The model-computed peak storm wave heights and water levels were ranked in descending order from the highest to lowest to identify statistically independent storm events. An empirical probability distribution was estimated from [

36]

where

Q is the probability of exceedance,

m is the rank of the peak storm wave height and water level, and

N is the total number of observed peak storm wave heights and water levels.

The Generalized Pareto Distribution (GPD) is an asymptotic distribution often used to estimate extreme values above a specific threshold, and the combination of a Poisson distribution and GPD in a peak-over-threshold (POT) framework results in a Generalized Extreme Value (GEV) distribution [

37]. This methodology produces distributions of annual maxima, which (in our case) are neither directly observed nor directly modeled. The Poission-GPD method produces the best fit among the generalized extreme value distribution types by minimizing the differences in the empirical distributions between the buoy/gauge observations and the modeled water level results. We applied the Poission-GPD method to the modeled peak water level and significant wave heights from 44 modeled extreme storm events using the MATLAB package WAFO [

38]. The individual peak values were obtained from defined storm events with a duration longer than an hour.

For each location, the GPD was fitted to modeled extreme values with a cumulative distribution function

where

u is the threshold which was determined using the dispersion index method implemented in the WAFO package,

$\sigma $ is the scale parameter, and

$\xi $ is the shape parameter. The maximum of a Poisson distributed number of independent GPD variables has a GEV distribution [

37,

39,

40]. Based on the assumption that the annual threshold exceedances

u is Poisson distributed, the cumulative distribution function for the annual maximum Poisson-GPD and GEV relationship and parameter transformation [

37,

39,

40] is:

where

$\lambda $ is the Poisson mean parameter that is estimated by dividing the total number of exceedances of threshold,

u, by the total number of years. Rearranging Equation (

5), we get the traditional form of the GEV cumulative distribution function:

where the scale parameter

$\psi ={\displaystyle \frac{\sigma}{{\lambda}^{\xi}}}$, and the location parameter

$\mu =u+\psi \left({\displaystyle \frac{{\lambda}^{\xi}-1}{\xi}}\right)$.

The return period is defined as the recurrence interval of a specific storm event water level or wave height (

X) that is equal to or exceeds the average of successive events (

x) within

T years. The exceedance probability is

Q, and the probability of occurrence

$Q(X>x)$ in a given year is

$Q=1/T$. Return periods were determined using the following relationship between the GEV return level

${s}_{T}$ and return period

T:

Return years of the empirical data points

Y were determined using the Weibull plotting position [

36,

41]