Joint Environment Design Parameters for Offshore Floating Wind Turbines in the Yangjiang Sea Area of China

Abstract

In recent years, the increasing frequency of strong and super typhoons has been attributed to rising sea surface temperatures due to global warming. This study utilized the Weather Research and Forecasting (WRF) and Simulating WAves Nearshore (SWAN) models to analyze 30 years of wind and wave data for the Yangjiang sea area in China. The accuracy of the numerical simulations was validated using observed data from typhoons Ty201213, Ty201522, Ty201822, and Ty202118, along with wind and wave data from December 2024. This study utilized the P-III distribution to analyze design wind parameters. At a height of 10 m, the 3 s and 10 min mean wind speeds for the 100- and 50-year return periods were 62.21 m/s, 47.85 m/s, 57.99 m/s, and 44.61 m/s, respectively. At hub height (170 m), the corresponding values were 80.27 m/s, 61.75 m/s, 74.84 m/s, and 57.57 m/s. Furthermore, this study successfully applied a 2D-KDE approach to construct a joint probability model and derive environmental contours for extreme environmental assessments. The HS and TP at project point P for the 100- and 50-year return periods are 13.61 m and 15.91 s, as well as 12.39 m and 15.07 s, respectively.

1. Introduction

In recent years, more and more countries have actively developed renewable energy to mitigate the effects of global warming. Numerous studies have indicated that the South China Sea holds abundant wind and wave energy [1,2]. China plans to achieve an offshore floating wind capacity of 200 GW by 2050 [3].

Due to rising sea surface temperatures from global warming, the frequency of typhoons in the Northwest Pacific has shown a gradual decline; however, their intensity has been increasing. The strong winds and waves associated with strong and super typhoons present significant challenges for the design and safety of ocean engineering and offshore structures [4]. While the fifth generation of European Centre for Medium-Range Weather Forecasts reanalysis (ERA5) data offers high resolution reanalysis data, many studies indicate that ERA5 data tend to underestimate wind fields during typhoon events [5]. This study employs the Weather Research and Forecasting (WRF) model to accurately simulate wind fields during typhoons. The WRF model supports parallel computing and is characterized by portability and efficiency. WRF can simulate both real atmospheric conditions and idealized scenarios, facilitating weather forecasting while serving as a platform for academic exchange among researchers. Currently, WRF is extensively used in studies of wind, precipitation, and air–sea coupling, showing particularly strong performance in simulating wind fields and precipitation with high accuracy compared to real-world observations [6,7]. Many scholars have used the WRF model to study wind fields during typhoon events, such as Typhoon Hagupit and Typhoon Hato, achieving favorable results [8,9]. The WRF model offers various schemes, with significant impacts on typhoon simulations primarily from microphysical (MP) schemes, cumulus (CU) schemes, and planetary boundary layer (PBL) schemes [10,11].

The Simulating WAves Nearshore (SWAN) model has been widely adopted internationally for simulating nearshore waves [12,13]. SWAN simulates processes such as wave generation, propagation, scattering, and dissipation. It employs linear wave equations and nonlinear energy balance equations to describe wave evolution. The SWAN model incorporates various physical parameterization schemes, such as wave generation source terms, wind-generated waves, nonlinear interactions, and whitecapping dissipation, to better simulate wave characteristics under different conditions. Wu et al. (2018) utilized SWAN to simulate waves during typhoons in the South China Sea, achieving satisfactory results [14]. Research by Zijlema et al. (2012) and Xu et al. (2019) indicates that the wave simulations in the SWAN model during typhoons are primarily influenced by parameterization schemes for wind input, whitecapping, and wind drag formula [15,16].

