Extracting independent and identically distributed samples from time series significant wave heights in the Yellow Sea

Coastal Engineering 158 (2020) 103693

Contents lists available at ScienceDirect

Coastal Engineering journal homepage: http://www.elsevier.com/locate/coastaleng

Extracting independent and identically distributed samples from time series significant wave heights in the Yellow Sea Zhuxiao Shao a, Bingchen Liang a, b, *, Huijun Gao a a b

College of Engineering, Ocean University of China, 238 Songling Road, Qingdao, 266100, China Shandong Province Key Laboratory of Ocean Engineering, Ocean University of China, 238 Songling Road, Qingdao, 266100, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Independence and homogeneity assumptions Automated method Region analysis Extreme significant wave heights Yellow sea

The assessment of extreme significant wave heights is crucial for the design of coastal defences and offshore infrastructures, which requires independent and identically distributed samples to be extracted. To extract in­ dependent samples at the regional scale, an automated method is proposed by extending an observational method. In addition to the initial threshold, a minimum interval is used to identify two consecutive storms with a long storm interval, and a minimum level is used to distinguish one storm with a small fluctuation around the initial threshold from two consecutive storms with a short storm interval. By using this method, independent samples in the Yellow Sea are extracted from a 40-year (1979–2018) hindcast of significant wave heights. In this area, storms during tropical cyclones are generally strong, and most storms are driven by winter storms. Comprehensively considering the storm intensity and frequency, the independent sample is divided by an automated method into homogenous samples 1 and 2, depending on the tracks and recorded times of tropical cyclones. Homogenous samples 1 and 2 represent the independent samples in the tropical cyclone and nontropical cyclone, respectively. Within homogenous sample 2, most independent samples (especially large inde­ pendent samples) are observed in oceanographic winter. In addition, the difference between the return signifi­ cant wave heights based on homogenous sample 2 and those based on the winter storm sample is very small. Thus, seasonal declustering is not performed on homogenous sample 2. Consequently, by utilizing the inde­ pendent and identically distributed samples extracted by two automated methods, a regional analysis of the extreme significant wave heights in the Yellow Sea is performed.

1. Introduction Extreme significant wave heights (SWHs) and their return periods play significant roles in determining the design criteria for coastal de­ fences and offshore structures (Cai and Reeve, 2013; Silva-Gonz� alez et al., 2013; Solari et al., 2018), which implies a balance between expenditure and security (Petrov et al., 2013; Reeve, 2016; Sartini et al., 2017). Many early studies focused on extreme value theory (Goda, 1989; Coles, 2001; Yan et al., 2020), including the use of sampling methods and probability distribution models. Typical sampling methods are the annual maxima (AM) method (de Haan, 1970), k-largest maxima method (Weissman, 1978) and peak over threshold (POT) method (Goda et al., 2001), which are employed to extract the annual maximal SWH, annual k-largest SWH and peak SWH over the threshold, respectively, as the extreme samples. By fitting these samples, the Gumbel model (Gumbel, 1958), generalized extreme value model (Sobey and Orloff,

1995) and generalized Pareto distribution (GPD) model (Pickands, 1975) are used to construct long-term distributions and then extrapolate the required return values (Soares and Scotto, 2001, 2004; Izaguirre et al., 2012, 2013). Such sampling methods and distribution models can assess the extreme SWH but do not explicitly consider the independence of the samples. In particular, although it is more robust than other methods, the AM method may not ensure this independence since an annual window (e.g., from January to December or from summer to summer) may split a storm. Two annual maximal SWHs obtained from neighbouring years may originate from one wave event (Shao et al., 2018a). Special care must be taken to ensure that the assumption of inde­ pendence is adequately satisfied. Mendoza and Jim� enez (2005) defined storms as events exceeding an SWH of 1.5 m, which is approximately twice the annually averaged SWH in the northwest Mediterranean Sea (Jim� enez et al., 1997). To exclude a large number of small storms,

* Corresponding author. College of Engineering, Ocean University of China, 238 Songling Road, Qingdao, 266100, China. E-mail address: [email protected] (B. Liang). https://doi.org/10.1016/j.coastaleng.2020.103693 Received 31 August 2019; Received in revised form 15 March 2020; Accepted 15 March 2020 Available online 19 March 2020 0378-3839/© 2020 Elsevier B.V. All rights reserved.

Z. Shao et al.

Coastal Engineering 158 (2020) 103693

Mendoza and Jim�enez (2006) increased the SWH threshold used to define a storm inducing a significant beach response; they also defined the maximum separation (3 days) and secondary wave height threshold (1.5 m) to separate storms in a situation involving two peaks. Mazas and Hamm (2011) proposed a double threshold method, in which the low threshold was required to be sufficiently high to discriminate two consecutive storms and sufficiently low to underlie extreme samples extracted by the high threshold. Their experience in extratropical areas was that the annual mean number of storms was approximately between 5 and 10. Lerma et al. (2015) revisited this double threshold method and believed that two consecutive storms should be discriminated depend­ ing on the characteristics of observable weather conditions (Lemm et al., 1999; Zhang et al., 2017; Wu et al., 2018). Li et al. (2014) defined a storm as exhibiting consecutive SWHs exceeding 3 m; to guarantee the independence of storms, the minimum interval was set as 6 h. Kapelonis et al. (2015) used a simple declustering algorithm to obtain samples of individual waves. This algorithm sweeps through a time series database and rejects events in small proximity to a large event based on a pre­ defined minimum separation time (96 h). More recently, Gao et al. (2018) analysed the meteorological characteristics in the Yellow Sea and used a fixed window to ensure the independence of samples; they sug­ gested that the time window should range from 3 days to 5 days and recommended a fixed window of 5 days. Extreme waves of different physical processes have different parent distributions (Gumbel, 1958). Considering that a probability distribu­ tion model does not implicitly consider homogeneity assumption (Morton et al., 1997), samples must be considered to be homogeneous before extracting extreme samples (Mathiesen et al., 1994; Sartini et al., 2018). One common method is direction declustering, which separates samples into directional bins. By using this method, extreme samples can be extracted from the main directional bins, and different extreme dis­ tributions can be associated with different wave directions. Thus, a design can be optimized to protect a structure in a special direction (Jonathan et al., 2008). However, uncertainties exist in selecting the number and width of directional bins, and direction declustering cannot completely distinguish storms of different origins in the same directional bin. To reduce the influences of these uncertainties, Mazas and Hamm (2011) selected a relatively large directional width around the dominant direction covering 86% of the initial data. Lerma et al. (2015) analysed samples in different directional bins and studied whether a storm was associated with a single directional bin or with several discontinuous directional bins; in the latter case, they suggested that storms in different directional bins should be treated separately because their distributions may not be identical. In addition to direction declustering, seasonal declustering and weather declustering are widely used for homogeneous samples. Sea­ sonal declustering isolates samples into seasons, which can reflect the seasonality of the climate. Morton et al. (1997) separated samples by date into four subsets corresponding to the calendar seasons; they employed seasonal thresholds rather than adopting a single annual threshold to ensure that appropriate models were fitted to the sample in each season. Li et al. (2014) divided samples into oceanographic winter (Oct. to Mar. in the Northern Hemisphere) and oceanographic summer (Apr. to Sep. in the Northern Hemisphere). Because the representative characteristics of storms among the oceanographic seasons are similar, they analysed the extreme waves in these seasons. Weather declustering distinguishes samples under different weather conditions, which sat­ isfies the homogeneity assumption in theory. Solari and Alonso (2017) proposed a homogenous methodology based on a weather-pattern classification method and identified weather patterns that resulted in extreme wave conditions; they found that homogenous subsets are fairly insensitive to the number of employed weather patterns. These patterns are typical synoptic conditions for a given area that are usually regarded as the average fields of some atmospheric variables, such as pressure and wind (Camus et al., 2014; Pringle et al., 2014, 2015; Rueda et al., 2016a, 2016b). Recently, Liang et al. (2019a) analysed storms at high latitudes

