1. Introduction
Eutrophication-induced algal blooms in freshwater ecosystems have become a critical global issue in water quality management [1–3], prompting extensive efforts to address the problem. In response, water resource management agencies have attempted to analyze environmental changes in watersheds and to predict algal blooms using integrated systems that link watershed, river, and reservoir models.
Among the tools used to predict algal blooms, the CE-QUAL-W2 model has been adopted widely and is recognized for its high predictive performance [4–6]. CE-QUAL-W2 is a two-dimensional (longitudinal and vertical) hydrodynamic water quality model commonly applied to lakes, reservoirs, and river systems [7, 8]. It effectively simulates vertically stratified environmental variables, including density, temperature distribution, and dissolved oxygen concentration [9, 10]. The capacity of the model to simulate multiple water quality constituents simultaneously, including nutrients, chlorophyll-a (Chl-a), and suspended solids, facilitates the comprehensive analysis of physical, chemical, and biological interactions within a water body [4].
Although CE-QUAL-W2 is a powerful tool for simulating complex hydraulic and water quality dynamics, it presents several notable challenges. The model requires extensive datasets and complex parameter configurations. Accurate simulations demand diverse input data, such as water temperature, flow rate, and nutrient concentrations, which in turn require considerable time and financial resources for field investigations and data collection [11, 12]. Furthermore, setting the initial and boundary conditions, calibration, and validation are complex processes, and obtaining reliable results without sufficient expertise is challenging [13, 14]. Therefore, validating model predictions against observed data is an essential step [15, 16]. This process has typically relied on trial-and-error methods, requiring experts to manually identify optimal parameter combinations [17–19].
While the trial-and-error method is effective, it requires time and effort. In response, various optimization techniques have been developed to automate the calibration process [13, 20]. However, most of these approaches have been limited to one-dimensional models with relatively few parameters [21–23]. In general, the number of model evaluations required is approximately 10–15 times the number of parameters to be optimized [21, 24]. Therefore, applying optimization techniques to water body models that require many parameter combinations and long run times poses practical limitations in terms of efficiency [25].
Recent advances in computing power, artificial intelligence, and deep learning have introduced new opportunities for overcoming these limitations. For instance, Pyo et al. [26] developed a model that integrated a convolutional block attention module into a convolutional neural network to predict cyanobacteria concentrations, achieving high performance with a Nash–Sutcliffe efficiency exceeding 0.76. Similarly, Kim et al. [27] applied various machine learning and deep learning techniques, including random forest, artificial neural networks, recurrent neural networks, long short-term memory, and gated recurrent unit, to forecast harmful cyanobacterial occurrences. Their results demonstrated strong forecasting capability. Wang et al. [28] also applied the random forest technique to predict the relative abundance and cell density of dominant cyanobacterial genera (Cyanobium and Microcystis) in river systems, achieving coefficient of determination values above 0.75.
As data-driven approaches, however, deep learning models often lack the hydrological and ecological interpretability provided by physical models and have limitations in generalization and long-term prediction. Consequently, recent studies have attempted to integrate the computational efficiency of deep learning with the interpretability of physically based models.
Several studies have integrated optimization techniques with CE-QUAL-W2. Afshar et al. proposed an automatic calibration approach using the particle swarm optimization (PSO) algorithm [29]. Subsequently, the multiobjective PSO (MOPSO) algorithm was introduced to expand the solution space and improve calibration accuracy [20]. Shabani et al. [13] noted that PSO-based calibration often prematurely converges on local optima and proposed an improved global-best harmony search algorithm, which enhanced solution diversity and calibration accuracy for water temperature simulations.
Recent studies have also demonstrated the utility of ensemble approaches for water quality modeling. For example, Ortiz-Lopez et al. [30] applied ensemble tree-based methods to predict raw water quality parameters such as turbidity and UV254 with high accuracy, while Zhu et al. [31] successfully employed ensemble learning with remote sensing data to estimate chlorophyll-a and other key indicators. These findings highlight the potential of ensemble frameworks to improve predictive reliability in complex aquatic environments.