In fields such as ocean engineering, wind turbine design, and environmental risk assessment, the accurate assessment of the probability of extreme environmental events (e.g., typhoons, giant waves, strong winds) is paramount. The environmental contour method serves as a core statistical technique for this purpose. Its goal is to construct the joint probability distribution of multivariate environmental parameters as well as subsequently derive extreme design conditions for a specified return period (e.g., 50-year or 100-year event) [17]. Traditional univariate frequency analysis fails to capture the complex dependencies between variables, potentially leading to significant underestimation or overestimation of the risk associated with joint extreme events. Consequently, the development of robust and efficient multivariate joint probability models has become a key research focus. At the core of the classic Inverse First-Order Reliability Method (IFORM) framework lies the conditional joint distribution model, and it relies on the Rosenblatt transformation. This approach develops environmental contours by breaking down the joint probabilistic distribution into a product of conditional distributions, thereby transforming the multivariate extreme value problem into a series of univariate problems [18]. Copula theory provides a revolutionary tool for multivariate joint distribution modeling. It separates the marginal distributions of individual variables from the dependence structure that links them together via a copula function [19,20]. The strength of this approach lies in its flexibility to choose the most suitable marginal distributions while independently modeling complex, nonlinear dependencies, particularly tail dependence, which is critical for risk assessment. In contrast to parametric models, kernel density estimation (KDE) is a non-parametric method that requires no prior assumptions about the underlying distribution form. It estimates the probability density function directly from data using a kernel function, offering superior adaptability to complex or multi-modal data distributions [21]. The core of its performance hinges on the optimal selection of the smoothing parameter (bandwidth). Recent research efforts are dedicated to developing more advanced bandwidth selection algorithms. For example, the diffusion-based kernel density estimator (diffKDE) proposed by the University of Kiel provides a suite of estimates with different smoothing intensities, thereby better handling multi-modal data and boundary problems [22]. International research on environmental contour methods demonstrates a clear evolution from traditional parametric models towards more flexible and robust frameworks. The conditional joint distribution model has enhanced its practical utility in engineering through integration with Bayesian methods. The copula model, with its powerful capacity for characterizing dependence structures, especially tail correlation, has become a mainstream tool for multivariate extreme value analysis. Meanwhile, the non-parametric kernel density model exhibits unique value in addressing uncertainties within complex systems due to its inherent distribution-free flexibility. Future research trends are expected to focus on the comparative coupling of multiple models, optimizing computational efficiency for higher-dimensional problems.

This paper investigates the environmentally design parameters of deep-sea floating wind turbines using the frequently typhoon-affected Yangjiang sea area project in Guangzhou as a case study. Section 2 introduces the data sources and outlines the configuration of the SWAN and WRF models. Section 3 presents the model validation process, along with the computational methods and results for design wind speeds and design waves. Section 4 serves as the conclusion.

2. Data and Methodology

2.1. Data

To validate wind and wave conditions, observational buoy B1 near the engineering project was utilized, selecting typhoon data from Ty201213, Ty201522, Ty201822, and Ty202118 to verify wind, significant wave heights (HS), and mean periods (TM) during typhoon conditions. Temporary buoy B2 data were used for validation of wind and waves during normal periods. The initial and boundary conditions for the WRF model are sourced from ERA5 (https://cds.climate.copernicus.eu/datasets/reanalysis-era5-single-levels?tab=overview, accessed on 10 August 2024) reanalysis data. ETOPO1 (https://ngdc.noaa.gov/mgg/global/relief/ETOPO1/tiled/, accessed on 15 August 2024) is used to provide bathymetric data for SWAN. Water depth, along with buoy observation stations B1 and B2, the project location P, and typhoon tracks, are illustrated in Figure 1 and

Table 1. The latitude and longitude along with the water depths for buoy points B1 and B2, as well as project point P.

Buoy	Latitude	Longitude	Depth (m)
B1	21.12° N	112.62° E	50.1
B2	20.97° N	112.11° E	46
P	20.96° N	112.14° E	46.66

.

2.2. Numerical Simulation

2.2.1. WRF Model

A 30-year wind field simulation of the Guangdong sea area was conducted using the WRF 4.3 model, employing a triple-nested domain. The time integration steps were set at 135 s, 45 s, and 15 s, respectively. The Mercator projection was utilized with vertical stratification comprising 37 layers. The simulation range of the WRF model extends from 7.85° S to 35.67° N and from 91.94° E to 138.06° E.

According to Siavash et al. (2021) and Jiang et al. (2025) [23,24], during WRF model computations, a 30 h loop was used, with the first 6 h for spin-up and the remaining 24 h for analysis. Following Yi et al. (2023) [25], we applied a 400 km radius around the Yangjiang engineering project to distinguish typhoon and normal periods. During typhoon periods, grid-nudging assimilation was employed to reduce typhoon track errors. The sea surface roughness and related air–sea flux parameters followed the default WRF model settings.

Based on relevant literature and prior work [26,27,28], we selected the Thompson MP scheme, KF CU scheme, and YSU PBL scheme for wind field simulations during typhoon events [29,30,31]. For normal weather conditions, we selected the Thompson MP scheme, Tiedtke CU scheme, and MYJ PBL scheme based on the literature and recommendations from the WRF user manual [32,33]. Other parameterization schemes remain unchanged, including the RRTMG [34] longwave and shortwave schemes. A modified MM5 scheme is used with the YSU PBL scheme for Surface Layer Options, while the Eta Similarity scheme is employed with the MYJ PBL scheme [35]. Specific WRF configuration settings are detailed in

Table 2. Specific settings of the WRF model.