in the South China Sea, where storms that occur during tropical cyclones are generally stronger than those during other weather events and the number of storms is sufficient to identify extreme samples. In the present study, an independent and identically distributed sample in the Yellow Sea is extracted based on a 40-year (1979–2018) time series hindcast of SWHs. By using visual observations at 18 study sites, the minimal SWH in the storm interval and the initial threshold are used to identify storms. Extending this method, an automated inde­ pendent sample selection method (AISSM) is proposed with the initial threshold, minimum interval and minimum level. By analysing the tracks and recorded times of tropical cyclones, an automated homoge­ nous sample selection method (AHSSM) is also proposed. To make suf­ ficient use of homogenous samples (Arns et al., 2013; Sartini et al., 2014, 2015a; Ruggiero et al., 2010; Silva-Gonz� alez et al., 2015; Laface et al., 2016; Davies et al., 2017), the POT method is used to extract extreme samples. Considering that the selection of the threshold is sensitive to the subjectivity of practitioners (Neelamani et al., 2007; Thompson �lez et al., 2017), an automated threshold se­ et al., 2009; Silva-Gonza lection method (ATSME) based on the characteristic of extrapolated SWHs (Liang et al., 2019a) is employed (see Appendix A). Then, as a natural method (Alves and Young, 2003; Men�endez et al., 2009; You, 2011; Solari and Losada, 2012a, 2012b; Vanem, 2015; Samayam et al., 2017), the GPD model is used to extrapolate the return SWH. Benefitting from the AISSM, AHSSM and ATSME, the spatial distribution of return SWHs is comparable. The remainder of this article is structured as follows. In the next section, the study region and initial database are presented. In Section 3, independent samples are extracted by the observational method and automated method. Section 4 shows an automated method for extract­ ing homogenous samples. Section 5 provides a regional analysis of extreme SWHs. Finally, the conclusions are given in Section 6. 2. Study region and initial database 2.1. Study region Storms are frequently associated with severe weather conditions such as winds circulating directly to or far from a site of interest (Sartini et al., 2015b). In other words, the study of storms should consider wind waves (Innocentini and Dos Santos Caetano Neto, 1996; Vitart et al., 1997; Heimbach and Hasselmann, 2000) and swells (Forget et al., 1995). These wind waves and swells account for more than half of the energy carried by all waves at the surface, surpassing the contributions of tides, tsunamis, coastal surges, and other phenomena (Semedo et al., 2011). Previous studies (Donelan et al., 1997; Ardhuin et al., 2009; Collard et al., 2009) showed that the swells generated from the westerly belt of the Southern Hemisphere propagate through most of the South­ ern Hemisphere and even to the Northern Hemisphere (such as the Northeast Pacific Ocean). However, the influence of such swells on waves in the Northwest Pacific Ocean is very small due to the shadowing effects of Southeast Asia and Australia. Liang et al. (2019b) analysed the swells generated from the westerly belt of the Northern Hemisphere and found that these swells influence waves throughout most of the North­ east Pacific Ocean and Northeast Atlantic Ocean. However, the in­ fluences of these swells on waves in the Northwest Pacific Ocean are still small due to the location of the wind wave zone and the propagation direction of the swells. Therefore, storms in the Northwest Pacific Ocean may be more associated with the local wind conditions. Solari and Alonso (2017) suggested that storms should be analysed during hurricanes or tropical cyclones in tropical areas. Young et al. (2012), Shao et al. (2018b) and Takbash et al. (2019) found that storms in the Northwest Pacific Ocean are always driven by tropical cyclones. By conducting a meteorological analysis, Shao et al. (2017) studied tropical cyclone waves in the South China Sea to assess extreme waves. However, storms at middle latitudes are typically generated by synoptic scale processes (Solari and Alonso, 2017). Upon performing a 2

Z. Shao et al.

Coastal Engineering 158 (2020) 103693

meteorological analysis, Gao et al. (2018) associated extreme waves with winter storms in the Yellow Sea. Mazas and Hamm (2011) studied the association between the num­ ber of extreme samples and the database length and suggested that the annual mean number of extreme samples ranges between approximately 2 and 5. If the database length is short, a number of approximately 5 is advisable to ensure that the annual mean number of extreme samples is sufficiently large; in contrast, if the database length is excessively long (approximately 40–50 years), a number of approximately 2 is more appropriate. In this study, a tropical cyclone in the Northwest Pacific Ocean is analysed. When the distance between the study site and the centre of a tropical cyclone is within 300 km, the corresponding tropical cyclone is recorded at the study site. In Fig. 1, the number of recorded tropical cyclones from 1979 to 2018 is shown. It can be observed that the middle latitudes display more than 80 recorded tropical cyclones, which means that extreme waves may be analysed during tropical cyclones. However, tropical cyclones rarely occur in the Yellow Sea, and thus, other weather events should be considered. Consequently, the Yellow Sea is selected as the study region (32� 37� N and 119� -127� E, as shown in Fig. 2) to analyse the extreme SWHs in an area that infrequently experiences tropical cyclones. In this region, 18 study sites (shown in Fig. 2) are employed to illustrate the detailed analysis process.

Fig. 2. The study region and 18 study sites.

non-stationary mode with spherical coordinates ranging from 3.5� N to 50� N and from 99� E to 150� E (shown in Fig. 1). Both the linear growth and the exponential growth of waves by winds are included (Cavaleri and Rizzoli, 1981; Snyder et al., 1981). Dissipation due to depth-induced wave breaking is treated according to the spectral formulation of Battjes and Janssen (1978); dissipation due to bottom friction is modelled using the Collins form (Collins, 1972); dissipation due to white-capping is applied with the formulations of Komen et al. (1984). Quadruplet and triad wave-wave interactions are activated using the discrete interaction approximation of Hasselmann et al. (1985) and the lumped triad approximation of Eldeberky (1996), respectively. A regular grid is employed with a mesh spacing of 0.1� in both longitude and latitude. The directional space is resolved in 48 equal directions. In the frequency space, the number of frequencies is 36, ranging from a minimum frequency of 0.03 Hz to a maximum frequency of 1 Hz. The modelling period spans from 1979 to 2018 with a calcu­ lation time step of 30 min and an output time step of 1 h. The bathy­ metric data, which have a spatial resolution of 0.1� in both longitude and latitude, are obtained from the General Bathymetric Chart of the Oceans. The winds used to drive the waves are provided by the ERAInterim (Dee et al., 2011) from the European Centre for Medium-Range Weather Forecasts, with a spatial resolution of 0.125� . Considering the influences of the boundary conditions, the wave pa­ rameters (namely, the SWH, peak period and wave direction) along the eastern boundary are extracted from a global wave hindcast (Liang et al., 2019b). In the study region, 5 buoys (shown in Fig. 1) are located along the coastlines of Korea and China. In Fig. 3, the hourly SWHs obtained via numerical modelling are compared with the hourly SWHs observed in 2014 at buoy B2, in 2014 at buoy B3 and in 2012 at buoy B4. The simulated SWHs match the observed data well, demonstrating the reli­ ability of the modelling results. Furthermore, validation statistics (i.e., the root mean square error (RMSE), scatter index (SI) and bias) for the simulated SWHs are presented in Table 1. These results also confirm the reliability of the simulated SWHs.

2.2. Initial database Considering that the return SWH is highly dependent on the initial database, this database (consisting of, for example, buoy measurements, satellite measurements and numerical hindcasts) must carefully be selected. In this study, a 40-year hindcast of SWHs is employed as the initial database, which is simulated by the Simulating WAves Nearshore (SWAN) model (Booij et al., 1999; Ris et al., 1999). This model is widely used for simulating waves at the global (Liang et al., 2019b), regional (Dragani et al., 2008; Codignotto et al., 2012) and local (Mortlock et al., 2014; Rusu et al., 2015) scales, which now runs in two-dimensional

3. Extraction of independent samples 3.1. Observational method The time series SWHs from 1979 to 2018 at location #9 are shown in Fig. 4. Evidently, the SWHs are excessively large, and the wave process is complex, especially for small wave events. To reduce the analysed SWHs and avoid the influences of small wave events, an initial threshold is needed. Mazas and Hamm (2011) suggested that the physical threshold

Fig. 1. The number of tropical cyclones recorded in the Northwest Pacific Ocean (the rectangle represents the study region, and the circles represent the buoy locations). 3

Z. Shao et al.

Coastal Engineering 158 (2020) 103693

Fig. 3. Time series comparisons between the simulated and measured SWHs in 2014 at buoy B2, in 2014 at buoy B3 and in 2012 at buoy B4.

storms are two individual wave events. In case 2 (Fig. 5 (b)), the storm interval is very short (4 h), which means that two consecutive storms belong to one wave event. In cases 3 (Fig. 5 (c)) and 4 (Fig. 5 (d)), the storm intervals (38 h and 40 h, respectively) are relatively short and very similar. To distinguish case 3 from case 4, the minimal SWH in the storm interval is analysed with the SWHs at the initiation of the previous storm (IPS) and the end of the following storm (EFS). When the minimal SWH is similar to the SWHs at the IPS or EFS, two consecutive storms can be regarded as two individual wave events (case 3). When the minimal SWH is obviously larger than the SWHs at the IPS and EFS, two consecutive storms can be regarded as one wave event (case 4). Consequently, the initial threshold cannot absolutely ensure sample independence due to possible fluctuations around this threshold (cases 2 and 4). In these cases, only one maximal SWH is extracted as the inde­ pendent sample. In the other cases (cases 1 and 3), the maximal SWHs of both consecutive storms are extracted as the independent sample. At location #9, 26 groups of neighbouring storms present characteristics similar to cases 2 and 4; thus, 374 maximal SWHs are extracted (Fig. 6 (a)). Visual observations are also employed at the other 17 study sites. For example, 375 and 369 independent samples are extracted at

Table 1 Observation periods and validation statistics at 5 buoys. Location

Observation Period (Hour Day/ Month/Year)

RMSE (m)

SI

Bias (m)

B1 B2 B3 B4 B5

00 01/01/2012-23 31/12/2012 00 01/01/2014-23 31/12/2014 00 01/03/2014-23 31/12/2014 00 01/01/2012-23 26/08/2012 00 01/09/2014-23 31/08/2015

