Quantifying the Uncertainties in Projecting Extreme Coastal Hazards: The Overlooked Role of the Radius of Maximum Wind Parameterizations

Abstract

Parametric tropical cyclone models are widely used to generate large wind field ensembles for assessing extreme storm tides and wave heights. The radius of maximum wind (RMW) is a key model parameter and is commonly estimated using empirical formulas. This study shows that uncertainty introduced by the choice of RMW formulas has been largely overlooked in tropical cyclone risk assessments. Using the Pearl River Estuary as a case study, historical wind fields (1981–2024) were generated with a parametric tropical cyclone model combined with eight empirical RMW formulas. Storm tides and wave heights during tropical cyclone events were simulated using a coupled wave–current model (ROMS–SWAN) and analyzed with extreme value theory. The results indicate that, for estuarine nearshore zones, the 100-year return period of water level and significant wave height vary by up to 1.26 m and 1.54 m, respectively, across all the selected RMW formulas. Joint probability analysis further shows that RMW uncertainty can shift the joint return period of the same compound storm tide and wave event from 100 years to 10 years. For an individual extreme event, differences in the RMW formula alone can produce deviations up to 2.11 m in peak storm tide levels and 3.8 m in significant wave heights. Such differences can also change the duration of extreme sea states by 13 h. These results highlight that RMW formula selection is a critical uncertainty factor, and related uncertainty should be considered in large-sample tropical cyclone hazard assessment and engineering design.

1. Introduction

Tropical cyclones (TCs) are among the most consequential meteorological hazards for coasts worldwide, generating extreme winds, storm tides, and extreme waves that threaten coastal infrastructure and public safety [1,2,3]. The Western North Pacific (WNP) is the most active basin for TCs. The South China Sea lies close to the WNP core activity region and rank among the most TC-affected areas globally, with an average of about 7.5 TCs making landfall along China’s southeastern coast each year [4,5,6]. Statistics from the China Marine Disaster Bulletin (1989–2015) show that marine hazards along China’s coast caused average direct economic losses of about CNY 18.8 billion per year and about 256 fatalities per year, with TC-driven storm tides and extreme waves as major contributors [7]. More critically, under ongoing global warming, TCs over WNP exhibit increases in frequency and intensity and a pronounced northward shift in tracks [8,9,10,11]. These changes shorten the return periods of extreme sea states and markedly increase the failure probability of coastal protection infrastructure [12,13]. Therefore, reliable assessments of extreme storm tide and wave heights are essential for coastal safety and socio-economic development.

Parametric tropical cyclone (TC) models are widely used to reconstruct TC wind fields for risk assessment [14,15]. Simulating storm tides and waves with hydrodynamic and wave models requires spatiotemporally continuous atmospheric forcing [16,17]. For long periods or large domains, generating such forcing with numerical weather models for every TC is often computationally prohibitive [18,19]. In addition, reanalysis products (e.g., ERA5) commonly underestimate TC intensity [20,21]. Consequently, the best track-based parametric TC models have become one of the most commonly adopted and broadly transferable approaches for wind field reconstruction. These models represent the radial distributions of near-surface wind speed and pressure balancing physical realism and computational efficiency [22]. Early parametric TC wind models range from simple analytic vortices (e.g., Rankine-type profiles) to more physically constrained gradient wind formulations such as the Holland model, and further to refined piecewise or thermodynamic frameworks that better represent radial structure [23,24,25,26,27,28,29,30]. This parameterization has further been extended to wave prediction through a two-dimensional parametric wave model based on conservation laws, enabling rapid wave estimation for large TC ensembles [31]. Across these representative parametric schemes, maximum wind speed and RMW remain fundamental parameters for describing TC intensity and size [32].

When large ensembles of TC wind fields are generated using parametric models, the RMW is typically estimated using empirical formulas [33]. The RMW is defined as the radial distance from the TC center to the region of maximum winds. Because the RMW is located near the eyewall and is difficult to observe, it is often absent from best track datasets, and empirical formulas are commonly used to infer it from central pressure, pressure deficit, or maximum wind speed for use in simulations [34]. For example, Gao et al. [35] combined Holland parametric winds for 192 TCs with ERA5 winds to assess extreme significant wave height and storm surge along the southeast coast of China, estimating the RMW using a NOAA central-pressure relationship calibrated with aircraft observations. Using an ADCIRC + SWAN coupled model, Imani et al. [36] simulated waves from 107 TCs affecting Zhejiang and Fujian during 1987–2018, with RMW prescribed from an empirical formula of central pressure. Liu et al. [37] applied an ADCIRC + SWAN model and the Jelesnianski model for storm surge risk assessment and evacuation planning in Daya Bay, determining RMW from an empirical relationship with the pressure deficit. For synthetic TCs, Lin et al. [38] coupled a stochastic track generator with a dynamical model to produce tens of thousands of events and parameterized RMW as an intensity-dependent variable. Similarly, Zhuge et al. [39] used Monte Carlo synthetic TC ensembles to improve the robustness of long return period estimates of water level and wave height, computing RMW from a relationship involving central pressure and latitude. In compound hazard assessments driven by synthetic TCs, empirical RMW relationships derived from aircraft reconnaissance and H*Wind datasets are also commonly used [40].

