The Optimization of Four Key Parameters in the XBeach Model by GLUE Method: Taking Chudao South Beach as an Example

Abstract

When the XBeach model is used to simulate beach profiles, the selection of four sensitive parameters—facua, gammax, eps, and gamma—is crucial. Among these, the two key parameters, facua and gamma, are particularly sensitive. However, the XBeach model does not specify the exact choice of these four key parameters, offering only a broad range for each one. In this paper, we investigate the applicability of tuning these four parameters within the XBeach model. We employ Generalized Likelihood Uncertainty Estimation (GLUE) to optimize the model settings. The Brier Skill Score (BSS) for each parameter combination is calculated to quantify the likelihood probability distribution of each parameter. The optimal parameter set (facua = 0.20, gamma = 0.50) was ultimately determined. Here, the facua parameter represents the degree of influence of wave skewness and asymmetry on the direction of sediment transport, while the gamma parameter represents the equivalent random wave in the wave dissipation model and is used to calculate the probability of wave breaking. Six profiles of the southern beach on Chudao Island are selected to validate the results, establishing the XBeach model based on profile measurement data before and after Typhoon “Lekima”. The results indicate that after parameter optimization, the simulation accuracy of XBeach is significantly improved, with the BSS increasing from 0.3 and 0.17 to 0.68 and 0.79 in P1 and P6 profiles, respectively. This paper provides a recommended range for parameter values for future research.

1. Introduction

Beaches serve as a material prerequisite for nearshore sediment transport; their profiles are influenced by waves, tides, currents, and sediments [1]. Storm events significantly impact the short-term evolution of beach geomorphology, defined by researchers as a marine condition where the significant wave height exceeds 1.5 times the annual average significant wave height for over 12 h [2]. During storms, interactions between waves, ocean currents, and beach and dune sediments help dissipate wave energy and act as natural defenses against storm surges. On the other hand, erosion and excessive scouring can have destructive effects on coastlines. Extensive research has been conducted on beach morphological changes during storms. The correlation between elevated dune erosion rates during storm conditions and the observation that the increase in offshore-directed sediment transport is only partially counterbalanced by a rise in wave-induced onshore transport capacities is evidenced by flume experiments [3]. Dune profile erosion is dominated by low-frequency waves [4].

Recent advancements in coupled hydrodynamic–morphodynamic numerical models, such as XBeach, have enhanced the accuracy and computational efficiency. XBeach, a two-dimensional depth-averaged (2DH) model, functions in both one-dimensional and two-dimensional modes to address integrated cross-shore and alongshore dynamics, encompassing wave propagation, water flow, sediment transport, and bed alterations. It has effectively demonstrated its capability to simulate various severe hydrodynamic conditions [5]. The XBeach model is often used for predicting beach morphology during storms and simulating beach recovery after storms [6,7]. Recently, more scholars have been expanding the applications of the XBeach model, such as studying the response of beach profile evolution to artificial reefs [8]. Upgrades and improvements to the XBeach model are ongoing [9]. The use of XBeach and other widely used numerical models to simulate beach morphology evolution involves some uncertainties, including boundary conditions and input data, model formulations, and manual calibration processes. Many studies have quantified prediction uncertainties due to inaccuracies in storm surges, wave elements, and storm duration by providing probabilistic estimates of storm erosion [10,11]. Different boundary conditions and input data are often used by researchers to compare beach morphological changes under different conditions (e.g., hydrodynamic intensity) [12]. Researchers have also discussed the uncertainties in simulating beach morphology changes under different dimensions (one-dimensional and two-dimensional) [13]. Parameter uncertainty is also a part of model prediction uncertainty, caused by variations in various free parameters during the model calibration process, and is rarely quantified. XBeach can accurately reproduce dune toe retreat and changes in dry beach volume; however, the sensitivity of the process-based XBeach model to parameter adjustments remains a concern, necessitating calibration for specific locations [14].