	DOM1	DOM2	DOM3
Resolution	27 km	9 km	3 km
Map projections	Mercator
Time_steps	135 s	45 s	15 s
Shortwave radiation	RRTMG
Longwave radiation	RRTMG
MP	Thompson
CU	KF (typhoon events) Tiedtke (normal events)		None
PBL	YSU (typhoon events) MYJ (normal events)
Land surface options	Unified Noah land surface model [36]
Surface Layer Options	Revised MM5 (typhoon events) Eta Similarity (normal events)
Boundary update frequency	1 h
Sea-surface roughness	default
Grid nudging	guv gt gq = 0.0003 s−1

.

2.2.2. SWAN Model

The fundamental equations and source terms of SWAN are similar to those of WAM, but it incorporates an implicit formulation for shallow water wave propagation. During computation, SWAN allows the choice between a Cartesian coordinate system (in meters) and a spherical coordinate system (in degrees). The equation is as follows:

$${\displaystyle \frac{\partial N}{\partial t}}+{\displaystyle \frac{\partial }{\partial x}}{C}_{x}N+{\displaystyle \frac{\partial }{\partial y}}{C}_{y}N+{\displaystyle \frac{\partial }{\partial \sigma }}{C}_{\sigma }N+{\displaystyle \frac{\partial }{\partial \theta }}{C}_{\theta }N={\displaystyle \frac{S_{\mathrm{tot}}}{\sigma }}$$

(1)

where $x$ and $y$ represent Cartesian coordinates, $t$ is time, and $\theta$ denotes the propagation direction of the dynamic density component. $N$ represents the dynamic density spectrum. The model accounts for all physical processes affecting wave dynamics from deep water to coastal areas, including wind growth, interactions between individual waves, wave breaking, and energy dissipation due to bottom friction. $C_x$, $C_y$, $C_{\sigma }$ and $C_{\theta }$ denote the propagation velocities of dynamic density in different directions. In shallow water conditions, the source term includes six distributions:

$$S_{\mathrm{tot}}=S_{\mathrm{in}}+S_{\mathrm{nl}3}+S_{\mathrm{nl}4}+S_{\mathrm{ds},\mathrm{w}}+S_{\mathrm{ds},\mathrm{b}}+S_{\mathrm{ds},\mathrm{br}}$$

(2)

where $S_{\mathrm{in}}$ represents the wind input, $S_{\mathrm{ds},\mathrm{b}}$ denotes the bottom friction term, and $S_{\mathrm{ds},\mathrm{w}}$ signifies the whitecapping dissipation term. The term $S_{\mathrm{nl}4}$ indicates the four-wave interaction. The depth-induced breaking term $S_{\mathrm{ds},\mathrm{br}}$ and the three-wave interaction term $S_{\mathrm{nl}3}$ are included in the SWAN model to better simulate wave conditions in shallow water regions, distinguishing it from other numerical wave models.

In this study, the SWAN model adopts a dual-nested domain, and the simulation range spans from 5° N to 25° N and from 105° E to 125° E. Based on prior experience and the relevant literature, we utilize the Westhuysen wind input [37], FIT wind drag formula, AB [38] whitecapping, and Jonswap bottom friction [39] physical parameterization schemes to simulate HS and TM in this region. The specific settings of the SWAN model are shown in

Table 3. Specific settings of the SWAN model.

	Domain1	Domain2
Resolution	10 km	1 km
Time_steps	1200 s	300 s
Wind input	Westhuysen
Wind drag formula	FIT
Whitecapping	AB
Bottom friction	Jonswap
Water depth	ETOPO1
Frequencies	0.05–1.0 HZ

.

3. Result and Discussion

3.1. Validation of WRF and SWAN

To validate the simulated data in the engineering sea area during typhoon conditions, we selected the B1 buoy for verification using observed data from typhoons Ty201213, Ty201522, Ty201822, and Ty202118. Figure 2, Figure 3, Figure 4 and Figure 5 present comparisons between observed and simulated wind speed, wind direction, HS, and TM at the B1 buoy during typhoon events. The results indicate that the simulated data align well with the observed data, with the simulated maximum wind speed and HS showing high consistency with the observed values. Additionally, the temporary buoy B2 recorded the observed wind speed, wind direction, HS, TM, and peak periods (TP) data from 1 November 2024 to 1 December 2024. Figure 6 presents a comparison of observed wind speed, wind direction, HS, and TP data from the B2 buoy under normal weather conditions with simulated data. The results demonstrate that the WRF and SWAN models can effectively simulate the wind and waves in the engineering sea area during normal weather conditions.