However, selecting a single empirical RMW formula can introduce substantial uncertainty in TC size and strongly modulate coastal hydrodynamic responses. Multiple studies show that radial structure can markedly affect peak storm tide and wave height [41,42]. In Hurricane Katrina experiments, storm tide varied by up to 30% as RMW increased [43]. North Atlantic wave simulations reveal a significant positive link between maximum significant wave height and TC size [44]. In addition, RMW characteristics depend on observational coverage, predictor choice, and regression methods, which can also introduce bias into empirical estimates [45,46]. Consequently, the adopted RMW formula in large-sample simulations directly affects extreme storm tides and significant wave heights, propagating into inundation mapping, vulnerability assessment, and loss estimation. Modern risk management further emphasizes compound extremes of water level and wave height [47,48], for which the joint dependence structure controls event frequency and severity. Biased datasets can distort both univariate extremes and cross-variable dependence, misidentifying hotspots and misallocating mitigation resources [49]. For design, underestimation reduces protection capacity, whereas overestimation inflates costs [50]. Under a 100-year standard, a 0.5 m error in extreme water level can raise seawall costs by 20–30% or lower the effective protection to only a few decades [51].

Taken together, a key research gap remains: the TC size uncertainty introduced by adopting a single empirical RMW formula is often neglected, particularly when large ensembles of TC tracks are modeled. Its impact on long return period storm tide, wave height, and their compound risk is still poorly quantified. To address this gap, this study quantifies the uncertainty in extreme storm tide and significant wave height estimates attributable to RMW formulas. We emphasize that TC size is as critical as the more commonly emphasized TC track and intensity in robust hazard assessment.

2. Materials and Methods

2.1. Study Area and Data

The study area covers the Pearl River Estuary (PRE) and the offshore sea area. Observations of water levels and significant wave heights in the shallow waters surrounding Hong Kong were compiled. For each TC season from 1981 to 2024, the TC producing the largest mean storm tide and significant wave height at Hong Kong was defined as the extreme TC event for that year. The annual maxima approach directly characterizes the most adverse storm tide and wave impacts on PRE during each TC season, aligning with risk assessment objectives that focus on the dominant hazard event in a given year. TC parameters were obtained from the best track dataset provided by the Joint Typhoon Warning Center (JTWC). TC tracks and intensities are shown in Figure 1a. Observed storm tides and wave heights were collected from multiple tide gauges, wave monitoring stations, and buoys across the PRE and nearby waters (Figure 1b).

2.2. Parametric TC Model

Holland (2010) [25] parametric TC model (H10) was used to reconstruct the wind and pressure fields of translating TCs and to provide atmospheric forcing for the coupled ROMS–SWAN system. H10 proposed a revised form of the classical Holland parametric wind pressure profile that allows observations from the outer region of TCs to be incorporated when constructing the radial wind speed structure. This formulation is better suited for reconstructing TC wind fields from limited storm parameters and for subsequent ocean modeling applications [25]. The analytical profiles produced by H10 yield dynamically consistent wind pressure fields that are well suited for coupled simulations of storm tides and waves. The governing equations for the radial pressure and wind speed structure of TC are given below.

$$p(r)=p_c+(p_n-p_c)\cdot e^{-{\left({\displaystyle \frac{RMW}{r}}\right)}^B}$$

(1)

$$V(r)={\left[V_{\max }^2{\left({\displaystyle \frac{RMW}{r}}\right)}^B{e}^{\left[1-{\left({\displaystyle \frac{RMW}{r}}\right)}^B\right]}\right]}^x$$

(2)

$$B={\displaystyle \frac{V_{\max }^2\rho \mathrm{e}}{p_{\mathrm{n}}-p_{\mathrm{c}}}}$$

(3)

$$x=\left\{\begin{array}{ll}0.5,\hfill & r\le RMW\hfill \\ 0.5+(r-RMW){\displaystyle \frac{x_n-0.5}{r_n-RMW}},\hfill & r>RMW\hfill \end{array}\right.$$

(4)

$$x_n={\displaystyle \frac{\ln V_n}{\ln \left[V_{\max }^2{\left(\frac{RMW}{r_n}\right)}^B{\mathrm{e}}^{{\left(1-\frac{RMW}{r_n}\right)}^B}\right]}}$$

(5)

where $V\left(r\right)$ and $p\left(r\right)$ denote the wind speed and pressure at a radial distance r from the TC center. $V_{\mathrm{m}\mathrm{a}\mathrm{x}}$ represents the maximum wind speed. $p_c$ and $p_n$ are the central and environmental pressures of TC, respectively. $B$ is the shape parameter controlling the radial wind distribution. $V_n$ and $r_n$ represent the wind speed at an outer observation point and its distance from the TC center, respectively. The 34-knot wind speed and its corresponding radius are used in this context.

To represent the full-domain wind field, the H10-computed winds were decomposed into radial and tangential components with respect to the TC center. The degree of inflow toward TC center was quantified using Equation (6). The translation-induced wind component reflects the large-scale environmental steering flow. TC translation can enhance winds on one side of the storm [52]. To account for this asymmetry, Equation (7) specifies how the large-scale translation wind is superimposed on the H10 wind field.

$$\beta =\left\{\begin{array}{cc}10°\left(1+r/RMW\right),& 0\le r<RMW\hfill \\ 20°+25°\left(r/RMW-1\right),& RMW\le r<1.2RMW\hfill \\ 25°,& r\ge 1.2RMW\hfill \end{array}\right.$$

(6)

$${\overrightarrow{V}}_{ts}(r)={\displaystyle \frac{RMW\times r}{RM{W}^2+r^2}}{\overrightarrow{V}}_t$$

(7)

where $\beta$ denotes the TC inflow angle, ${\overrightarrow{V}}_{ts}$ denotes the translation-induced wind vector at radius r from the TC center, and ${\overrightarrow{V}}_t$ denotes the TC translation velocity vector.