0.22 0.22 0.25 0.30 0.28

0.35 0.27 0.27 0.22 0.39

0.09 0.04 0.04 0.01 0.04

Fig. 4. Time series SWHs from 1979 to 2018 at location #9 (the red line represents the initial threshold).

should be associated with the annual mean number of storms, approx­ imately between 5 and 10, which is approximately twice the annual mean number of extreme samples. In the Yellow Sea, Gao et al. (2018) assessed the extreme SWHs at 4 study sites; the annual mean number of extreme samples ranged from 4.24 to 5.12 based on a 22-year hindcast. In this study, the database length is 40 years, which means that an annual mean number of extreme samples of approximately 2 and an annual mean number of storms of approximately 5 may be appropriate. To ensure that the initial threshold is sufficiently small to be below the threshold, the annual mean number of storms is extended to 10. At location #9, 2.84 m is determined and 400 storms are extracted from 0.35 million time series SWHs. In Section 4, the extreme sample is analysed. The number of extreme samples is obviously smaller than the number of storms, which validates the definition of the initial threshold. Benefitting from the initial threshold, visual observations are possible and reveal 4 cases of extracted storms. In case 1 (Fig. 5 (a)), the storm interval is very long (1793 h), which means that two consecutive

Fig. 5. Four cases of neighbouring storms at location # 9 (the red line repre­ sents the initial threshold, and the green line represents the minimum level). 4

Z. Shao et al.

Coastal Engineering 158 (2020) 103693

similar to case 1, and no instances of case 4 are observed, which means that two consecutive storms with a storm interval longer than this long limitation can be regarded as two individual wave events. Long limita­ tions and short limitations also exist at the other 17 study sites, as shown in Table 2. In Fig. 7, the distribution of minimal SWHs is disorderly (case 3 exists together with case 4) when the storm interval ranges from 33 h to 106 h. Therefore, a quantitative criterion is needed to distinguish case 3 from case 4; accordingly, this criterion can replace the qualitative criterion in the observational method. The initial thresholds and mean SWHs at the 18 study sites are shown in Table 2. The initial threshold is approxi­ mately 2.43–3.44 times the mean SWH. As the mean SWH reflects the mean wave condition, this value may be set as an identification index. When the minimal SWH in the storm interval is smaller than the mean SWH, two consecutive storms can be regarded as two individual wave events. In contrast, two consecutive storms can be regarded as one wave event. According to the above analyses, long limitations and short limita­ tions can be used to identify cases 1 and 2, respectively; furthermore, the mean SWH can be used to distinguish case 3 from case 4. Cases 1 and 3 represent two individual wave events with a long storm interval and a short storm interval, respectively, whereas cases 2 and 4 represent one wave event with a small fluctuation around the initial threshold. The mean SWH can also be used to distinguish cases 1 and 3 from cases 2 and 4. Thus, analyses of long and short limitations can be omitted. By comparing the 399 minimal SWHs in the storm interval with the mean SWH, the independent sample can be extracted. However, the analysis process is still complex. To simplify this process, a long limitation is reserved. Under this condition, only 44 minimal SWHs in the storm in­ terval need to be compared with the mean SWH at location #9. Consequently, the AISSM is proposed based on the observational method and can be summarized as follows:

Fig. 6. Independent samples extracted by visual observations at locations #9 (a), #1 (b) and #18 (c).

locations #1 (Fig. 6 (b)) and #18 (Fig. 6 (c)), respectively.

(1) Initial threshold. Extract the storm from the initial database; (2) Minimum interval. Identify two consecutive storms with a long storm interval, and extract their maximal SWHs as the indepen­ dent sample; (3) Minimum level. Identify two consecutive storms with a short storm interval, and extract their maximal SWHs as the indepen­ dent sample. Identify one storm with a small fluctuation around the initial threshold, and extract its maximal SWH as the inde­ pendent sample.

3.2. Automated method By using the observational method, 7 200 storms are analysed to distinguish cases 1 and 3 from cases 2 and 4 at the 18 study sites. Hence, it is difficult to apply this method to multiple study sites. To improve the extraction efficiency, the storm interval and the minimal SWH within this interval are analysed. In Fig. 7, the storm intervals are sorted in ascending order at location #9, and the corresponding minimal SWHs generally decrease. When the storm interval is smaller than 32 h (short limitation), the minimal SWH is slightly smaller than the initial threshold. Under this condition, all neighbouring storms are similar to case 2, and no instances of case 3 are observed, which means that two consecutive storms with a storm interval smaller than this short limi­ tation can be regarded as one wave event. When the storm interval is larger than 107 h (long limitation), the minimal SWH is obviously smaller than the initial threshold. Here, all neighbouring storms are

In the AISSM, the definition of the initial threshold is the same as that Table 2 Statistics for the initial thresholds, mean SWHs, short limitations and long lim­ itations at the 18 study sites.

Fig. 7. The storm intervals and corresponding minimal SWHs at location #9. 5

Location

Initial Threshold (m)

Mean SWH (m)

Short Limitation (h)

Long Limitation (h)

#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14 #15 #16 #17 #18

1.83 2.43 2.76 2.96 2.89 2.41 1.93 2.67 2.84 2.99 3.04 2.72 1.02 2.16 2.78 3.06 3.15 3.12

0.68 0.83 0.91 0.93 0.87 0.70 0.69 0.84 0.95 0.99 0.99 0.90 0.42 0.83 1.01 1.07 1.11 1.10

35 24 25 29 27 39 25 36 32 39 38 28 32 36 29 35 34 27

100 96 116 102 95 101 96 111 107 103 103 102 102 114 112 105 92 109

Z. Shao et al.

Coastal Engineering 158 (2020) 103693

in the observational method, and the minimum level is set as the mean SWH (shown in Fig. 5). As shown in Table 2, the long limitations at the 18 study sites are all smaller than 120 h. As the weather window in the Yellow Sea is approximately 3–5 days (Gao et al., 2018), the minimum level is set as 120 h. To validate the definitions of the minimum interval and minimum level, their sensitivities in extracting the independent sample are stud­ ied. At location #9, the sensitivity analysis of the minimum interval is shown in Fig. 8 (a). When the minimum interval increases from 0 to 107 h, the number of independent samples gradually decreases. If the min­ imum interval is set as a value in this range, the number of independent samples will be larger than that extracted by the observational method, and some neighbouring storms similar to cases 2 or 4 will be neglected. When the minimum interval is larger than 107 h, the number of inde­ pendent samples will be constant, and if the minimum interval is obvi­ ously larger than 107 h, superfluous analyses will reduce the efficiency. Thus, 120 h is reasonable as a minimum interval. The sensitivity analysis of the minimum level is shown in Fig. 8 (b). When the minimum level increases from 0.22 m to 2.84 m, the number of independent samples gradually increases. If the minimum level is set as 0.22 m, all neighbouring storms with a storm interval smaller than 120 h are regarded as one wave event; the number of independent samples is smaller than that extracted by the observational method, and 24 groups of neighbouring storms similar to case 3 are neglected. If the minimum level is set as 2.84 m, 400 storms exceeding the initial threshold are all regarded as individual wave events; the number of independent samples is larger than that extracted by the observational method, and 26 groups of neighbouring storms similar to cases 2 or 4 are neglected. To specifically analyse the sensitivity of the minimum level to the identifications of cases 2, 3 and 4, the lost sample percentage and false sample percentage at location #9 are studied (Fig. 9). Compared with the independent sample extracted by the observational method, the lost sample percentage and false sample percentage are defined as the

Fig. 9. Lost sample percentage (blue circle) and false sample percentage (red circle) at location #9.

relative numbers of lost samples and false samples, respectively. When the minimum level increases from 0.22 m to 0.89 m, the lost sample percentage decreases; if the minimum level is set as a value in this range, some neighbouring storms similar to case 3 are neglected. When the minimum level is larger than 0.89 m, the lost sample percentage is equal to zero. The false sample percentage is zero when the minimum level is smaller than 1.01 m. However, when the minimum level increases from 1.01 m to 2.84 m, the false sample percentage gradually increases; if the minimum level is set as a value in this range, some neighbouring storms similar to cases 2 or 4 are falsely regarded as case 3. These analyses reveal that the minimum level should be set as a value between 0.89 m and 1.01 m. Thus, a mean SWH of 0.95 m as a minimum level is reasonable. Sensitivity analyses of the minimum interval and minimum level conducted at the other 17 study sites show a similar conclusion. The above definitions are suitable for extracting independent samples, and the samples excellently match those extracted by visual observations in subsection 3.1. In Appendix B, the excluded SWHs in cases 2 and 4 are analysed with the independent samples shown in Fig. 6, and this analysis further demonstrates that these samples are reasonable. 3.3. Comparison with other methods 3.3.1. Fixed window method In the fixed window method, a fixed window associated with the meteorological characteristics of a given region is used to extract the independent sample. In the Yellow Sea, Gao et al. (2018) suggested that this window could be selected from 3 days to 5 days through association with the weather windows of winter storms. To reduce the analysed data, they also suggested employing an initial threshold. At location #9, three windows of 3, 4 and 5 days are used to extract the independent sample with an initial threshold of 2.84 m, and 470, 422 and 420 independent samples, respectively, are obtained. The number of samples in the first group is obviously larger than the cor­ responding numbers in the second and third groups, which means that the extracted independent samples are highly dependent on the de­ cisions made by practitioners. Nevertheless, compared with the inde­ pendent samples extracted by the AISSM (or the observational method), these numbers are obviously large. During a storm, the period of SWHs exceeding the initial threshold may be very long. In this condition, a fixed window may extract two SWHs from an individual wave event. This deficiency can be overcome by increasing the initial threshold, necessitating a higher-quality defi­ nition. In addition, the fixed window method cannot identify case 3; in other words, if two individual wave events have a short storm interval, this method may extract only one SWH.

