基于贝叶斯公式的地下水污染源识别

将贝叶斯公式与地下水二维水质对流-扩散方程相耦合，建立依靠监测井监测值的地下水污染源参数（污染源强度M、排放位置（X₀，Y₀）和排放时刻T₀）反演模型.针对监测井监测值信息量不充分或者监测值与模型参数关联性较弱的问题，提出了一种基于贝叶斯公式与信息熵的监测井优化设计方法.构建一个污染物在承压含水层中瞬时排放的算例，在确定单井监测及监测次数条件下，以监测井位置D及监测频率Dt的优化为目标，分别进行模型参数后验分布信息熵最小的单目标监测方案优化，以及信息熵最小且监测耗时最短的多目标监测方案优化.依据优化后的监测方案采用延迟拒绝自适应Metropolis算法进行污染源参数反演识别.算例研究表明：在预设定单井监测，且监测次数为5次条件下，单目标优化后的监测方案为D=（830.2，199.8），△t=2.7，在此监测方案下，4个污染源参数M，X₀，Y₀，T₀的反演均值误差分别为19.5%、13.2%、3.4%、1.3%；多目标优化后的监测方案为D=（807.9，199.4），△t=1.2，在此监测方案下，4个污染源参数M，X₀，Y₀，T₀的反演均值误差分别为19.9%、13.4%、3.7%、4.2%.与基于单目标优化的监测方案的反演结果相比，基于多目标优化的监测方案条件下，污染源参数的反演均值误差虽分别增加了0.4%、0.2%、0.3%、2.9%，但监测时间却显著缩短了55.6%.

Identification of groundwater pollution sources based on Bayes’ theorem

Coupling Bayes’ Theorem with a two-dimensional (2D) groundwater solute advection-diffusion transport equation, it is possible to establish an inverse model based on monitoring well data to identify a set of contamination source parameters including source intensity (M), release location (X₀,Y₀) and release time (T₀). To address the issues of insufficient monitoring data from the wells or weak correlation between monitoring data and model parameters, a monitoring well design optimization approach was developed based on the Bayesian formula and information entropy. To demonstrate how the model works, an example with an instantaneous release of a contaminant in a confined groundwater aquifer was employed. Under the condition of single well monitoring and determined monitoring counts, with the target of optimization of monitoring location D and monitoring frequency Dt, both the single-objective monitoring scheme with the minimum information entropy of the model parameter posterior distribution and the multi-objective monitoring scheme with the smallest information entropy and the shortest monitoring time were optimized respectively. According to the optimized monitoring scheme, the delayed rejection adaptive Metropolis algorithm was used to identify the pollution source parameters. The case study results showed that under the condition of pre-set single well monitoring and 5monitoring times, the single-objective optimized monitoring scheme was D=(830.2,199.8),△t=2.7. Under this monitoring scheme, the mean errors of inverse 4pollution source parameters M,X₀,Y₀,T₀ were 19.5%, 13.2%, 3.4%, and 1.3%, respectively. The multi-objective optimization monitoring scheme was D=(807.9,199.4),△t=1.2. Under the monitoring scheme, the mean errors of inverse 4parameters M,X₀,Y₀,T₀ were 19.9%, 13.4%, 3.7%, and 4.2%, respectively. Compared with the monitoring scheme based on the single-objective optimization, while the inversion mean error of the pollution source parameters based on the multi-objective optimization monitoring scheme increased by 0.4%, 0.2%, 0.3%, 2.9% respectively, the monitoring time has been significantly reduced by 55.6%.