2.3. Empirical Formulas of RMW

As a key inner-core scale parameter in parametric TC models, RMW is difficult to observe directly because of limitations in observing techniques and the complex dynamics of TCs. Many studies have therefore analyzed large datasets to identify statistical relationships between RMW and more readily observed parameters such as maximum wind speed, latitude, and central pressure. For example, previous studies have documented eyewall contraction during the intensification stage and a general tendency for the inner-core size to increase with latitude ($\varphi$) [53,54,55]. Eight empirical RMW formulas widely used for WNP are applied in this study.

Table 1. Sources, sample regions, and explicit expressions of the eight empirical RMW formulas (RMW in km; $V_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $V_t$ in m/s; $p_c$ and $p_n$ in hPa; $\varphi$ in degrees).

Formula Source	Region of Sample Origin	Formula Form
G1959 [56]	U.S. East Coast and Gulf of Mexico	$RMW=28.52\tanh [0.0873(\varphi -28)]+12.22{\mathrm{e}}^{{\displaystyle \frac{p_c-1013.2}{33.86}}}+0.2V_t+37.22$
W2004 [26]	Atlantic and Eastern Pacific	$RMW=51.6e^{-0.0223V_{\max }+0.0281\varphi }$
K2005 [57]	Japanese coast	$RMW=80-0.769(950-p_c)$
K2007 [58]	Atlantic and Pacific	$RMW=66.785-0.1769V_{\max }+1.0619(\varphi -25.0)$
V2008 [59]	Atlantic	$RMW=e^{3.015-0.00006291{(p_n-p_c)}^2+0.0337\varphi }$
L2013 [60]	Western North Pacific	$RMW=-18.04\ln (p_n-p_c)+110.22$
F2017 [61]	Western North Pacific	$RMW=(-37.82+0.11\varphi )\cdot \ln \left((2.86-0.0029\varphi ){(p_n-p_c)}^{0.7}\right)+178.2$
K2018 [62]	Western North Pacific	$RMW=105.4158-0.5548V_{\max }+1.3653(\varphi -25)$

summarizes the sources (hereafter, the empirical RMW formulas are denoted by the first author’s initial followed by the publication year), sampling regions used for empirical relationship fitting, and functional forms of the formulas, with RMW reported in kilometers (km).

2.4. ROMS–SWAN Model

A two-way coupled wave–current modeling system combining the Regional Ocean Modeling System (ROMS) and the Simulating WAves Nearshore model (SWAN) was applied to simulate TC-induced extreme storm tides and waves in the PRE. ROMS and SWAN share an orthogonal curvilinear grid that spans most of the northern South China Sea, ensuring coverage of the selected historical TC tracks. The core study area is located in the PRE and offshore deep-water region and is sufficiently distant from the open boundaries to minimize sensitivity to boundary forcing (Figure 1a). The grid comprises 440 × 420 cells, with a 500 m resolution in the PRE that smoothly coarsens offshore to 8 km, balancing accuracy and computational efficiency. Bathymetry in deep waters was taken from the Earth Topography 2022 global DEM, while shallow estuarine bathymetry was obtained from high-resolution electronic navigational charts. Tidal forcing was prescribed using 15 major constituents from TPXO9-atlas (MM, MF, Q1, O1, P1, K1, S1, N2, M2, S2, K2, MN4, M4, MS4, and 2N2). Previous evaluations have shown that the performance of global tidal models depends on both tidal constituents and location, and no single model consistently outperforms others across all sites and metrics [63]. TPXO9-atlas was selected for its assimilation of recent satellite altimetry and ~1/30° resolution, enabling improved representation of tidal water-level variability and spatial gradient. Initial and lateral boundary conditions were derived from the HYCOM global reanalysis. Because this study focuses on the influence of RMW on ocean state variables under parametric wind forcing, and baroclinic effects driven by temperature and salinity are secondary for this purpose, temperature and practical salinity were held constant at 25 °C and 35, respectively. SWAN was run in non-stationary mode and included standard source terms for whitecapping dissipation, depth-induced breaking, bottom friction, and nonlinear wave–wave interactions. The JONSWAP spectral shape was applied to the open boundaries in the SWAN model. ROMS and SWAN exchanged fields every 600 s via the Model Coupling Toolkit to achieve wave–current coupling.

The model’s performance in simulating astronomical tide levels and significant wave height was validated by comparing the simulated and observed data at multiple PRE stations during each year’s TC window. The results from TC “Mangkhut” (2018) are presented as an example. The model closely reproduces the observed tide (Figure 2). Black circles denote observations, and the red line denotes the simulation. Root mean square error (RMSE), mean bias, correlation coefficient, and the mean absolute phase error (MAE of phase) of the astronomical tide level are summarized in

Table 2. Evaluation metrics for the model performance in simulating astronomical tides.

