TAO Shanshan, DONG Sheng, *, LIN Yifan, and GUEDES SOARES Carlos
Joint Return Value Estimation of Significant Wave Heights and Wind Speeds with Bivariate Copulas
TAO Shanshan1), DONG Sheng1), *, LIN Yifan2), and GUEDES SOARES Carlos3)
1) College of Engineering, Ocean University of China, Qingdao 266100, China 2) Huadian Heavy Industries Co., Ltd., Beijing 100070, China 3) Instituto Superior Técnico, Universidade de Lisboa, Lisboa 1049-001, Portugal
The joint design criteria of significant wave heights and wind speeds are quite important for the structural reliability of fixed offshore platforms. However, the design method that regards different ocean environmental variables as independent is conservative. In the present study, we introduce a bivariate sample consisting of the maximum wave heights and concomitant wind speeds of the threshold by using the peak-over-threshold and declustering methods. After selecting the appropriate bivariate copulas and univariate distributions and blocking the sample into years, the bivariate compound distribution of annual extreme wave heights and concomitant wind speeds is constructed. Two joint design criteria, namely, the joint probability density method and the conditional probability method, are applied to obtain the joint return values of significant wave heights and wind speeds. Results show that (28.5 ± 0.5) m s?1is the frequently obtained wind speed based on the Atlantic dataset, and these joint design values are more appropriate than those calculated by univariate analysis in the fatigue design.
joint design; wave height; wind speed; compound distribution; copula
Reasonable design parameters of ocean environmental variables can ensure the structural reliability and economy of the fixed offshore platform. The return values of these variables are estimated separately before they are combined as the design criterion. In general, these extreme return values of different variables never occur simultaneously, thereby indicating that this design method is conservative.
Considering different variables to be dependent is a suit- able approach to assessing joint design parameters. American Petroleum Institute (API) presented two solutions. The first report of API (API, 1993) suggested that metocean loads should be combined consistently with the probability of their simultaneous occurrence under loading conditions. In addition, the second report (API, 2000) recommended a consequence-based criterion that considers the joint effect of different metocean conditions on platform structures.
The annual maximum is commonly used when the sample size is large, and these sample points are usually considered independent, which follow a univariate distribution such as lognormal distribution, generalized extreme value (GEV) distribution, and modified maximum entropy distribution (MMED) (Isaacson and MacKenzie, 1981; Muir and El-Shaarawi, 1986; Zhang and Xu, 2005; Guedes So- ares and Scotto, 2011; Dong, 2013a). However, only applying annual maxima in the presence of extreme data is a wasteful approach (Coles, 2001). In improving this situation, the-largest values within a block were used (Gu- edes Soares and Scotto, 2004), but this method can also be wasteful if one block contains more extreme events than the other.
The peak-over-threshold (POT) method can lead to a series of exceedances above a specified threshold value, which can be modeled using the generalized Pareto distribution (GPD) (Ferreira and Guedes Soares, 1998; Jonathan, 2021). However, for dependent series, the threshold exceedances can occur in groups, implying that one extreme event is likely to be followed by another (Coles, 2001). Asymptotics suggest that the distribution of any one of the threshold exceedances might be modeled by the GPD, but clustering induces dependence in the series. In addition, no general theory has been proposed to eliminate this dependence. Declustering classifies the dependent threshold exceedances into clusters and assumes that the cluster maxima of these exceedances are independent and derived from the same GPD.
If univariate data are recorded for short years, and the extreme meteorological processes (such as typhoons) are prevailing in this study area, then the compound extreme value distribution (Ma and Liu, 1979; Korolev, 2019) can be applied to calculate the return values by combining the process maxima with the annual occurrence frequency. This method is another attempt to obtain more sample points apart from the POT method.
The original data sequences are usually divided into blocks based on time. The vector of component-wise max- ima in these blocks (such as annual extreme wave heights and annual extreme wind speeds) can be selected as multivariate extreme value samples, and their limiting behavior is theoretically modeled by the multivariate extreme value distribution (Jiang, 2021). Morton and Bowers (1996) accounted for the dependence between the wave heights and wind speeds by using the bivariate logistic model. Za- chary(1998) calculated the joint co-occurrence prob- ability of wave height, period, and wind speed based on the trivariate logistic model. The sample with component- wise maxima only considers extreme conditions of all ocean environmental variables but neglects the fact that such con- ditions are not likely to occur simultaneously.