Fig. 8. Sensitivity analyses of the minimum interval (a) and minimum level (b) at location #9: (a) from 0 to 200 h with intervals of 1 h; (b) from 0.22 m to 2.84 m with intervals of 0.01 m.

3.3.2. Double threshold method In the double threshold method, a physical threshold is used to 6

Z. Shao et al.

Coastal Engineering 158 (2020) 103693

extract the independent sample. At location #9, six physical thresholds are determined depending on the suggestion of Mazas and Hamm (2011). Compared with the independent samples extracted by the AISSM (or the observational method) shown in Table 3, the numbers of independent samples extracted by the double threshold method are large. In fact, the role of the physical threshold in the double threshold method is similar to that of the initial threshold in the AISSM. However, the double threshold method lacks a definition for the minimum level, and the use of a physical threshold cannot result in the identification of cases 2 and 4. In Table 3, when the physical threshold ranges from 2.84 m to 3.23 m, the difference in the numbers of independent samples extracted by the double threshold method and AISSM decreases. To reduce the in­ fluences of cases 2 and 4, two other physical thresholds (3.35 m and 3.49 m) are determined. When the physical threshold is set as 3.49 m, the independent samples in the double threshold method and AISSM are the same. However, as shown in Section 4, the threshold at location #9 is 3.50 m, which is slightly higher than 3.49 m. Mazas and Hamm (2011) suggested that the physical threshold needs to be sufficiently lower than the threshold. A sufficiently low physical threshold can ensure the number of storms required by the selection of the threshold (Li et al., 2012). To increase the number of independent samples, a lower physical threshold is needed. However, Appendix B presents an SWH of 3.48 m. When the physical threshold decreases, a falsely selected independent sample is unavoidable, which reveals the significant role of the mini­ mum level.

Fig. 10. Tracks of tropical cyclones: (a) within the study region (the black rectangle); (b) at location #9 by a fixed distance; (c) for homogenous sample 1 at location #9.

sample is within the recorded time of the identified tropical cyclone, this sample is extracted as homogenous sample 1. For example, at location #9, the occurrence time of 29 independent samples matches the recor­ ded time of 29 tropical cyclones (Fig. 10 (c)). These independent sam­ ples are extracted as homogenous sample 1 (Fig. 11). Comparing this sample with the independent sample shown in Fig. 6 (a), the largest 6 independent samples occur within the tropical cyclone, which means that the tropical cyclone dominates the strong storm. 4.2. Homogenous sample in a non-tropical cyclone Excluding homogenous sample 1 from the independent sample, the remaining samples are analysed in every calendar month. This analysis is particularly necessary considering that these samples may come from multiple weather events. The remaining 345 independent samples at location #9 are shown in Fig. 12. These samples are generally smaller than homogenous sample 1, as shown in Fig. 11. Table 4 describes the numbers and largest values of the remaining samples in the 12 calendar months. The number of the remaining samples from Oct. to Mar. is obviously larger than that from Apr. to Sep.; the number of remaining samples in oceanographic winter is 281, accounting for more than 81% of the remaining independent samples. In addition, the largest value of the remaining samples from Oct. to Mar. is obviously larger than that from Apr. to Sep.; relatively strong storms occur mainly in oceanographic winter. These phenomena match the meteorological analysis in the Yellow Sea (Gao et al., 2018); that is, winter storms dominate large waves. Considering that the remaining samples (especially the large sam­ ples) are observed mainly in oceanographic winter, seasonal decluster­ ing may not be necessary for these samples. To validate this hypothesis, the extreme SWHs are extrapolated under the non-tropical cyclone and winter storm conditions based on the remaining independent sample and this sample in oceanographic winter, respectively. In the remaining independent sample, the extreme sample is extracted by the ATSME. At location #9, a common stable threshold range (from 3.03 m to 3.50 m) is identified for 50-year, 100-year, 150-

4. Extraction of homogenous samples 4.1. Homogenous sample in a tropical cyclone Fig. 1 illustrates that the number of tropical cyclones in the Yellow Sea is relatively small. Thus, the number of independent samples in the tropical cyclone may be insufficient for assessing extreme waves. However, considering that tropical cyclones always drive strong storms, the independent samples during such weather events cannot be neglected. From 1979 to 2018, 149 tropical cyclones were recorded in the study region (Fig. 10 (a)). Within these tropical cyclones, not all cyclones can influence waves at a specified location. To estimate this influence pre­ liminarily, a fixed distance (300 km) is employed. This distance allows some small influences to be identified. Under this condition, the track of the tropical cyclone is far from the specified location; alternatively, the intensity of the tropical cyclone is weak. However, no large influences are neglected (Liang et al., 2019a). Under this condition, the track of the tropical cyclone is close to the specified location, and the intensity of the tropical cyclone is strong. For example, 49 tropical cyclones are deemed to influence the waves at location #9 (Fig. 10 (b)). After identifying the tropical cyclones influencing waves at specified locations, temporal matching is used to extract the independent sample in this weather (the time lag does not influence the temporal matching, as shown in Appendix C). When the occurrence time of the independent Table 3 Statistics for the physical threshold and number of independent samples at location #9. Physical Threshold (m)

Number of Independent Samples by the Double Threshold Method

Number of Independent Samples by the AISSM

2.84 2.89 2.96 3.03 3.13 3.23 3.35 3.49

400 360 320 280 240 200 160 120

374 340 307 271 234 197 158 120

Fig. 11. Homogenous sample 1 at location #9. 7

Z. Shao et al.

Coastal Engineering 158 (2020) 103693

To study the necessity of the sample homogeneity, the return SWH is extrapolated based on the independent sample. At location #9, the GPD model is directly fitted to this sample exceeding 3.50 m. The corre­ sponding return level plot and quantile plot are shown in Fig. 14 (a) and 14 (b), respectively. Considerable differences are observed between Fig. 13 (a) and 14 (a) and between Fig. 13 (b) and 14 (b). The fitting result is unsatisfactory based on the independent sample, and the largest 5 extreme samples obviously deviate from the model results. These phenomena reveal that the extreme samples may originate from more than one population; for example, the largest 5 extreme samples may be sourced from the tropical cyclone, while the other extreme samples may originate from the winter storm. Table 6 provides the 50-year, 100-year, 150-year and 200-year return SWHs with their confidence intervals. These return values are obviously larger than the return SWHs in the non-tropical cyclone because the large sample in the tropical cyclone condition is included in the extrapolation. Moreover, these confidence intervals are obviously wider than the confidence intervals in the non-

Fig. 12. Homogenous sample 2 at location #9.

year and 200-year return periods. By fitting the GPD model with the remaining independent sample exceeding 3.50 m, the return SWHs are extrapolated (Fig. 13 (a)). The likelihood method (Schendel and Thongwichian, 2017) reparametrizes the likelihood in terms of the un­ known quantile and uses profile likelihood arguments to construct a 95% confidence interval. Table 5 provides the 50-year, 100-year, 150-year and 200-year return SWHs with their confidence intervals. These confidence intervals indicate that the variance in the extrapolated SWHs is acceptable. To estimate the asymptotic tail approximation, the quantile plot at location #9 (Fig. 13 (b)) is analysed, which shows that there are generally few differences between the empirical and fitted quantiles, indicating a good fit for the return SWH. To compare the return SWH in the winter storm with that in the nontropical cyclone, a threshold of 3.50 m is used to extract the extreme sample in the winter storm. By fitting the GPD model with this sample, the return SWHs are extrapolated (Fig. 13 (c)), and the corresponding quantile plot is obtained (Fig. 13 (d)). Few differences are observed between Fig. 13 (a) and 13 (c) and between Fig. 13 (b) and 13 (d). To analyse these differences quantitatively, the 50-year, 100-year, 150-year and 200-year return SWHs with their confidence intervals in the winter storm are assessed (Table 5). Compared with these values in the nontropical cyclone, the differences in the return SWHs and confidence in­ tervals are very small. This outcome can be explained by the fact that the remaining independent sample in oceanographic summer is observed at the bottom of this sample, which has a weak influence on the extrapo­ lation. Consequently, to simplify the sample homogeneity analysis process, seasonal declustering is not performed on the remaining inde­ pendent sample, and all remaining independent samples are directly extracted as homogenous sample 2.

Fig. 13. Return level plots ((a) and (c)) and quantile plots ((b) and (d)) at location #9: (a) and (b) in the non-tropical cyclone; (c) and (d) in the winter storm condition.

