An improved method of groundwater model structural uncertainty analysis
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摘要: 高斯过程回归（GPR）是一种基于贝叶斯理论的监督学习算法，在基于数据驱动（DDM）的模型结构不确定性分析中具有广泛应用。目前研究中通常假设物理参数和超参独立并进行联立识别，这会导致参数补偿。文章提出两步识别DDM量化模型结构误差，并通过2个地下水模型案例，分别在不考虑模型结构误差、考虑模型结构误差（联立识别DDM、两步识别DDM）的情况下，对比分析了参数识别和模型预测结果。结果表明，不考虑模型结构误差直接进行参数识别时，为补偿结构误差，物理参数会过度拟合，从而影响模型预测效果。基于DDM刻画模型结构偏差时，物理参数和超参的独立性假设会影响参数识别结果。提出的两步识别DDM法没有假设物理参数和超参独立，能够减少参数过度拟合效应，从而更准确刻画结构误差，有效提高了模型的预测性能。
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关键词:
- 模型结构不确定性 /
- 数据驱动 /
- 高斯过程回归 /
- 马尔科夫链蒙特卡洛模拟 /
- 溶质运移
Abstract: Gaussian Process Regression (GPR) is a supervised learning algorithm based on Bayesian theory, which is widely used in model structural uncertainty analysis based on data-driven method (DDM). In this study, it is usually assumed that the physical parameters and hyperparameters are independent and identified jointly, which will lead to parameter compensation. In this paper, a two-stage based DDM method is proposed to quantify the model structural errors, and two case studies are used to compare and analyze the results of parameter identification and model prediction with considering the model structural errors (joint calibration based DDM and two-stage based DDM) and without considering the model structural errors. The results show that when the parameters are identified directly without considering the model structural errors, the parameters will be overfitted and compensate the model structural errors, thereby affecting the model prediction performance. When considering the model structure deviation based on DDM, the independence assumption of physical parameters and hyperparameters will affect the parameter estimation results. The proposed two-stage based DDM method does not assume that the physical parameters and hyperparameters are independent, and can reduce parameter overfitting caused by the independence assumption of physical parameters and hyperparameters, portraying more accurate structural errors and effectively improving the model prediction performance. -
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图 2 模型识别和验证数据点的位置
Figure 2. Location of model observation wells during calibration and validation periods
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图 3 识别得到的模型参数和超参边缘后验分布
Figure 3. Identified marginal posterior distributions of model parameters and hyperparameters
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图 4 定流量非完整井流模型示意图
Figure 4. Diagrammatic representation of partially penetrating well pumping at a constant rate
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图 5 模型识别和验证数据点的位置
Figure 5. Location of model observation wells during calibration and validation periods
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图 6 识别得到的模型参数和超参边缘后验分布
Figure 6. Identified marginal posterior distributions of model parameters and hyperparameters
表 1 模型参数的先验分布
Table 1 Prior distributions of model parameters
参数 先验分布 Co /（mol·L−1） Uniform on [45.0,52.0] V/（cm·d−1） Uniform on [45.0,52.0] λ Gamma, k=5, θ=0. 2, on [0.1,0.8] σ Exponential, μ=0. 25, on [3.0,10.0] σδ Uniform on [0.1,0.5] 
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表 2 模型预测性能指标统计结果
Table 2 Statistics of model prediction performance
识别期 验证期 RMSE MAE MRE RMSE MAE MRE 不考虑结构误差 4.6056 4.1056 0.1974 8.9583 6.7699 0.2279 联立识别DDM 5.3165 4.9521 0.2225 8.2356 6.3889 0.2240 两步识别DDM 4.6500 4.2916 0.2094 7.7770 5.6255 0.2068
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表 3 模型参数的先验分布
Table 3 Prior distributions of model parameters
参数 先验分布 Krr/（m·d−1） Uniform on [8.0,14.0] M/d Uniform on [75.0,90.0] λ Gamma, k=5, θ=0. 2, on [0.2,0.6] σ Exponential, μ=0. 25, on [5.0,10.0] σδ Uniform on [0.05,0.5]
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表 4 模型预测性能指标统计结果
Table 4 Statistics of model prediction performance
识别期 验证期 RMSE MAE MRE RMSE MAE MRE 不考虑结构误差 5.2084 3.5591 0.2226 3.5764 3.1728 0.4488 联立识别DDM 8.1380 7.9843 0.6576 1.8912 1.5249 0.2253 两步识别DDM 8.0390 7.7821 0.6315 1.7184 1.3589 0.1954
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