In addition, the bagging method adopted in this study is known to enhance model performance by reducing the variance of unstable estimators, as explained by Bühlmann [32]. Furthermore, Cassales et al. [33] note that the independent nature of the classifiers in bagging allows for efficient parallel processing, which can significantly reduce overall computing time. Thus, the proposed integration of CE-QUAL-W2 with a bagging ensemble not only addresses the computational challenges of parameter optimization but also contributes to more robust predictions of algal bloom dynamics.
Despite these developments, most studies have focused primarily on general water quality parameters and considered only a limited range of variables. Research specifically targeting algal bloom prediction remains limited. Therefore, this study developed an optimization algorithm based on the bagging ensemble method, differing from previous approaches. The proposed algal bloom prediction algorithm integrated this optimization technique with the CE-QUAL-W2 model, and was constructed for eight weir sections in the Nakdong River, Korea. Sensitivity analysis was performed to identify and prioritize influential parameters and define appropriate optimization ranges. The applicability of the proposed algorithm to the Nakdong River system was evaluated. The method is proposed as a foundational framework for optimizing complex two-dimensional water quality models.
2. Materials and Methods
2.1. Study Area
The study area comprises the main stem of the Nakdong River in South Korea, which includes eight multipurpose weirs constructed in 2011 as part of the Four Major Rivers Restoration Project to enable artificial hydraulic regulation in key sections of the river system. The eight weirs included in this study are the Sangju (36°25’58”N, 128°14’57”E), Nakdan (36°21’36” N, 128°18’21” E), Gumi (36°14’12” N, 128°20’44” E), Chilgok (36°00’56” N, 128°23’53” E), Gangjeong-Goryeong (35°50’25” N, 128°27’38” E), Dalseong (35°44’03” N, 128°25’00” E), Hapcheon-Changnyeong (35°35’27” N, 128°21’22” E), and Changnyeong-Haman (35°22’47” N, 128°33’06” E) weirs.
2.2. Model Construction
The CE-QUAL-W2 model was constructed for the main stem of the Nakdong River, which consists of nine segments: eight sections delineated by the multipurpose weirs and one downstream segment. The computational grid in CE-QUAL-W2 was established based on longitudinal distance, vertical depth, average channel width, and riverbed slope.
To construct the model, cross-sectional and longitudinal survey data provided by the Han River Flood Control Office and River Master Plan [34] were used to generate volume–area–elevation tables. Based on these data, depth, average width by depth, and slope were defined for each segment.
Meteorological data were obtained from the Korea Meteorological Administration [35], while water temperature and water quality data were from the Water Quality Monitoring Network of the Ministry of Environment [36] and used as input parameters. Flow data were collected from the real-time hydrological and discharge databases provided by the Water Management Information System [37]. The input dataset was configured to distinguish inflows and discharges from movable weirs, small hydropower plants, and sand-flush gates.
2.3. Sensitivity Analysis for Parameter Selection
To improve the optimization efficiency, a sensitivity analysis was conducted to identify parameters influencing the concentrations of four algal groups (diatoms, cyanobacteria, green algae, and other algae) simulated using CE-QUAL-W2. To this end, parameters influencing algal concentrations in CE-QUAL-W2 were first classified and categorized into three groups: algal rates (RATE), algal stoichiometry (STOI), and algal temperature rate coefficients (TEMP) [8].
The RATE group consists of coefficients embedded in the model equations that reflect processes such as algal growth, mortality, excretion, and settling. The STOI group includes coefficients that define the relationships between algal biomass and other water quality components, such as phosphorus, nitrogen, carbon, and silica. The TEMP group comprises coefficients related to temperature effects on algal growth. All parameters in these groups are input separately for the four algal groups: diatoms (ALG1), cyanobacteria (ALG2), green algae (ALG3), and other algae (ALG4).
For these parameters, the sensitivity analysis was conducted on 17 parameters for each of the four algal groups: nine RATE parameters and eight STOI parameters. The TEMP parameters were not included in the analysis, as growth coefficients corresponding to water temperature for each algal species were adopted from case studies provided in the CE-QUAL-W2 user manual [8].
