Bayesian Updating Method of Excavation Considering Various Uncertainties and Stage Correlation
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摘要:
基坑开挖响应预测模型的准确性受多种不确定性的影响，贝叶斯理论可以有效融合先验信息和观测数据，为降低土体参数不确定性和提高变形预测准确度提供了新途径. 然而，传统的贝叶斯更新方法对不确定性的考虑有限，因此本文提出了一种能够考虑土体参数、模型偏差、观测误差不确定性以及阶段相关性的贝叶斯更新方法. 通过两个实际案例的验证结果标明：所提方法能够有效降低模型参数的不确定性，提高模型对于不同土体类型基坑开挖响应预测的准确度.
Abstract:The accuracy of excavation response prediction models is generally influenced by various uncertainties, including those related to soil parameters, model uncertainties, measurement errors. Bayesian methods provide a novel way to reduce and/or quantify these uncertainties, and is a natural framework for improving model predictions by systematically integrating prior knowledge with observational data. However, existing Bayesian updating methods typically addressed the uncertainties with soil parameters or/and model biases, while the measurement errors are ignored. Besides, correlations between different excavation stages are also overlooked for mathematical convenience. These simplifications may lead to unreliable predictions in practice. In this study, a novel Bayesian updating method is proposed, which simultaneously incorporates uncertainties in soil parameters, model bias, observational errors, and stage correlations. Two case studies are used to illustrate and validate the method. The results demonstrate that the proposed approach significantly enhances the accuracy of semi-empirical models in predicting excavation responses across different soil types.
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图 2 阶段二参数后验采样散点图
a.土体参数；b.模型偏差系数；c.观测误差标准差
Fig. 2. Posteriori sampling scatterplots of parameters in stage 2
图 3 不同参数阶段二后验与先验分布的区别
a. 归一化不排水抗剪强度su/σ'v；b. 归一化初始杨氏模量Ei/σ'v；c.水平变形模型偏差系数αh；d. 竖向沉降模型偏差系数αv；e. 水平变形观测误差标准差σε，h；f. 沉降观测误差标准差σε，v
Fig. 3. The difference between posterior distributions in stage 2 and priors distributions of different parameters
图 4 参数的先验和后验分布
a.归一化不排水抗剪强度su/σ'v；b.归一化初始杨氏模量Ei/σ'v；c.水平变形模型偏差系数αh；d.竖向沉降模型偏差系数αv；e.水平变形观测误差标准差σε，h；f.竖向沉降观测误差标准差σε，v
Fig. 4. Prior and Posterior distribution of parameters
图 5 不同阶段最大墙体变形与地表沉降预测值与观测值
a. 地连墙最大水平位移；b.地表竖向最大沉降
Fig. 5. The predictions of the maximum wall deflections and ground settlement versus the observations at various stages
图 6 波动范围对最终阶段开挖响应预测的影响
a.地连墙最大水平位移；b.地表竖向最大沉降
Fig. 6. Influence of the scale of fluctuation on the predicted excavation response at the final excavation stage
图 7 不同方法预测最终阶段开挖响应结果对比
a.地连墙最大水平位移; b. 地表竖向最大沉降
Fig. 7. Comparison of results of different methods for predicting final stage excavation response
图 9 参数的先验和后验分布
a. 归一化不排水抗剪强度su/σ'v；b.归一化初始杨氏模量Ei/σ'v；c. 水平变形模型偏差系数αh；d.竖向沉降模型偏差系数αv；e. 水平变形观测误差标准差σε，h；f.竖向沉降观测误差标准差σε，v
Fig. 9. Prior and Posterior distribution of parameters
图 10 不同阶段最大墙体变形与地表沉降预测值与观测值
a. 地连墙最大水平位移；b. 地表最大竖向沉降
Fig. 10. The predictions of the maximum wall deflections and ground settlement versus the observations at various stages
图 11 不同方法预测最终阶段开挖响应结果对比
a. 地连墙最大水平位移; b. 地表最大竖向沉降
Fig. 11. Comparison of results of different methods for predicting final stage excavation response
表 1 X转换函数多项式系数
Table 1. The coefficients in transformation function
变量 适用范围 b1 b2 b3 He 0~30 -0.4 24 -50 ln(EI/γwhavg4) ≥0 11.5 -295 2 000 B/2 0≤B≤100 -0.04 4 90 su/σ'v 0.2~0.4 3 225 -2 882 730 Ei/σ'v 200~1 200 0.000 41 -1 500 
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表 2 案例一不同阶段确定性参数与观测数据
Table 2. Deterministic parameters and field observations at different stages in Example 1
参数 开挖阶段 2 3 4 He(m) 8.5 13.5 18 B(m) 22.5 22.5 22.5 ln(EI/γwhavg4) 4.52 4.88 4.88 ∑Hi/Hwall 1.0 1.0 1.0 β(m) 34 34 34 yh(mm) 2.6 5.6 7.1 yv(mm) 6.5 7.5 8.0
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表 3 案例一先验分布
Table 3. Prior distribution in Example 1
参数 分布类型 均值 变异系数 su/σ'v 正态分布 0.25 0.20 Ei/σ'v 正态分布 500 0.20 αh 截断正态分布 1.00 0.25 αv 截断正态分布 1.00 0.34 σɛ, h 对数正态分布 1.35 0.35 σɛ, v 对数正态分布 1.35 0.35
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表 4 案例二不同阶段确定性参数与观测数据
Table 4. Deterministic parameters and field observations at various stages of Example 2
参数 开挖阶段 3 4 5 6 7 He(m) 6.9 10.2 13.2 16.2 18.5 B(m) 33.4 33.4 33.4 33.4 33.4 ln(EI/γwhavg4) 7.47 7.62 7.28 7.22 7.19 ∑Hi/Hwall 0.87 0.87 0.87 0.87 0.87 β(m) 31 31 31 31 31 yh(mm) 12 25 31 40 47 yv(mm) 25 42 49 59 62
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表 5 案例二先验分布
Table 5. Prior distribution in Example 2
参数 分布类型 均值 变异系数 su/σ'v 正态分布 0.25 0.16 Ei/σ'v 正态分布 500 0.16 αh 截断正态分布 1 0.25 αv 截断正态分布 1 0.34 σɛ, h 对数正态分布 1.35 0.20 σɛ, v 对数正态分布 1.35 0.20
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