According to the Intergovernmental Panel on Climate Change (IPCC, 2007), average global temperature increased by approximately 0.74 ∘C (i.e. at least 0.56 ∘C and up to 0.92 ∘C) between 1906 and 2005 (Hwang, 2013). The IPCC (2007) Fourth Assessment Report (AR4) noted that since 1961, world mean sea level (MSL) has increased by around 1.8 mm (i.e. 1.3–2.3 mm) per year; when melting polar ice is taken into account, this figure increases to 3.1 mm (2.4–3.8 mm). Moreover, the overall area of Arctic ice has decreased by an average of 2.7 % annually since 1978, and the amount of snow on mountains has also declined (Kim and Cho, 2013). These observations have sparked growing interest in how much sea levels will increase, including research into how changes in the climate can best be coped with (Radic and Hock, 2011; Schaeffer et al., 2012). Most industrial facilities on the Korean Peninsula, including plants, ports, roads, and shipyards, are located near the shore – as indeed are most residential buildings. These topographical characteristics make the cities of South Korea especially vulnerable to damage caused by sea-level rise and the associated large socioeconomic losses.

Yoon and Kim (2012) investigated 51 years' worth of sea-level changes using data from tidal gauges at 17 stations located around the Korean Peninsula. They utilized regression analysis to calculate the general trend in MSL for 1960–2010 at each station and found that around South Korea MSL rose more quickly than it did globally. The linear rising trend of MSL was relatively small along South Korea's western coast (averaging 1.3 mm yr-1) but large on the southern and eastern coasts (3.2 and 2.0 mm yr-1, respectively) and very large around Jeju Island (5.6 mm yr-1, i.e. more than 3 times the global average).

According to AR4, the rate of sea-level rise may accelerate after the 21st century, and this should be taken into consideration when designing coastal structures if disasters are to be avoided. Therefore, places most likely to be affected by current and future climate change need more accurate predictions of sea-level variation and surge heights, with a “surge” being defined as the difference between observed and predicted sea level. In the present work, Busan, a major metropolitan area on the southeastern coast of South Korea, has been used as a case study. According to the calculations of Yoon and Kim (2012) , the sea level around Busan rose by an average 1.8 mm yr-1 from 1960 to 2010, i.e. roughly the same as the global trend over the same period.

The Korean Peninsula is bounded by three distinct sea systems, generally known in English as the Yellow Sea, the Korea Strait, and the East Sea or Sea of Japan. This characteristic has often led to severe damage to its coastal regions. According to the Korea Ocean Observing and Forecasting System (KOOFS), Typhoon Maemi in September 2003 had a maximum wind speed of 54 m s-1 (metres per second), and these strong gusts caused an unexpected storm surge. This event caused USD 3.5 billion in property damage, as shown in Table 1. All three of the highest peaks ever recorded by South Korea's tidal-gauge stations also occurred in that month.

The most typhoon-heavy month in South Korea is August, followed by July and September, with two-thirds of all typhoons occurring in July and August. Tables 2 and 3 present statistics about typhoons in South Korea over periods of 68 and 10 years ending in 2019, respectively, and Fig. 1 shows the track of Typhoon Maemi from 4–16 September 2003. As can be seen from Fig. 1, Typhoon Maemi passed into Busan from the southeast, causing direct damage upon landfall, after which its maximum 10 min sustained wind speed was 54 m s-1. Typhoon Maemi prompted the insurance industry, the South Korean government, and many academic researchers to recognize the importance of advance planning and preparations for such storms, as well as for other types of natural disasters.

Effective measures for reducing the damage caused by future typhoons, especially the design and re-design of waterfront infrastructure, will require accurate prediction of storm-surge height. When Typhoon Maemi struck the Korean Peninsula in 2003, South Korea was operating 17 tidal-gauge stations, of which 8 had been collecting data for 30 years or more. They were located on the western (n=5), southern (n=10), and eastern coasts (n=2).

