhang J.Bayesian monitoring design and parameter inversion for groundwater contaminant source identification[D].Hangzhou:Zhejiang University,2017.
[31]Lenhart L,Eckhardt K,Fohrer N,et al.Comparison of two different approaches of sensitivity analysis[J].Physics and Chemistry of the Earth,2002,27:645-654.
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