Assessment of uncertainty in environmental contours due to parametric uncertainty in models of the dependence structure between metocean variables

A key factor for computing environmental contours is the appropriate modeling of the dependence structure among the environmental variables. It is known that all the information on the dependence structure of a set of random variables is contained in the copulas that define their multivariate probability distribution. Provided that copula parameters are estimated by means of statistical inference using observations, recordings, numerical or historical data, uncertainty is unavoidably introduced in their estimates. Parametric uncertainty in the copulas parameters then introduces uncertainty in the environmental contours. This study deals with the assessment of uncertainty in environmental contours due to parametric uncertainty in the copula models that define the dependence structure of the environmental variables. A point estimation approach is adopted to estimate the statistics of the uncertain coordinates of the environmental contours considering they are given in terms of inverse functions of conditional copulas. A case study is reported using copulas models estimated from storm hindcast data for the Gulf of Mexico. Uncertainty in environmental contours of significant wave height, peak period and wind speed is assessed. The accuracy of the point estimation of the mean and variance of the contour coordinates is validated based on Monte Carlo simulations. A parametric study shows the manner in which greater parametric uncertainty induces larger variability in the environmental contours. The influence of parametric uncertainty for different degrees of association is also analyzed. The results indicate that variability between contours considering parametric uncertainty can be meaningful.     

Environmental contours using copulas

A procedure is proposed for constructing environmental contours using copula theory. Copulas are functions that define the multivariate probability distribution of a random vector or a set of random variables, and, thus, also determine their dependence structure. Constructing environmental contours requires knowledge of the joint probability distribution of the environmental variables. In many practical applications, the available statistical data is used to estimate the marginal distributions and the linear correlation matrix, and then the Nataf distribution model is employed to obtain the multivariate probability distribution. It turns out that such an approach implies a particular model of dependence structure defined by a Gaussian copula, which might not always be the appropriate one. In this work, some classes of bivariate copulas are considered for modeling the dependence structure of the environmental variables. We examine measures of association, rank-based methods for estimation of copulas, goodness of fit tests for copulas, and copula selection criteria, and apply them to metocean data from hindcasts of tropical storms and extra-tropical events in the Gulf of Mexico. A formulation is proposed for expressing the variates that define the environmental contours as functions of copulas. It is then applied for computing environmental contours of significant wave height, peak spectral period and wind velocity using the estimated copula models.     

海洋重现期波高统一化计算方法

重现期波高是港口海岸及海洋工程设计中不可回避的一个重要设计参数,尤其对深水海港、海上平台、海底油气管道、沿海核电站等重大涉海工程设计具有巨大的经济价值和深远的社会效益。但是,现有重现期波高推算缺乏统一的计算方法,导致计算结果相差悬殊。研究重现期波高的统一化计算方法,分析重现期波高计算中存在的各种不确定因素,提出减少这些不确定因素的新方法,建立误差小、应用方便、方法统一的重现期波高计算方法。基于澳大利亚悉尼站的长期连续观测波浪数据,研究发现:广义帕累托函数(generalized Pareto distribution III,GPD-III)和威布尔(Weibull)是重现期波高计算的最佳候选极值分布函数,新推导的函数形状参数计算公式较好提高重现期波高的计算精度,极值波高数据的分析方法和样本大小是影响重现期波高计算精确度的两个重要因素,短期波浪资料和年极值法可能高估重现期波高值。逐个风暴的极值波高数据分析法及最佳候选极值分布函数GPD-III和Weibull建议应用于涉海工程设计的重现期波高推算。     

Modelling the long-term distribution of significant wave height with the Beta and Gamma models

The paper suggests modelling the long-term distribution of significant wave height with the Gamma, Beta of the first and second kind models. The three models are interrelated, flexible and cover the three different tail types of Extreme Value Theory. They can be used simultaneously as a means of assessing the uncertainty effects that result from choosing equally plausible models with different tail types. This procedure is intended for those applications that require the long-term distribution of significant wave height as input rather than the prediction of extreme values. The models are fitted to some significant wave data as an illustration. Details about maximum likelihood estimation are given in A.     

