Extreme significant wave height of tropical cyclone waves in the South China Sea

Zhuxiao Shao, Bingchen Liang , Huajun Li , Ping Li, Dongyoung Lee 4 1 College of Engineering, Ocean University of China, 238 Songling Road, Qingdao 266100, China 2 Shandong Province Key Laboratory of Ocean Engineering, Ocean University of China, 238 Songling Road, Qingdao 266100, China 5 3 China Classification Society, Beijing 100007, China 4 Korea Institute of Ocean Science & Technology, Ansan, Korea


Introduction
Reasonable assessments of extreme significant wave heights are highly important for the security and expense of coastal defence and offshore structures (Ojeda andGuillén, 2006, 2008;Ojeda et al., 2010Ojeda et al., , 2011;;Mortlock andGoodwin, 2015, 2016;Mortlock et al., 2017).To obtain this assessment, a sample is extracted from an accurate initial database, the extreme sample is identified by a reliable sampling method and an appropriate probability distribution model is fitted.
The initial database highly influences the assessment of extreme significant wave heights (Godoi et al., 2017;Lucas et al., 2017;Li et al., 2018;Ribal and Young, 2019).In previous studies, the long-term continuous database is usually employed as the initial database, such as a 32-year measured significant wave height in the Gulf of Maine (Viselli et al., 2015), a 44-year hindcasted significant wave height in the North Atlantic Ocean (Muraleedharan et al., 2016) and a 22-year hindcasted significant wave height in the Yellow Sea (Gao et al., 2018).Considering that the extreme significant wave height should be extrapolated based on an independent and identically distributed database required for the extreme value theory (EVT) (Coles, 2001;Sobradelo et al., 2011), these time series buoy measurements and numerical hindcasts should be processed.The homogenous methodology is used to extract homogenous significant wave heights via separation in carefully chosen directional sectors and seasonal analyses, as well as separation of the sea state into independent wave systems (Lerma et al., 2015;Solari and Alonso, 2017).Declustering methodology, such as the double-threshold approach (Mazas and Hamm, 2011) and minimum separation time method (Kapelonis et al., 2015), is used to differentiate the individual wave event.However, these methodologies may introduce uncertainty in the sample (such as the subjectivity of practitioners in the selections of initial threshold and time window), which influences the extreme sample selection.
In the South China Sea (SCS), time series wave parameters have been simulated (Zheng et al., 2012;Mirzaei et al., 2015;Yaakob et al., 2016) and extreme waves have been investigated based on long-term continuous data (Zheng et al., 2015;Chen et al., 2017;Wang et al., 2018).In addition, Shao et al. (2018a) and Liang et al. (2019) analysed the extreme significant wave height in a tropical cyclone.Shao et al. (2018a) compared the annual maxima (AM) method (Tawn, 1988) with the POT method.The AM method is an easy sampling method that does not require additional work, as the method directly extracts the annual maximum significant wave height for extrapolation.However, the AM method has limitations in a fixed time window (i.e. 1 year), which cannot guarantee the independence and number of samples.The annual maximum significant wave height obtained from neighbouring years may originate from the same extreme wave; some maximum significant wave heights may be neglected (i.e. the annual maximum significant wave height may be smaller than some unselected peaks of significant wave heights in other years), resulting in an insufficient num-ber of samples, especially for a relatively long return period.In a tropical cyclone, the AM method's limitation is further exacerbated, even if the return period is close to the database size.The annual frequency, intensity and track of recorded tropical cyclones vary greatly, and corresponding waves have obvious differences.Shao et al. (2018a) found that the minimum sample may be much lower than the maximum sample, and the minimum sample may be too small to represent the extreme wave (i.e. the minimum sample in the AM method is obviously smaller than the extreme sample in the POT method).