4.3. Concluding remarks

Table 5 Statistics for the thresholds and return SWHs with 95% confidence intervals at location #9 for the non-tropical cyclone and winter storm.

According to the above analyses, the AHSSM is proposed and can be summarized as follows:

Weather

(1) Fixed distance. Identify the tropical cyclone influencing the waves at a specified location; (2) Temporal matching. Extract the independent sample as homog­ enous sample 1 when the occurrence time of the independent sample matches the recorded time of the identified tropical cyclone; then, extract the rest of the remaining independent samples as homogenous sample 2.

NonTropical Cyclone Winter Storm

Threshold (m)

Return SWHs (m) with 95% Confidence Intervals 50-year

100-year

150-year

200-year

3.50

4.50 (4.40, 4.72) 4.51 (4.40, 4.72)

4.57 (4.46, 4.86) 4.57 (4.47, 4.86)

4.60 (4.49, 4.93) 4.60 (4.50, 4.94)

4.62 (4.51, 4.98) 4.61 (4.51, 4.99)

3.50

Table 4 Statistics for the remaining independent samples in the 12 calendar months at location #9. Month

Jan.

Feb.

Mar.

Apr.

May.

Jun.

Jul.

Aug.

Sep.

Oct.

Nov.

Dec.

Number of Independent Samples Largest Value of Independent Samples (m)

52 4.19

49 4.27

33 4.58

19 3.66

11 3.45

5 3.80

8 3.62

11 3.99

10 3.83

33 4.27

49 4.56

65 4.25

8

Z. Shao et al.

Coastal Engineering 158 (2020) 103693

than the selection of the threshold. Therefore, the sample homogeneity is essential for the study of extreme waves. 5. Spatial analysis of extreme waves In the study region, the time series SWHs at all grid nodes are eval­ uated by the AISSM. Fig. 16 (a) depicts the distribution of the initial threshold, which gradually decreases from the offshore zone to the nearshore zone because storms are fundamentally influenced by the water depth, which is obviously deeper in the offshore zone. In the eastern nearshore zone, the initial threshold is generally larger than that in the western nearshore zone; in the southern study region, the initial threshold is generally larger than that in the northern study region. These phenomena can also be associated with the water depth, which is obviously greater in the eastern nearshore zone and southern study re­ gion. In addition to the water depth, these phenomena can be associated with the exposure of each site to storms. On the one hand, the study region is partly protected by the Korean Peninsula. The fetch is relatively narrow in the northern study region, and swells coming from the east have difficulty reaching this region. On the other hand, a relatively large number of tropical cyclones affects the eastern nearshore zone and southern study region (Fig. 1). During such tropical cyclones, storms are generally stronger than during other weather events. Fig. 16 (b) illus­ trates the distribution of the number of independent samples. At most locations, this number exceeds 350. However, the number of indepen­ dent samples at some nearshore locations is relatively small because the storms therein are relatively moderate. Considering that this number is still larger than 200, however, the definition of the initial threshold is not adjusted at these locations. By applying the AHSSM at all grid nodes in the study region, ho­ mogenous samples 1 and 2 are extracted. The distributions of the

Fig. 14. Return level plot (a) and quantile plot (b) at location #9 based on the independent sample exceeding 3.50 m. Table 6 Statistics for the thresholds and return SWHs with 95% confidence intervals at location #9 based on the independent sample. Threshold (m)

Return SWHs (m) with 95% Confidence Intervals 50-year

100-year

150-year

200-year

3.50

5.69 (5.20, 6.85) 6.33 (5.48, 9.71)

6.21 (5.54,8.06) 7.16 (5.84, 13.52)

6.53 (5.74, 8.92) 7.72 (6.04, 16.73)

6.77 (5.84, 9.60) 8.16 (6.23, 19.57)

3.80

tropical cyclone. In Fig. 14 (a), more samples are used to extrapolate the return SWHs than in Fig. 13 (a). However, the uncertainty in Fig. 14 (a) is obviously greater than that in Fig. 13 (a). These phenomena reveal that the extrapolation demonstrated in Fig. 14 (a) is unreliable. To avoid the influence resulting from threshold selection, the ATSME is applied to the independent sample. At location #9, a threshold of 3.80 m is identified to extract extreme samples. By fitting the GPD model with this sample, the return SWHs are extrapolated (Fig. 15 (a)), and the corresponding quantile plot is obtained (Fig. 15 (b)). Comparing these figures with Fig. 14 (a) and 14 (b), the fitting result is relatively acceptable because some small independent samples are excluded from the extreme sample. Comparing these figures with Fig. 13 (a) and 13 (b), however, the fitting result is still unsatisfactory because the extreme sample still originates from multiple populations. Table 6 provides the 50-year, 100-year, 150-year and 200-year return SWHs with their con­ fidence intervals. These return values are larger than the return SWHs based on the independent sample exceeding 3.50 m because the large sample plays a significant role in the extrapolation. Furthermore, these confidence intervals are obviously wider than the confidence intervals based on the independent sample exceeding 3.50 m. When the number of extreme samples decreases, the uncertainty in the extrapolation in­ creases, and the deficiency resulting from multiple populations in the distribution is amplified. These phenomena reveal that the main issue plaguing the assessment is the population of extreme samples rather

Fig. 15. Return level plot (a) and quantile plot (b) at location #9 based on the independent sample exceeding 3.80 m.

Fig. 16. Spatial distributions of the initial threshold (a) and number of inde­ pendent samples (b) in the study region. 9

Z. Shao et al.

Coastal Engineering 158 (2020) 103693

maximal SWHs under tropical cyclone and non-tropical cyclone condi­ tions are shown in Fig. 17. Under tropical cyclone conditions, relatively large maximal SWHs are observed throughout most of the southeastern study region. The pattern shown in Fig. 17 (a) highly matches the pattern of recorded numbers of tropical cyclones shown in Fig. 1. Because the frequency and intensity of tropical cyclones vary consid­ erably, the influence of the weather on the distribution of maximal SWHs is generally stronger than that of the water depth. Under nontropical cyclone conditions, relatively large maximal SWHs are observed mostly in the centre and southeastern extents of the study re­ gion. Because the variation in winter storms is relatively small, the in­ fluence of the water depth on the distribution is generally stronger than that of the weather. Comparing the maximal SWHs under non-tropical cyclone conditions with those under tropical cyclone conditions, the maximal SWH in homogenous sample 2 is generally smaller than that in homogenous sample 1. However, in parts of the western and eastern nearshore zones (especially the western nearshore zone), the maximal SWH of homogenous sample 1 is relatively small due to the relatively weak intensity of tropical cyclones therein. The regional analysis of extreme SWHs requires a uniform approach to obtain spatially comparable results (Chini et al., 2010; Izaguirre et al., 2010, 2011; Perez et al., 2017; Vanem, 2017; Polnikov et al., 2017; Li et al., 2018). In the study region, the number of homogenous sample 1 limits the assessment. However, this limitation does not exist for ho­ mogenous sample 2. By using the ATSME, the extreme samples under non-tropical cyclone conditions can be identified at every grid node. Then, the GPD model can be fitted for extrapolation. Fig. 18 depicts the distributions of the 50-year, 100-year, 150-year and 200-year return SWHs. These distributions generally match the pattern shown in Fig. 17 (b). The maximal SWH in a 40-year period is slightly smaller than the

Fig. 18. Spatial distributions of the 50-year (a), 100-year (b), 150-year (c) and 200-year (d) return SWHs under non-tropical cyclone conditions.

50-year return SWH, which validates the rationality of this extrapolation. 6. Conclusions In this study, an independent and identically distributed sample in the Yellow Sea is analysed. In this area, the influence of swells is rela­ tively small, and storms are strongly coupled to the local wind condi­ tions. To extract the sample under multi-weather conditions (including tropical cyclones and winter storms), the initial database needs to be consecutive rather than intermittent during a single weather event (e.g., a tropical cyclone). Thus, a 40-year time series hindcast of SWHs is employed. To extract the independent sample from a consecutive initial data­ base, a declustering method is needed. At specified locations, an observational method can be employed. The initial threshold is used to reduce the analysed data and emphasize the characteristics of storms. To ensure a sufficient number of independent samples, this threshold is set as a flexible and reliable value corresponding to an annual mean number of storms of 10. The minimal SWH in the storm interval is analysed with the SWHs at the IPS and EFS to distinguish cases 1 and 3 from cases 2 and 4. According to this criterion, the fluctuations around the initial threshold are excluded from the selected storms, and the independence of the remaining storms is guaranteed. At the regional scale, the AISSM, which is proposed by extending the observational method, can be employed. The definition for the initial threshold is the same, and the independent samples are highly consistent. In the AISSM, the minimum interval, which is set as 120 h, is used to simplify the analysis process (identify case 1). In addition, the minimum level, which is set as the mean SWH, is used to distinguish case 3 from cases 2 and 4. Comparing the AISSM with other declustering methods (e.g., the fixed window method and double threshold method), the significant roles of the flexible sampling window and minimum level are revealed. With the independent sample, the AHSSM is proposed to extract the homogenous sample. By analysing the tracks and recorded times of tropical cyclones, the independent sample in the tropical cyclone is extracted as homogenous sample 1, and the remaining independent samples are extracted as homogenous sample 2. In homogenous sample 2, the samples are analysed in every calendar month. The results show that these samples (especially large samples) occur primarily in ocean­ ographic winter. As the difference between the return SWHs based on homogenous sample 2 and those based on the winter storm sample is

