Bayesian modeling and characterization of underwater explosion shock wave loads with parameter uncertainty
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摘要: 水下爆炸冲击波荷载具有显著的变异性和不确定性，为克服经典确定性经验模型忽略这种不确定性而导致预测偏差的问题，通过收集682组水下爆炸试验数据，对压力峰值pm、时间常数θ、冲量I及冲击波比能密度es关键荷载模型进行模型参数和模型误差的不确定性分析，并在Cole经验模型框架下构建水下爆炸冲击波荷载的Bayesian概率模型，采用Bayesian推断方法对模型参数进行更新和校准，实现爆炸冲击波荷载的概率化表征。结果表明：Cole模型计算参数变异系数介于0.03~0.48之间，模型误差变异系数介于0.19~0.38之间，其中仅压力峰值模型误差近似服从Normal分布，时间常数、冲量及比能密度模型误差呈明显偏态分布，且模型误差随比例爆距的增大逐渐趋于稳定；在有限试验样本条件下，Bayesian概率模型能够显著提升参数估计精度，有效降低模型不确定性，实现模型精度与试验成本之间的合理平衡。研究表明，所建水下爆炸冲击波荷载Bayesian概率模型能够合理描述荷载的不确定性特征，为水下结构抗爆可靠性设计提供考虑荷载变异性的随机输入，并为工程风险评估与概率分析提供更全面的依据。
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关键词:
- 水下爆炸 /
- 经验公式 /
- 不确定性量化 /
- Bayesian推断 /
- 概率模型
Abstract: The shock wave load generated by underwater explosions exhibits significant variability and uncertainty. To address the prediction bias caused by classical deterministic empirical models that ignore this uncertainty, an uncertainty analysis of both model parameters and model errors was conducted for key load model parameters—peak pressure pm, time constant θ, impulse I, and shock wave specific energy density es, based on 682 sets of underwater explosion test data. Within the framework of the empirical model Cole, a Bayesian probabilistic model for underwater explosion shock wave loads was developed. Bayesian inference methods were employed to update and calibrate the model parameters, enabling a probabilistic characterization of the explosion shock wave load. The results show that the coefficient of variation for the calculated parameters of the model Cole ranges from 0.03 to 0.48, while the coefficient of variation for model errors lies between 0.19 and 0.38. Among these, only the modelling error for peak pressure approximately follows a normal distribution. In contrast the modelling errors for the time constant, impulse, and specific energy density exhibit distinctly skewed distributions. Moreover, the model errors gradually stabilize as the scaled distance increases. Under the condition of limited experimental samples, the Bayesian probabilistic model significantly improves parameter estimation accuracy, effectively reduces model uncertainty, and achieves a reasonable balance between model precision and experimental cost. The analysis demonstrates that the developed Bayesian probabilistic model for underwater explosion shock wave loads can reasonably characterize the uncertainty of the loads. It provides stochastic inputs that explicitly account for load variability for the reliability-based blast-resistant design of underwater structures, and offers a more comprehensive basis for engineering risk assessment and probabilistic analysis. -
图 2 水下爆炸荷载特征参数拟合结果
Figure 2. Fitting results for underwater explosion loads parameters
图 3 荷载表征参数的柱状图及对应概率分布
Figure 3. Histograms and probability distribution of load characterization parameters
图 4 水下爆炸荷载参数试验值与Cole经验模型值对比
Figure 4. Comparison of experimental results and predicted ones by the Cole empirical model for parameters of underwater explosion load
图 5 不同比例爆距模型误差均值及变异系数
Figure 5. Mean errors and coefficients of variation at different scaled distances
图 6 Bayesian概率模型参数后验密度分布
Figure 6. Posterior distributions of parameters in Bayesian probabilistic model