On the long-term response of marine structures

This paper addresses some important issues related to the estimation of long-term extreme responses of marine structures. Several convolution models to establish the long-term distribution of a marine structure response parameter are available in the literature. These methods are typically based either on all short-term peaks, all extreme short-term peaks or all short-term upcrossing rates. The main assumptions and simplifications of the five models most usually found in the literature are discussed in this paper. A linear single-degree-of-freedom (SDOF) system along with a bi-lognormal probability model for significant wave heights and zero-crossing wave periods have been used for numerical tests. An improved approach to efficiently evaluate the long-term convolution integrals is also proposed in this paper. It is shown that a combination of the Inverse First Order Reliability Method (IFORM) and an Importance Sampling Monte Carlo Simulation (ISMCS) approach can be used to obtain a very good result for the exact solution of long-term integrals.     

Joint modelling of wave spectral parameters for extreme sea states

Characterising the dependence between extremes of wave spectral parameters such as significant wave height (H_S) and spectral peak period (T_P) is important in understanding extreme ocean environments and in the design and assessment of marine structures. For example, it is known that mean values of wave periods tend to increase with increasing storm intensity. Here we seek to characterise joint dependence in a straightforward manner, accessible to the ocean engineering community, using a statistically sound approach.Many methods of multivariate extreme value analyses are based on models which assume implicitly that in some joint tail region each parameter is either independent of or asymptotically dependent on other parameters; yet in reality the dependence structure in general is neither of these. The underpinning assumption of multivariate regular variation restricts these methods to estimation of joint regions in which all parameters are extreme; but regions where only a subset of parameters are extreme can be equally important for design. The conditional approach of Heffernan and Tawn (2004), similar in spirit to that of Haver (1985) but with better theoretical foundation, overcomes these difficulties.We use the conditional approach to characterise the dependence structure of H_S and T_P. The key elements of the procedure are: (1) marginal modelling for all parameters, (2) transformation of data to a common standard Gumbel marginal form, (3) modelling dependence between data for extremes of pairs of parameters using a form of regression, (4) simulation of long return periods to estimate joint extremes. We demonstrate the approach in application to measured and hindcast data from the Northern North Sea, the Gulf of Mexico and the North West Shelf of Australia. We also illustrate the use of data re-sampling techniques such as bootstrapping to estimate the uncertainty in marginal and dependence models and accommodate this uncertainty in extreme quantile estimation.We discuss the current approach in the context of other approaches to multivariate extreme value estimation popular in the ocean engineering community.     

Non-stationary extreme value models to account for trends and shifts in the extreme wave climate due to climate change

The extreme values of wave climate data are of great interest in a number of different ocean engineering applications, including the design and operation of ships and offshore structures, marine energy generation, aquaculture and coastal installations. Typically, the return values of certain met-ocean parameters such as significant wave height are of particular importance. There exist many methods for estimating such return values, including the initial distribution approach, the block maxima approach and the peaks-over threshold approach. In a climate change perspective, projections of such return values to a future climate are of great importance for risk management and adaptation purposes. However, many approaches to extreme value modelling assume stationary conditions and it is not straightforward how to include non-stationarity of the extremes due to for example climate change. In this paper, various non-stationary GEV-models for significant wave height are developed that account for trends and shifts in the extreme wave climate due to climate change. These models are fitted to block maxima in a particular set of wave data obtained for a historical control period and two future projections for a future period corresponding to different emission scenarios. These models are used to investigate whether there are trends in the data within each period that influence the extreme value analysis and need to be taken into account. Moreover, it will be investigated whether there are significant inter-period shifts or trends in the extreme wave climate from the historical period to the future periods. The results from this study suggest that the intra-period trends are not statistically significant and that it might be reasonable to ignore these in extreme value analyses within each period. However, when it comes to comparing the different data sets, i.e. the historical period and the future projections, statistical significant inter-period changes are detected. Hence, the accumulated effect of a climatic trend may not be negligible over longer time periods. Interestingly enough, such statistically significant shifts are not detected if stationary extreme value models are fitted to each period separately. Therefore, the non-stationary extreme value models with inter-period shifts in the parameters are proposed as an alternative for extreme value modelling in a climate change perspective, in situations where historical data and future projections are available.     