3.2. Design Wind Speed

Using the WRF and SWAN models, we conducted a 30-year (1994–2023) reanalysis of marine environmental element data for the project sea area. Based on the 30-year reanalyzed marine environmental element data, we applied the Weibull, log-normal, Gumbel, and P-III models to fit the probability distribution of annual wind speed extremes. The fitting results are shown in Figure 7.

The four probability distributions, Gumbel, log-normal, Weibull, and P-III, were employed for fitting the distribution of annual extreme wind speeds at project point P. To determine the optimal distribution for the extreme value sequence, the goodness of fit was assessed using the K-S test, root-mean-square error method, and AIC criteria. This comprehensive evaluation aimed to select the most suitable marginal distribution type for ocean environmental factor annual extremes. The fitting test results for each distribution are presented in

Table 4. Statistical test results for annual extreme values at project.

Probability Distributions		K-S		RMSE	AIC
		Dn	Dn,α		v	Value
Wind (m/s)	Weibull	0.1086	0.2417	0.0441	3	−181.26
	Log-normal	0.0829	0.2417	0.0295	2	−207.46
	Gumbel	0.0763	0.2417	0.0379	2	−192.29
	P-III	0.0745	0.2417	0.0228	3	−220.86

.

The D_n values for the distribution fittings of annual extreme wind speeds at project point P are all less than D_n,α, indicating that all distributions pass the K-S test and demonstrate good fit. In the statistical distribution curve, the fitted tail of the P-III distribution closely aligns with the empirical points, indicating a good fit. Similarly, in the fitting process of the P-III distribution, the empirical frequency points and the theoretical frequency curves obtained through different fitting methods are plotted on the same probability paper. First, it is ensured that the estimated curve passes through the center of the data cluster to avoid overall overestimation or underestimation. Special attention is given to the fitting in the high recurrence period (low frequency) region. The selected curve should align with the tail behavior of the data in terms of its extension trend at high quantiles, ensuring the accuracy of the extreme value analysis. Finally, a quantitative comparison is conducted using statistical tests to calculate the AIC values for different empirical fitting methods that meet the aforementioned criteria. The method corresponding to the smallest statistical value is selected. Based on the comprehensive results of extreme value statistical tests for various marine factors, the P-III distribution is identified as the optimal distribution type for calculating the design recurrence values of wind parameters at project point P.

For the design values of other wind parameters, calculations can be made according to relevant design standards.

(1)

Wind Speed at Hub Height

Assuming a mathematical expression for wind speed variation with height, a commonly used model is the power law curve:

$$V(z)=V\left(z_{\mathrm{r}}\right)\cdot {\left({\displaystyle \frac{z}{z_{\mathrm{r}}}}\right)}^{\alpha }$$

(3)

where $V\left(z_{\mathrm{r}}\right)$ and $z_{\mathrm{r}}$ represent wind speed at an arbitrary height and arbitrary height, respectively. $z$ is the hub height, and $V(z)$ is the wind speed at the hub height. α is the coefficient, which is set to 0.09 based on data from nearby meteorological towers.

(2)

Conversion of 3s Wind Speed

The American Petroleum Institute (API) standards provide empirical formulas for converting average wind speeds over different time periods to the 1 h average wind speed $U_{1hr}$:

$${\displaystyle \frac{U_t}{U_{1hr}}}=1.277+0.296\tanh [0.39\ln ({\displaystyle \frac{45}{t}})],1s<t<3600s$$

(4)

where $U_{1hr}$ is the 1 h average wind speed, and $U_t$ is the average wind speed over any given time within one hour.

The mean wind speeds at project point P for different return periods, heights, and t time periods are shown in

Table 5. Mean wind speeds at project point P under different conditions and return periods.