The sensitivity analysis revealed that most parameters exhibited linear relationships between parameter adjustments and changes in simulated results, allowing for linear regression analysis. However, two RATE parameters, the maximum algal growth rate (AG) and light saturation intensity (ASAT), did not show clear linearity, necessitating additional sensitivity analysis. During this extended analysis, the concentration variations for ALG3 and ALG4 were minimal, and their influence on Chl-a concentrations, which serve as key algal indicators in South Korea, was negligible. Therefore, these two algal groups were excluded from the optimization targets.
Accordingly, the additional sensitivity analysis focused on AG and ASAT for ALG1, ALG2, and Chl-a. Based on the results, the priority of parameters was determined, and the final selection included maximum algal growth rate (AG), maximum algal respiration rate (AR), maximum algal excretion rate (AE), maximum algal mortality rate (AM), algal settling rate (AS), algal half-saturation for phosphorus limited growth (AHSP), and light saturation intensity (ASAT) from the RATE group, and stoichiometric equivalent between algal biomass and phosphorus (ALGP) and Fraction of algal biomass that is converted to particulate organic matter when algae die (APOM) from the STOI group. A detailed analysis of the mathematical formulations in the model was performed for each selected parameter to verify their respective roles, functions, and influences.
2.4. Incorporating Temporal Algal Trends
Informed by the results of an additional sensitivity analysis, this study focused on the temporal trends of diatoms (ALG1) and cyanobacteria (ALG2). This focus is also consistent with the distinct seasonal succession pattern in Korean river systems, where diatoms typically dominate during the cooler winter and spring, while cyanobacteria become dominant in the warmer summer and autumn periods [38, 39]. The temporal analysis of the study watershed’s algal communities confirmed this seasonal succession, showing that cyanobacteria were predominant from May to September, while diatoms were more influential during the remaining months. Based on this, the simulation period was divided into segments, and the relative error (%Difference) was evaluated for each segment to ensure that the overall %Difference across the full simulation period could be minimized.
Segments 1–3 included, January–April, May–September; and October–December, respectively. For each segment, greater weights were assigned to parameters associated with the algal species exerting the most influence during the corresponding time period. Diatom-related parameters were prioritized in Segments 1 and 3, while cyanobacteria-related parameters were emphasized in Segment 2. Through this weighted approach, the optimization process was designed to reflect both temporal patterns and overall simulation accuracy.
2.5. Evaluation Method for Optimization Targets
Chl-a was set as the optimization target. The objective of the optimization algorithm was to minimize the %Difference between observed and simulated Chl-a values, calculated using Eq. (1).
where Qi represents the observed value, Pi the simulated value, and n the number of data points. The efficiency ranges shown in Table 1 are generalized from the criteria suggested by Donigian [40] while Moriasi et al. [41] also proposed performance ratings using the mathematically identical Percent Bias (PBIAS) metric.
2.6. Development of the Optimization Algorithm
The reliability of the optimization results and the time required to complete the optimization process are critical factors when applying optimization algorithms. This study applied the bagging ensemble technique to minimize the number of model executions and effectively reproduce the temporal trends of algal blooms [42, 43].
As illustrated in Fig. 1, the bagging ensemble technique typically involves generating bootstrap samples (test samples), which are used to train multiple learning algorithms (classifiers). Collectively, these classifiers form an ensemble and the results from individual classifiers are then combined to construct a final model (combined classifier). This process improved the optimization performance and predictive accuracy, while also improving computational efficiency.
In this study, the number of bootstrap samples (test samples) was not based on the number of observed events within the simulation period. While bootstrap samples are generally created through random resampling with replacement from the raw dataset, the observed data in this study were regarded as arbitrarily extracted bootstrap samples. This approach is conceptually similar to active learning techniques [44], in which data are selected based on specific criteria rather than random sampling, as commonly seen in machine learning models such as those using the bagging method. Studies have also adopted selective sampling approaches to enhance training efficiency. For example, Settles [45] demonstrated that active learning improved model generalization by prioritizing samples with high uncertainty. A more recent study showed that applying active learning to pretrained models could maintain high performance while reducing the amount of training data required [25]. Conceptually, these findings support the approach adopted here, in which observed data were used as bootstrap samples.