Fig. 17. Spatial distributions of maximal SWHs under tropical cyclone (a) and non-tropical cyclone (b) conditions. 10

Z. Shao et al.

Coastal Engineering 158 (2020) 103693

very small, seasonal declustering is not applied to homogenous sample 2. Benefitting from both the AISSM and the AHSSM, the spatial distri­ butions of extreme waves under tropical cyclone and non-tropical cyclone conditions are comparable and effectively match the meteoro­ logical and bathymetric conditions, respectively. The results show that the maximal value in homogenous sample 1 is generally larger than that in homogenous sample 2. According to these studies, two criteria are developed from a wave impact perspective in the context of coastal and offshore engineering. To guarantee the design security in the study re­ gion, a large SWH in the tropical cyclone and non-tropical cyclone conditions is employed. To reduce engineering expenditures (especially in the open ocean), the return SWHs under non-tropical cyclone con­ ditions can be employed because winter storms occur frequently. In other sea areas, the AISSM and AHSSM may still be applicable to the extraction of independent and identically distributed samples. For example, swells may have considerable influences. Under this condition, wind seas and swells can be separated by a numerical simulation or an empirical method (Zheng and Li, 2017). With the swell parameters, the AISSM can be used to directly extract the sample. In addition, the weather in the target sea may be complex. Hence, to identify different populations, homogenous sample 2 extracted by the AHSSM should be

further analysed by implementing direction declustering or seasonal declustering because little information is historically recorded during non-tropical cyclone weather. Declaration of competing interest There are no conflicts of interest to declare. CRediT authorship contribution statement Zhuxiao Shao: Writing - original draft. Bingchen Liang: Writing original draft. Huijun Gao: Writing - original draft. Acknowledgments The authors would like to acknowledge the support of the National Science Fund (Grant No. 51739010, 51679223, 51609224), the 111 Project (No. B14028), the Shandong Provincial Natural Science Key Basic Program (Grant No. ZR2017ZA0202), a grant from the 7th Gen­ eration Ultra-Deep-water Drilling Rig Innovation Project, the Funda­ mental Research Funds for the Central Universities (201513056) and the Shandong Provincial Natural Science Foundation (ZR2015PE019).

Appendix A. Threshold selection method �nchez-Arcilla et al., 2008; Bernardara et al., 2014), empirical methods (Lang et al., Many methods, such as graphical diagnostics (Coles, 2001; Sa 1999; Ferreira et al., 2003; Reiss and Thomas, 2007), probabilistic-based techniques (Hill, 1975; Beirlant et al., 2006; Goegebeur et al., 2008), computational approaches (Beguería, 2005; Solari et al., 2017; Liang et al., 2019a) and mixture models (Carreau and Bengio, 2009; Eastoe and Tawn, 2010; Solari and Losada, 2012b) have been proposed in previous studies for the selection of thresholds. Considering that a suitable threshold can be automatically and uniquely determined with the ATSME (Liang et al., 2019a), this method is employed in this study. The terms u1 , …, um are equally spaced with an increasing candidate threshold. Hsi;j represents the i-year return SWH based on the threshold of uj . The difference, ΔHsi;j , in i-year return SWHs (Hsi;j and Hsi;j 1 ) for neighbouring thresholds (uj and uj 1 ) is defined as follows: ΔHsi;j ¼ Hsi;j

Hsi;j

(1)

1

(1) Candidate threshold. Identify the suitable range for equally spaced and increasing candidate thresholds, (u1 , um ), and the threshold interval, Δu ¼ umNtotu1 . u1 is set as the minimal sample, um is set as the maximal sample, and Ntot is set as the number of samples.

(2) Return period and value. Choose the order of i (i ¼ i1 ; :::; ini ) for different return periods, which are dependent on NT and the requirements of practitioners. Extrapolate the i-year return SWH, Hsi;j , which corresponds to every candidate threshold, uj . (3) Stable threshold range. Calculate the difference, ΔHsi;j , in the return SWHs for neighbouring thresholds. Define a characteristic parameter, chi;j , to record the stable characteristics of return SWHs. Find the uniquely stable threshold range for the i-year return period. (4) Suitable threshold. Determine the suitable threshold within the stable threshold range.

The ATSME selects a unique threshold within a uniquely stable threshold range for a specific return period. In theory, every threshold within the stable threshold range can be selected as a suitable threshold because the variation in return SWHs in this range is generally small. To guarantee the security of the design wave height, the highest threshold in the common stable threshold range is selected as the suitable threshold in this study. Appendix B. Excluded SWHs in cases 2 and 4 At locations #9, #1 and #18, 26, 25 and 31 groups of neighbouring storms similar to cases 2 or 4 are identified by the AISSM. The corresponding excluded maximal SWHs, which are located at the bottom of the independent samples shown in Fig. 6, are presented in Fig. 19.

11

Coastal Engineering 158 (2020) 103693

Z. Shao et al.

Fig. 19. Excluded maximal SWHs at locations #9 (a), #1 (b) and #18 (c).

Appendix C. Time lag A time lag exists between a tropical cyclone and a tropical cyclone wave. To study the influence of this time lag on the extraction of homogenous sample 1, the time series SWHs during the recorded time of the identified tropical cyclone are analysed with an independent storm exceeding the initial threshold. At location #9, the largest independent sample (6.82 m) is observed during tropical cyclone Bolaven in 2012 (Fig. 20). The duration of the tropical cyclone (from 08 20/08 to 14 29/08) covers the exceeding duration of the storm (from 08 27/08 to 14 28/08). It can be concluded that this independent sample in the tropical cyclone can be extracted by temporal matching because the recorded time covers the whole period of the tropical cyclone rather than covering only the strong wind period.

Fig. 20. Time series SWHs at location #9 during tropical cyclone Bolaven in 2012 (the red line represents the initial threshold).

References

Arns, A., Wahl, T., Haigh, I.D., Jensen, J., Pattiaratchi, C., 2013. Estimating extreme water level probabilities: a comparison of the direct methods and recommendations for best practise. Coast. Eng. 81, 51–66. Battjes, J.A., Janssen, J.P.F.M., 1978. Energy loss and set-up due to breaking of random waves. In: Proceedings of 16th International Conference on Coastal Engineering. ASCE, pp. 569–587. Beguería, S., 2005. Uncertainties in partial duration series modelling of extremes related to the choice of the threshold value. J. Hydrol. 303 (1–4), 215–230. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.L., 2006. Statistics of Extremes: Theory and Applications. John Wiley & Sons.

Alves, J.H.G.M., Young, I.R., 2003. On estimating extreme significant wave heights using combined Geosat, Topex/Poseidon and ERS-1 Altimeter Data. Appl. Ocean Res. 25, 167–186. Ardhuin, F., Chapron, B., Collard, F., 2009. Observation of swell dissipation across oceans. Geophys. Res. Lett. 36 (6).