	RMSE (m)	Mean Bias (m)	Correlation Coefficient	MAE of Phase (°)
Chiwan	0.204	−0.079	0.968	52.81
Dawanshan	0.155	−0.023	0.952	12.59
Guangzhou	0.194	−0.036	0.880	43.40
Nansha	0.205	−0.008	0.966	34.01
Quarry Bay	0.168	−0.044	0.922	28.29
Zhuhai	0.202	−0.066	0.906	58.17

. Figure 3 summarizes the performance of SWAN in simulating significant wave height, with data points corresponding to the WLC, QF303, QF305, and QF306 stations; time windows strongly influenced by the RMW were excluded (24 h before and after the peak). Most significant wave height data points fall within the acceptable ±1 m range, with an RMSE of 0.553 m and no evident overall bias. As shown in Figure 2 and Figure 3, the model provides a robust tidal baseline and reliable significant wave height simulations across the estuary, which are essential for credible nearshore modeling under TC forcing.

2.5. Extreme Value Analysis

During the extrapolation of extremes, the generalized extreme value (GEV) distribution was applied. Through its shape parameter, the GEV distribution can flexibly represent a wide range of tail behaviors and has therefore been widely used in the statistical analysis of marine hydrological extremes [64,65]. The GEV function is defined as follows:

$$G(z)=\exp \left\{-{\left[1+\xi \left({\displaystyle \frac{z-\mu }{\sigma }}\right)\right]}^{-1/\xi }\right\},\hspace{1em}1+\xi {\displaystyle \frac{z-\mu }{\sigma }}>0$$

(8)

where $G\left(z\right)$ is referred to as the GEV distribution family, $\sigma$ is the scale parameter, $\mu$ is the location parameter, and $\xi$ is the shape parameter. The parameters were estimated using the maximum likelihood estimation method. When applying the GEV method for extreme value analysis, the return level ($z_p$), corresponding to a return period of $1/p$, years can be determined by using Equation (9). By arranging multi-year observed data in descending order, with each datum assigned a rank $m$, the empirical return period $T_e$ can be calculated using Equation (10).

$$z_p=\left\{\begin{array}{ll}\mu -{\displaystyle \frac{\sigma }{\xi }}\left[1-{\{-\log (1-p)\}}^{-\xi }\right],\hfill & \xi \ne 0\hfill \\ \hspace{1em}\mu -\sigma \log \{-\log (1-p)\},\hfill & \xi =0\hfill \end{array}\right.$$

(9)

$$T_e={\displaystyle \frac{n+1}{m}}$$

(10)

To evaluate the applicability of the GEV distribution in the statistical analysis of extreme water levels and wave heights, the Kolmogorov–Smirnov (K–S) test is used for assessing the goodness of fit. When the p-value of the K–S test exceeds the prescribed significance level 5%, there is insufficient evidence to reject the null hypothesis, indicating that the estimates obtained from the extreme value distribution can be regarded as reliable.

To assess the compound risk arising from storm tides and waves, a joint distribution model for water level and wave height is developed based on copula theory. According to Sklar’s theorem, there exists a copula function $C(\cdot )$ that links the marginal distributions of multiple variables. The Gumbel copula is employed in this study because it accurately quantifies the probability of concurrent extreme high values and is widely used in joint extreme analyses of waves and sea level [66]. Its bivariate form can be expressed as follows:

$$C(u,v;\alpha )=\exp \left\{-{\left[{(-\ln u)}^{\alpha }+{(-\ln v)}^{\alpha }\right]}^{1/\alpha }\right\},\hspace{1em}0<u,v<1$$

(11)

where $u$ and $v$ denote the marginal cumulative probabilities of water level and wave height, respectively. The parameter $\alpha$ represents the degree of dependence. The relationship between $\alpha$ and rank correlation coefficient $\tau$ is given by $\tau =1-1/\alpha$, which is used to estimate the copula parameter.

3. Results

3.1. Impact of RMW Formulas on Long Return Period Estimates

Using different empirical RMW formulas, the peak water level and peak significant wave height associated with the selected TCs (1981–2024) were extracted. Long-term records of water level and wave height were obtained from the Quarry Bay and WLC stations, and the annual maxima were derived for each series. To remove the influence of long-term sea-level rise, all water-level series were detrended by subtracting the corresponding monthly mean. For the observation data and model results, annual extreme water levels at the Quarry Bay station and annual extreme significant wave heights at the WLC station were fitted with a GEV distribution, and all fits passed the K–S test. Based on the GEV results, return period curves were constructed for water level and significant wave height (Figure 4), where colored lines denote the different formulas and the gray line denotes observation. The set of return period curves obtained from the different RMW formulas defined an uncertainty envelope.

Table 3. RP100 water level and wave height extrapolated from the GEV fits, together with their 95% confidence intervals and GEV shape parameter.

	RP100 Value (m)		RP100 95% CI Width (m)		GEV Shape Parameter
	Water Level	Wave Height	Water Level	Wave Height	Water Level	Wave Height
G1959	1.93	3.65	0.50	2.04	−0.187	−0.083
W2004	2.02	3.86	0.60	1.78	−0.164	−0.151
K2005	3.19	5.19	2.62	2.53	0.123	−0.140
K2007	2.56	4.59	1.42	2.23	−0.005	−0.140
V2008	2.07	4.06	0.71	2.14	−0.119	−0.105
L2013	2.16	4.23	0.83	2.07	−0.096	−0.132
F2017	2.29	4.31	1.04	2.08	−0.055	−0.140
K2018	3.01	4.92	2.17	2.24	0.073	−0.170

summarizes the predicted 100-year return period (RP100) water level and significant wave height, along with the widths of their 95% confidence intervals (CI) and the GEV shape parameter.