The ensemble and final model were constructed as follows. First, based on the expected variation in algal species concentrations derived from the sensitivity analysis for each parameter adjustment, the %Difference between the simulated and observed Chl-a values was evaluated for each observed event. To meet the conditions of a “Good” rating, nine key parameters were adjusted. The optimal parameter combinations predicted through this process were designated as the classifiers for each ensemble. The final model (combined classifier) was derived by applying weights that reflected the frequency of parameter usage within each ensemble, the magnitude of parameter adjustments, and the temporal characteristics of algal species.
In deriving the final model, individual ensembles yielded both positive and negative %Difference values. Ensembles whose %Difference direction was inconsistent with that of the entire simulation period were excluded. Based on the derived weights, the overall %Difference during the simulation period was set as the optimization target, completing one full cycle of the optimization process. The optimization was repeated iteratively until the %Difference between the simulated and observed Chl-a values satisfied a “Good” rating, resulting in a fully calibrated outcome.
The sequence of the optimization algorithm can be summarized as follows. First, the rate of change in the Chl-a concentration was estimated based on the parameters selected through sensitivity analysis. Then, calibration was performed for each observed event to ensure that the %Difference for Chl-a met or exceeded a “Good” rating. Next, the method developed based on the bagging ensemble technique was applied to derive the optimal parameter combination, thereby completing one cycle of optimization. The process was repeated until the %Difference over the entire simulation period for Chl-a satisfied a “Good” rating, resulting in the final optimized model output.
3. Results and Discussion
3.1. Sensitivity Analysis
The sensitivity analysis was conducted by varying each parameter from 50% to 150% of its initial value and evaluating the resulting changes in algal concentration. For the eight weir sections of the Nakdong River, sensitivity analysis was performed on 17 parameters (excluding those in the TEMP group); Table 2 summarizes the average annual rate of change in algal concentration for the six most influential parameters.
The overall average of the annual rate of change in algal concentration for each algal group (ALG1–ALG4) across all eight weir sections was calculated. ALGP had the greatest influence on ALG1, while AG had the most significant influence on ALG2, ALG3, and ALG4. Then, these most influential parameters were adjusted to 50% and 150% of their initial values to assess the changes in algal concentrations. For ALG1, setting ALGP to 50% of its initial value resulted in a +53.11% change in algal concentration, whereas setting it to 150% resulted in a −37.66% change. Adjusting AG to 50% and 150% of its initial value caused ALG2 to change by −58.53% and +169.14%, ALG3 to change by −61.23% and +174.14%, and ALG4 to change by −58.13% and +161.20%, respectively.
Overall, all algal groups (ALG1, ALG2, ALG3, ALG4) showed consistent response patterns to parameter changes. The parameter AG exhibited a positive relationship, with algal concentrations decreasing when AG was reduced to 50% of the initial value and increasing when raised to 150%. In contrast, ALGP, AM, AR, ASAT, and AHSP demonstrated negative relationships, where algal concentrations increased at 50% and decreased at 150%. This is because AG directly amplifies the growth term, whereas the other parameters act as limiting or loss-related factors, such as yield reduction, enhanced losses, or intensified constraints on growth.
Afshar et al. [46] conducted a sensitivity analysis by varying the parameters AG and AHSP between 50% and 150% of their calibrated values, and evaluated the resulting changes in the Chl-a, dissolved oxygen, ammonium nitrogen, and phosphate phosphorus concentrations. In their study, reducing AG to 50% of the calibrated value led to a decrease in Chl-a, while increasing it to 150% caused an increase. Conversely, reducing AHSP to 50% increased Chl-a, while increasing AHSP to 150% decreased Chl-a. These trends were consistent with our results when AG and AHSP were varied from 50% to 150% of their initial values.