Compared with the AM method, the POT method is a natural sampling method without additional limitations.When the threshold is suitable, the POT method can guarantee the representativeness and number of extreme samples.However, the threshold selection process is relatively complex.Shao et al. (2018a) and Liang et al. (2019) analysed the sensitivity of the return-significant wave height to the threshold.The researchers found that the suitable threshold should be determined within the stable threshold range.When the return-significant wave height is insensitive to the threshold (i.e. the variation in return-significant wave heights is generally small), the corresponding return-significant wave height is defined as the stable return-significant wave height and the corresponding range of thresholds is defined as the stable threshold range.Using the sensitivity analysis, Shao et al. (2018a) defined the largest threshold within the common stable threshold range as the suitable threshold and Liang et al. (2019) proposed an Automated Threshold Selection Method based on the characteristic of Extrapolated significant wave heights (ATSME).The ATSME employs the differences in extrapolated significant wave heights for neighbouring thresholds as the diagnostic parameters to identify the uniquely stable threshold range via an automated method and selects the largest threshold within the stable threshold range as the suitable threshold for different return periods.
In this study, the assessment of extreme significant wave heights is further studied in the SCS.Before the assessment, the meteorological characteristics are analysed to identify extreme weather.In the SCS, the tropical cyclone always drives the storm wave (Anoop et al., 2015;Hithin et al., 2015;Sanil Kumar and Anoop, 2015;Ojeda et al., 2017;Wang et al., 2017;Mortlock et al., 2018;Sanil Kumar et al., 2018) and the number of tropical cyclones is sufficiently large.Thus, it is possible to study the extreme significant wave height in a tropical cyclone (Young et al., 2012;Young and Vinoth, 2013;Young, 2017).To achieve the assessment, a 40-year (1975-2014) hindcasted significant wave height of tropical cyclone waves is employed as the initial database.Considering that the hindcast is independently simulated during the tropical cyclone recorded in the SCS, the maximum significant wave height of the tropical cyclone wave can be directly extracted as the sample when the tropical cyclone influences the wave at the targeted location.Based on the sample, the POT method threshold is studied.By analysing the sensitivity of the return-significant wave heights and the characteristics of the tropical cyclone waves, the sample distribution presents a separation within the stable threshold range.As validated by the asymptotic tail approximation and estimation uncertainty, the high sample shown in the distribution of the sample is suitable for extrapolating extreme significant wave heights in the SCS.
The article is structured as follows.In the next section, the POT-GPD and ATSME are introduced.The initial data and study sites are presented in Sect.3. In Sect.4, the sampling method is described.In Sect.5, the characteristics of tropical cyclone waves are discussed.Finally, the discussions and conclusions are presented in Sect.6.