12

Z. Shao et al.

Coastal Engineering 158 (2020) 103693 Jonathan, P., Ewans, K., Forristall, G., 2008. Statistical estimation of extreme ocean environments: the requirement for modelling directionality and other covariate effects. Ocean Eng. 35 (11–12), 1211–1225. Kapelonis, Z.G., Gavriliadis, P.N., Athanassoulis, G.A., 2015. Extreme value analysis of dynamical wave climate projections in the Mediterranean Sea. Procedia Comput. Sci. 66, 210–219. Komen, G.J., Hasselmann, S., Hasselmann, K., 1984. On the existence of a fully developed wind-sea spectrum. J. Phys. Oceanogr. 14, 1271–1285. Laface, V., Malara, G., Romolo, A., Arena, F., 2016. Peak over threshold vis-� a-vis equivalent triangular storm: return value sensitivity to storm threshold. Coast. Eng. 116, 220–235. Lang, M., Ouarda, T., Bob� ee, B., 1999. Towards operational guidelines for over-threshold modeling. J. Hydrol. 225 (3–4), 103–117. Lemm, A.J., Hegge, B.J., Masselink, G., 1999. Offshore wave climate, perth (western Australia), 1994-96. Mar. Freshw. Res. 50 (2), 95–102. Lerma, A.N., Bulteau, T., Lecacheux, S., Idier, D., 2015. Spatial variability of extreme wave height along the Atlantic and channel French coast. Ocean Eng. 97, 175–185. Li, F., Bicknell, C., Lowry, R., Li, Y., 2012. A comparison of extreme wave analysis methods with 1994-2010 offshore Perth dataset. Coast. Eng. 69, 1–11. Li, F., Van Gelder, P., Ranasinghe, R., Callaghan, D.P., Jongejan, R.B., 2014. Probabilistic modelling of extreme storms along the Dutch coast. Coast. Eng. 86, 1–13. Li, J., Pan, S., Chen, Y., Fan, Y.M., Pan, Y., 2018. Numerical estimation of extreme waves and surges over the northwest Pacific Ocean. Ocean Eng. 153, 225–241. Liang, B.C., Shao, Z.X., Li, H.J., Shao, M., Lee, D.Y., 2019a. An automated threshold selection method based on the characteristic of extrapolated significant wave heights. Coast. Eng. 144, 22–32. Liang, B.C., Gao, H.J., Shao, Z.X., 2019b. Characteristics of global waves based on the third-generation wave model SWAN. Mar. Struct. 64, 35–53. Mazas, F., Hamm, L., 2011. Amulti-distribution approach to POT methods for determining extreme wave heights. Coast. Eng. 58 (5), 385–394. Mathiesen, M., Goda, Y., Hawkes, P.J., Mansard, E., Martín, M.J., Peltier, E., 1994. Recommended practice for extreme wave analysis. J. Hydraul. Res. 32 (6), 803–814. Mendoza, E.T., Jim� enez, J.A., 2005. Factors controlling vulnerability to storm impacts along the Catalonian coast. Coast. Eng. 4, 3087–3099. Mendoza, E.T., Jim� enez, J.A., 2006. Storm-induced beach erosion potential on the Catalonian coast. J. Coast Res. SI48, 81–88. Men�endez, M., M�endez, F.J., Izaguirre, C., Luce~ no, A., Losada, I.J., 2009. The influence of seasonality on estimating return values of significant wave height. Coast. Eng. 56, 211–219. Mortlock, T.R., Goodwin, I.D., Turner, I.L., 2014. Nearshore SWAN model sensitivities to measured and modelled offshore wave scenarios at an embayed beach compartment, NSW, Australia. Aust. J. Civ. Eng. 12 (1), 67–82. Morton, I.D., Bowers, J., Mould, G., 1997. Estimating return period wave heights and wind speeds using a seasonal point process model. Coast. Eng. 31 (1–4), 305–326. Neelamani, S., Al-Salem, K., Rakha, K., 2007. Extreme waves for Kuwaiti territorial waters. Ocean Eng. 34 (10), 1496–1504. Perez, J., Menendez, M., Losada, I.J., 2017. GOW2: a global wave hindcast for coastal applications. Coast. Eng. 124, 1–11. Petrov, V., Guedes Soares, C., Gotovac, H., 2013. Prediction of extreme significant wave heights using maximum entropy. Coast. Eng. 74, 1–10. Pickands, J., 1975. Statistical inference using extreme order statistics. Ann. Stat. 3, 119–131. Polnikov, V.G., Sannasiraj, S.A., Satish, S., Pogarskii, F.A., Sundar, V., 2017. Estimation of extreme wind speeds and wave heights along the regional waters of India. Ocean Eng. 146, 170–177. Pringle, J., Stretch, D.D., B� ardossy, A., 2014. Automated classification of the atmospheric circulation patterns that drive regional wave climates. Nat. Hazards Earth Syst. Sci. 14 (8), 2145–2155. Pringle, J., Stretch, D.D., B� ardossy, A., 2015. On linking atmospheric circulation patterns to extreme wave events for coastal vulnerability assessments. Nat. Hazards 79 (1), 45–59. Reeve, D., 2016. Preface for special section on coastal flood risk. Water Sci. Eng. 1 (9), 1–2. Reiss, R.D., Thomas, M., 2007. Statistical Analysis of Extreme Values. Birkh€ auser, Basel. Ris, R.C., Holthuijsen, L.H., Booij, N., 1999. A third-generation wave model for coastal regions: 2 Verification. J. Geophys. Res. 104, 7667–7681. Rueda, A., Camus, P., M� endez, F.J., Tom� as, A., Luce~ no, A., 2016a. An extreme value model for maximum wave heights based on weather types. J. Geophys. Res.: Oceans 121 (2), 1262–1273. Rueda, A., Camus, P., Tom� as, A., Vitousek, S., M�endez, F.J., 2016b. A multivariate extreme wave and storm surge climate emulator based on weather patterns. Ocean Model. 104, 242–251. Ruggiero, P., Komar, P.D., Allan, J.C., 2010. Increasing wave heights and extreme value projections: the wave climate of the US Pacific Northwest. Coast. Eng. 57 (5), 539–552. Rusu, L., Ponce De Le� on, S., Soares, C.G., 2015. Numerical modelling of the north Atlantic storms affecting the west Iberian coast. Marit. Technol. Eng. 1365–1370. Samayam, S., Laface, V., Annamalaisamy, S.S., Arena, F., Vallam, S., Gavrilovich, P.V., 2017. Assessment of reliability of extreme wave height prediction models. Nat. Hazards Earth Syst. Sci. 17 (3), 409–421. S� anchez-Arcilla, A., Gomez Aguar, J., Egozcue, J.J., Ortego, M.R., Galiatsatou, P., Prinos, P., 2008. Extremes from scarce data: the role of Bayesian and scaling techniques in reducing uncertainty. J. Hydraul. Res. 46 (S2), 224–234. Sartini, L., Mentaschi, L., Besio, G., 2014. Sub-mesoscale wave height return levels on the basis of hindcast data: the North Tyrrhenian Sea. Coast. Eng. Proc. 1 (34), 39.

Bernardara, P., Mazas, F., Kergadallan, X., Hamm, L., 2014. A two-step framework for over-threshold modelling of environmental extremes. Nat. Hazards Earth Syst. Sci. 14 (3), 635. Booij, N., Holthuijsen, L.H., Ris, R.C., 1999. A third-generation wave model for coastal regions: 1. Model description and validation. J. Geophys. Res. 104, 7649–7666. Cai, Y., Reeve, D.E., 2013. Extreme value prediction via a quantile function model. Coast. Eng. 77, 91–98. Camus, P., Men� endez, M., M� endez, F.J., Izaguirre, C., Espejo, A., C� anovas, V., et al., 2014. A weather-type statistical downscaling framework for ocean wave climate. J. Geophys. Res.: Oceans 119 (11), 7389–7405. Carreau, J., Bengio, Y., 2009. A hybrid Pareto model for asymmetric fat-tailed data: the univariate case. Extremes 12 (1), 53–76. Cavaleri, L., Rizzoli, P.M., 1981. Wind wave prediction in shallow water: theory and applications. J. Geophys. Res. Oceans 86 (C11), 10961–10974. Chini, N., Stansby, P., Leake, J., Wolf, J., Roberts-Jones, J., Lowe, J., 2010. The impact of sea level rise and climate change on inshore wave climate: a case study for East Anglia (UK). Coast. Eng. 57 (11–12), 973–984. Codignotto, J.O., Dragani, W.C., Martin, P.B., Simionato, C.G., Medina, R.A., Alonso, G., 2012. Wind-wave climate change and increasing erosion in the outer Río de la Plata, Argentina. Continent. Shelf Res. 38, 110–116. Coles, S., 2001. An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics. Springer-Verlag London, London, pp. 78–91. Collard, F., Ardhuin, F., Chapron, B., 2009. Monitoring and analysis of ocean swell fields from space: new methods for routine observations. J. Geophys. Res.: Oceans 114 (C7). Collins, J.I., 1972. Prediction of shallow-water spectra. J. Geophys. Res. 77 (15), 2693–2707. Davies, G., Callaghan, D.P., Gravois, U., Jiang, W.P., Hanslow, D., Nichol, S., et al., 2017. Improved treatment of non-stationary conditions and uncertainties in probabilistic models of storm wave climate. Coast. Eng. 127, 1–19. de Haan, L.F.M., 1970. On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Amsterdam. Dee, D.P., Uppala, S.M., Simmons, A.J., Berrisford, P., Poli, P., Kobayashi, S., et al., 2011. The ERA-Interim reanalysis: configuration and performance of the data assimilation system. Q. J. R. Meteorol. Soc. 137, 553–597. Donelan, M.A., Drennan, W.M., Katsaros, K.B., 1997. The air-sea momentum flux in conditions of wind sea and swell. J. Phys. Oceanogr. 27 (10), 2087–2099. Dragani, W.C., Garavento, E., Simionato, C.G., Nu~ nez, M.N., Martín, P., Campos, M.I., 2008. Wave simulation in the outer Rio de la Plata estuary: evaluation of SWAN model. J. Waterw. Port, Coast. Ocean Eng. 134 (5), 299–305. Eastoe, E.F., Tawn, J.A., 2010. Statistical models for overdispersion in the frequency of peaks over threshold data for a flow series. Water Resour. Res. 46 (2). Eldeberky, Y., 1996. Nonlinear Transformation of Wave Spectra in the Nearshore Zone. Ph.D. Thesis. The Delft University of Technology, Department of Civil Engineering, The Netherlands. Ferreira, A., de Haan, L., Peng, L., 2003. On optimising the estimation of high quantiles of a probability distribution. Statistics 37 (5), 401–434. Forget, P., Broche, P., Cuq, F., 1995. Principles of swell measurement by SAR with application to ERS-1 observations off the Mauritanian coast. Int. J. Rem. Sens. 16 (13), 2403–2422. Gao, H.J., Wang, L.Q., Liang, B.C., Pan, X.Y., 2018. Estimation of extreme significant wave heights in the Yellow Sea, China. 28th Int. Ocean Polar Eng. Conf. 387–391. Goda, Y., 1989. On the methodology of selecting design wave height. Int. Conf. Coast. Eng. 899–913. Goda, Y., Konagaya, O., Takeshita, N., Hitomi, H., Nagai, T., 2001. Population distribution of extreme significant wave heights estimated through regional analysis. Int. Conf. Coast. Eng. 276, 1078–1091. Goegebeur, Y., Beirlant, J., de Wet, T., 2008. Linking Pareto-tail kernel goodness-of-fit statistics with tail index at optimal threshold and second order estimation. Rev. Stat. 6 (1), 51–69. Gumbel, E.J., 1958. Statistics of Extremes. Columbia University Press, New York, 1958. Hasselmann, S., Hasselmann, K., Allender, J.H., Barnett, T.P., 1985. Computations and parameterizations of the linear energy transfer in a gravity wave spectrum, Part II: parameterizations of the nonlinear transfer for application in wave models. J. Phys. Oceanogr. 15, 1378–1391. Heimbach, P., Hasselmann, K., 2000. Development and application of satellite retrievals of ocean wave spectra. Elsevier Oceanogr. Ser. 63, 5–33. Hill, B.M., 1975. A simple general approach to inference about the tail of a distribution. Ann. Stat. 3 (5), 1163–1174. Innocentini, V., Dos Santos Caetano Neto, E., 1996. A case study of the 9 August 1988 South Atlantic storm: numerical simulations of the wave activity. Weather Forecast. 11 (1), 78–88. Izaguirre, C., Mendez, F.J., Menendez, M., Luce~ no, A., Losada, I.J., 2010. Extreme wave climate variability in southern Europe using satellite data. J. Geophys. Res.: Oceans 115 (C4). Izaguirre, C., M� endez, F.J., Men� endez, M., Losada, I.J., 2011. Global extreme wave height variability based on satellite data. Geophys. Res. Lett. 38 (10). Izaguirre, C., Men� endez, M., Camus, P., M�endez, F.J., Mínguez, R., Losada, I.J., 2012. Exploring the interannual variability of extreme wave climate in the Northeast Atlantic Ocean. Ocean Model. 59, 31–40. Izaguirre, C., M� endez, F.J., Espejo, A., Losada, I.J., Reguero, B.G., 2013. Extreme wave climate changes in Central-South America. Clim. Change 119 (2), 277–290. Jim�enez, J.A., S� anchez-Arcilla, A., Valdemoro, H.I., Gracia, V., Nieto, F., 1997. Processes reshaping the Ebro delta. Mar. Geol. 144 (1–3), 59–79.