This study focuses on the 15 tidal-gauge stations located on the southern and western coasts (Fig. 2). The reason for excluding the remaining two stations is that the majority of typhoons do not arrive from the east or make landfall on that coast. The hourly tidal data for this study have been provided by the Korea Hydrographic and Oceanographic Agency (KHOA, 2019) and are used with that agency's permission.

The western tidal-gauge stations are located at Incheon, Gyeongin, Changwon, Gunsan, and Mokpo. These five stations have operated for different lengths of time, ranging from 2 to 61 years. Collection of the sea levels observed hourly by each station throughout their respective periods of operation revealed the top three sea-level heights at each. These heights, which are shown in Table 4, are clearly correlated with the dates of arrival of typhoons.

The same approach was applied to the data from the 10 tidal-gauge stations on the south coast, as shown in Tables 5 and 6.

One of the focal tidal-gauge stations has observation records covering more than half a century. It is located on the south coast at Busan, South Korea's second-largest city. Thanks to its location near the sea, Busan's international trade has boomed, and as a consequence it now boasts the largest port in South Korea. The Nakdong, the longest and widest river in South Korea, also passes through it. Due to these geographical characteristics, Busan has been very vulnerable to natural disasters, and the importance of accurately predicting the characteristics of future storms is increasingly recognized by its government and other stakeholders. The top three sea-level heights at the tidal-gauge station there are shown in Table 6. As this table indicates, all of the top three water heights recorded in the long history of this station occurred during Typhoon Maemi's passage through the area.

KHOA makes hourly observations of water height at the Busan tidal-gauge station, and the annual means presented in this paper have been calculated from that hourly data. As can be seen in Fig. 3, plotting MSL for each year confirms that short-term water-level variation merely masks the long-term trend of sea-level increase. Therefore, on the assumption that MSL variation was a function of time, a linear regression was performed, with the resulting coefficient of slope indicating the rate of increase (Yoon and Kim, 2012). The data utilized to estimate MSL for the tidal-gauge station in Busan were provided by KHOA, which performed quality control on the data before releasing it to us. Additionally, however, a normality test was performed, and the results (as shown in Table 7) indicated that the hourly sea-level data followed a normal distribution at a significance > 0.05. The Kolmogorov–Smirnov normality test was adopted as it is well suited to datasets containing more than 30 items.

As can be seen in Fig. 3, the average rate of increase in MSL at Busan's tidal-gauge station from 1962 to 2019 was 2.4 mm yr-1, yielding a difference of 16.31 cm between the beginning and the end of that period. This finding is broadly in line with the Yoon and Kim (2012) finding that the rate of MSL increase around the Korean Peninsula as a whole between 1960 and 2010 was about 2.9 mm yr-1. In addition, linear-regression analysis of the sea-level fluctuation data for 1965–2019 was utilized to discern the MSL trend. The significance level of 0.000 (<0.05) obtained via analysis of variance (ANOVA; Table 8) indicates that the regression model of sea-level fluctuations was significant. Its correlation coefficient (0.96) also indicated a strong positive relationship between sea-level rise and recentness. The coefficient of determination (R2) was utilized to describe how well the model explained the collected data. The closer R2 is to 1, the better the model can predict the linear trend: here it was 0.74, as shown in Table 9. This means that the linear-regression model explained 74 % of the sea-level variation. While this result suggests that the linear-regression analysis for sea-level fluctuation at the tidal-gauge station in Busan is reliable, such results may not be generalizable because variation in the data could have been due to several factors, including geological variation and modification of gauge points.

When a storm occurs, surge height tends to increase, and these larger surges can cause natural disasters such as floods. In this study, before calculating the height of a surge, we took account of the dates and times when the three greatest sea-level heights were observed, as well the dates and times when typhoons occurred. These data are presented side by side in Table 10.