To preserve the uncertainty introduced by the empirical RMW formulas while largely eliminating the influence of the SWAN model itself, the maximum RP100 difference across the tested schemes was computed. The results indicate that the choice of empirical formula for RMW exerts a strong influence on estimates of extreme water levels and wave heights at long return periods. For extreme water levels at return periods shorter than 10 years, the estimates from different formulas are relatively consistent, spanning 1.0–2.2 m. However, the inter-formula divergence increases rapidly with increasing return period. At the RP100, K2005 yields the largest prediction of 3.19 m, whereas G1959 provides the most conservative estimate of 1.93 m. The difference reaches 1.26 m, corresponding to a relative discrepancy exceeding 65%. For the RP500, the range broadens to approximately 2.2–4.1 m, with the maximum estimate nearly twice the minimum. The observational return period curve lies mainly near the middle of the predicted envelope, suggesting that the characteristics of real extremes fall between the estimates from different formulas. K2005 and K2018 occupy the upper part of the envelope; K2007 and F2017 provide intermediate estimates; and G1959, W2004, V2008, and L2013 form the lower-bound predictions. Extreme significant wave height exhibits a similar pattern but with more pronounced uncertainty. Within the 1–10-year return period range, predictions already show clear dispersion, varying from 2.6 m to 3.7 m. When extended to RP100, the predicted extreme values diverge by up to 1.54 m across the RMW formulas, with K2005 predicting 5.19 m and G1959 predicting only 3.65 m. Consistent with the extreme water-level results, the observational curve remains bracketed by the formulas, and the relative ranking of high and low predictions mirrors that observed for water level. Although all datasets passed the K–S test, supporting the reliability of tail extrapolation, the stability of the extrapolated estimates remains a key concern. Wider confidence intervals indicate greater uncertainty in the estimates. As summarized in Table 3, the 95% CI for the RP100 water level is narrowest under G1959, at 0.50 m, whereas K2005 yields a markedly broader interval of 2.62 m, approximately a fourfold increase. Likewise, for RP100 significant wave height, the 95% CI under K2005 is 42% wider than that under W2004. The choice of the empirical RMW formula can also alter the fitted extreme value distribution type. For extreme water levels, K2005 and K2018 yield positive GEV shape parameters, indicating a Fréchet-type distribution, whereas the other cases correspond to a Weibull-type distribution. In contrast, the extreme significant wave height distributions are Weibull-type in all cases.

After fitting the water level and wave height series at all grid points with the GEV distribution, the spatial distributions of RP100 water level and significant wave height were mapped (Figure 5). For extreme water levels, the spatial patterns are broadly similar among the RMW formulas, yet the magnitudes differ substantially. Under G1959, nearshore RP100 water levels are generally around 3 m. Under W2004, V2008, and L2013, RP100 water levels reach about 4 m on the western side of the estuary and exceed 4 m within multiple embayments along the western coast of the PRE. In the F2017 case, extreme water levels increase further within the estuary, with the western shoreline from Zhuhai to Nansha facing RP100 levels of about 4.5 m. K2005, K2007, and K2018 produce even more severe water level extremes. Over most of the inner estuary, RP100 water levels exceed 5 m. Meanwhile, RP100 water levels exceed 5 m along the western PRE coast and rise above 4.5 m within bays on the eastern side. These results imply an uncertainty of approximately 2 m in surge-related water-level extremes attributable to the choice of RMW formulation. Overall, inter-scheme differences in extreme water level are most pronounced in shallow estuarine zones and are relatively smaller in offshore deep-water areas. Compared with the water level, wave height uncertainty is more evident in deep-water regions deeper than 50 m. Under G1959, RP100 significant wave heights are only 10–15 m across most deep-water areas, and the orientation of the high-wave zone aligns with the TC track. Under W2004, V2008, and L2013, deep-water RP100 significant wave heights are mainly within 12–17 m. By contrast, K2005, K2007, and K2018 generate extensive areas exceeding 15 m over the same domain, with maxima surpassing 20 m, and the high-wave zone becomes displaced relative to the TC track. Notably, near the 50–100 m isobaths offshore of the PRE, RP100 significant wave height often differs by 3–5 m among schemes, far exceeding typical model uncertainty.

3.2. Impact of Empirical RMW on Joint Distribution

A Gumbel copula was used to couple the GEV marginal distributions of water level and significant wave height at the WLC station. The resulting joint probability density functions and joint return period contours for RMW formulas are presented in Figure 6. Red dots represent the original water level and significant wave height, the colored shading indicates the joint probability density, and the black curves denote joint return period contours. As shown in Figure 6, the joint distribution of annual extreme water level and significant wave height at WLC differs markedly among the RMW formulas. Under K2005, K2007, and K2018, the high-density region shifts toward larger water levels and significant wave heights and exhibits a more elongated shape. In contrast, the distribution centers derived from G1959 and W2004 are lower by approximately 0.5 m in both water level and significant wave height. These differences indicate that the choice of RMW formula modulates the typical levels at which extreme water level and significant wave height occur jointly.

In addition to the shift in distribution centers, differences in dependence between extreme water level and significant wave height are reflected by the dependence parameter α. In Figure 6, G1959 exhibits the weakest dependence, representing a weak-to-moderate positive association in which higher water levels increase the likelihood of larger significant wave heights but with limited joint variability. By contrast, K2018 shows very strong dependence, indicating that extreme water levels and significant wave heights tend to increase simultaneously. As a result, different RMW formulas can lead to inconsistent pairing of water level and significant wave height intensities, thereby undermining the robustness of the joint distribution.