The distribution of zero-crossing wave heights and periods in a stationary sea state

Existing theoretical distributions of wave height and period do not reflect measured joint distributions from field data. A simulation methodology is introduced to retain the essential features of the theoretical background in Gaussian random noise but to avoid further compromising assumptions in the interpretation of height and period in the amplitude domain. A joint distribution can be associated directly with an empirical or measured variance spectrum. Spectral shape appears to dominate the detail of predicted joint distributions. There is generally a much sharper decay in probability levels at higher periods than is predicted by theoretical models. For Jonswap spectra, there is a dominant central ridge and a distinct bimodal structure in the joint distribution, features that are not evident in symmetric Gaussian spectral forms. The wave height distributions for Jonswap spectra differ little from the Rayleigh distribution, except at extreme wave heights where Rayleigh overpredicts. The period distributions are strongly sensitive to spectral shape. In the conditional distribution of periods, given the height, the asymptotic median period at extreme wave heights is significantly longer than the mean period for Jonswap spectra, but not for symmetric Gaussian forms.     

长期极值统计理论及其在海洋环境参数统计分析中的应用

海洋环境极值参数(如风速、流速、波高、周期等)在海洋工程设计中具有重要意义。利用次序统计和极值理论方面的较新研究成果，从理论上证明了多种统计分布中Weibull分布是最优的，使长期极值统计建立在一个更坚实的基础上；同时引入基于序列统计的最大似然估计方法。利用大量数据．对最小二乘估计方法和最大似然估计法进行对比分析，指出最大似然估计法是精确估计．而最小二乘估计方法是保守估计。     

基于WW3的南海北部有效波高时空变化及其极值重现期估算方法分析

本文基于第3代海浪模式WAVEWATCH Ⅲ (WW3）模拟的1996–2015年海浪后报数据,分析了南海北部有效波高及其极值的时空变化特征,并采用Pearson-Ⅲ和Gumbel两种极值分布方法对该区极值波高重现期进行了估算。结果表明,南海北部有效波高的季节变化和空间分布与季风风场基本一致,呈现秋冬高春夏低,并自吕宋海峡西侧向西南降低的特征,与ERA5再分析数据结果高度相似。有效波高极值（简称极值波高）的时空分布特征受时间分辨率强烈影响,采用极值数据的分辨率越高（如逐小时）,所展现的台风型波浪特征越显著。扣除季节变化信号后的有效波高和年极值波高均体现出较强的线性增高趋势,近20年升高的比例分别为7.7%和31.6%,值得警惕和关注。该区多年一遇极值波高存在若干个大值区,且与台风的路径、强度有直接联系,表明台风是引发该区域极端大浪的最主要机制。对比Pearson-Ⅲ和Gumbel极值分布估算结果发现：若极值波高较低,频率随极值波高升高缓慢降低,此时两种极值分布的估算都比较准确,差异极小,可忽略不计;但当研究时间范围内,某年极值波高远超其他年份时,Pearson-Ⅲ极值分布估算结果明显高...     