Extrapolation theory
The POT method extracts the maximum significant wave heights above a selected value (i.e.threshold), u, as the extreme sample.For u, which is sufficiently large, the distribution function of peak exceedances can be approximated by a member of the GPD (Pickands, 1975;Embrechts et al., 1997): where Hs * represents the peak exceedance over the threshold, σ represents the scale parameter and k represents the shape parameter.These GPD parameters (σ and k) are estimated using the maximum likelihood estimation method, which is recommended by Mazas and Hamm (2011): where N represents the number of events exceeding the threshold (i.e. the number of extreme samples) and Hs represents the maximum significant wave height.The i-year return-significant wave height, Hs i , is defined as follows: Thus, the value can be calculated with the following equation: where N T represents the size of the dataset.

Automated threshold selection method
The ATSME determines a unique threshold within a uniquely stable threshold range for a specific return period.Liang et al (2019) found that the stable threshold range shows a pattern associated with the return period.The minimum threshold of the stable threshold range controls the representativeness of the extreme sample; thus, the samples over the minimum threshold can represent extreme waves well, and the minimum thresholds for different return periods remain constant.The maximum threshold of the stable threshold range controls the number of extreme samples, and a longer return period requires more extreme samples; thus, the maximum thresholds gradually decrease when the return period increases.Consequently, excluding the sample within the stable threshold ranges does not obviously influence the returnsignificant wave heights, and a suitable threshold should be determined within the stable threshold range.The terms u 1 , . . ., u m are candidate thresholds.Hs i,j represents the i-year return-significant wave height based on the threshold of u j .The difference, Hs i,j , in i-year returnsignificant wave heights (Hs i,j and Hs i,j −1 ) for neighbouring thresholds (u j and u j −1 ) is defined as follows: Hs i,j = Hs i,j − Hs i,j −1 . (5) To study the influence of the excluded samples on the returnsignificant wave height with an increasing threshold and to select a suitable threshold, the ATSME is defined as follows (Liang et al., 2019).
1. Sample.Take the sample from the initial database under an independent and identically distributed assumption.
2. Candidate threshold.Identify the suitable range for the equally spaced, increasing candidate thresholds (u 1 , u m ) and the threshold interval u = u m −u 1 N tot .u 1 is set as the minimum sample, u m is set as the maximum sample and N tot is set as the number of samples.
3. Return period and value.Choose the order of i (i = i 1 , . . ., i n i ) for different return periods, which is dependent on N T and the requirement of practitioners.Extrapolate the return-significant wave height for the i-year, Hs i,j , which corresponds to every candidate threshold, u j .
4. Stable threshold range.Calculate the difference, Hs i,j , in the return-significant wave height for neighbouring thresholds.Define a characteristic parameter, ch i,j , to record the stable characteristics of the return-significant wave heights.Find the uniquely stable threshold range for the i-year return period.5. Suitable threshold.Determine the suitable threshold within the stable threshold range, such as the maximum threshold.3 Initial data and study sites

Initial data
Significant wave heights from a 40-year hindcast of tropical cyclone waves (Shao et al., 2018a) are adopted as the initial database, which is simulated using the third-generation spectral wind wave model SWAN (Simulating WAves Nearshore) (Booij et al., 1999;Mortlock et al., 2014;Amrutha et al., 2016).This model is forced by the blended wind, which is obtained by combining the European Centre for Medium-Range Weather Forecasts reanalysis wind data (ECMWF, covering the ERA-40; Uppala et al., 2005 and ERA-Interim;Dee et al., 2011) and Holland model wind data (Shao et al., 2018b).The spatial resolution is 0.0625 • for both longitude and latitude, and the temporal resolution is 1 h.From 1975 to 2014, waves are simulated only during 974 independent tropical cyclones.

Study sites
To analyse the extreme significant wave height, 22 locations were selected as the study sites (Fig. 1).When the distance between the centre of the tropical cyclone and the study site is within 300 km, this tropical cyclone is recorded, and hourly significant wave heights simulated during this tropical cyclone are adopted as the initial database at the study site.At the 22 study sites, the number of recorded tropical cyclones is 247 to 403, and the annual mean number of recorded tropical cyclones is 6.175 to 10.075.The corresponding tropical cyclone waves are sufficient for assessing extreme significant wave heights in the SCS (Mazas and Hamm, 2011).
4 Study of the POT method

Sample
As required by the EVT, the extreme significant wave height should be extrapolated based on the independent wave un-der the same type of meteorological event (Lerma et al., 2015;Solari and Alonso, 2017).Considering that the initial database is simulated only during the independent tropical cyclone, the maximum significant wave height of recorded tropical cyclone waves can be directly extracted as the sample at the study site.For example, 328 tropical cyclones are recorded at location no.1; thus, 328 maximum significant wave heights during these tropical cyclones are extracted as the sample.

Sensitivity of return values to threshold
Sensitivity of the return-significant wave height to the threshold can be used in threshold selection.This method fits the GPD over a range of candidate thresholds and selects the suitable threshold by identifying the stability of the returnsignificant wave heights.If return-significant wave heights are insensitive to the threshold, the corresponding threshold can be selected as the suitable threshold.The benefit of this method is that it requires practitioners to graphically inspect and comprehend the data features and assess the uncertainty of the candidate thresholds (Scarrott and MacDonald, 2012).The drawback of this method is that the threshold is not uniquely selected and another criterion is needed to identify the optimal threshold (Lerma et al., 2015).

