多维最大熵模型及其在海岸和海洋工程中的应用研究
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摘要
海岸及海洋工程所处的环境条件非常恶劣，它们经常遭受台风、寒潮等严重天气过程的影响。海堤是防御海洋灾害的主要建筑物，我国沿海地区经济发达，一旦遭遇台风而导致海堤损毁，将产生难以估量的经济损失和人员伤亡；海洋平台造价昂贵，灾害性的天气过程将导致平台无法正常工作，严重时甚至会使平台倾覆、石油泄漏，污染海洋环境、造成生态破坏。影响海岸及海洋工程的环境要素主要有风、浪、潮、流、海冰、海雾等，如何有效地构造其设计标准，对于工程结构的安全性和可靠性非常重要。传统的设计标准采用单因素方法，将每种海洋环境要素视为独立变量，进行单独设计。这种方法过于保守，无法反映实际的海洋环境条件。因而有必要进行多维环境要素联合设计标准的研究，这就需要有足够准确的多元概率分布模型作为理论支撑。目前建立的模型类型单一，有时无法反映实际海况，拟合时误差较大。
本文在最大熵原理的基础上，推导了二维及多维最大熵模型，将其应用于海堤和海洋平台等的多维环境荷载联合设计中，并将其引入多维复合极值理论、结构可靠度分析以及风险分析中。本文的主要工作如下。
一维最大熵分布函数可以涵盖海洋环境设计的多种分布，因而可以避免线型选择的问题。本文总结并提出了一维最大熵分布函数的7种参数估计方法，即三参数矩法、四参数矩法、经验适线法、最大似然法、概率权重法、L-矩法和粒子群算法；对这些参数估计方法进行了实例验证和Monte-Carlo模拟，结果表明最大似然法和经验适线法的拟合效果较优，因而推荐使用。
由于海洋环境设计参数的估计误差较大，因而有必要引入置信区间的概念。本文提出了5种设计重现值的区间估计方法，即Woodruff法、最大似然法、样本分位数渐近法、顺序统计量法和符号检验法。根据数值模拟和实例验证得出，对一维最大熵分布函数，重现值区间估计的参数方法优于非参数方法；而在参数方法中，推荐使用最大似然法区间估计。
鉴于一维最大熵分布函数在应用中的优越性，类似对二维情形，给出适用于海洋环境要素的约束组合，建立了二维最大熵分布函数，并推导出其参数的矩估计方法。基于二元Copula函数，推导出边缘分布均服从最大熵分布函数的各种二维最大熵模型，并将这一类Copula模型统一在最大熵原理之下，实现了其与二维最大熵分布函数的统一。利用二维最大熵模型，进行了渤海年极值波高和伴随风速的联合概率设计，以及营口、葫芦岛海冰的同现概率分析。计算结果表明，二维最大熵分布模型对数据的拟合优度良好。
针对当前多维概率模型大多限于低维（二维或三维），很难向更高维推广的缺点，根据多维联合信息熵和最大熵原理，建立了一般的多维最大熵模型、矩约束下的多维最大熵模型，以及适合海洋环境约束的三维最大熵分布函数；鉴于建立的三维最大熵分布函数难于计算，利用多元Copula函数构造了边缘均服从一维最大熵分布函数的多维最大熵模型，并将其统一于最大熵原理之下。利用黄海某岛25年的年极值波高、周期、增水数据，根据三维最大熵模型，提出了一种海堤顶高程和越浪率的联合设计方法。
针对灾害性天气过程中多维环境要素的两组样本（过程中同为极值的要素及荷载最大时伴随的环境要素），提出了两种新型的多维复合极值模型，进而获得多维复合最大熵模型。利用这些模型对青岛风暴潮的强度进行了分级，对台风过程下导管架平台的环境条件设计进行了计算。结果表明，多维复合最大熵模型描述台风/寒潮等灾害性天气过程的统计分布是合适的；并且，其中最大荷载对应环境要素组合下的多维复合极值分布对应的设计值更为安全。
将多维最大熵模型引入静态结构可靠度计算中，结合直接积分法，使得结构可靠度计算的全概率法得以实现。鉴于静态可靠度计算无法考虑结构与荷载随时间的变化，无法真正保证结构的耐久性和安全性，因而对结构的时变可靠度进行了相应讨论，给出了相应服役期内时变可靠度的计算公式。最终，结合渤海某平台，就较为简单的情形进行了工程验证。
利用标值点过程理论，将灾害性天气过程视为随机点事件，将过程中各种环境荷载作为该点事件的标值，建立了灾害性天气过程的Poisson多元标值点过程模型。将多维最大熵分布作为多维标值的联合分布，利用以上点过程模型，推导出了海洋平台的综合风险率，为其安全防护提供了参考。
The surrounding environmental conditions of marine and offshore structureswhich are influenced frequently by severe synoptic processes including typhoon andcold wave are very harsh. Sea dikes play an extremely important role in marinedisaster prevention. Once they are damaged by typhoon, the economic losses andcasualties are not overestimated because of high developed economy and populationdensity along the China’s coastline. In addition, disastrous synoptic processes severelyimpact the regular work of offshore platforms with high cost. At worst, these poorconditions would result in platforms overturning and oil spill which cause pollution tothe ocean circumstance and destruction of the ecological environment. Actually, theprimary environmental elements that affect the coastal and ocean structures includewind, wave, tide, current, sea ice, sea fog, and so forth. The establishment of jointdesign criteria of these loads is crucial for the safety and reliability of marinestructures. However, the traditional design standard is too conservative to reflect thereal ocean circumstance, since it regards all the environmental elements asindependent variables. Therefore, it is necessary to study the joint design criteria ofmultivariate environmental elements which demands proper multivariate probabilitydistributions. While, there are several shortages for existing models, such as singletype and large errors in data fitting.