表 1 试验工况及测试结果统计
Table 1. Experimental conditions and statistical results
变量 符号 单位 总数 均值 标准差 最小值 中位数 最大值 变异系数 装药质量 me g 682 990.25 1602.84 0.05 200 6000 1.62 装药密度 ρe g/cm3 639 1.62 0.26 0.60 1.65 2.10 0.16 爆炸深度 D m 614 3.94 2.54 0.10 4 12 0.64 比例爆距 Z m/kg1/3 682 4.26 3.75 0.32 3.25 29.55 0.88 时间常数 θ μs 593 95.13 83.54 5.50 71.64 608.70 0.88 压力峰值 pm MPa 677 16.14 12.77 1.02 13 138.70 0.79 冲量 I Pa·s 593 1652.72 1852.73 28.22 788.97 9665 1.12 比能密度 es kJ/m2 553 14.07 32.54 0.03 3.59 392.88 2.31 注：总数指该参数有效数据组的数量；变异系数=标准差/均值，用于衡量数据的相对离散程度。 
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表 2 水下爆炸荷载特征参数拟合结果
Table 2. Fitting for underwater explosion loads parameters
模型工况 模型参数 kp αp kθ αθ kI αI $ {k}_{{{e}_{\text{s}}}} $ $ {\alpha }_{{{e}_{\text{s}}}} $ 试验拟合结果 49.64 1.10 102.54 −0.23 5546.38 0.91 82.09 1.98 TNT炸药模型 52.40 1.13 84 −0.23 5760 0.89 84.40 2.04
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表 3 水下爆炸荷载Cole经验公式参数取值统计
Table 3. Statistical parameters of Cole empirical formula
模型参数 总数 均值 标准差 最小值 中位数 最大值 变异系数 kp 75 51.46 12.88 17.36 53.30 74.40 0.25 αp 75 1.07 0.16 0.55 1.11 1.37 0.15 kθ 34 103.31 21.57 44.11 99.96 154.00 0.21 αθ 34 −0.22 0.10 −0.72 −0.22 −0.10 0.48 kI 23 6880.70 1322.61 4910 6455 9575 0.19 αI 23 0.95 0.07 0.80 0.93 1.09 0.07 $ {k}_{{{e}_{\text{s}}}} $ 22 100.10 17.79 69.57 105.51 128.25 0.18 $ {\alpha }_{{{e}_{\text{s}}}} $ 22 2.06 0.06 1.97 2.06 2.26 0.03
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表 4 各荷载模型参数Anderson-Darling拟合优度检验结果
Table 4. Anderson-Darling goodness-of-fit test results for load parameters
分布函数 模型参数 kp αp kθ αθ kI αI $ {k}_{{{e}_{\text{s}}}} $ $ {\alpha }_{{{e}_{\text{s}}}} $ Normal 3.37 6.56 0.98 2.58 1.15 0.37 0.43 0.76 Lognormal 6.91 9.40 0.93 0.81 0.83 0.36 0.49 0.68 Weibull 2.66 3.56 1.22 2.44 1.32 0.54 0.42 2.02 Gamma 5.65 8.43 0.85 1.07 0.96 0.36 0.50 0.67
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表 5 水下爆炸荷载Cole经验模型评估结果
Table 5. Evaluation of Cole empirical model for underwater explosion load
θ pm I es εRMSE/μs R2 βCov εRMSE/MPa R2 βCov εRMSE/(Pa·s) R2 βCov εRMSE/(kJ·m−2) R2 βCov 41.13 0.75 0.42 5.37 0.84 0.22 504.92 0.92 0.57 6.86 0.95 1.03
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表 6 Cole经验模型误差统计描述
Table 6. Statistical of Cole empirical model errors
模型误差 总数 均值 标准差 变异系数 最小值 中位数 最大值 εME(θ) 593 1.08 0.40 0.38 0.29 0.99 2.71 εME(pm) 677 1.02 0.19 0.19 0.46 1.02 1.55 εME(I) 593 1.06 0.33 0.31 0.20 1.06 2.12 εME(es) 553 1.07 0.36 0.34 0.11 1.02 2.49
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表 7 Cole经验模型误差Anderson-Darling拟合优度检验结果
Table 7. Anderson-Darling goodness-of-fit test for Cole empirical model errors
分布函数 模型误差 εME(θ) εME(pm) εME(I) εME(es) Normal 10.42 1.59 2.39 3.20 Lognormal 1.63 3.55 8.32 5.63 Weibull 9.09 5.54 2.01 4.37 Gamma 2.76 2.24 4.84 2.39
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表 8 不同比例爆距下各荷载表征量模型误差的Anderson-Darling拟合优度检验结果
Table 8. Anderson-Darling goodness-of-fit test for load parameter errors at different scaled distances