As Table 10 indicates, the top three recorded sea levels at each south coast tidal-gauge station corresponded with the occurrence of typhoons in 20 out of 30 cases. Moreover, the dates and times of the three highest sea levels observed during all 57 years' worth of data from Busan all coincided with Typhoon Maemi passing out of the area.

As well as USD 3.5 billion in property damage, Typhoon Maemi caused 135 casualties in Busan and nearby cities (National Typhoon Center, 2011). However, other typhoons – notably including Thelma, Samba, and Megi – also caused considerable damage, as shown in Table 1.

Bermúdez et al. (2019) studied flood drivers in coastal and riverine areas as part of their approach to quantifying flood hazards, using 2D shallow-water models to compute the correlation between extreme events and flood drivers. They also adopted ordinary least-squares regression analysis to construct a 10 000-year time series and computed water levels' exceedance probabilities for comparison. However, the possibility of river discharges, sea-wave trends, and tidal fluctuations were not considered in their study.

The wrecking of wind farms by extreme windstorms is of considerable concern in the North Sea region, which is home to 38 such farms belonging to five different countries. According to the Monte Carlo simulation-based risk-management study of Buchana and McSharry (2019), the total asset value of these wind farms is EUR 35 billion. It used a log-logistic damage function and Weibull probability distribution to assess the risks posed to wind farms in that region by extreme strong wind and exceedance probability to predict the extent of financial loss from such damage in terms of solvency capital requirement (SCR). The same study also simulated the results of various climate change scenarios, and the results confirmed that higher wind speed and higher storm frequency were correlated with rises in SCR: a finding that could be expected to help emergency planners, investors, and insurers reduce their asset losses.

According to a study by Catalano et al. (2019) of high-impact extratropical cyclones (ETCs) on the northeastern coast of the Unites States, limited data caused by these storms' rarity made it difficult to predict the damage they would cause or analyse their frequency. To overcome this, they utilized 1505 years' worth of simulations derived from a long coupled model, GFDL FLOR, to estimate these extreme events' exceedance probabilities and compared the results against those of short-term time series estimation. This not only revealed that the former was more useful for statistical analysis of ETCs' key characteristics – which they defined as maximum wind speed, lowest pressure, and surge height – but also that the use of a short time series risked biassing estimates of ETCs' return levels upwards (i.e. underestimating their actual frequency). While these results regarding return levels and time series were valuable, Catalano et al. (2019) did not distinguish between the cold season and the warm season of each year, which could also have led to biased results.

A joint-probability methodology was used to analyse extreme water heights and surges on China's coast by Chen et al. (2019). They obtained the sea-level data from nine gauge stations, and utilized 35 years' worth of simulation data with a Gumbell distribution and a Gumbell–Hougaard copula. The three major sampling methods proposed in the study were structural-response, wave-dominated, and surge-dominated sampling. The first was utilized to assess structures' performance in response to waves and surges. Joint-probability analysis revealed that such performances were correlated with extreme weather events in the target region and that such correlations became closer when wave motion was stronger. In addition, based on their finding that joint exceedance probability tended to overestimate return periods for certain water levels, Chen et al. (2019) recommended that offshore defence facility designers use joint-probability density to estimate return levels of extreme wave heights. However, while their study provided a useful methodology, particularly with regard to sampling methods and probability modelling of return periods and structural performance, they only looked at China's coast, and therefore their findings are unlikely to be generalizable to the Korean Peninsula.