The XBeach model has a large number of free parameters and faces practical challenges due to the time-consuming nature of individual model runs, often leading to less stringent calibration. XBeach generally depends on the expertise of researchers who manually adjust one parameter at a time to pinpoint those most sensitive for optimization during calibration. Typically, only a limited number of parameter values are tested, employing heuristic approaches to evaluate performance and establish the best parameter configuration [15,16]. Recent studies have developed a reference approach, the Generalized Likelihood Uncertainty Estimation (GLUE) method, to systematically evaluate the sensitivity and uncertainty of parameter changes in the XBeach model due to storm impacts [17,18]. The application of GSA in the GLUE method has recently appeared in field studies for calibrating parameters of the 2D XBeach model. This method evaluates the sensitivity order of selected parameters, but it only manually tunes the highly sensitive parameters to determine the optimal parameter set, lacking systematic evaluation of the applicability range for each parameter value.

This paper establishes a one-dimensional in situ XBeach model, using the beach profile elevations on the southern coast of Chudao Island collected before and after Typhoon “Lekima”, and water level and wave data from nearby marine stations as input conditions. Since the GLUE method and concept have been well established in other fields, this paper, building on the work of Simmons et al. [19] applies this method to another real-world case study (Chudao Island) to explore its advantages and disadvantages when applied to the relatively complex coastal numerical model XBeach. This expands the practical application of the method. It also investigates the applicability of the method to different beaches, comparing parameter sensitivities in various modeling regions and input conditions, and discussing the generality and error ranges of optimal parameter sets. This provides an initial parameter tuning recommendation range for future XBeach modeling, enhancing the study of optimal parameter sets while reducing workload.

2. Data and Methods

2.1. Study Area

The study area, Chudao Island, is located in the Ludong Jiaonan uplift area of the Shandong Peninsula in China, bordered by the Yellow Sea to the east, Shidao Bay to the south, and Sangou Bay to the north. Chudao Island covers an area of approximately 0.75 km² and has about 2.7 km of sandy beaches. The sediments are primarily fine sand, characteristic of high-quality natural sandy coasts. The research area is characterized by a temperate monsoon climate, with dominant N–NW winds and average wind speeds ranging from 6.4 to 6.6 m/s. The precipitation throughout the year is unevenly distributed, with around 80% of it falling during the summer months [20]. The waters around Chudao Island are primarily influenced by wind waves, with the main wave directions being SW and SEE. The annual average significant wave height is 0.41 m, and the average wave period is 5.02 s. The tides are predominantly irregular semidiurnal, characterized by noticeable diurnal inequalities. At Chudao Island’s southern beach, the dune crest elevation is 3.6 m at the westernmost profile and 2.4 m at the easternmost profile, according to the national 1985 elevation datum. The sediments comprise gravel, medium sand, and fine sand. The geographical location of Chudao Island and the research profiles discussed in this paper are shown in Figure 1.

Typhoon “Lekima” was a severe typhoon that occurred in the Western Pacific region during the summer of 2019, making it the strongest typhoon to land in China that year. It made landfall on the coast of Taizhou, Zhejiang at 01:45 on 10 August, and again in Qingdao, Shandong at 20:00 on 11 August, thereafter entering the influence range of Chudao Island until it was decommissioned. The central minimum pressure of “Lekima” was 925 hPa, with a maximum wind speed of 54 m/s [21].

2.2. Introduction of XBeach

XBeach is a model that simulates wave groups using short-wave averaging and now offers users options to select from stationary wave mode, surfbeat mode, and non-hydrostatic mode. The governing equations of the model are based on the incompressible Navier–Stokes (N–S) equations in Cartesian coordinates, as proposed by Batchelor [22], and the wave motion equations are derived using the methods of Holthuijsen et al. [23]. The shallow water equations employ the Generalized Lagrangian Mean (GLM) [24] formula to transform wave-induced currents and undertows. The relationships between Lagrangian velocity $u^L$, Eulerian velocity $u^E$, and Stokes velocity $u^S$ in the momentum equation and continuity equation are as follows:

$$\begin{array}{l}{u}^L=u^E+u^S\\ v^L=v^E+v^S\end{array}$$

(1)

where $u^S$ and $v^S$ are the components of the Stokes drift in the x and y directions, respectively.