渤海和黄海北部波高与周期联合分析

分析波高与周期的联合分布特征对于海洋平台设计、海洋工程建筑等有着重要的意义。基于SWAN模型模拟的波浪后报数据对渤海和黄海北部1999~2018年的波浪特征进行了统计分析。分别对20年的波高和周期数据进行了统计分析,得到了研究区域20年有效波高和波周期的季平均值和最大值的区域分布特征。然后以散布图的形式刻画了整个区域20年波高和周期的联合分布特征。为了更深入地研究波高和周期的联合分布规律,选择了两个研究点A1和A2,A1在渤海内部相对近岸,A2在黄海北部深海区。统计结果表明,在A1和A2,波高与周期的联合分布特征较为相似,均呈现斜三角形的分布特征,然而大波高大周期的波浪却呈现不同的分布特征。最后,利用20年的波浪后报数据,在A1和A2点构建了有效波高和谱峰周期的联合概率模型,并采用IFORM法得到了50年、100年和200年重现周期的环境等值线,为研究海域海上结构物的可靠性设计提供了参考。     

Joint modelling of extreme ocean environments incorporating covariate effects

Characterising the joint distribution of extremes of ocean environmental variables such as significant wave height (H_S) and spectral peak period (T_P) is important for understanding extreme ocean environments and in the design and assessment of marine and coastal structures. Many applications of multivariate extreme value analysis adopt models that assume a particular form of extremal dependence between variables without justification. Models are also typically restricted to joint regions in which all variables are extreme, but regions where only a subset of variables is extreme can be equally important for design. The conditional extremes model of Heffernan and Tawn (2004) provides one approach to overcoming these difficulties.Here, we extend the conditional extremes model to incorporate covariate effects in all of threshold selection, marginal and dependence modelling. Quantile regression is used to select appropriate covariate-dependent extreme value thresholds. Marginal and dependence modelling of extremes is performed within a penalised likelihood framework, using a Fourier parameterisation of marginal and dependence model parameters, with cross-validation to estimate suitable model parameter roughness, and bootstrapping to estimate parameter uncertainty with respect to covariate.We illustrate the approach in application to joint modelling of storm peak H_S and T_P at a Northern North Sea location with storm direction as covariate. We evaluate the impact of incorporating directional effects on estimates for return values, including those of a structure variable, similar to the structural response of a floating structure. We believe the approach offers the ocean engineer a straightforward procedure, based on sound statistics, to incorporate covariate effects in estimation of joint extreme environmental conditions.     

The joint long-term distribution of design environmental factors for offshore structures Multivariate probabilities and simulating method

With the development of ocean engineering, it is one of the most important factors which determine the structural safety, cost and suitable forms of engineerings to select the ocean environmental design criteria. Owing to the complexity , variation and randomness of ocean environmental conditions, the commonly used methods for determining design criteria cannot consider the joint occurring probabilities of several environmental factors ,therefore, lead to overestimate design criteria of them and result in an unnecessary overspend invest in engineering. On the basis of the measured and hindcasting data and the multi-demension joint probability theory, this paper presented the study of the joint loads of wind , wave and current on the offshore structures and its responsible joint probability level with the application of random simulation techniques, and presented the joint design criteria of environmental loads for the realistic design of engineerings.     

The influence of seasonality on estimating return values of significant wave height

A time-dependent generalized extreme value (GEV) model for monthly significant wave heights maxima is developed. The model is applied to several 3-hour time series from the Spanish buoy network. Monthly maxima show a clear non-stationary behavior within a year, suggesting that the location, scale and shape parameters of the GEV distribution can be parameterized using harmonic functions. To avoid a possible over-parameterization, an automatic selection model, based on the Akaike Information Criterion, is carried out. Results show that the non-stationary behavior of monthly maxima significant wave height is adequately modeled, drastically increasing the significance of the parameters involved and reducing the uncertainty in the return level estimation. The model provides new information to analyze the seasonal behavior of wave height extremes affecting different natural coastal processes.     