Characteristics of tropical cyclone waves
To further analyse the candidate thresholds within the stable threshold range, the characteristics of tropical cyclone waves are investigated.The track and intensity of tropical cyclones affect the waves at the study site.When the tropical cyclone track is close to the study site and the intensity of the tropical cyclone is strong, the corresponding wave is sufficiently strong for representing the extreme wave at the study site.In this case, the maximum significant wave height of this tropical cyclone wave should be extracted as the extreme sample.For example, at location no. 1, the maximum significant wave heights during tropical cyclones Pabuk in 2007, Linfa In contrast, when the track of the tropical cyclone is far from the study site or the intensity of the tropical cyclone is weak, the corresponding wave is insufficiently strong to represent the extreme wave at the study site.In this case, the maximum significant wave height of this tropical cyclone wave should not be extracted as the extreme sample.For example, at location no. 1, the maximum significant wave heights during tropical cyclones Maria in 2000 and Toraji in 2001 are 2.59 and 1.57 m, respectively.Although the intensities of these tropical cyclones are strong when they influence the waves at location no. 1, the tracks of these tropical cyclones are too far from location no. 1 (shown in Fig. 3).The maximum significant wave heights during tropical cyclones Trami in 2001 and Wutip in 2007 were 2.47 and 2.20 m, respectively.Although the tracks of these tropical cyclones are close to location no. 1, the intensities of these tropical cyclones are weak when these tropical cyclones influence the waves at location no. 1 (shown in Fig. 4).The maximum significant wave heights during tropical cyclones Kaitak in 2005 and Kammuri in 2008 were 1.11 and 2.36 m, respectively.The tracks of these tropical cyclones are far from location no. 1, and the intensities of these tropical cyclones are weak when they influence the waves at location no. 1 (shown in Fig. 5).
The track and intensity influences of the tropical cyclones are reflected in the sample distribution (i.e. the distribution of the maximum significant wave height).In Fig. 6, the distribution of the sample at location no. 1 is presented.The sample is counted from 0 to 15 m with an interval of 0.05 m,   sponding annual mean number of extreme samples is 3.425) is within the stable threshold range shown in Sect.4.2.
To further validate the separation, the asymptotic tail approximation and estimation uncertainty are analysed.The asymptotic tail approximation can be estimated by the quantile plot, which is discussed by Coles (2001) and produced by a free package running in R. In Fig. 7, the quantile plot for the threshold of 4.15 m is presented, which shows that there are generally few differences between the empirical and fitted quantiles, indicating a good fit for the selected threshold.In Table 1, the return-significant wave height is shown with the confidence interval.The likelihood method (Schendel and Thongwichian, 2017) reparameterizes the likelihood in terms of the unknown quantile and uses profile likelihood arguments to construct an approximate 95 % confidence interval.At location no. 1, the confidence intervals indicate that  the variance in the extrapolated significant wave heights is acceptable.
The same conclusion can be reached at the other 21 study sites.For example, the sample distributions at location nos.7 and 10 (Fig. 8) present separation values of 3.35 and 4.1 m, respectively.Based on these separation values, the GPD model is used to extrapolate the return-significant wave heights for return periods of 50, 100, 150 and 200 years (Table 2).To validate the reliability of the return-significant wave heights, the asymptotic tail approximation and estimation uncertainty are analysed.For example, the quantile plots at location nos.7 and 10 are presented in Fig. 9, and the confidence intervals at 21 study sites are shown in Table 2.The fits of the results are good, and the uncertainties of the returnsignificant wave heights are acceptable.