Davies et al. (2017) proposed a framework for probability modelling of coastal storm surges, especially during non-stationary extreme storms and tested it using the El Niño–Southern Oscillation (ENSO) on the east coast of Australia. Importantly, they applied their framework to ENSO and seasonality separately. This is because while ENSO affects storm-wave direction, mean sea level, and storm frequency, seasonality is mostly related to storm-surge height, storm-surge duration, and total water height. This separation has the advantage of allowing all storm variables of non-stationary events to be modelled, regardless of their marginal distribution. Specifically, Davies et al. (2017) applied non-parametric distribution to storm-wave direction and steepness and parametric distribution to duration and surge using mixture-generalized extreme value probability modelling, which they argued was more useful than standard models like generalized Pareto distribution (GPD). They said this was because the statistical threshold in an extreme mixture model can be integrated into the analysis, whereas a GPD model should be given an unbiased threshold: if it is low, too many normal data may be included. Accordingly, they utilized bootstrapping for the confidence interval to show the uncertainty of the non-stationary aspects of the extreme events. They also added a Bayesian method to provide wider confidence intervals with less bias. Their findings are mainly beneficial to overcoming the challenges of GPD threshold selection; however, robust testing of their approach will require that it be applied to a wider range of abnormal climate phenomena.

Similar research was conducted by Fawcett and Walshaw (2016), who developed a methodology for estimating the return levels of extreme events such as sea surges and high winds of particular speeds, with the wider aim of informing practical applications such as design codes for coastal structures. They reported that two of the most popular existing methods for doing so, block maxima (BM) and POT, both have shortcomings and concluded that a Bayesian approach would be more accurate. Specifically, they argued that BM and POT methods tend to waste valuable data and that considering all exceedance via accurate estimation of the extremal index (reflecting uncertainty's natural behaviour) could compensate for this disadvantage. They further proposed that the seasonal variations should be taken into consideration with the all exceedance data, where possible.

In response to Japanese government interest in unexpected flooding caused by extreme storm surges during typhoons and other high-wind events, Hisamatsu et al. (2020) simulated typhoons as a means of predicting the cost of the damage they would cause in Tokyo Bay, which is very vulnerable to such events due to its geographic and socio-economic characteristics. Using stochastic approaches, they modelled future typhoons over a 10 000-year period and calculated flooding using a numerical surge model based on the probability of historical typhoons. These flooding calculations, in turn, were utilized to create a storm-surge inundation map, representing exceedance probabilities derived from stochastic hazard calculations pertaining to 1000 typhoons. Next, the completed map was overlaid on government-provided values of Tokyo Bay's buildings and other infrastructural elements to assess the spatial extent and distribution of the likely damage. The results showed that Chiba and Kanagawa would be the most damaged areas and would suffer financial losses of JPY 158.4 billion and 91.5 billion, respectively, with an exceedance probability of 0.005 (as commonly used to estimate damage in the insurance industry). However, the real estate values they used were 2 decades out of date at the time their study was conducted, meaning that further validation of their approach will be needed.

Another effort to estimate return periods was made by McInnes et al. (2016), who created a stochastic dataset on all cyclones that occurred near Samoa from 1969 to 2009. That dataset was utilized to model storm tides using an analytic cyclone model and a hydrodynamic model, which also took into account prevailing climate phenomena such as La Niña and El Niño when estimating return periods. The authors found that tropical cyclones' tracks could be affected by La Niña and El Niño and, more specifically, that the frequency of cyclones and storm tides during El Niño was consistent across all seasons, whereas La Niña conditions make their frequency considerably lower in the La Niña season. Additionally, McInnes et al. (2016) proposed that sea-level rise had a more significant influence on storm tides than future tropical cyclones did, based on their finding that future cyclones' frequency would be reduced as the intensity of future cyclones increased. Lastly, they found that the likelihood of a storm tide exceeding a 1 % annual exceedance probability (i.e. a once a century tide) was 6 % along the entire coastline of Samoa. However, other effects such as sea-level fluctuations and meteorological factors were not included in their calculations.