Some multivariate sampling methods of conditional extreme values are proposed to avoid the conservative joint design. For example, a certain ocean environmental variable such as wave height can be regarded as the dominant variable, and then its block maxima and concomitant data of other variables are gathered to be a multivariate sample. The margins of conditional extremes often cannot be modeled by Gumbel, Fréchet, Weibull, or GEV distributions; therefore, the multivariate extreme value theory is not suit- able for these samples.
Copula functions proposed by Sklar (1959) can combine the marginal distribution of different ocean environmental parameters with some correlations among them, which will lead to a joint probability distribution (Nelsen, 2006; Lee and Joe, 2018). The margins can be selected from different types of univariate distributions, which is a suitable way to construct the bivariate probability distribution of samples with conditional extreme values. Hanne(2004) obtained the joint distribution of the wave height and wave period based on a bivariate normal copula. Muhaisen(2010) established the bivariate probability model of significant wave height and storm duration based on a copula for the optimum design of gravel breakwaters.
Similar to the univariate case, selecting a sample point from each block for multivariate samples is a wasteful approach, and the POT method and declustering can be applied to obtain more effective multivariate sample points. In the case of the original data recorded for a short period of time in the area where typhoons are prevailing, Liu(2006) extended the compound extreme value distribution from one dimension to multidimensions.
This paper proposes a method to predict the joint return values of the significant wave height and wind speed, which is demonstrated in a data set based on three-hourly data from a North Atlantic Ocean location. The content of this paper includes the selection of bivariate data samples, the construction of joint probability distributions, and a discussion of the joint design criteria.
Bivariate samples are necessary to calculate the joint design values of the wave height and wind speed. The paired three-hourly significant wave height and wind speed sequence are (1,1) and (2,2). If they are blocked into years, then we can obtain different bivariate samples, such as the data vector of the annual component-wise max- ima, the data combination of annual maxima of the dominant variable, and the concomitant values of the other vari- able, such as the annual maximum wave height and corresponding wind speed in a data pair of (H,W). However, only one group of data pairs each year is identified in these bivariate samples, as mentioned in Section 1. In the present study, we utilize the POT method and declustering to acquire more sample points. The bivariate data sampling procedures of the wave height (1,2, ···) and wind speed (1,2, ···) in this paper are as follows.
We consider wave height as the dominant variable of structural dynamic response as the offshore structures are primarily affected by waves. First, we focus on the original data sequence of wave heights (three-hourly,), which may be a nonstationary and seasonal time series. Selecting a group of extreme wave heights that are independently and identically distributed (i.i.d.) plays an important role in the following analysis.
Similar to the method and procedure given by Coles (2001), we first apply the POT method to obtain a series of wave heights (zero-crossing peak-to-trough amplitudes) exceeding a fixed threshold0. These exceedances of wave heights over a selected section can be defined as one cluster. Subsequently, the maximum of each cluster, namely, the cluster maxima, can be identified. However, two adjacent clusters, wherein the time span between their cluster maxima is less than a half day (12 h), will be merged. After identifying all clusters, we can acquire a series of bivariate samples composed of the cluster maxima and their corresponding wind speeds derived from the original data sequences, which are termed threshold process maximum wave heights (data setC) and concomitant wind speeds (data setC), respectively, hereafter in the paper.
Selecting the wave height threshold0during data sampling is important. The use of the mean-residual life plot, which is based on the univariate extreme value theory and the GPD, is a common practice in threshold selection. In the mean-residual life plot, the horizontal axis represents the threshold0, and the vertical axis represents the mean excess function(0) =(C-0|C>0). A proper threshold0must be selected to make(0) approximately linear to0.