In this paper, several bivariate and multivariate maximum entropy modelsdeduced based on maximum entropy theory are applied in the joint design ofmultivariate environmental loads on sea dikes and ocean platforms. Also, thesemodels are introduced into multivariate compound extreme value theory, reliabilityanalysis of structure and risk analysis. The main work of this paper is as follows.
One-dimensional maximum entropy distribution (OMED) function covers almostall the distributions adopted in ocean environmental design, so fitting the observationswith it can avoid the choice of fitting curves. This paper proposes or summarizes seven parameter estimation methods for OMED, including method of moment withthree parameters (MMT), method of moment with four parameters (MMF), empiricalcurve-fitting method (ECFM), maximum likelihood method (MLM), probabilityweighted method (PWM), L-moment method (LMM) and particle swarmoptimization method (PSOM). These methods are tested and validated using field dataand corresponding Monte-Carlo simulations. Results indicate that MLM and ECFMare both recommended due to their best fit to data.
To avoid large errors of point estimation for design values of marineenvironmental elements, the concept of confidence intervals is introduced in thispaper. It contains five interval estimation methods for design return values, such asWoodruff method, maximum likelihood method, sample quantile asymptotic method,order statistic method and sign test method. According to the results of simulation andcase studies, with respect to OMED, parameter methods, among which maximumlikelihood is recommended firstly, are always better than non-parameter methods.
Exactly as OMED, bivariate maximum entropy distribution (BMED) is deducedwith proper constraints for two-dimensional joint Shannon entropy. Meanwhile, themoment method for its parameters estimation is derived. Based on bivariate Copulafunctions, several types of BMED models with OMED as marginal distributions areconstructed. More importantly, they would be consistent with maximum entropytheory. These models are utilized to establish the joint probability design of waveheight and corresponding wind speed, analyze corresponding probability of sea icebetween Yingkou and Huludao. The conclusion is that BMED models can fit bivariatedata well.
Presently, most multivariate models are limited in two or three dimensions anddifficult to be generalized to higher dimensions. On the basis of multivariate jointinformation entropy and maximum entropy theory, this paper proposes generalmultivariate maximum entropy models (MMEDs), MMEDs with moment constraints,and trivariate maximum entropy distribution (TMED) function with proper constraintsfor marine environments. Since the calculation for TMED function is very hard, TMED models with OMED margins based on trivariate Copuls are constructed.Similarly it could coincide with maximum entropy theory. As an example, a jointdesign method for the crest level and overtopping rate of sea dikes is proposed usingthe data of25a annual extreme wave height, wave period and storm surge observatedin an island of Huanghai Sea.
This paper proposes two new multivariate compound extreme models fordifferent data samples (all extreme values in the same process, and concomitantvalues when the load is the largest in one process) in severe weather processes. Andthen multivariate compound maximum entropy models are easily derived. Applyingthese models, the classification criteria of storm intensity in Qingdao is establishedand the environmental element design values for a jacket platform in severe weatherprocesses are calculated.