Items		Return Period
		2	5	10	50	100	500
Wind (10 m)	3 s	34.31	42.37	47.48	57.99	62.21	71.53
	10 min	26.39	32.59	36.52	44.61	47.85	55.02
Wind (170 m)	3 s	44.27	54.67	61.27	74.84	80.27	92.3
	10 min	34.05	42.06	47.13	57.57	61.75	71

. Figure 8 presents the design levels (ordinate) corresponding to different return periods (abscissa, logarithmic scale), where blue solid points denote the design values, the light purple shaded area represents the 95% confidence interval, and the red dashed lines mark the lower/upper bounds of the interval. Notably, the underlying data sample of this study has a length of 30 years. When extrapolating to ultra-long return periods (e.g., 500 years) based on such a short sample, the uncertainty of statistical extrapolation will be further magnified, leading to an even broader confidence interval (inferred from the trend in this figure). Such a wide interval implies a large fluctuation range of design values for ultra-long return periods, which poses challenges to the safety margin assessment of marine engineering structures. This also highlights the practical significance of optimizing modeling methods (e.g., non-parametric approaches like KDE) to mitigate extrapolation uncertainty.

3.3. Design Waves

3.3.1. IFORM Theory

Marine engineering applications such as offshore wind turbines, drilling platforms, ship design, and coastal structures exhibit strong coupling effects and inherent randomness. Conventional univariate extreme value analysis neglects the interdependencies among variables, often resulting in overly conservative designs. By considering only the individual maximum of each variable, this approach yields occurrence probabilities significantly lower than those of joint extremes. Moreover, it may lead to non-conservative risks due to the omission of unfavorable combinations, such as high wave heights coupled with high wind speeds.

The IFORM integrated with the environmental contour concept effectively addresses these limitations. Recognized in international standards such as IEC 61400-3-2 [40] and DNV-RP-C205 [41], IFORM provides a robust framework for defining extreme environmental conditions in design. It employs probability space transformations to construct a hypersurface of constant failure probability.

The environmental variables are represented by a vector X = (X1, X2, …, Xn), where, for instance, X1 denotes significant wave height (Hs) and X2 represents 10 m wind speed (U10). The joint cumulative distribution function (CDF) is expressed as follows:

F_X(x) = P(X₁ ≤ x₁, …, X_n ≤ x_n)

(5)

A major challenge lies in accurately capturing the complex correlations among marine environmental parameters. Deriving an analytical solution for the high-dimensional joint extreme value distribution remains generally infeasible.

The structural failure probability Pf is mapped to a hypersphere in standard normal space via the following relation:

P_f = P(g(X) ≤ 0) = Φ(−β)

(6)

where g(X) denotes the limit state function, and β is the reliability index.

For a target return period, designated T_R (measured in years), the annual failure probability can be determined by the following:

P_f,annual = 1/(T_R⋅N)

(7)

where N represents the number of independent environmental events per year.

The corresponding reliability index is expressed as follows:

β = −Φ⁻¹(P_f,annual) = Φ⁻¹(1 − 1/(T_R⋅N))

(8)

The construction of environmental contours primarily involves transforming the original correlated non-normal variables X into independent standard normal variables U. The principal methodologies include the following:

(1)

Rosenblatt Transformation

This method applies a chain-rule factorization to the joint CDF:

$$\left\{\begin{array}{l}{U}_1={\Phi }^{-1}(F_{X_1}(x_1))\\ U_2={\Phi }^{-1}(F_{X_2|X_1}(x_2|x_1))\\ ⋮\\ U_n={\Phi }^{-1}(F_{X_n|X_1,\dots ,X_{n-1}}(x_n|x_1,\dots ,x_{n-1}))\end{array}\right.$$

(9)

While the Rosenblatt transformation can accurately capture any dependency structure, it demands an understanding of the conditional distributions associated with environmental variables. Its computational complexity increases significantly in high-dimensional settings with intricate correlations.

(2)

Nataf Transformation

Based on a Gaussian copula assumption, the Nataf transformation converts the marginal distributions as follows:

$$Z_i={\Phi }^{-1}(F_{X_i}(x_i))$$

(10)

The linear correlation matrix R_Z is adjusted from the original correlation matrix using Hermite polynomial integration:

$${\rho }_{Z_i{Z}_j}=G({\rho }_{X_i{X}_j},{\sigma }_{X_i},{\sigma }_{X_j})$$

(11)

(3)

Copula Function Approach

This technique directly models the joint probability distribution of environmental variables using a copula function:

$$F_X(x)=C(F_{X_1}(x_1),\dots ,F_{X_n}(x_n))$$

(12)

Commonly used copula families include Gaussian, t, Clayton, and Gumbel.