The joint return period contours differ substantially in both location and shape among formulas at the same nominal return level. For moderately frequent extremes, the 50-year joint water level–significant wave height combination predicted by G1959 is more than 1 m lower than that predicted by K2018. This implies that a compound event labeled as 50 years under G1959 would be classified as approximately a 10-year event under K2018. The discrepancy becomes even larger for rarer extremes, such as the 100-year and 200-year events. For a given return period, the predicted thresholds of extreme water level or significant wave height can differ by several meters across RMW formulas. Specifically for the 100-year event, some formulas produce peak water levels below 3 m (G1959, W2004, V2008, L2013, F2017), whereas others reach 4–5 m (K2005, K2007, K2018). Wave height differences are of a similar magnitude, with some formulas yielding about 4 m for the 100-year significant wave height and others indicating 5–6 m.

3.3. RMW-Induced Uncertainty in Representative TC Modeling

As an illustrative case, TC “Mangkhut” (2018) shows substantial spread in RMW derived from different RMW formulas. In September 2018, TC “Mangkhut” made landfall in the Philippines and then tracked westward toward South China, where storm surge and extreme waves triggered coastal overtopping, urban inundation, and infrastructure damage. Tide-gauge records indicate a maximum storm surge of approximately 1.91 m in Hong Kong and up to 2.5 m in Shenzhen. The direct economic loss in Hong Kong was approximately HKD 4.6 billion. The Philippines reported more than 100 fatalities, whereas no deaths were recorded in Hong Kong, although over 450 people were injured [67]. Figure 7a presents the RMW time series, and

Table 4. Mean values and ranges of the RMW estimates from each empirical formula before and after landfall over Luzon Island.

	Mean RMW (km)		Variation Range of RMW (km)
	Before Luzon Landfall	After Luzon Landfall	Before Luzon Landfall	After Luzon Landfall
G1959	16.8	25.8	4.4	15.6
W2004	18.8	34.9	2.9	42.1
K2005	49.2	81.1	0.5	60.0
K2007	45.4	54.0	3.9	12.7
V2008	20.3	34.7	3.3	20.5
L2013	28.7	42.1	0.7	39.0
F2017	26.6	47.5	2.6	55.8
K2018	56.6	74.4	5.1	30.4

summarizes the mean and range for each formula for the periods before and after the Luzon landfall. Even before the Luzon landfall, the mean RMW differed by up to 39.8 km among formulas, while the within-formula ranges remained relatively narrow. As “Mangkhut” crossed the northern South China Sea toward Guangdong, the between-formula spread in mean RMW increased to 55.3 km, accompanied by a pronounced widening of the ranges. After the Guangdong landfall, the inter-formula spread further increased, reaching a maximum of 71.94 km.

Figure 7b illustrates the differences in the radial wind speed distributions of TC Mangkhut at 00:00 on 16 September 2018 (UTC) when different empirical RMW formulas are applied. As the RMW increases, the region of strong winds shifts outward to larger radii from the TC center. This outward shift weakens the radial wind speed gradient outside the TC center and yields a broader radial wind profile. Notably, Figure 7a shows that at 00:00 on 16 September 2018 (UTC), the K2005 and K2018 formulas yield very similar estimates of the RMW, differing by only 3.3 km. Consequently, the radial wind speed distributions associated with K2005 and K2018 in Figure 7b are nearly indistinguishable. For a translating TC, the near-surface wind is the vector sum of the rotational wind and the storm translation, producing a right-of-track enhancement and a left-of-track reduction. Wind speeds greater than 30 m/s are defined as strong winds. The left-of-track strong-wind radius increases from 98.4 km to 187.5 km across the tested RMW range, whereas the right-of-track radius increases from 120.8 km to 230.8 km. The increase on the right-hand side is 23.5% larger than that on the left-hand side.

The simulated total water level and significant wave height at the stations in the PRE are shown in Figure 8 and Figure 9. Apart from the differences induced by the RMW formulas, the simulations agree well with the observations. The peak values of total water level and significant wave height, together with the durations of extreme conditions, are summarized in

Table 5. Maximum inter-formula differences in peak water level and peak significant wave height, together with the maximum differences in the duration of extreme conditions. Total water level is reported for Quarry Bay, Dawanshan, Chiwan, Guangzhou, Nansha, and Zhuhai, whereas significant wave height is reported for QF303, QF305, QF306, and WLC.

Station	Peak Value Difference (m)	Duration Differences (h)
Quarry Bay	1.36	4
Dawanshan	1.13	4
Chiwan	1.71	3
Guangzhou	2.04	2
Nansha	2.06	3
Zhuhai	2.11	3
QF303	3.80	8
QF305	3.25	13
QF306	1.87	13
WLC	1.52	4

. Extreme sea states are defined here as total water level exceeding 2 m, significant wave height exceeding 6 m at estuarine offshore zones, and 3 m in estuarine nearshore zones. Across the different RMW formulas, the spread in peak total water level exceeds 1 m at Quarry Bay, Dawanshan, and Chiwan, and exceeds 2 m at Nansha, Zhuhai, and Guangzhou. For waves, the inter-formulation spreads in significant wave height at the three deep-water stations are 3.8 m, 3.25 m, and 1.87 m, respectively. Even at the estuarine nearshore WLC station, a difference of 1.52 m is obtained. In addition to peak magnitudes, the duration of extreme total water level and significant wave height is also a key metric. In the estuarine region, the duration of extreme total water level varies by up to 4 h among the RMW formulas. At Guangzhou, located upstream within the estuary, the corresponding difference is only 2 h. For significant wave height, the three deep-water stations exhibit 8–13 h differences, whereas the estuarine nearshore WLC station shows a smaller difference of 4 h.