模型误差 Z/(m·kg-1/3) 统计结果 分布函数 区间 中位数 均值 变异系数 Normal Lognormal Weibull Gamma εME(θ) 0.32～1.71 1.21 1.13 0.31 3.31 1.67 3.36 2.17 1.74～2.32 2.32 0.95 0.50 5.37 1.14 3.68 2.00 2.34～3.00 2.71 1.07 0.37 7.23 3.61 6.72 4.71 3.02～3.50 3.25 1.11 0.33 1.19 0.54 1.12 0.62 3.51～4.10 3.90 1.15 0.32 2.01 1.27 1.90 1.38 4.11～5.27 4.60 0.96 0.36 0.46 0.67 0.60 0.38 5.27～6.90 6.10 1.22 0.34 0.64 1.85 0.68 1.36 6.91～29.55 9.50 1.08 0.36 0.98 0.99 1.14 0.81 εME(pm) 0.32～1.71 1.21 1.00 0.18 0.46 0.68 0.64 0.58 1.74～2.32 2.32 1.02 0.15 0.60 1.53 0.77 1.10 2.34～3.00 2.71 1.06 0.19 1.09 2.94 0.69 2.17 3.02～3.50 3.25 1.03 0.23 1.96 0.73 2.53 1.05 3.51～4.10 3.90 1.03 0.16 0.70 0.62 1.70 0.58 4.11～5.27 4.60 1.00 0.17 1.65 2.81 0.98 2.36 5.27～6.90 6.10 0.93 0.14 0.70 0.98 1.19 0.82 6.91～29.55 9.50 1.01 0.20 0.35 0.48 0.72 0.36 εME(I) 0.32～1.71 1.21 1.20 0.18 0.68 1.00 0.55 0.90 1.74～2.32 2.32 0.93 0.42 1.13 0.80 0.67 0.52 2.34～3.00 2.71 1.10 0.23 0.50 0.74 0.51 0.58 3.02～3.50 3.25 1.02 0.25 1.13 0.75 1.37 0.75 3.51～4.10 3.90 1.12 0.28 1.49 1.48 1.47 1.41 4.11～5.27 4.60 0.93 0.33 0.35 1.60 0.40 0.89 5.27～6.90 6.10 1.11 0.31 2.32 3.25 2.52 2.93 6.91～29.55 9.50 1.13 0.36 0.64 1.19 0.60 0.88 εME(es) 0.32～1.71 1.21 1.15 0.21 0.31 0.41 0.42 0.33 1.74～2.32 2.32 1.04 0.48 2.19 1.78 1.45 1.09 2.34～3.00 2.71 1.10 0.28 0.62 0.46 0.73 0.33 3.02～3.50 3.25 1.02 0.33 0.32 1.34 0.34 0.52 3.51～4.10 3.90 1.12 0.26 0.28 0.18 0.46 0.13 4.11～5.27 4.60 0.99 0.34 0.18 1.36 0.22 0.67 5.27～6.90 6.10 0.97 0.25 0.31 0.75 0.35 0.53 6.91～29.55 9.50 1.12 0.36 2.72 1.45 2.39 1.85
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表 9 荷载表征参数的均值与标准差
Table 9. Mean values and standard deviations of load parameters
荷载参数 样本量(占比) k α σ 均值 标准差 均值 标准差 均值 标准差 θ 50(8.43%) 81.05 6.02 −0.32 0.029 32.03 2.87 100(16.86%) 113.4 8.55 −0.24 0.035 57.67 3.84 200(33.73%) 98 5.69 −0.27 0.03 54.35 2.60 400(67.45%) 105.4 4.38 −0.22 0.026 52.02 1.80 593(100%) 105.8 3.48 −0.26 0.022 53.93 1.53 pm 50(7.39%) 62.14 1.27 1.18 0.036 2.64 0.26 100(14.77%) 61.63 1.17 1.22 0.027 2.60 0.18 200(29.54%) 56.09 0.94 1.17 0.024 3.47 0.17 400(59.08%) 45.98 0.40 0.95 0.0095 3.91 0.14 677(100%) 46.28 0.33 0.97 0.0078 3.70 0.097 I 50(8.43%) 4518 208.2 0.75 0.055 489.80 40.42 100(16.86%) 4653 181.8 0.69 0.038 490.80 30.99 200(33.73%) 6166 177.2 0.97 0.035 653.70 30.85 400(67.45%) 6141 66.64 0.99 0.012 640.60 21.97 593(100%) 6209 65.01 0.98 0.012 719.4 20.04 es 50(9.04%) 96.85 5.32 1.96 0.18 9.22 0.87 100(18.08%) 97.15 3.64 1.99 0.096 6.83 0.47 200(36.17%) 94.40 1.86 2.05 0.051 5.85 0.29 400(72.33%) 87.52 1.12 1.96 0.013 7.71 0.27 553(100%) 88.97 0.92 1.95 0.011 8.06 0.24
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表 10 荷载表征参数模型评估
Table 10. Model evaluation of load parameters
荷载参数 样本量(占比) 后验模型评估指标 先验模型评估指标 εRMSE R2 βCov εRMSE R2 βCov θ 50(8.43%) 43.78 μs 0.76 0.39 41.13 μs 0.75 0.42 100(16.86%) 39.84 μs 0.80 0.52 200(33.73%) 39.68 μs 0.81 0.43 400(67.45%) 39.84 μs 0.80 0.44 593(100%) 39.11 μs 0.81 0.47 pm 50(7.39%) 8.50MPa 0.70 0.27 5.37 MPa 0.84 0.22 100(14.77%) 8.73 MPa 0.69 0.24 200(29.54%) 6.52 MPa 0.83 0.22 400(59.08%) 3.72 MPa 0.94 0.28 677(100%) 3.70 MPa 0.94 0.26 I 50(8.43%) 781.30 Pa·s 0.82 0.53 504.92 Pa·s 0.92 0.57 100(16.86%) 740.21 Pa·s 0.84 0.57 200(33.73%) 518.62 Pa·s 0.92 0.53 400(67.45%) 525.94 Pa·s 0.92 0.53 593(100%) 516.72 Pa·s 0.92 0.53 es 50(9.04%) 6.33 kJ/m2 0.96 0.81 6.86 kJ/m2 0.95 1.03 100(18.08%) 6.80 kJ/m2 0.96 0.77 200(36.17%) 7.42 kJ/m2 0.95 0.68 400(72.33%) 6.07 kJ/m2 0.96 0.70 553(100%) 5.94 kJ/m2 0.97 0.73
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