In the U-space, the failure boundary is defined by a hypersphere of radius β:

$$B=\{u\in R^n:\parallel u\parallel =\beta \}$$

(13)

By inversely transforming points from this hypersphere in standard normal space back to the physical variable space, the environmental contour is obtained:

$$C=\{x\in R^n:x=T^{-1}(u), \parallel u\parallel =\beta \}$$

(14)

A parametric approach can be employed for contour generation. For a bivariate case involving significant wave height Hs and spectral peak period Tp, the contour is derived as follows:

$$\left\{\begin{array}{l}{x}_1=F_{X_1}^{-1}(\Phi (\beta \cos \theta ))\\ x_2=F_{X_2|X_1}^{-1}(\Phi (\beta \sin \theta )|x_1)\end{array}\right.$$

(15)

3.3.2. Two-Dimensional Kernel Density Estimation

Two-dimensional kernel density estimation (KDE) provides a flexible non-parametric alternative, capable of effectively capturing the correlation structure of wave parameters and other variables, provided that sufficient data are available.

Consider nn independent and identically distributed two-dimensional sample points ${\{X_i=(X_i,Y_i)\}}_{i=1}^n$ drawn from an unknown joint probability density function f(x). The objective is to estimate f(x) using a non-parametric approach. The core idea of kernel density estimation is to center a kernel function at each sample point, thereby transforming the discrete sample set into a continuous density distribution.

The estimated density is given by the weighted average of all kernel functions:

$${\widehat{f}}_h(x)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{K}_{\mathbf{H}}(x-X_i)}$$

(16)

where K_H(⋅) denotes the kernel function controlled by the bandwidth matrix H.

The kernel function K(⋅) must satisfy the following conditions: non-negativity, K(u) ≥ 0; normalization, $\underset{{ℝ}^2}{{\displaystyle \int }}K(u)du=1$; symmetry, $K(-u)=K(u)$.

In two-dimensional KDE, a bandwidth matrix HH (symmetric positive definite) is introduced to accommodate potential anisotropy in the data:

$${\widehat{f}}_h(x)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{K}_H(x-X_i)}$$

(17)

$$K_H(u)={\left|H\right|}^{-1/2}K(H^{-1/2}u)$$

(18)

H governs the shape and scale of the kernel: if H = h²I, the kernel is isotropic; if H contains off-diagonal elements, it can capture correlations between variables. Through the spectral decomposition H = RDR^T, the rotation matrix R controls the orientation of the kernel, while the scaling matrix D determines the axis lengths.

Bandwidth selection is a central challenge in KDE, as it directly influences the bias–variance trade-off of the estimator. The theoretical criterion for bandwidth selection is to minimize the mean integrated squared error (MISE):

$$MISE(H)=E\left[{\displaystyle \int {\left({\widehat{f}}_H(x)-f(x)\right)}^{2}dx}\right]$$

(19)

Common bandwidth selection algorithms include the following:

(1)

Smoothed Cross-Validation (SCV)

This method minimizes the cross-validation score:

$$SCV(H)={\displaystyle \int {\widehat{f}}_H^2(x)dx}-{\displaystyle \frac{2}{n}}{\displaystyle \sum _{i=1}^n{\widehat{f}}_{H,-i}(X_i)}$$

(20)

where ${\widehat{f}}_{H,-i}$ denotes the density estimate obtained after excluding the i-th sample.

(2)

Plug-In Method

This approach approximates the MISE via Taylor expansion to solve for the optimal H. First, the Hessian matrix is estimated, leading to the following optimal bandwidth formula for the Gaussian kernel:

$$\Psi =\int ({\nabla }^{2}f)fdx$$

(21)

$$H_{\mathrm{op}t}=n^{-1/3}{\left({\displaystyle \frac{4}{3}}\right)}^{1/3}{\widehat{\Psi }}^{-1/3}$$

(22)

(1)

Rule-of-Thumb Method

A simplified bandwidth for isotropic Gaussian kernels can be expressed as follows:

$$h={\left({\displaystyle \frac{4}{3n}}\right)}^{1/5}\widehat{\sigma }$$

(23)

where $\widehat{\sigma }$ represents the sample standard deviation. For two-dimensional data, this should be adjusted according to the characteristics of each variable.

This study adopts the Rule-of-Thumb method to determine the bandwidth, which is calculated based on the sample standard deviation and sample size referring to Silverman’s rule to balance computational efficiency and estimation accuracy for extreme value intervals.