The sediment transport model utilizes the depth-averaged convection–diffusion equation [24]:

$${\displaystyle \frac{\partial hC}{\partial t}}+{\displaystyle \frac{\partial hC{u}^E}{\partial x}}+{\displaystyle \frac{\partial hC{v}^E}{\partial y}}+{\displaystyle \frac{\partial }{\partial x}}\left(D_{h}h{\displaystyle \frac{\partial C}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(D_{h}h{\displaystyle \frac{\partial C}{\partial y}}\right)={\displaystyle \frac{hCeq-hC}{T_s}}$$

(2)

where $C$ represents the depth-averaged sediment concentration. C_eq refers to the equilibrium sediment concentration (either suspended load or bedload). $D_h$ is the sediment diffusion coefficient, $Ts$ is the adaptation time used to represent the amount of sediment entrainment, and it can be approximated by the depth $h$ and the sediment settling velocity:

$$Ts=\max (0.05{\displaystyle \frac{h}{{\omega }_s}},0.2)s$$

(3)

where $h$ represents the water depth and ${\omega }_s$ represents the sediment settling velocity.

In the XBeach model, there are several parameters, and for the subsequent analysis in this study, we have selected a few important parameters related to wave overtopping and beach erosion processes from previous research [25,26,27].

Table 1. The definitions, default values, and recommended ranges of the parameters.

Parameters	Meanings	Default Values	Recommended Ranges
facua	The degree to which wave tilt and asymmetry affect sediment transport direction	0.1	0–1
gamma	The breaking parameter in wave dissipation models	0.55	0.4–0.9
gammax	The maximum allowable ratio between wave height and water depth	2	0.4–5
eps	Threshold water depth above which cells are considered wet (m)	0.005	0.001–0.1

displays the physical descriptions, default values, and recommended ranges for these four parameters.

2.3. Model Setting

The XBeach simulation of the Chudao Island beach storm surge erosion one-dimensional model employs a graded mesh, with a unit grid spatial scale of 5 m perpendicular to the shoreline at the hydrodynamic boundary. The nearshore beach is designated as the focal research area, with a grid resolution enhanced to 0.5–1 m in the direction perpendicular to the shoreline. Based on the water depths recorded at the Yellow Sea Buoy Station No. 7 and the Shidao Marine Observatory, the bed elevation at the hydrodynamic boundary is set at approximately −11 to −12.5 m. Designated grid areas including the terrestrial and offshore boundary regions are set as non-erosive zones, with the nearshore beach established as the focus area for this study and the simulation duration set to 72 h. The model employs dynamic tidal levels and wave spectra for boundary conditions, with wave conditions updated hourly using JONSWAP spectrum files. The chosen model computation mode is “surfbeat”. This section provides a detailed introduction to the topographic measurements before and after the storm, as well as the establishment of boundary conditions.

2.3.1. Terrain Condition

Nearshore topography was surveyed by using the Beidou Haida TS7 RTK that produced by Guangzhou HiTarget Navigation Technology Co., Ltd. (Guangzhou, China) before and after the storm (15:00 on 10 August to 15:00 on 13 August 2019). Hardened road surfaces near the shore facilitated the driving of fixed stakes, and a layout operation was conducted before each measurement session to guarantee consistent profiling. Measurements started from the fixed stakes by the shore, conducted at the lowest tide, with distances between points set at 2 m and reduced to 50 cm in eroded areas; pre-storm underwater topography was sourced from single-beam echo sounder regional depth surveys. Throughout the measurement procedure, surface sediment samples were gathered from the upper, middle, and lower portions of each beach profile, which included the backshore, foreshore, and areas undergoing erosion. Each sample, weighing around 300 g, was sieved and processed to serve as grain size data for the XBeach model. In the XBeach model, the median grain size (D50) used is the pre-typhoon monthly average measurement of 0.3 mm, and D90 is the average measured value of 0.47 mm.