Silva-González et al. (2017) studied threshold estimation for analysis of extreme wave heights in the Gulf of Mexico and argued that appropriate thresholds for this purpose should consider exceedances. They applied the Hill estimator method, an automated threshold-selection method, and the square-error method for threshold estimation in hydrological, coastal engineering, and financial scenarios with very limited data and found that the square-error method had the most advantages because it did not consider any prior parameters that could affect thresholds. The authors went on to propose improvements to that method, i.e. the addition of differences between quantiles of the observed samples and median quantiles from GPD-aided simulation. When GPD was utilized to estimate observed samples, it effectively prevented convergence problems with the maximum-likelihood method when only small amounts of data were available. The key advantage of the approach of Silva-González et al. (2017) is that the choice of a threshold can be made without reliance on any subjective criteria. Additionally, no particular choice of marginal probability distribution is required to estimate a threshold. However, to be of practical value, their method will need to incorporate more meteorological factors.

Lastly, the Wahl et al. (2015) study of the exceedance probabilities of a large number of synthetic and a small number of actual storm-surge scenarios utilized four steps: parameterizing the observed data, fitting different distribution models to the time series, Monte Carlo simulation, and recreating synthetic storm-surge scenarios. Specifically, projected 40 and 80 cm sea-level rises were used as the basis for investigating the effects of climate change on flooding in northern Germany. Realistic joint exceedance probabilities were used for all parameters with copula models, and the exceedance probabilities of storm surges were obtained from the bivariate exceedance probability method with two parameters, i.e. the highest total water level with the tidal fluctuations and intensity. The findings of Wahl et al. (2015) indicated that extremely high water levels would cause substantial damage over a short time period, whereas relatively small storm surges could inflict similar levels of damage but over a much longer period. However, like various other studies cited above, Wahl et al. (2015) did not take seasonal variation into account.

Extreme events are hard to predict because data points are so few, and predicting their probability is particularly difficult due to their asymptotic nature. Extreme value probability theory deals with how to find outlier information, such as maximum or minimum data values, during extreme situations. Examining the tail events in a probability distribution is very challenging. However, it is considered very important by civil engineers and insurers due to their need to cope with low-probability, high-consequence events. For example, the designs and insurance policies of bridges, breakwaters, dams, and industrial plants located near shorelines or other flood-prone areas should account for the probability, however low, of major flooding. Various probability models for the study of extreme events could potentially be used in the present research, given that its main topic is the extreme high water levels caused by typhoons. Extreme value theories can be divided into two groups, according to how they are defined. In the first, the entire interval of interest is divided into a number of subintervals. The maximum value from each subinterval is identified as the extreme value, and following this the entirety of these extreme values converge into a GEV distribution. In the second group, values that exceed a certain threshold are identified as extreme and converge to a GPD. The following two subsections discuss the BM and POT methods as illustrations of these two groups, respectively (Coles, 2001).

The BM approach relies on the distribution of the maximum extreme values in the following equation, 1Mn=maxX1,…,Xn, where the Xn series, comprising independent and identically random variables, occurs in order of maximum extreme values, n is the number of observations in a year, and Mn is the annual maximum.

Data is divided into blocks of specific time periods, with the highest values within each block collectively serving as a sample of extreme values. One limitation of this method is the possibility of losing important extreme value data because only the single largest value in each block is accounted for, and thus the second-largest datum in one block could be larger than the highest datum in another.

The POT method can address the above-mentioned limitations of BM, insofar as it can gather all the data points that exceed a certain prescribed threshold and use limited data more efficiently because it relies on relatively large or high values instead of the largest or highest ones. All values above the threshold – known as exceedances – can be explained by the differentiated tail data distribution. The basic function of this threshold is to assort the larger or higher values from all data, and the set of exceedances constitutes the sample of extreme values. This means that although POT can capture potentially important extreme values even when they occur close to each other, selecting a threshold that will yield the best description of the extreme data can be challenging (Bommier, 2014); i.e. if it is set too high, key extreme values might be lost, but if it is set too low, values that are not really extreme may be included unnecessarily. Determining appropriate threshold values thus tends to require significant trial and error, and various studies have proposed methods for optimizing such values (Lopeman et al., 2015; Pickands, 1975; Scarrott and Macdonald, 2012). Pickands (1975), for instance, suggested that independent time series that exceed high enough thresholds would follow GPD asymptotically, thus avoiding the inherent drawbacks of BM. The distribution function F of exceedance can be computed as 2Fθ(x)=PX-θ≤x|X>θ,x≥0, where θ is the threshold and X is a random variable.