However, this tool may not be suitable for nonstationary and seasonal data sequences. Thus, we apply the bootstrap technique, as a comparison to the mean-residual life plot method, to test the i.i.d. characteristic of the threshold process maximum wave heights and concomitant wind speeds. We can obtain a set ofCfor each given0.Cis randomly divided into two groups (C1,C2; the corresponding groups1and2ofCcan be obtained), and K-S tests are applied to determine whether the two groups are from the same population. The process is repeatedNtimes, and the numbernof test results is not counted from the same population. Finally, the test value is calculated as follows:=n/N. The smaller theTvalue, the better the i.i.d. of the sample.
The bivariate sample selected in Section 2 is not a vector of component-wise maxima in each block; thus, it cannot be directly modeled using the bivariate extreme value distributions.
Hereafter, we use the random vector (,) to represent the i.i.d. pair of the threshold process maximum wave heights and the corresponding concomitant wind speeds at each point. The distribution of the marginal variablethat denotes a special threshold exceedance can be well-fitted by the GPD (Coles, 2001) and not by the univariate extreme value distribution functions (such as Gumbel, Fré-chet, Weibull, or GEV). In theory,is not a pure GPD variable, and it can be fitted by other probability distributions such as the lognormal distribution and MMED (which can be transformed into many common probability distributions in coastal and ocean engineering, Table 1). Meanwhile, the concomitant wind speedis usually not the block maximum. Therefore, it cannot be described by the univariate extreme value distribution. Similarly, the lognormal distribution, MMED, and GPD are potential distribution functions for.
The probability density function of lognormal, MMED, GPD, and GEV distributions is provided in Appendix A. The parameters can be estimated by using the maximum likelihood method (Rao and Hamed, 2000; Coles, 2001; Dong, 2013b). In addition, cross-validation is applied to compare the generalized errors of curve fittings to select the best univariate probability distributionF(for) orF(for). The final probability distribution types ofandmay be different.
Table 1 Relationship between MMED and other distributions
The copula theory can be used to combine different types of marginal distributions into a joint probability distribution based on the theorem of Sklar (1959).
Lettingbe a joint probability distribution function with marginsFandF, a copulais obtained, such as that for all (,) in (-∞, ∞) × (-∞, ∞)
is unique ifFandFare continuous; otherwise,is uniquely determined by the product of the ranges ofFandF.Conversely, ifis a copula andFandFare distribution functions, then the functiondefined by Eq. (1) is a joint probability distribution function with marginsFandF. Here we defineandas=F() and=F(), respectively.
Copula has various types, such as elliptical copulas (nor- mal, student), Archimedean copulas (Frank, Gumbel- Hougaard, Clayton,), extreme value copulas, and Fralie- Gumbel-Morgenstern copulas (Nelson, 2006; Gudendorf and Segers, 2010). In the present study, we construct the joint distribution of the random vector (,) using the bi- variate normal, Frank, Gumbel-Hougaard, and Clayton cop- ulas based on Eq. (1). Their probability distribution func- tions are given in Appendix B.
We applied the joint probability distribution proposed in Section 3.1 to describe the bivariate sample of the threshold process maximum wave heights and the concomitant wind speeds. The design return values are generally measured in years; thus, we divide the bivariate sample into blocks of years. Consequently, the annual extreme wave heights and concomitant wind speeds can be extracted from the bivariate sample, and their joint probability distributions can be obtained on the basis of the principle of multivariate compound extreme value distributions.
We use (,) to denote the annual extreme wave height and concomitant wind speed for the joint probability dis- tribution(,), the random variableto denote the num- ber of threshold process maximum wave heights each year for the probability distribution(=) =P, and {(ξ,η):= 1, 2, ···,} to denote the sample pairs of the threshold process maximum wave heights and the corresponding con- comitant wind speeds. In addition, we use (ξ,η) to represent the i.i.d. realization of (,), where its joint prob- ability distribution is given by(,) based on Eq. (1). If= 0, then (,) becomes (,), and its joint probability distribution is(,). Then, we obtain the following equation:
and
For arbitrary≥ 1, we have:
where(,) is the joint probability density function of the random vector (,), andF() is the marginal probability distribution function of random variable. Substituting Eq. (4) with Eq. (3), we can construct various bivariate compound distributions described by Eq. (5) once we specify the distributions forF(),(,), and(,) from different types of univariate and bivariate probability distributions (Eqs. (A.1 – A.4) and Eq. (1)):
Iffollows a Poisson distribution,
then(,) becomes
Eq. (7) refers to the Poisson bivariate compound distribution, which is generally replaced by
The2test and AIC method are given in Appendix C.