2.3.2. Tide Level and Wave Boundary

The hydrodynamic boundary conditions for the XBeach model utilize wave data from the Yellow Sea Buoy Station No. 7 (122.6° E, 37.1° N) and tidal data from the State Oceanic Administration Shidao Marine Observatory (122.4° E, 36.9° N) collected during the typhoon (Figure 2 and Figure 3). In this study, the marine boundaries for the XBeach model were selected according to the water depths noted at these marine stations. The No. 7 Directional Wave Rider Buoy, which records free surface gravity waves, operates within an amplitude range of −2000 to +2000 cm, has an amplitude resolution of 1 cm, and measures wave periods ranging from 1.6 to 30 s. The data are recorded at a frequency of 1.28 Hz and are outputted hourly. Tidal measurements are conducted using an DCX-25-type automatic water level recorder that produced by Beijing Hydrosurvey Sway Technology Co., Ltd. (Beijing, China), which continuously records changes in sea level.

3. GLUE Method

The steps of the GLUE (Generalized Likelihood Uncertainty Estimation) method include generating parameter sets, constructing likelihood functions, setting thresholds to analyze parameter sensitivity and selecting calibration parameter sets, and analyzing simulation results. This section provides a detailed description of each step. Research indicates that using the GLUE technique instead of manual calibration can significantly enhance predictive capabilities and reveals marked differences in XBeach model performance controlled by the wave–dune impact process [28]. The GLUE methodology adopted in this paper can be briefly described as follows: Initially, a multitude of parameters are selected using the GSA method, with each parameter’s values equidistantly distributed within recommended ranges to form parameter sets for simulation. The BSS method is then used to evaluate the simulation results, analyzing which parameter changes have the most significant impact on the outcomes, thus ranking the sensitivity of each parameter. Subsequently, the two most sensitive parameters are chosen to narrow the range of parameter values and finely partition these into parameter sets for further simulation. Based on the magnitude of the BSS values, the distribution of parameter values is analyzed at the peak BSS value (i.e., when the simulation effect is optimal), thereby assessing the uncertainty in shoreline morphology evolution due to parameter changes. Specific details of the method are discussed in the various subsections of this chapter.

3.1. Parameter Sets Generation and Evaluation

According to Table 1, the first step involves uniformly distributing values for the selected four parameters within the model-recommended range, ultimately generating a large set of unique parameter sets. Given that the number of simulations needed for GLUE analysis varies with the number of parameters considered, this paper initially undertakes a comprehensive preliminary sampling analysis of four parameters. Based on the results of sensitivity and probability distributions, parameters with higher sensitivity are selected to save computational time and allow for more precise sampling analysis.

The Brier Skill Score (BSS) is a suitable and widely utilized metric for evaluating model skill and probability in GLUE analysis. The higher the BSS value, the better the predictive performance [29]. The range of BSS values and their corresponding predictive performances are shown in

Table 2. Range of BSS evaluation scores and the predictive effect they signify.

The Range of BSS Values	The Predictive Effect
1~0.8	Excellent
0.8~0.5	Good
0.5~0.3	Reasonable
0.3~0	Poor
<0	Bad

. The BSS assessment approach is appropriate for simulating coastal dynamics and is now broadly applied [27]. The BSS used for predicting beach erosion during storms can be expressed as follows:

$$BSS=1.0-{\displaystyle \frac{MSE(Y,X)}{MSE(I,X)}}$$

(4)

$$MSE(Y,X)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{(Y_i-X_i)}^2}$$

(5)

where MSE represents the mean squared error, $X_i$ represents a series of observed seabed elevations after the storm, $Y_i$ represents the simulated seabed elevations, and $I$ represents the observed seabed elevations before the storm. To preliminarily evaluate the posterior distribution of parameters, a BSS threshold must be set to indicate which parameter values can better estimate the morphology of the beach after the storm. If the morphological evolution effect simulated by the parameter set does not reach the BSS threshold, the probability of that run is assigned a value of 0. The likelihood measure L_BSS is defined simply as follows [30]:

$$L_{BSS}={\displaystyle \frac{{BSS}_i}{{\displaystyle {\sum }_{i=1}^n{BSS}_i}}}$$

(6)

where ${BSS}_i$ denotes the $n$ value for each parameter set, and n is the total count of model runs surpassing the threshold. This likelihood function is valid as long as the BSS value of the threshold is not less than 0. More complex likelihood estimation methods have been developed by researchers [31,32], but the simple likelihood described above has been proven to be capable of verifying the distribution of modeling parameters.