Combined with direct integration method, MMEDs are introduced into staticstructure reliability analysis. Consequently, total probability method becomes feasible.However, time-varying condition of resistances and loads does not take into accountin this analytical procedure. In order to overcome this difficulty, time-variantreliability analysis is discussed, and several calculation equations of time-variantreliability in the tour of duty are deduced here.
In fact, severe synoptic processes could be treated as stochastic point eventsaccording to scale point process theory. Poisson multivariate scale point processmodel for severe synoptic processes could be established if considering the extremevalues of environmental elements in one process as the scale values of the point event.Adopting MMED to fit the joint distribution of multivariate scale values, the syntheticrisk of ocean platforms is derived, which might provide references for safetyprotection.
本文在最大熵原理的基础上，推导了二维及多维最大熵模型，将其应用于海堤和海洋平台等的多维环境荷载联合设计中，并将其引入多维复合极值理论、结构可靠度分析以及风险分析中。本文的主要工作如下。
一维最大熵分布函数可以涵盖海洋环境设计的多种分布，因而可以避免线型选择的问题。本文总结并提出了一维最大熵分布函数的7种参数估计方法，即三参数矩法、四参数矩法、经验适线法、最大似然法、概率权重法、L-矩法和粒子群算法；对这些参数估计方法进行了实例验证和Monte-Carlo模拟，结果表明最大似然法和经验适线法的拟合效果较优，因而推荐使用。
由于海洋环境设计参数的估计误差较大，因而有必要引入置信区间的概念。本文提出了5种设计重现值的区间估计方法，即Woodruff法、最大似然法、样本分位数渐近法、顺序统计量法和符号检验法。根据数值模拟和实例验证得出，对一维最大熵分布函数，重现值区间估计的参数方法优于非参数方法；而在参数方法中，推荐使用最大似然法区间估计。
鉴于一维最大熵分布函数在应用中的优越性，类似对二维情形，给出适用于海洋环境要素的约束组合，建立了二维最大熵分布函数，并推导出其参数的矩估计方法。基于二元Copula函数，推导出边缘分布均服从最大熵分布函数的各种二维最大熵模型，并将这一类Copula模型统一在最大熵原理之下，实现了其与二维最大熵分布函数的统一。利用二维最大熵模型，进行了渤海年极值波高和伴随风速的联合概率设计，以及营口、葫芦岛海冰的同现概率分析。计算结果表明，二维最大熵分布模型对数据的拟合优度良好。
针对当前多维概率模型大多限于低维（二维或三维），很难向更高维推广的缺点，根据多维联合信息熵和最大熵原理，建立了一般的多维最大熵模型、矩约束下的多维最大熵模型，以及适合海洋环境约束的三维最大熵分布函数；鉴于建立的三维最大熵分布函数难于计算，利用多元Copula函数构造了边缘均服从一维最大熵分布函数的多维最大熵模型，并将其统一于最大熵原理之下。利用黄海某岛25年的年极值波高、周期、增水数据，根据三维最大熵模型，提出了一种海堤顶高程和越浪率的联合设计方法。
针对灾害性天气过程中多维环境要素的两组样本（过程中同为极值的要素及荷载最大时伴随的环境要素），提出了两种新型的多维复合极值模型，进而获得多维复合最大熵模型。利用这些模型对青岛风暴潮的强度进行了分级，对台风过程下导管架平台的环境条件设计进行了计算。结果表明，多维复合最大熵模型描述台风/寒潮等灾害性天气过程的统计分布是合适的；并且，其中最大荷载对应环境要素组合下的多维复合极值分布对应的设计值更为安全。
将多维最大熵模型引入静态结构可靠度计算中，结合直接积分法，使得结构可靠度计算的全概率法得以实现。鉴于静态可靠度计算无法考虑结构与荷载随时间的变化，无法真正保证结构的耐久性和安全性，因而对结构的时变可靠度进行了相应讨论，给出了相应服役期内时变可靠度的计算公式。最终，结合渤海某平台，就较为简单的情形进行了工程验证。
利用标值点过程理论，将灾害性天气过程视为随机点事件，将过程中各种环境荷载作为该点事件的标值，建立了灾害性天气过程的Poisson多元标值点过程模型。将多维最大熵分布作为多维标值的联合分布，利用以上点过程模型，推导出了海洋平台的综合风险率，为其安全防护提供了参考。
The surrounding environmental conditions of marine and offshore structureswhich are influenced frequently by severe synoptic processes including typhoon andcold wave are very harsh. Sea dikes play an extremely important role in marinedisaster prevention. Once they are damaged by typhoon, the economic losses andcasualties are not overestimated because of high developed economy and populationdensity along the China’s coastline. In addition, disastrous synoptic processes severelyimpact the regular work of offshore platforms with high cost. At worst, these poorconditions would result in platforms overturning and oil spill which cause pollution tothe ocean circumstance and destruction of the ecological environment. Actually, theprimary environmental elements that affect the coastal and ocean structures includewind, wave, tide, current, sea ice, sea fog, and so forth. The establishment of jointdesign criteria of these loads is crucial for the safety and reliability of marinestructures. However, the traditional design standard is too conservative to reflect thereal ocean circumstance, since it regards all the environmental elements asindependent variables. Therefore, it is necessary to study the joint design criteria ofmultivariate environmental elements which demands proper multivariate probabilitydistributions. While, there are several shortages for existing models, such as singletype and large errors in data fitting.