Fu, meanwhile, can be defined by conditional probabilities 3Fθ(x)=F(θ+x)-F(θ)1-F(θ)ifx≥00else. According to Bommier (2014), the distribution of exceedances (Y1, …, Ynθ) can be generalized by GPD with the following assumption: when Y=X-θ for X>θ, and X1, …, Xn, Yj=Xi-θ can be described with i, which is the jth exceedance, i=1, …, nθ.

The GPD can be expressed as 4Gx(x;ξ,σ,θ)=1-1+ξ(x-θ)σ1/ξξ≠01-exp⁡-(x-θ)σξ=0, with x being independent and identically random variables, σ the scale, ξ the shape, and θ the threshold. All values above θ are considered tail data (extreme values). The probability of exceedance over a threshold when calculating a return level that is exceeded once every N years (N-year return periods xN) is calculated as follows: 5PX>x|X>θ=1+ξx-θσ-1/ξ. If the exceedances above the threshold are rare events λ (as measured by number of observations per year), we can expect P(X>θ) to follow Poisson distribution. The mean of exceedance per unit of time (γ^) describes that distribution. 6P(X>θ)=γλ In other words, γ can be estimated by dividing the number of exceedances by the number of years in the observation period.

Combining the POT and Poisson processes with GPD allows us to describe the conditional probability of the extreme values that exceed the designated threshold, as per Eq. (7) (Lopeman et al., 2015): 7P(A|B)=P(A∩B)P(B). In addition, when Bayes' theorem is applied to the role of GPD in conditional probability, we can rewrite Eq. (7) as follows: 8GX(x)=P(θ<X<x)P(X>θ).

To determine the height of surges from publicly available KHOA data, it was first necessary to predict sea levels. Equation (9) explains the interrelationship of observed water level, predicted water level, tidal-fluctuation height, and residual (surge) at time ti, 9Yi=Xi+Si, where i=1, 2, …, n, n is the time series of the input dataset, Xi is the predicted water height at ti, Yi is the observed water height at ti, and Si is the surge height.

A standard harmonic analysis was performed to calculate predicted sea-level height based on hourly tidal-gauge data. First, this technique was used to estimate the tidal components of all seawater-level data, allowing residuals to be isolated so that surge data could be calculated once sea-level rise had been estimated. Second, the estimated constituents were used to predict tidal fluctuations in the years simulated via Monte Carlo. Then, the TideHarmonics package in R (Stephenson, 2017) was used to estimate tidal components, as detailed below.

Given a time series Y(t) of total water levels, with t denoting time in hours, the tidal component with M harmonic constituents is computed as 10Y^(t)=Z+∑m=1MAmcos⁡π180ωmt-ψm, where ωm is the angular frequency of the mth component in degrees per hour. The 2M+1 parameters to be estimated are the amplitudes Am, the phase lag ψm in degrees, and the MSL Z.

To account for long astronomical cycles (LACs), nodal-correction functions for both the amplitude and phase are used. With these corrections, the tidal component takes the following form: 11Y^LAC(t)=Z+∑m=1MHmfm(t)cos⁡π180ωmt-gm+um(t)+Vm, where fm(t) and um(t) represent the nodal corrections for the amplitude and phase, respectively. In this new formulation, the amplitude and phase parameters to be estimated are denoted by Hm and gm (in degrees). Finally, Vm is the reference signal, by which the phase lag gm is calculated and set to refer to the origin t=0.