3.2. Parameter Optimization Method

3.2.1. Parameter Sensitivity Processing

In the present study, a method called Generalized Sensitivity Analysis (GSA) has been formulated to systematically evaluate the variations in parameter sensitivity in the XBeach model induced by storm impacts [33]. The GSA method can systematically rank the relative sensitivity of the model’s experimental parameters. Utilizing the GSA method, researchers can rank the sensitivity of each experimental parameter through straightforward visualization techniques. This involves plotting the cumulative distribution function of behavioral parameter values (e.g., BSS > 0.5) against non-behavioral parameter values (with an assumed uniform distribution of BSS values) on the same graph. The greater the difference between these two distributions, the more sensitive the model is to that particular parameter. Conversely, a smaller disparity suggests a lower sensitivity to that parameter [34]. The Kolmogorov–Smirnov D statistic [35] can be employed to further quantify the difference between these two distributions, providing a measure of the maximal cumulative frequency difference between the two cumulative distribution curves. The D statistic ranges from 0 to 1, where 0 indicates an insensitive parameter and 1 indicates a highly sensitive parameter.

3.2.2. Parameter Optimization Selection

To help select the most reliable parameter sets for specific modeling objectives, the likelihood values of each parameter set are calculated first and then distributed. For each individual parameter, the values extracted from each behavioral parameter set are simply combined with the corresponding likelihood values of that parameter set during the distribution process. That is, the process forms a depiction of the behavioral distribution by selecting each individual parameter from the overall parameter set. This paper visualizes the likelihood distribution of individual parameters by plotting probability density functions, determining the range of adaptability of parameters under these modeling conditions by observing the probability density distributions of each parameter.

3.3. GLUE Establishment

This paper selects six profiles from the study area for GLUE analysis, as shown in Figure 1. Chudao Island beach morphology is primarily divided into two types: the southwest part has wider berms, higher dune base elevations, and gentler slopes; the northeast part lacks berms and has steep slopes. Therefore, the initial application and analysis of the GLUE method are first conducted on the P1 profile in the southwestern part of the study area. A broad sampling range for the parameters, determined from the XBeach user manual [35], is first used to conduct an initial GLUE analysis on the selected set of four parameters. This analysis provides a sensitivity ranking of the parameters within this modeling effort and narrows down the range of parameters that conform to behavioral values. Based on the sensitivity ranking, the two most sensitive parameters are selected, and a more accurate range of parameter values is established. Subsequently, profile P6, which is located far from profile P1, is chosen. These two profiles can represent the sectional characteristics of the study area. A new set of parameters is compiled to conduct a more detailed GLUE analysis on these two profiles, thereby determining the optimal parameter combination for the modeling. Selecting a threshold that is too low can hinder the determination of the parameter adjustment range, whereas a threshold that is too high might lead to fewer instances surpassing the threshold. Considering these characteristics, the initial behavioral threshold for this study is set at BSS > 0, and the secondary behavioral threshold is set at BSS > 0.5.

