Inference methods and uncertainty assessment of extreme wave heights
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摘要: 针对提升海洋工程中波高极值推算精度的需求，本文系统对比分析了年极值法（AM）与超阈值法（POT）的适用性及其不确定性。基于杭州湾两站点的多年再分析波浪数据，分别构建年极值序列与超阈值样本，并采用广义极值分布（GEV）和广义Pareto分布（GPD）进行建模；其中，POT法通过最小化尾部误差准则优化阈值选择。进一步结合Delta法与Bootstrap法，量化了模型参数与重现期水平波高估计值的置信区间。结果表明，对于高重现期水平，POT法给出的波高估计值更高，置信区间更窄，更适用于对极端事件敏感的工程设计场景；在不确定性分析方面，Bootstrap法较Delta法更能全面反映模型不确定性。本研究为波高极值建模建立了更为稳健的分析框架与参数推算依据。
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关键词:
- 波高极值 /
- 不确定性 /
- 超阈值法 /
- Bootstrap法
Abstract: To enhance the accuracy and reliability of extreme wave height inference in ocean engineering, this study systematically compares the applicability and uncertainty of the annual maxima (AM) method and the peak over threshold (POT) method. Utilizing reanalysis wave data from two representative sites in Hangzhou Bay, annual maxima series and POT samples were constructed. These were modeled using the generalized extreme value (GEV) distribution and the generalized Pareto distribution (GPD), respectively; for the POT method, threshold selection was optimized via a tail least squares error (TLSE) criterion. Confidence intervals for model parameters and return period level wave height estimates were further quantified using both the Delta method and the Bootstrap method. The results demonstrate that for high return periods, the POT method yields higher wave height estimates with narrower confidence intervals, rendering it more suitable for engineering design scenarios sensitive to extreme events. In uncertainty analysis, the Bootstrap method more comprehensively captures model uncertainty compared to the Delta method. This work establishes a more robust analytical framework and inferential basis for extreme wave height modeling.-
Key words:
- extreme wave height /
- uncertainty /
- peak over threshold method /
- Bootstrap method
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图 2 年极值和超阈值波高样本选取对比
Fig. 2 Comparison between the annual maximum model and peak-over-threshold model
图 7 波高估计值（100 a）和尾部误差随阈值变化图
Fig. 7 Variation of 100-year return period wave heights estimates and the TLSE with thresholds
图 8 Bootstrap法计算的平均尾部误差随阈值变化图
Fig. 8 The TLSEB by Bootstrap varies with the thresholds
图 10 不同方法推算的重现期波高对比
Fig. 10 Comparison of return period wave heights derived from different methods
图 11 不同置信区间不确定性对比（#3）
Fig. 11 Comparison of uncertainty of different confidence intervals (#3)
图 12 不同方法推算的重现期波高对比（#3）
Fig. 12 Comparison of return period wave heights derived from different methods (#3)
表 1 研究点信息
Tab. 1 Information of research stations
研究点 纬度/°N 经度/°E 水深/m 波高最大值/m 波高最小值/m 波高平均值/m #1 30.84 121.92 5.00 1.95 0.05 0.45 #2 30.73 121.99 5.00 2.35 0.07 0.55 #3 30.00 123.00 50.00 10.72 0.15 1.32 
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表 2 不同极值理论方法参数估计结果及重现期波高比较
Tab. 2 Comparison of parameter estimation and return period wave height using different extreme value theories
研究点 方法 参数 重现期波高/m μ/u ξ σ 2 a 5 a 10 a 20 a 50 a 100 a #1 AM法 1.297 −0.046 0.199 1.37 1.61 1.77 1.93 2.15 2.32 POT法 0.88 0.106 0.182 1.37 1.57 1.74 1.93 2.23 2.49 #2 AM法 1.576 −0.030 0.243 1.67 1.95 2.14 2.33 2.58 2.77 POT法 1.06 0.154 0.136 1.69 1.91 2.11 2.33 2.66 2.93
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表 3 不同B值对应的TLSEB（u = 0.88 m）
Tab. 3 TLSEB corresponding to different Bootstraps (u = 0.88 m)
研究点 参数 原始样本TLSE B值 50 100 200 500 1000 2 000 #1 TLSEB 0.145 0.228 0.264 0.240 0.241 0.242 0.242 ξ 0.182 0.173 0.176 0.180 0.178 0.180 0.180 σ 0.106 0.106 0.106 0.105 0.106 0.106 0.106 #2 TLSEB 0.263 0.386 0.405 0.397 0.392 0.395 0.396 ξ 0.154 0.162 0.151 0.153 0.155 0.153 0.153 σ 0.136 0.135 0.137 0.136 0.136 0.137 0.137
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表 4 GPD 参数置信区间比较
Tab. 4 Comparison of confidence intervals of GPD parameters
研究点 方法 ξ置信区间 σ置信区间 ξ置信区间宽度 σ置信区间宽度 #1 Bootstrap法 [0.101, 0.263] [0.094, 0.119] 0.162 0.025 Delta法 [0.090, 0.274] [0.093, 0.118] 0.184 0.025 #2 Bootstrap法 [0.081, 0.224] [0.125, 0.148] 0.143 0.023 Delta法 [0.068, 0.240] [0.121, 0.152] 0.172 0.031
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