LettingF() andF() represent the marginal distribution of the annual extreme wave height () and the concomitant wind speed (), respectively, we can derive the design return values using the following equation:
whereTandTare the return periods ofand, respec- tively. However, Eq. (11) presents an overly conservative method because the extreme return values of different en- vironmental variables rarely occur simultaneously. There- fore, we introduce two bivariate joint design criteria, name- ly, the joint probability density method (JPDM) and the conditional probability method (CPM).
JPDM uses the sets of the wave height and wind speed with the largest joint probability density, that is, the combination that occurs the most frequently. These combinations may cause the greatest fatigue failure during the life-time of offshore and ocean structures.
Based on the joint probability distribution ofand, as shown in Eq. (8), the corresponding probability density function using JPDM (x,y) is presented as follows:
In this case study, we used the hindcast significant wave heights and wind speeds sampled at every 3 h between 1958 and 2001 from a point (19?W, 59?N) in the North Atlantic Ocean (Pilar, 2008) to verify the efficiency of the abovementioned joint distributions and design criteria. This original dataset consists of 128568 pairs of significant wave heights and wind speeds, accounting for 2920 or 2928 pairs each year. As shown in Fig.1, the hindcast data demonstrate strong nonstationary, and seasonal characteristics.
Fig.1 Three-hourly wave heights.
Based on the sampling method presented in Section 2, given the different threshold0, we can obtain the maximum valueCof the wave height extreme process and cor- responding wind speedC. Fig.2(a) shows the mean- residual life curve of (0,(0)). The mean-residual life curve is maintained as a parallel straight line in0= [12.2 m, 12.8 m]. Fig.2(b) gives the results of the i.i.d. tests by using the bootstrap method. In the interval of0= [12.2 m, 12.8 m], when0= 12.4 m, theTvalue is the minimum. There- fore,0= 12.4 can be applied as the actual threshold.
The histograms of the corresponding bivariate sample are shown in Figs.3(a – b), which indicate that the wave heights are right skewed, and the wind speeds are approximately and normally distributed.
The bivariate sample grouped by the annual extreme wave heights and annual extreme wind speeds is also selected, and its scatter plots are shown in Fig.4. These annual extreme sample pairs are applied to calculate the joint design values of wave height and wind speed to compare the design values obtained by the peak over threshold meth- od.
Fig.2 Mean-residual life plot and i.i.d. test results.
Fig.3 Histograms of threshold process maximum wave heights and concomitant wind speeds.
Fig.4 Scatter plots of annual extreme wave heights and wind speeds. (a), annual extreme wave heights; (b), annual extreme wind speeds.
As mentioned in Section 3.1, the abovementioned bivariate sample with 134 pairs of the threshold process maximum wave heights and concomitant wind speeds can be fitted by the lognormal distribution, MMED, and GPD sep- arately (Appendix A). We apply the maximum likelihood estimation to fit the parameters of these univariate distributions, and such parameters are examined by 10-fold cross-validation to determine the type of distribution that gives the smallest generalized error.
Fig.5 Quantile-quantile (Q-Q) plots to show the GPD model fit to the threshold process maximum wave heights (a), and the MMED model fit to the concomitant wind speeds (b).
As mentioned in Section 3.1 and Section 5.2, we construct the joint probability distribution of this bivariate sample (Sample I) using four bivariate copulas (normal, Gumbel-Hougaard, Frank, and Clayton, see Appendix B) and two marginal distributions (GPD for the threshold process maximum wave heights and MMED for the concomitant wind speeds). These probability models are denoted as BN-GM (ivariate model based on theormal copula and thePD andMED margins), BGH-GM, BF-GM, and BC-GM.
By contrast, we apply the symmetric logistic model (Ap- pendix D) with two GEV margins to fit the sample pairs of the annual extreme wave heights and annual extreme wind speeds (Sample II). Similarly, this joint probability model is denoted as BL-GEVs (ivariate model based on the symmetricogistic model and twomargin).