4. Results

4.1. Sensitivity Analysis

Before assessing the best parameter values, it is essential first to establish the sensitivity order of the selected parameters. This preliminary step helps understand how changes in these four free parameters influence the XBeach model in the region and assess their suitability for this application. From Figure 4, it is apparent that the parameters eps and gammax are relatively insensitive, as the differences between the behavioral and uniform distributions are minor, and the Kolmogorov–Smirnov D statistical values are low, at 0.12 and 0.05, respectively. The parameters facua and gamma exhibit higher sensitivity, with statistical values of 0.51 and 0.39, respectively. Sensitivity can be categorized from highest to lowest, either visually or quantitatively, in the order of facua, gamma, eps, and gammax. Parameters with lower sensitivity are not suitable for GLUE analysis. Therefore, profiles P1 and P6 were selected to form a new set of over 200 parameters with facua and gamma for a more detailed secondary GLUE analysis. In the graph, the x-intervals where the behavioral likelihood curves have a steeper slope indicate the ranges where each parameter is more applicable. Based on this, the initial sampling ranges for the parameters facua and gamma were narrowed to reduce the workload during the secondary analysis. The finalized ranges for the secondary analysis are facua from 0.1 to 0.4, and gamma from 0.4 to 0.7.

4.2. Parameter Optimization Analysis

Figure 5 shows the probability density distribution of two experimental XBeach parameters on profiles P1 and P6, based on the specified precise sampling parameters and ranges, with each 0.05 parameter value interval serving as a distribution range. According to the statistics of Formula (3), the practical significance of the probability density histogram is as follows: when a certain parameter setting value results in the BSS value calculated by the model being closest to 1 (a higher value) and the erosion volume is close to the observed value, the likelihood that the simulation result is close to the observed value is greater, and the probability density value is highest (i.e., the highest in the histogram). As waves propagate into shallow water and approach the coast, compared to deep-water waves, they exhibit short, high crests and long, shallow troughs. When waves enter the surf zone and approach breaking, they assume a sawtooth shape, with the wave front becoming steeper and the back gentler. These waveform changes are typically defined as skewness and asymmetry. Asymmetric waveforms lead to skewed near-bottom trajectory movements and cause sediment transport [35]. In the XBeach model, the parameter facua controls both of these phenomena. The PDF indicates that for the two selected profiles, when the facua parameter exceeds the default value of 0.1, there is a higher probability, indicating that increased shoreward sediment transport correlates with improved performance. However, the probability gradually decreases or even becomes zero as facua exceeds 0.25. The probability density is highest around a facua value of 0.2 for both profiles P1 and P6. Gamma is the breaking index in the wave dissipation model. The gamma parameter is used to calculate the probability of wave breaking. The likelihood of breaking depends solely on local and instantaneous wave parameters. As energy increases or water depth decreases, the probability of breaking should monotonically increase towards 1. As the gamma parameter increases, the probability of wave breaking and the release of energy also increase, resulting in a lesser tendency for terrain evolution compared to default parameter values. The PDF shows that the probability density is highest for both profiles when gamma is around 0.5. This indicates that the model performs better when wave breaking occurs earlier (i.e., when there is a higher probability of wave breaking), resulting in increased energy dissipation before the waves reach the shoreline.

The GLUE analysis data confirmed that the model performance of the selected profiles P1 and P6 is similar. Using the probability density distribution shown in Figure 5, the facua and gamma parameters were finally calibrated, and the determined parameter value combination was identified as the optimal parameter set for the simulation. Based on the analysis results, this paper selects facua and gamma parameters of 0.2 and 0.50, respectively, for all measured profile simulations. All other parameters retain the model’s default settings. The elevation changes in the two profiles after simulating storm surge effects, before and after parameter tuning, are shown in Figure 6. The calculated BSS values before and after calibration are shown in

Table 3. Calculated BSS values before and after calibration for P1 and P6 profiles.

Profiles	BSS Before Parameter Calibration	BSS After Parameter Calibration
P1	0.30	0.68
P6	0.17	0.79

. The experimental findings of this study suggest that the default XBeach model parameters tend to overestimate both the height and width of erosion in the profile simulations relative to the actual topographical changes observed before and after the typhoon. Through parameter optimization analysis, it is possible to limit the erosion height and width at the beach shoulder, making the process and rate of profile evolution more realistic and closer to actual conditions, thus more accurately simulating the evolution of profiles under hydrodynamic actions.