3.3.3. H_s-T_p Environmental Contours

The H_s-T_p environmental contour defines a joint exceedance boundary in terms of a specified return period, serving as a critical input for the design loads of marine structures. For instance, the 50-year contour encompasses all combinations of H_s and T_p that may induce structural responses corresponding to a 50-year return period. In practice, marine environmental parameters such as H_s and T_p often exhibit non-Gaussian marginal distributions and complex nonlinear dependencies. Conventional parametric assumptions (e.g., log-normal or Weibull distributions) or simplistic correlation coefficients may fail to capture their true joint probabilistic structure. Two-dimensional kernel density estimation (2D KDE) provides a non-parametric approach to estimate a smooth, data-driven joint probability density function directly from scattered observational data, without presupposing any specific distribution family. It effectively captures the underlying distribution and dependence characteristics. The IFORM is then employed to map the joint probability problem into an independent U-space, within which a limit state surface corresponding to a specified failure probability can be determined.

The integrated KDE–IFORM procedure for constructing environmental contours comprises the following key steps:

Step 1: Joint Probability Modeling via 2D KDE

Long-term hourly hindcast data of H_s and T_p from the project sea area (1994–2023) are used. To improve normality and numerical stability, logarithmic transforms are applied, (x, y) = (log(H_s), log(T_p)). The joint probability density at any point (x, y) is estimated as follows:

$$\widehat{f}(x,y)={\displaystyle \frac{1}{n{h}_x{h}_y}}{\displaystyle \sum _{i=1}^{n}K(\frac{x-X_i}{h_x})K(\frac{y-Y_i}{h_y})}$$

(24)

where n is the sample size, K(⋅) is the Gaussian kernel, h_x and h_y are bandwidths optimized via cross-validation, and (X_i, Y_i) are the log-transformed samples. The output is a smooth joint probability density function f_{{Hs, Tp}}(log(h), log(t)) that accurately reflects the empirical dependence structure.

Step 2: Transformation to Standard Normal Space

Using the Rosenblatt transformation, the correlated non-Gaussian variables (log(H_s), log(T_p)) are mapped into independent standard normal variables (U₁, U₂) via conditional cumulative distribution functions. First, transform log(H_s) to U₁:

$$\Phi (u_1)=F_{\log (H_s)}\log ((h))$$

(25)

$$u_1={\Phi }^{-1}(F_{\log (H_s)}\log ((h)))$$

(26)

where the marginal CDF F_{log(Hs)} is obtained by integrating the KDE:

$$F_{\log (H_s)}\log ((h))={\displaystyle {\int }_{-\infty }^{\log (h)}{\displaystyle {\int }_{-\infty }^{\infty }\widehat{f}(x,y)dydx}}$$

(27)

Then, transform log(T_p) conditioned on H_s to U₂:

$$\Phi (u_2)=F_{\log (T_p)|\log (H_s)}(\log (t)|\log ((h))$$

(28)

$$u_2={\Phi }^{-1}(F_{\log (T_p)|\log (H_s)}(\log (t)|\log ((h)))$$

(29)

where the conditional CDF F_{{log(Tp)|log(Hs)}} is derived from the KDE as follows:

$$F_{\log (T_p)|\log (H_s)}(\log (t)|\log ((h))={\displaystyle \frac{{\displaystyle {\int }_{-\infty }^{\log (t)}\widehat{f}(\log (h),y)dy}}{{\displaystyle {\int }_{-\infty }^{\infty }\widehat{f}(\log (h),y)dy}}}$$

(30)

Each point (H_s, T_p) in the physical space thus corresponds uniquely to a point (U₁,U₂) in U-space.

Step 3: Contour Definition in U-Space

In the independent standard normal space, the limit state surface corresponding to an annual exceedance probability is defined by a circle centered at the origin, u₁² + u₂² = β². The radius β is directly related to the annual exceedance probability. In the two-dimensional independent standard normal space, the probability that a point is located outside the circle that is centered at the origin and has a radius of β.

Step 4: Inverse Mapping to Physical Space

Points located on the circle in U-space are converted back to the physical (H_s, T_p) space through the inverse Rosenblatt transformation. The resulting set of points forms the environmental contour for the specified return period.

Traditional parametric methods require pre-specified marginal distributions and copula functions (e.g., Gumbel, Weibull, or Gumbel copula). Incorrect assumptions can introduce bias, and such models often struggle to capture complex nonlinear dependencies, particularly under extreme extrapolation. In contrast, the KDE-IFORM approach is fully data-driven, flexibly adapts to the true joint distribution, inherently incorporates all interdependencies, and generally yields more accurate extreme environmental estimates.