The summation term in Y^LAC(t) can alternatively be written as follows: 12∑m=1Mβm,1fm(t)⋅cos⁡π180ωmt+um(t)+Vm+∑m=1Mβm,2fm(t)⋅sin⁡π180ωmt+um(t)+Vm, where βm,1=Hmcos⁡(gm) and βm,2=Hmsin⁡(gm). What is gained from this new representation is a linear function with respect to the parameters βm,1 and βm,2 that need to be estimated, and hence linear regression can be used. Given the large time span covered by the data, M=60 harmonic tidal constituents were estimated, and a constant mean sea level Z was assumed across all years of available data.

Because observed sea level usually differs from predicted sea level, Fig. 5 depicts the former (as calculated through harmonic analysis) in blue. Predicted sea levels are shown in green, and surge height is shown in red. As the figure indicates, the highest overall water level coincided with the highest surge during Typhoon Maemi, i.e. at 21:00 GMT+9 on 12 September 2003. Given a total water height of 211 cm, the surge height was calculated as 73.35 cm. The unexpectedly large height of the surge induced by Typhoon Maemi caused USD 3.5 billion in property damage and many causalities in Busan, as mentioned in Table 1.

At a given annual target rate – i.e. number of storms per year – the algorithm proposed by Lopeman et al. (2015) (Fig. 6) computes the threshold such that this rate approximates the resulting yearly number of “exceedance clusters”, i.e. consecutive surge observations that lie above the threshold. Hence, rather than choosing an “ideal” threshold according to some other criterion, the algorithm simply finds the threshold that forces a chosen target rate to occur. Accordingly, a study of this kind could set its target rate as the average rate observed over a given period or as a value that the researchers find reasonable in light of their knowledge of historical data for their focal area.

Next, the algorithm iteratively updates the threshold to allow a computationally intensive (but not exhaustive) exploration of possible threshold values between its minimum value (i.e. here the minimum observed surge height) and its maximum value (i.e. maximum observed surge height). Specifically, it first sets the threshold to 0 cm and then iteratively overwrites it according to the following steps.

The exceedance clusters produced at a given iteration and given threshold are identified, and the resulting annual storm rate computed.

As shown in Figs. 7–9, the threshold algorithm (Fig. 6) achieved convergence relatively quickly for all three target rates selected, with the number of iterations required for convergence ranging from three (with a target rate of 3.0) to five (with a target rate of 10).

Figure 10 displays six possible thresholds. The first, 31.2 cm, was based on a target rate of 3.5 and 189 clusters and is shown in red. The dark blue line represents the second threshold of 30.54 cm, (target rate = 4.0; clusters = 217), the purple line shows a threshold of 29.56 cm (target rate = 4.5; clusters = 246), the green line shows a threshold of 29.15 cm (target rate = 5.0; clusters = 274), the sky blue line shows a threshold of 28.33 cm (target rate = 6.0; clusters = 324), and the orange line shows a threshold of 26.53 cm (target rate = 8.0; clusters = 431).

Figures 7–9 show that, as expected, when the target rate increases, the threshold decreases, and as the threshold decreases, the number of clusters (i.e. storm events) increases. Conversely, the lower the target rate, the lower the number of clusters and the higher the threshold. Thus, if the desired number of storms is three per year, the algorithm will converge in three iterations and set the threshold level to 32.01 cm; this results in a total of 164 storm events over the time span covered by our data. Conversely, if the desired target rate is 10 storms per year, the threshold is significantly lower (25.43 cm), and the total number of storm events more than trebles to 539 clusters (Fig. 10). As can be seen in Fig. 11, we chose only one maximum value to represent each cluster. Figures 12–14 show the stages of the clustering of surges when the target rate is set to 5.0, the threshold is 29.15 cm, and the number of clusters is 274 (though it should be noted that Fig. 12 indicates only the number of surges, due to the difficulty of visually representing all surge dates and times from the period 1962–2019). The surge data above the designated threshold, arrived at via the threshold-selection method above, are shown in Fig. 12. Here, the data above the threshold are clustered based on their start and end times, and again only one maximum value was chosen at each cluster. Figure 14 presents all maxima obtained from a cluster separately.