The profile erosion and deposition above the beach intertidal zone are represented by calculating the Unit Erosion Depth (UED). The calculation formula is as follows:

$$UED={\displaystyle {\int }_{X0}^{X1}\Delta ZdX}$$

(7)

The UED represents the erosion–deposition volume per unit width, where X₁ and X₀ are the horizontal coordinates of the farthest and starting measurement points of the profile, and $\Delta Z$ is the elevation difference at the same horizontal coordinate. The UED values for P1 and P6, simulated before and after parameter tuning, and measured after the typhoon, are shown in

Table 4. Calculated UED values for P1 and P6 profiles before and after parameter tuning, and measured after the typhoon.

Profile	Measured Unit Erosion–Deposition Volume/m3	Imulated Unit Erosion–Deposition Volume with Default Parameters/m3	Simulated Unit Erosion–Deposition Volume After Parameter Tuning/m3
P1	−7.2	−17.4	−8.5
P6	−5.3	−17.2	−8.8

(positive values indicate deposition, negative values indicate erosion). This demonstrates to readers the quantitative effective erosion changes of the profiles before and after parameter tuning. The results show that the simulated erosion volumes of both profiles with default parameters were more than twice the measured values, while the simulated erosion volumes after GLUE method tuning were very close to the measured values.

The optimal parameter value ranges for the XBeach simulation during Typhoon “Lekima” for profiles P1 and P6 were determined. To further ascertain the applicability of the optimal parameter set, both the default parameter set and the determined optimal parameter set were applied to profiles P2–P5. Figure 7 shows the simulated elevation changes before and after parameter optimization for these profiles. The figure illustrates that under default parameters, all profiles exhibited shoulder erosion heights and widths greater than the actual topographical changes measured after the typhoon. After parameter optimization, the simulations of the profiles generally more closely matched the actual measured forms.

Table 5. The BSS values calculated using default and optimized parameters for P2–P5 profiles.

Profiles	P2	P3	P4	P5
Calculated BSS values using default parameters	0.64	−0.22	0.82	0.22
Calculated BSS values using optimized parameters	0.80	0.33	0.76	0.81

displays the BSS values calculated using both default and optimized parameter sets for these four profiles. From the table, it is evident that the BSS value for profile P3, calculated after parameter optimization, is 0.33, indicating that the optimization did not significantly increase the BSS to a higher value. The BSS values for profiles P2 and P5 significantly increased after parameter optimization, achieving good results. For profile P4, the BSS value slightly decreased rather than increased after parameter optimization, though the reduction was minimal. The slight decrease in the BSS value, as shown in the figure, may be due to the optimized parameters imposing a greater restriction on shoulder erosion. The UED values for P2–P5, simulated before and after parameter tuning, and measured after the typhoon, are shown in

Table 6. Calculated UED values for P1–P6 profiles before and after parameter tuning, and measured after the typhoon.

Profile	Measured Unit Erosion–Deposition Volume/m3	Simulated Unit Erosion–Deposition Volume With Default Parameters/m3	Simulated Unit Erosion–Deposition Volume After Parameter Tuning/m3
P2	−8.5	−18.1	−11.5
P3	−10.0	−25.1	−17.0
P4	−15.0	−24.1	−12.5
P5	−16.4	−32.7	−19.7

. The results similarly show that the simulated erosion volumes for these four profiles with default parameters were also more than twice the measured values. After tuning with the GLUE method, the simulated erosion volumes for all profiles became much closer to the measured values. By integrating the data from Figure 7 and Table 5 and Table 6, it can be concluded that the optimized parameter set derived for profiles P3 and P6 is suitable for these six profiles. Through this parameter optimization, it is possible to effectively simulate the evolution of all profiles of Chudao Island’s beach under the impact of Typhoon Lekima, facilitating further research analysis. Moreover, this successful optimization demonstrates that the derived optimal parameter set values can provide direction and reference for future research. The variation in BSS values after parameter optimization reminds modelers that GLUE analysis can be conducted from other uncertainties based on different conditions.