Based on 30 years (1994–2023) of hourly hindcast data for the project sea area, the H_s-T_p environmental contours derived from the KDE-IFORM method are presented in Figure 9.

The extreme environmental boundary, derived from hindcast marine environmental data, defines an envelope encompassing the specified return period and accurately reflects the dependency between variables. Discrete points along this contour provide the requisite inputs for the reliability design and evaluation of marine engineering structures. The resulting design values of the extreme H_S and the associated T_P in the project sea area are listed in

Table 6. Joint design wave parameters at project point P.

Items	Return Period (Year)
	1	5	10	50	100	500
Hs (m)	5.49	8.31	9.54	12.39	13.61	16.46
Tp (s)	10.74	12.82	13.68	15.07	15.91	17.52

.

In Figure 10, the blue scatter points represent the original observational data, while the red, black, pink, and green curves denote the 100-year return period environmental envelopes fitted by the Gumbel, Clayton, Frank, and Gaussian copula models, respectively. Visual inspection of the plot yields the following key observations: all four copula models exhibit poor agreement between their fitted envelopes and the distribution pattern of the original data, particularly the clustering characteristics of scatter points in the region of large Hs. These models fail to effectively capture the actual correlation of extreme environmental parameters (i.e., large H_s and large T_p). Traditional copula models rely on the assumptions of parametric marginal distributions and fixed correlation structures, which are incompatible with the heavy-tailed and asymmetric distribution characteristics of marine environmental parameters. This inconsistency leads to significant fitting deviations of the envelopes in the extreme value region.

4. Conclusions

To estimate the design wind speeds and waves for project point P in the coastal waters near Guangzhou for different return periods, this study employed the WRF and SWAN to simulate wind and wave conditions over a 30-year period (1994–2023). The WRF model utilized Thompson MP, KF CU, and YSU PBL for typhoon wind field simulations, while Thompson MP, Tiedtke CU, and MYJ PBL were applied for normal conditions. For the SWAN model, the Westhuysen wind input FIT wind drag formula AB whitecapping and JONSWAP bottom friction were utilized for wave simulations in the project sea area.

Validation through buoy B1’s measured data (wind speed, wind direction, HS, and TM) during typhoons Ty201213, Ty201522, Ty201822, and Ty202118 demonstrated good consistency between simulated and observed data. Additionally, comparisons of buoy B2’s measured values under normal weather conditions for wind speed, wind direction, HS, and TP confirmed that the WRF and SWAN models effectively simulate normal conditions in the project area. The design wind speeds at project point P for different return periods were fitted using Gumbel, log-normal, Weibull, and P-III probability distributions. The optimal distribution was selected based on the K-S test, RMSE, and AIC. Results indicated that the P-III distribution closely approximated the tail of the empirical data, yielding a good fit. At a height of 10 m, the 3 s and 10 min mean wind speeds for 100-year and 50-year return periods were 62.21 m/s, 47.85 m/s, 57.99 m/s, and 44.61 m/s, respectively. At hub height (170 m), the corresponding values were 80.27 m/s, 61.75 m/s, 74.84 m/s, and 57.57 m/s.

This study successfully applied a two-dimensional KDE approach to construct a joint probability model and derive environmental contours for extreme environmental assessment. The HS and TP at project point P for 100-year and 50-year return periods are 13.61 m and 15.91 s, as well as 12.39 m and 15.07 s, respectively. The results demonstrate that this non-parametric method offers significant advantages in characterizing the joint distribution of key environmental parameters, specifically wave height and wave period. A major strength of KDE is its independence from prior assumptions regarding the underlying distribution forms, enabling it to adaptively capture complex dependence structures and multi-modal features present in the data, thereby avoiding potential biases caused by model misspecification. This capability is particularly critical for accurately describing environmental parameters under complex meteorological conditions, such as during typhoon passages overlapping with normal sea states, ensuring that the derived contours genuinely reflect the intrinsic characteristics of the data. The environmental contour method based on two-dimensional KDE proves to be a powerful and flexible tool, especially suitable for situations where parametric model assumptions are questionable or data structures are complex. The validation under normal weather conditions relies solely on one month of measured data from the B2 temporary buoy, which may result in the model not accurately simulating long-term design parameters for the engineering location. In future work, we will collect more measured data under normal weather conditions at the engineering site to enhance the long-term simulation accuracy of the model, and we will focus on the development of more computationally efficient algorithms and on exploring the integration of KDE with extreme value theory, to further enhance the characterization of tail risks associated with joint extreme events.