Based on the simulations associated with each RMW formula, the maximum values were extracted to make an envelope at every grid point. The maximum envelope maps of storm surge and significant wave height induced by TC “Mangkhut” are shown in Figure 10. The storm surge was computed by subtracting the astronomical tide-only water level from the simulated total water level driven jointly by tide and wind forcing. With storm intensity and track held constant, Figure 10 demonstrates that the spatial distributions of storm surge and wave height differ markedly among the RMW formulas. For storm surge, pronounced residuals exceeding 2 m are primarily concentrated along the windward coastline from the PRE to Daya Bay. Both the magnitude and the spatial extent of this high-surge zone are highly sensitive to the prescribed RMW. Under lower-magnitude formulas such as G1959, W2004, and V2008, the high-surge zone does not extend into the Daya Bay sector. In addition, storm surge within the estuary is weaker and does not produce water-level increases exceeding 3 m. When higher-magnitude RMW formulas are used, including K2005, K2018, and F2017, the estuarine surge increases by more than 1 m, and the high-surge zone expands seaward and extends into deep water toward the Wanshan Archipelago.

Regarding high waves, the maximum wave heights consistently occur in the deep-water region in the right-front quadrant of the track and decay toward the coast. This indicates that for a given wind field and translation path, the large-scale structure of the simulated wave field is broadly consistent across RMW formulas. Nevertheless, the magnitude and affected area of extreme wave height differ substantially among formulas. With G1959 and W2004, the area exceeding 10 m is relatively compact, with local maxima of approximately 13–15 m and a limited spatial footprint of the peak region. In contrast, under K2005, K2007, and K2018, the area exceeding 15 m expands markedly, and the spatial coverage of extreme waves nearly doubles. The V2008, L2013, and F2017 cases fall between these two groups, producing extreme wave heights of intermediate intensity and extent.

4. Discussion

4.1. Why Relying on a Single RMW Formula Introduces Systematic Bias

This study indicates that, in large-sample risk assessments driven by parametric TC wind fields, empirical estimation of RMW is not a secondary sensitivity factor but can be a dominant source of error in evaluating extreme water level and wave height. The key reason is that different empirical RMW formulas exhibit pronounced long-term tendencies toward systematic overestimation or underestimation. The empirical RMW statistics for the 1981–2024 TCs are summarized in Figure 11. The box shows the interquartile range and median, with whiskers extending to 1.5 × IQR and outliers marked as blue circles. Box colors indicate the empirical RMW formulas. Significant differences are observed among the formulas in the median, distributional range, and tail dispersion of the RMW distribution. The relative ranking of formulas is consistent between representative case studies and long-term sample statistics. This implies that adopting a single formula imposes a structural bias on a key TC-scale dimension, thereby systematically altering the radial wind field characteristics within the constructed TC ensemble. When propagated through large ensembles of track modeling, these differences directly modify the TC radial wind structure and the footprint of strong winds. For TC “Mangkhut” (2018), the mean RMW differences among formulas reach 55.3 km during key stages, with a maximum difference of 71.94 km, shifting the maximum wind ring and modifying the outer wind speed decay, which broadens the radial profile overall (Figure 7). The outward expansion of the strong-wind footprint increases effective fetch and forcing duration, resulting in substantial differences in the peaks and durations of storm surge and extreme wave heights. For “Mangkhut” (2018), differences in the RMW formula alone produce a 2.11 m spread in nearshore peak water level, a 3.8 m spread in significant wave height, and up to a 13 h difference in the duration of extreme waves at deep-water stations. Spatially, both the magnitude and areal extent of high-surge and huge-wave zones increase markedly with the expansion of the strong wind belt (Figure 10).

4.2. Consequences for Extreme Value and Compound Hazard Assessments

With limited sample lengths, uncertainty in extreme value extrapolation is governed primarily by the estimation of the tail shape and scale parameters, which are highly sensitive to observations in the distribution tail. Our results confirm that RMW strongly influences annual extremes, leading to unacceptably large discrepancies in long return period design values across different RMW formulas. At the RP100, extreme water levels in estuarine nearshore zones differ by 1.26 m, and extreme significant wave heights differ by 1.54 m (Table 3). The coastline-wide exposure to extreme water level and wave height hazards likewise varies substantially among formulas (Figure 5). Consequently, in risk assessments based on historical TC samples, using a single RMW formula can systematically bias regional hazard estimates toward overly aggressive or overly conservative levels, leading to overinvestment and misallocation of resources. For assessments based on synthetic TC tracks, uncertainty introduced by the RMW formula may mask the robustness gains expected from large-sample approaches and reduce confidence in identifying potential hazard hotspots. Moreover, in compound risk assessments, the RMW formula affects not only the marginal distributions but also the dependence structure between variables, thereby distorting return period classification of compound events (Figure 6). The same 100-year water level-wave height compound event under one formula may correspond to only a 10-year event under another. Given the growing emphasis on compound hazards, the pronounced uncertainty in joint probability density shapes and joint return period contours induced by RMW formulas represents a non-negligible concern that should no longer be overlooked.