Discussion and conclusions
In general, Shao et al. (2018a) and Liang et al. (2019) select the suitable threshold within the stable threshold range.Benefiting from the stable characteristics of return-significant wave heights, their threshold selection criteria can be used to assess the extreme significant wave height.The criterion of Shao et al. (2018a) is relatively simple, and the cri-   2019) is relatively stable, due to a diagnostic process of return-significant wave heights.For example, at location no. 12, Shao et al. (2018a) extrapolate the return-significant wave heights for the return periods of 50, 100, 150 and 200 years, which are 9.59, 9.86, 9.99 and 10.06 m, respectively.However, under the criterion of Liang et al. (2019), the corresponding return-significant wave heights are 9.69, 9.89, 9.96 and 10.05 m, respectively.When the variation in few return-significant wave heights is relatively large in the stable threshold range, the returnsignificant wave heights of Liang et al. (2019) are more stable than those of Shao et al. (2018a), especially for a short return period.
To determine the suitable threshold within the stable threshold range without a subjective definition, the thresholds within the stable threshold range are further analysed and associated with the characteristics of the tropical cyclone wave.When studying the tropical cyclone wave, a fixed distance is used to identify the initial database at the study site.This fixed distance allows some small samples (the corresponding track is far or the intensity is weak) to be extracted; however, no large samples (the corresponding track is close and intensity is strong) are neglected.Associated with these influences (i.e.track and intensity influences) from the tropical cyclones, the sample distribution has a natural separation distinguishing the high sample (strong influence from the tropical cyclone) from the low sample (weak influence from the tropical cyclone).Linking this distribution with the stable threshold range, the separation is within the stable threshold range.Thus, this separation can be used to identify the extreme sample (i.e.high sample in the distribution).Note that in Table 9 of Shao et al. (2019) and Tables 1 and 2 in this study, the return-significant wave heights for the return periods of 50, 100, 150 and 200 years are similar at the same 22 study locations.However, the threshold selection criterion in this study is relatively simple and objective, and this criterion can reflect the characteristics of the tropical cyclone wave.In addition, under this criterion, the asymptotic tail approximation and estimation uncertainty show that the fits are good and the uncertainties of the return-significant wave heights are acceptable.

Figure 1 .
Figure 1.The study sites in the study region.
Shao et al. (2018a) andLiang et al. (2019) analysed the sensitivity of the return-significant wave height and provided threshold selection criteria to determine a unique threshold.Liang et al. (2019) diagnosed the return-significant wave height within the stable threshold range.If some returnsignificant wave heights within the stable threshold range are relatively different from the others, the corresponding candidate thresholds are rejected.Thus, the conclusions ofLiang et al. (2019) on the sensitivity of the return-significant wave height are employed in this study.For example, at location no. 1, the equally spaced, increasing candidate thresholds are identified by a threshold interval of 0.05 m, and the stable threshold ranges for the 50-, 100-, 150-and 200-year return periods are[3.3,5.75 m], [3.3, 5.25 m], [3.3, 4.6 m] and [3.3, 4.5 m)], respectively.

Figure 2 .
Figure 2. Tracks of the centres of tropical cyclones Pabuk, Linfa, Molave and Meranti (triangle represents location no. 1, curves represent tracks of centres and circles represent locations of centres).

Figure 3 .
Figure 3. Tracks of the centres of tropical cyclones Maria and Toraji (triangle represents location no. 1, curves represent tracks of centres and circles represent locations of centres).

Figure 4 .
Figure 4. Tracks of the centres of tropical cyclones Trami and Wutip (triangle represents location no. 1, curves represent tracks of centres and circles represent locations of centres).

Figure 6 .
Figure 6.Histogram of the maximum significant wave height from 0 to 15 m with intervals of 0.05 m at location no. 1.

Figure 7 .
Figure 7.The quantile plot for GPD-fitted maximum significant wave heights at location no. 1 for the threshold of 4.15 m.

Figure 8 .
Figure 8. Histograms of the maximum significant wave heights at location nos.7 and 10.

Figure 9 .
Figure 9. Quantile plots for GPD-fitted maximum significant wave heights: (a) for the threshold of 3.35 m at location no.7 and (b) for the threshold of 4.1 m at location no.10.

Table 1 .
Statistics for the return-significant wave heights and confidence intervals at location no. 1 for the threshold of 4.15 m.

Table 2 .
Statistics for thresholds (separations), extreme samples and return-significant wave heights with 95 % confidence intervals. ]