5. Discussions

Comparing the results of this study with the field XBeach model simulations of Harely et al. [16], which did not establish a systematic parameter tuning method, the default parameter simulations significantly overestimated the maximum waterline and underwater beach erosion during the event. In their study, after selecting and tuning parameters, 352 operational models were established. By statistically analyzing the beach volume changes of each model, the optimal parameter values were determined, showing that the common sensitive parameters were gamma and facua. The eps parameter, which showed low sensitivity in this study, was highly sensitive in their research, indicating that parameter sensitivity in modeling varies under different regional and hydrodynamic conditions. In contrast, the results from Simmon et al. [19] using the GLUE method for parameter selection showed that the eps parameter was also less sensitive. This may be due to the fact that, in the model’s source code for simulating sandy coastal morphology changes, the magnitude of this parameter is relatively small compared to parameters such as facua, which have a greater computational weight. The optimal parameter values for facua and gamma from their statistical analysis had an error of about 0.05 compared to the values in this study, indicating that the optimal parameter sets obtained using the GLUE method were more consistent.

The number of model runs and the scope of uncertainty related to parameters are significantly influenced by the choice of behavioral threshold. For instance, in the secondary GLUE analysis of this paper, the threshold was raised from 0 to 0.5, which narrowed the parameter analysis boundaries and reduced the number of parameter sets reaching the behavioral threshold, confirming that the choice of parameter ranges and behavioral thresholds are closely related [36]. This paper utilizes a one-dimensional beach profile model. Due to the existence of intricate two-dimensional processes in the area, it may be unrealistic to anticipate that this one-dimensional model can accurately forecast storm responses. In such cases, it is necessary to lower the behavioral threshold to generate a sufficient set of behavioral parameters. From the predictive results of this paper, the study area responds well to the XBeach one-dimensional beach profile simulation, and setting the behavioral threshold at 0.5 achieves the desired research objective. As for the reasons for the discrepancies between the simulated profile shapes of P3 and other profiles after tuning and their actual measurements, one possibility might be minor timing errors in profile measurement relative to before and after the typhoon; another could be the insufficient resolution of the underwater topography. These factors will be considered in greater detail in future studies to obtain more accurate results. One-dimensional models are computationally efficient and can achieve precise results when handling optimal parameter values but are less conducive to comparing differences in optimal parameter values across different profiles. Two-dimensional models are slower in computation but allow for the observation of suitable parameter value differences among different types of profiles under the same conditions, which is more advantageous for comparative analysis. The application of the GLUE method requires precise measurements of topography and hydrodynamic conditions, as well as support from numerous simulation iterations. Therefore, more conclusions from GLUE research are beneficial for future studies. The research presented in this paper enables an evaluation of the model’s capability to predict storm erosion before and after storm surges in the profiles of the study area.

6. Conclusions

This paper applies a model calibration method (GLUE) to the one-dimensional XBeach modeling of Chudao Island beach before and after Typhoon “Lekima”, which provides a comprehensive explanation of the method’s framework and each part of the modeling process. The optimized parameter set was then used to simulate the coastal evolution during Typhoon “Lekima” under different tidal conditions.

The paper primarily addresses the uncertainty in the type of profiles studied. In future research on the XBeach model, readers can use the optimal parameter set derived in this paper as a reference when selecting parameter values within this range for initial simulations, which will help narrow down the parameter range and reduce workload. Additionally, the optimal parameter set obtained after tuning can be compared and analyzed with the conclusions of this paper to identify similarities and differences, further refining the application of the XBeach model. The GLUE method has demonstrated its effectiveness as a calibration technique for coastal erosion modeling. In future studies utilizing the GLUE method, it is recommended to closely align with the hydrodynamic process measurements of beach elevations to obtain more precise optimal parameter values. Whether the optimal parameter range determined in this study is applicable to different storm processes and beach regions remains a key direction for future research. Building upon the findings of this study, we aim to establish a parameter calibration framework applicable to various storm events and coastal environments, thereby providing a valuable reference for researchers conducting simulations in their respective domains.