4.3. Practical Recommendations for Uncertainty-Aware RMW Parameterization

Taken together, the results and mechanistic analysis indicate that RMW cannot be treated as a region-independent parameter that can be prescribed statically using a single empirical formula. RMW is jointly modulated by regional boundary-layer dynamics, vertical wind shear, and air–sea interactions. Simple empirical relationships cannot reliably represent the associated radial structure of TCs. We recommend moving from deterministic RMW specification based on a single empirical formula to explicit quantification of RMW uncertainty in TC-driven storm tide and extreme wave height risk assessments. First, physically constrained and dynamically varying RMW estimation schemes should be developed, for example, by integrating storm intensity, latitude, environmental conditions, and lifecycle information within a unified framework to reduce systematic biases arising from regression form and sample dependence. Where observations are sparse, limited available RMW information can be used to regionalize and calibrate empirical relationships, thereby improving the reliability of large-sample simulations. Second, given the incomplete coverage of high-quality RMW observations and practical constraints on computational resources, we recommend an ensemble-based treatment of RMW for greater robustness. Figure 11 shows that the formulas exhibit persistently high-bias, low-bias, and intermediate-bias groups in long-term statistics, and case comparisons further suggest that observations or best track estimates often fall within the range spanned by the empirical formulas (Figure 7a). Accordingly, for large-sample construction of TC wind fields, RMW schemes can be grouped into representative low, intermediate, and high sets. The resulting envelope or the ensemble median can then be reported, and the envelope width can be used as an uncertainty metric in protection-level design and investment decision-making.

4.4. Limitations and Future Work

It should be noted that the parametric TC radial structure adopted here assumes a single-peaked profile, whereas double-peaked or multi-peaked structures are not uncommon in real TCs. Because secondary maxima are typically weaker than the primary maximum and rarely govern nearshore peak surge, this limitation does not alter our central conclusion that the choice of RMW formula systematically reshapes the strong-wind footprint and drives differences in risk estimates. However, multi-peaked structures may broaden the outer wind field. They may also prolong the duration of strong winds, which can modify the evolution of extreme significant wave heights and storm tides. As a result, the composition of annual maxima and the joint distribution characteristics may change, especially at deep-water sites and for compound hazard metrics. Coupled simulations spanning long TC time series and multiple empirical RMW formulations are computationally demanding. We therefore adopted pragmatic model configurations, including a moderate grid resolution and simplified representations of secondary processes. To isolate TC-scale responses, the background temperature and salinity were prescribed as constants to minimize baroclinic effects. Nevertheless, under strong wind forcing, a more realistic thermohaline structure may influence the development and persistence of local extremes. In addition, we fitted extremes using the annual maxima approach to ensure sample independence and methodological robustness, but this framework may overlook information from multiple severe events occurring within the same year. Future work can validate the findings using peaks-over-threshold methods or non-stationary extreme value models to improve the credibility of long return period extrapolation. Accordingly, we will consider more flexible radial-structure representations and alternative extreme value frameworks to robustly assess extreme water level and wave height hazards under the combined effects of multi-peaked structures and RMW uncertainty.

5. Conclusions

In this study, we quantify a previously underappreciated source of uncertainty in parametric tropical cyclone (TC) model assessments of extreme storm tide and wave hazards. This uncertainty arises when the radius of maximum wind (RMW) is prescribed using a single empirical formula. Using the Pearl River Estuary (PRE) as a case study, we implemented a coupled wave–current model (ROMS–SWAN) and simulated storm tides and extreme wave heights for historical TCs during 1981–2024 under an annual-maxima extreme value framework. TC wind fields were reconstructed using the Holland10 model, and eight empirical RMW formulas were incorporated to form a wind field ensemble. Generalized extreme value (GEV) models were used to fit and extrapolate long return period water levels and significant wave heights, and a Gumbel copula was applied to construct the joint water level and wave height distribution for compound hazard characterization.

The results show that the RMW formula is not a minor sensitivity factor. Instead, it exerts a strong structural influence on hazard estimates that has been largely overlooked in practice. Specifically, in estuarine nearshore zones, the 100-year return period (RP100) estimates of water level and significant wave height differ by up to 1.26 m and 1.54 m among formulas, and the spread increases further at longer return periods. Formulas yielding higher estimates are often associated with wider confidence intervals, indicating that RMW selection affects not only design magnitudes but also extrapolation uncertainty and reliability. Copula-based joint probability analysis further reveals that relying on a single empirical formula can distort the classification of compound hazard risk. For the same storm tide and wave compound event, the joint return period can shift from 100 years to 10 years depending on the formula used. Such dramatic shifts render hazard zoning highly fragile and can readily lead to misallocation of disaster mitigation resources. In a representative extreme TC case, differences in the RMW formula alone produce maximum deviations of 2.11 m in nearshore peak surge and 3.8 m in significant wave height and lead to up to a 13 h difference in the duration of extreme sea states. This implies that engineering design and emergency response may be biased in both intensity assessment and the available evacuation window if a single formulation is assumed by default.

Therefore, we highlight the substantial risk of subjectively adopting a single RMW formula in hazard assessments and argue that TC size deserves attention comparable to that given to intensity and track. From an applied perspective, we advocate physically based and dynamically varying RMW estimation to avoid dependence on any single empirical formula. Moreover, we recommend an ensemble-based treatment of RMW under data-limited conditions to enhance the robustness of hazard assessment and coastal protection design.