In this paper, several bivariate and multivariate maximum entropy modelsdeduced based on maximum entropy theory are applied in the joint design ofmultivariate environmental loads on sea dikes and ocean platforms. Also, thesemodels are introduced into multivariate compound extreme value theory, reliabilityanalysis of structure and risk analysis. The main work of this paper is as follows.
One-dimensional maximum entropy distribution (OMED) function covers almostall the distributions adopted in ocean environmental design, so fitting the observationswith it can avoid the choice of fitting curves. This paper proposes or summarizes seven parameter estimation methods for OMED, including method of moment withthree parameters (MMT), method of moment with four parameters (MMF), empiricalcurve-fitting method (ECFM), maximum likelihood method (MLM), probabilityweighted method (PWM), L-moment method (LMM) and particle swarmoptimization method (PSOM). These methods are tested and validated using field dataand corresponding Monte-Carlo simulations. Results indicate that MLM and ECFMare both recommended due to their best fit to data.
To avoid large errors of point estimation for design values of marineenvironmental elements, the concept of confidence intervals is introduced in thispaper. It contains five interval estimation methods for design return values, such asWoodruff method, maximum likelihood method, sample quantile asymptotic method,order statistic method and sign test method. According to the results of simulation andcase studies, with respect to OMED, parameter methods, among which maximumlikelihood is recommended firstly, are always better than non-parameter methods.
Exactly as OMED, bivariate maximum entropy distribution (BMED) is deducedwith proper constraints for two-dimensional joint Shannon entropy. Meanwhile, themoment method for its parameters estimation is derived. Based on bivariate Copulafunctions, several types of BMED models with OMED as marginal distributions areconstructed. More importantly, they would be consistent with maximum entropytheory. These models are utilized to establish the joint probability design of waveheight and corresponding wind speed, analyze corresponding probability of sea icebetween Yingkou and Huludao. The conclusion is that BMED models can fit bivariatedata well.
Presently, most multivariate models are limited in two or three dimensions anddifficult to be generalized to higher dimensions. On the basis of multivariate jointinformation entropy and maximum entropy theory, this paper proposes generalmultivariate maximum entropy models (MMEDs), MMEDs with moment constraints,and trivariate maximum entropy distribution (TMED) function with proper constraintsfor marine environments. Since the calculation for TMED function is very hard, TMED models with OMED margins based on trivariate Copuls are constructed.Similarly it could coincide with maximum entropy theory. As an example, a jointdesign method for the crest level and overtopping rate of sea dikes is proposed usingthe data of25a annual extreme wave height, wave period and storm surge observatedin an island of Huanghai Sea.
This paper proposes two new multivariate compound extreme models fordifferent data samples (all extreme values in the same process, and concomitantvalues when the load is the largest in one process) in severe weather processes. Andthen multivariate compound maximum entropy models are easily derived. Applyingthese models, the classification criteria of storm intensity in Qingdao is establishedand the environmental element design values for a jacket platform in severe weatherprocesses are calculated.
Combined with direct integration method, MMEDs are introduced into staticstructure reliability analysis. Consequently, total probability method becomes feasible.However, time-varying condition of resistances and loads does not take into accountin this analytical procedure. In order to overcome this difficulty, time-variantreliability analysis is discussed, and several calculation equations of time-variantreliability in the tour of duty are deduced here.
In fact, severe synoptic processes could be treated as stochastic point eventsaccording to scale point process theory. Poisson multivariate scale point processmodel for severe synoptic processes could be established if considering the extremevalues of environmental elements in one process as the scale values of the point event.Adopting MMED to fit the joint distribution of multivariate scale values, the syntheticrisk of ocean platforms is derived, which might provide references for safetyprotection.
引文
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