13

Z. Shao et al.

Coastal Engineering 158 (2020) 103693 � 2012a. Unified distribution models for met-ocean variables: Solari, S., Losada, M.A., application to series of significant wave height. Coast. Eng. 68, 67–77. � 2012b. A unified statistical model for hydrological variables Solari, S., Losada, M.A., including the selection of threshold for the peak over threshold method. Water Resour. Res. 48 (10). Solari, S., Alonso, R., 2017. A new methodology for extreme waves analysis based on weather-patterns classification methods. Coast. Eng. Proc. 1 (35), 23. � 2017. Peaks over Threshold (POT): a Solari, S., Egüen, M., Polo, M.J., Losada, M.A., methodology for automatic threshold estimation using goodness of fit p-value. Water Resour. Res. 53 (4), 2833–2849. Solari, S., Alonso, R., Teixeira, L., 2018. Analysis of coastal vulnerability along the Uruguayan coasts. J. Coast Res. 85, 1536–1540. Takbash, A., Young, I.R., Ø, Breivik, 2019. Global wind speed and wave height extremes derived from long-duration satellite records. J. Clim. 32 (1), 109–126. Thompson, P., Cai, Y., Reeve, D., Stander, J., 2009. Automated threshold selection methods for extreme wave analysis. Coast. Eng. 56 (10), 1013–1021. Vanem, E., 2015. Non-stationary extreme value models to account for trends and shifts in the extreme wave climate due to climate change. Appl. Ocean Res. 52, 201–211. Vanem, E., 2017. A regional extreme value analysis of ocean waves in a changing climate. Ocean Eng. 144, 277–295. Vitart, F., Anderson, J.L., Stern, W.F., 1997. Simulation of interannual variability of tropical storm frequency in an ensemble of GCM integrations. J. Clim. 10 (4), 745–760. Weissman, J., 1978. Estimation of parameters and large quantiles based on the k largest observations. J. Am. Stat. Assoc. 73, 812–815. Wu, G.X., Shi, F.Y., Kirby, J.T., Liang, B.C., Shi, J., 2018. Modeling wave effects on storm surge and coastal inundation. Coast. Eng. 140, 371–382. Yan, Z.D., Liang, B.C., Wu, G.X., Wang, S.Q., Li, P., 2020. Ultra-long return level estimation of extreme wind speed based on the deductive method. Ocean Eng. 197, 106900. You, Z.J., 2011. Extrapolation of historical coastal storm wave data with best-fit distribution function. Aust. J. Civ. Eng. 9, 73–82. Young, I.R., Vinoth, J., Zieger, S., Babanin, A.V., 2012. Investigation of trends in extreme value wave height and wind speed. J. Geophys. Res.: Oceans 117 (C11). Zheng, C.W., Li, C.Y., 2017. Propagation characteristic and intraseasonal oscillation of the swell energy of the Indian Ocean. Appl. Energy 197, 342–353. Zhang, C., Zhang, Q., Zheng, J.K., Demirbilek, Z., 2017. Parameterization of nearshore wave front slope. Coast. Eng. 127, 80–87.

Sartini, L., Mentaschi, L., Besio, G., 2015a. Comparing different extreme wave analysis models for wave climate assessment along the Italian coast. Coast. Eng. 100, 37–47. Sartini, L., Cassola, F., Besio, G., 2015b. Extreme waves seasonality analysis: an application in the Mediterranean Sea. J. Geophys. Res.: Oceans 120 (9), 6266–6288. Sartini, L., Besio, G., Cassola, F., 2017. Spatio-temporal modelling of extreme wave heights in the Mediterranean Sea. Ocean Model. 117, 52–69. Sartini, L., Weiss, J., Prevosto, M., Bulteau, T., Rohmer, J., Maisondieu, C., 2018. Spatial analysis of extreme sea states affecting Atlantic France: a critical assessment of the RFA approach. Ocean Model. 130, 48–65. Schendel, T., Thongwichian, R., 2017. Confidence intervals for return levels for the peaks-over-threshold approach. Adv. Water Resour. 99, 53–59. Semedo, A., Su�selj, K., Rutgersson, A., Sterl, A., 2011. A global view on the wind sea and swell climate and variability from ERA-40. J. Clim. 24 (5), 1461–1479. Shao, Z.X., Liang, B.C., Pan, X.Y., Gao, H.J., 2017. Analysis of extreme waves with tropical cyclone wave hindcast data. 27th Int. Ocean Polar Eng. Conf. 30–33. Shao, Z.X., Liang, B.C., Li, H.J., Lee, D.Y., 2018a. Study of sampling methods for assessment of extreme significant wave heights in the South China Sea. Ocean Eng. 168, 173–184. Shao, Z.X., Liang, B.C., Li, H.J., Wu, G.X., Wu, Z.H., 2018b. Blended wind fields for wave modeling of tropical cyclones in the South China Sea and East China Sea. Appl. Ocean Res. 71, 20–33. Silva-Gonz� alez, F.L., Heredia-Zavoni, E., Montes-Iturrizaga, R., 2013. Development of environmental contours using Nataf distribution model. Ocean Eng. 58, 27–34. Silva-Gonz� alez, F.L., V� azquez-Hern� andez, A.O., Sagrilo, L.V.S., Cuamatzi, R., 2015. The effect of some uncertainties associated to the environmental contour lines definition on the extreme response of an FPSO under hurricane conditions. Appl. Ocean Res. 53, 190–199. Silva-Gonz� alez, F.L., Heredia-Zavoni, E., Inda-Sarmiento, G., 2017. Square Error Method for threshold estimation in extreme value analysis of wave heights. Ocean Eng. 137, 138–150. Snyder, R.L., Dobson, F.W., Elliott, J.A., Long, R.B., 1981. Array measurement of atmospheric pressure fluctuations above surface gravity waves. J. Fluid Mech. 102, 1–59. Soares, C.G., Scotto, M.G., 2001. Modelling uncertainty in long-term predictions of significant wave height. Ocean Eng. 28 (3), 329–342. Soares, C.G., Scotto, M.G., 2004. Application of the r largest-order statistics for long-term predictions of significant wave height. Coast. Eng. 51 (5–6), 387–394. Sobey, R.J., Orloff, L.S., 1995. Triple annual maximum series in wave climate analyses. Coast. Eng. 26 (3–4), 135–151.

14