Storm surges are characterized by four major parameters: peak time, peak height, duration, and rise ratio. Peak time follows a gamma distribution because POT produces a Poisson process of exceedance occurrence and the waiting times between consecutive exceedances in a Poisson process are by definition exponentially distributed (Lopeman et al., 2015). For peak times (interarrival times), this study therefore uses a gamma (exponential) distribution.

On the other hand, for peak height GPD is typically used because some representation theorem results from extreme value statistics indicate that if the cluster maxima follow a Poisson process, then the intensity – in this case, height – of the cluster peaks follows a GPD distribution (Lopeman et al., 2015; Zhong et al., 2014). However, a Weibull distribution has been applied to peak storm-surge heights in this study because it fits the data better, especially with regard to the right-hand tail.

Because the rise ratio does not appear to be evenly distributed along the interval [0, 1], a beta distribution was used because the rise ratio is by definition between 0 and 1, and such a distribution is commonly used to model continuous random variables that occur within that range (Lopeman et al., 2015).

Duration follows a lognormal distribution, which was used for the following two reasons previously articulated by Lopeman et al. (2015). First, it models a continuous random variable, duration, which by definition is positive. And second, it is quite flexible: as it has two parameters, it can fit the data better than other distributions with just one, e.g. exponential distribution.

Figures 15–17 indicate the lack of any clear relationship between rise ratio, on the one hand, and either duration or exceedance, on the other hand. However, peak exceedance and cluster duration appear to have a linear relationship.

If we assume that an independent and identically distributed data sample (x1, …, xn) is observed from a population with a distribution of interest parameterized by an unknown variable θ that the researcher wants to estimate, the MLE estimator θ^MLE is defined as 13θ^MLEx1,…,xn=argmaxθ0∏i=1nfxi;θ0, where f(⋅;θ) denotes the probability density function of the distribution of interest, parameterized by θ0. The distributions of interest for the data in this study were chosen as follows.

First, Ti∼Exponential(λ) was chosen, where Ti denotes the interarrival time between the peak of the i-1th cluster and the peak of the ith cluster. This distributional assumption is equivalent to assuming that a Poisson process governs peak-surge arrivals.

Based on our simulations, exceedances of water height above the designated threshold were computed using MLE estimates. Table 11 presents the distribution parameters of the storm-surge parameters that were computed, each using a different probability model. These distribution parameters were based on the exceedance above the algorithmically designated threshold of 29.15 cm mentioned above.

Figure 18 shows the GPD cumulative distribution function as estimated by MLE, and the empirical distribution function, with the latter shown as dots. Each dot represents the observed proportion of exceedances below a certain height in a given season (blue: cold season; red: warm season), while the corresponding value on the fitted line of the same season gives the probability that the exceedances are below that height per the estimated GPD distribution.

We also fit our empirical data to five different probability distribution models – i.e. Fréchet, gamma, GPD, lognormal, and Weibull – as seen in Figs. 19 and 20, using the case of storm-surge data. Calculation of the mean squared error between the probability models and the empirical data revealed that the gamma and Weibull distributions had the best fit to the data for both cold and warm seasons when MLE was used for estimating parameters of the probability model. These findings support previous ones by Bardsley (2019) regarding the Weibull distribution's appropriateness to extreme value estimation. According to Bardsley, such a distribution could explain enough to enable extrapolation of the degree beyond the utilized data history, provided that the scale and shape parameter of the distribution are positive (meaning that the probability model has a good fit to the data). In the case of our own research, the shape and scale parameters were 1.87 and 5.21, respectively, indicating that the Weibull distribution model will likely have a good fit to large amounts of data beyond the dataset we used.