The correlated parametersin joint probability models BN-GM, BGH-GM, BF-GM, BC-GM, and BL-GEVs are estimated as 0.58 (?1££1), 1.68 (≥ 1), 1.36 (10), 4.22 (> 0), and 1.45 (≥ 1), respectively.
Fig.6 χ and plots for threshold process wave heights and concomitant wind speeds and corresponding bivariate copula models. (a), χ-plot; (b), -plot.
Fig.7 χ and plots for annual extreme wave heights and annual extreme wind speeds and the corresponding BL-GEV model. (a), χ-plot; (b), -plot.
Sample I has a total of 134 data points; thus, thevalues are taken from 10, 11, 12, 13, and 14, which correspond to 100, 121, 144, 169, and 196 equal-sized unit cells, for the2test of BN-GM, BGH-GM, BF-GM, and BC- GM models. Similarly, for the2test of BL-GEV, which is based on Sample II with a total of 44 data points, thevalues are taken from 6, 7, 8, 9, and 10. Thesevalues can ensure adequate unit cells for model evaluation and adequate sample points in each unit cell. The results are shown in Table 2, which implies that these models, except for the BC-GM model, can pass the2test. Notably, different se- lections ofmay lead to different ‘best’ fitting models that are solely based on the2test.
Table 2 c2 test results
Note:is calculated by using Eq. (A.10).
The bivariate compound distribution (Eq. (8)) for Sample I can be established by using the Poisson distribution with= 3.0455 and the BN-GM model for the threshold process maximum wave heights and concomitant wind speeds. This bivariate compound distribution is denoted as P-BN-GM.
Fig.8 Poisson fitting for the frequency of the annual threshold maxima.
The return values (Eq. (11)) calculated from these univariate distributions are listed in Table 3. They show that 1) the wave heights calculated by the Poisson GPD (Sample I) and the GEV distribution (Sample II) are similar because they both utilize the annual maximum values; 2) the wind speeds calculated by the GEV distribution (Sample II) are larger than those calculated by the Poisson MM- ED (Sample I) because the latter is based on concomitant wind speeds instead of annual extremes.
Table 3 Return values by univariate analysis
Based on the P-BN-GM model for Sample I and the BL-GEVs model for Sample II, the joint design values can be obtained by JPDM and CPM (Table 4 and Table 5). The results show that 1) the design wave heights and design wind speeds given by P-BN-GM are about 0.79 and 0.78 times those given by BL-GEVs for the joint design criterion JPDM; 2) under the same-year return wave height, the most likely wind speeds obtained by P-BN-GM (based on CPM) are less than 0.93 times the-year return wind speed obtained by P-MMED based on Sample I; 3) under the same-year return wave height, the most likely wind speeds obtained by BL-GEVs (based on CPM) are less than 0.98 times the-year return wind speed obtain- ed by GEV based on Sample II; 4) the most likely wind speeds calculated by P-BN-GM are smaller than those calculated by BL-GEVs (based on CPM), showing a slight change because of different return wave heights. Therefore, based on the Atlantic dataset, (28.5 ± 0.5) m s?1wind may occur the most frequently, which must be considered when studying fatigue damage.
Table 5 Design values by CPM
Notes: The data of P-BN-GM model are threshold process maximum wave heights and concomitant wind speeds, and the data of BL-GEVs are annual extreme wave heights and annual extreme wind speeds.
Sample I is blocked into years to obtain the joint design criteria, and the bivariate compound distributions are proposed to describe the joint behavior of the annual extreme wave heights and concomitant wind speeds extracted from this sample. For the data applied in Section 5, the P-BN- GM model is used to calculate the joint design variables using the JPDM and CPM models. By contrast, the BL- GEVs based on two GEV margins and the symmetric logis- tic model are constructed to model the data pairs of the an- nual extreme wave heights and annual extreme wind speeds. Based on this probability distribution, the joint design val- ues can be calculated using the JPDM and CPM models.