Prediction of Extreme Significant Wave Height from Daily Maxima

For prediction of the extreme significant wave height in the ocean areas where long term wave data are not available, the empirical method of extrapolating short term data (1-3 years) is used in design practice. In this paper two methods are proposed to predict extreme significant wave height based on short-term daily maxima. According to the da-a recorded by the Oceanographic Station of Liaodong Bay at the Bohai Sea, it is supposed that daily maximum wave heights are statistically independent. The data show that daily maximum wave heights obey log-normal distribution, and that the numbers of daily maxima vary from year to year, obeying binomial distribution. Based on these statistical characteristics, the binomial-log-normal compound extremum distribution is derived for prediction of extreme significant wave heights (50-100 years). For examination of its accuracy and validity, the prediction of extreme wave heights is based on 12 years' data at this station, and based on each 3 years' data respectively     

Uncertainty analysis on extreme value analysis of significant wave height at eastern coast of Korea

In this study, we considered the problem of estimating long-term predictions of design wave height based on the observation data collected over 10–15 years along the eastern-coast of the Korean peninsula. We adopted a method that combines Bayesian method and extreme value theory. The conventional frequency analysis methods must be reconsidered in two ways. First, the conventional probability distributions used in the frequency analysis should be evaluated to determine whether they can accurately model the variation in extreme values. Second, the uncertainty in the frequency analysis should also be quantified. Therefore, we performed a comparative study of the Gumbel distribution and GEV distribution to show the higher efficiency of the latter. Further, we compared the Bayesian MCMC (Markov Chain Monte Carlo) scheme and the MLE (Maximum Likelihood Estimation) with asymptotic normal approximation for parameter estimation to confirm the advantage of the Bayesian MCMC with respect to uncertainty analysis.     

A straightforward approach for using single time domain simulations to assess characteristic extreme responses

Short-term wave design approach of marine structures, using nonlinear time domain simulations, is a design procedure that is recognized by various modern standard codes. One of the most challenging points of this approach is the evaluation of the characteristic extreme values for response parameters used in the design check equations. The most straightforward and recommended way to evaluate a response characteristic value is by fitting an extreme value probability distribution to the N-sample of extreme values extracted from N independent time domain simulations with duration equal to the short-term period indicated by the code, which is usually taken as 3 h. However, this procedure would not be practical for some types of marine structures, such as risers and mooring lines, under numerous design load cases and demanding huge finite element models. A more feasible approach would be to assess the response extreme value distribution using only a single short-term time domain simulation with duration shorter than 3 h. But reduced time simulations always introduce some additional statistical uncertainty into the extreme values estimates. This paper discusses a workable way of properly taking into account the statistical uncertainty associated with the simulation length in the assessment of a characteristic short-term extreme response value based on a single time series.     

Long-term response analysis of FPSO mooring systems

The design of mooring systems for floating production units usually considers extreme environmental conditions as a primary design parameter. However, in the case of FPSO (Floating, Production, Storage and Offloading) units, the worst response for the mooring system may be associated with other sea state conditions due to the fact that its extreme response may be associated with a resonant period instead of an extreme wave height. The best way to deal with this problem is by performing long-term analysis in order to obtain extreme response estimates. This procedure is computationally very demanding, since many short-term environmental conditions, and their associated stochastic nonlinear time domain numerical simulations of the mooring lines, are required to obtain such estimates. A simplified approach for the long-term analysis is the environmental contour-line design approach. In this paper a Monte Carlo-based integration procedure combined with an interpolation scheme to obtain the parameters of the short-term response distribution is employed to hasten the long-term analysis. Numerical simulations are carried out for an FPSO at three different locations considering a North Sea joint probability distribution for the environmental parameters. The long-term analysis results are compared against those obtained using extreme environmental conditions and environmental contour-line methodology. These results represent the characteristic load effect for the design of mooring systems of floating units using the reliability analysis for mooring line. The results show that the long-term results are usually more critical than those obtained with the other approaches and even different mooring lines can be identified as the critical ones.     

海洋工程设计风速与波高的联合分布

选取我国渤海某处21a的风暴过程后报资料,考虑风暴发生频次的影响,提出泊松二维逻辑分布,并且将其用于海洋石油工程设计中极值风速与波高的联合概率计算,给出了计算海域的风浪设计参数,并与传统的设计标准进行了比较.计算结果表明,新的统计模式适用于受风暴影响海区的海洋工程结构设计,特别是边际油田的开发建设.