The results show that the design values of the wind speeds obtained using JPDM and CPM are smaller than those obtained from univariate analysis for the same return periods of wave heights. For the two models, P-BN- GM and BL- GEVs, corresponding to different bivariate samples in Section 5, the design wave heights and design wind speeds are different mostly because the samples are collected in different ways. In addition, the design wind speeds obtained by P-BN-GM are generally smaller than those obtained by BL-GEVs (JPDM and CPM). For the Atlantic dataset, (28.5 ± 0.5) m s?1is considered as the commonly occurred wind speed. Therefore, the design values obtained from joint distributions may be more appropriate than those calculated by univariate analysis in the fatigue design.
The study was supported by the National Natural Science Foundation of China (No. 52171284).
The probability density function of the lognormal distribution is calculated as follows:
whereμandσare two unknown parameters, and they correspond to the mean value and standard deviation of the logarithm of the original random variable, respectively.
The probability density function of the MMED is calculated as follows:
where,,, and0(the location parameter) are the parameters.
The probability density function of the GPD is calculated as follows:
where>-/when> 0, and<<-/when< 0.
The probability density function of the GEV distribution is calculated as follows:
where,, and> 0 are the location parameter, shape parameter, and scale parameter, respectively.
The probability distribution function of the bivariate nor- mal copula is calculated as follows:
where Φ(·) is the one-dimensional standard normal distribution function, and Φ?1(·) is the inverse function of Φ(·); ?1 ≤≤ 1 is the correlation coefficient of Φ?1() and Φ?1().andare independent when= 0 and completely correlated if || = 1.
The probability distribution function of the bivariate Gumbel-Hougaard copula is calculated as follows:
where≥ 1 is the correlated parameter, and the relationship betweenand the Kendall rankis= 1 ? 1/.andare independent when= 1, andandtend to be completely correlated if→ +∞.
The probability distribution function of the bivariate Frank copula is measured as follows:
where≠ 0 is the correlated parameter, and the relationship betweenand the Kendall rankis
> 0 denotes thatandhave a positive correlation, whereas< 0 denotes thatandhave a negative correlation. Moreover, when→ 0, the random variablesandtend to be independent.
The probability distribution function of the bivariate Clayton copula is measured as follows:
where 0 << +∞ is the correlated parameter, and the relationship betweenand the Kendall rankis=/ (+ 2).andtend to be independent when→ 0 and tend to be completely correlated when→ +∞.
1) The2test for bivariate probability distributions
Hu (2002) introduced a statistic, which follows a2distribution, and it can estimate the fitting quality between the bivariate copula and sample.
The sample point of (,) is {ξ,η}, and the marginal distributions areF() andF(). Letu=F(ξ),v=F(η),= 1, 2, ···,, thenuandvare obtained from uniform distribution [0, 1]. A grid of [0, 1] × [0, 1] is constructed, which consists of×equal-sized unit cells. Based on the sample sizeand distribution of the sample points (u,v), thevalue must ensure enough unit cells for model eval- uation and enough sample points in each unit cell.
The unit in theth row andth column is denoted as(,),,= 1, 2, ···,. For each {u,v}, if (–1)/≤u≤/and (– 1)/≤v≤/exist, then {u,v}?(,). LetAbe the number of the actual sample points that fall into the unit(,) andBbe the number of predicted points {u,v} produced by the bivariate probability model, which fall into the unit(,). Then
2) AIC for bivariate probability distributions
AIC is proposed by Akaike (1974) based on information entropy concepts, and it penalizes overly complex models by correcting the maximum log-likelihood for the number of parameters in the model. The AIC measure is defined as follows:
whereis the number of parameters in the bivariate probability model,is the sample size of {ξ,η}, and() is the likelihood function of the sample pairs defined as follows:
The probability distribution of the symmetric logistic model is calculated as follows:
whereG() andG() are the marginal distributions, and≥ 1 is the correlated parameter.
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(January 21, 2022;
May 31, 2022;
August 14, 2022)
? Ocean University of China, Science Press and Springer-Verlag GmbH Germany 2023
. E-mail: dongsh@ouc.edu.cn
(Edited by Xie Jun)
Journal of Ocean University of China2023年5期