For the sensitivity analysis, we adopted the parameter setting from Afshar et al. [46], varying each parameter from 50% to 150% of its initial value. However, whereas their study analyzed only two parameters (AG and AHSP), our research assessed the sensitivity of algal concentrations to a more comprehensive set of 17 parameters (e.g., AG, ALGP, AM, AR, ASAT, AHSP), excluding the TEMP group. To integrate the optimization methodology, we performed an additional sensitivity analysis.
To verify the roles and functions of the parameters, the correlation between the rate of parameter change and the rate of change in simulated algal concentrations was analyzed. The extent of the change in the simulated results required to meet the %Difference target within the optimization algorithm was also estimated. A reverse-calculation approach using regression equations derived from the sensitivity analysis was applied to estimate the appropriate parameter values. As a result, most parameters could be described using linear regression models; however, AG (maximum algal growth rate) and ASAT (light saturation intensity) did not conform to linear relationships. Accordingly, an additional sensitivity analysis was conducted using the same method as the initial analysis, but with more finely segmented parameter ranges. In additional sensitivity analysis, a trial-and-error approach was employed to identify the parameter ranges yielding predictable responses. The parameters were systematically adjusted upward from 10% of their default values without a predefined upper limit, which established the effective ranges as 10–700% for AG and 10–160% for ASAT. The results of this additional sensitivity analysis are visualized in Fig. 2.
This additional sensitivity analysis revealed that the relationship between the rate of change in AG and the annual average algal concentration was best modeled as a fourth-order polynomial regression equation for ALG1 (Fig. 2a), ALG2 (Fig. 2c), and Chl-a (Fig. 2e). This regression equation was then applied during the optimization process to estimate the parameter change rate necessary to achieve the target %Difference. The analysis of Fig. 2a, 2c, and 2e also indicated that stable changes occurred when AG ranged from 0.1 to 10 (corresponding to a 30–350% change in the parameter), leading to the definition of this “Application Range” for the optimization. Therefore, the applicable optimization range for AG was set to 0.1–10.
Similarly, the analysis of the rate of change in ASAT and algal concentrations revealed an inverse relationship, but only within specific thresholds, as shown in Fig. 2b, 2d, and 2f. Beyond these thresholds, the trend reversed. In the regression analysis, ALG1 showed a clear inverse relationship when the parameter change rate was below 20% (ASAT = 20), ALG2 below 60% (ASAT = 40), and Chl-a below 40% (ASAT = 30). Based on these findings visualized in the graphs, the applicable “Application Range” of ASAT was set to 25–110 for ALG1 and 45–110 for ALG2 to ensure a stable, linear relationship for the optimization process. The linear regression equations derived from these specific ranges were then incorporated into the optimization process.
Based on the results of the sensitivity analysis, parameters that showed a clear correlation between their rate of change and the variation in simulated algal concentrations were prioritized for optimization, as were those that had a significant influence on algal dynamics. Accordingly, nine key parameters were selected as optimization targets, and their respective ranges of application were used in the optimization algorithm (Table 3). The final set of optimization parameters included AG, AR, AE, AM, AS, AHSP, and ASAT from the RATE group, and ALGP and APOM from the STOI group.
3.2. Optimization Algorithm Efficiency Evaluation
When evaluating the performance of the developed optimization algorithm, the objective was to ensure that the %Difference between the simulated and observed values after calibration met at least the “Good” rating criteria. To assess its applicability, the evaluation was conducted on the Nakdong River system, which served as the target area for algorithm development. The model simulation period spanned 1 year, from January 1 to December 31, 2013. The reliability of the predictions of the optimized model was reviewed accordingly. After calibration using the optimization algorithm, the simulated Chl-a concentrations were compared with observed data, and model efficiency was evaluated for each weir section based on the %Difference.
The %Difference values for each weir section were as follows: Sangju Weir −7.89, Nakdan Weir −0.77, Gumi Weir 0.35, Chilgok Weir −9.32, Gangjeong Goryeong Weir −7.79, Dalseong Weir −8.90, Hapcheon Changnyeong Weir −10.16, and Changnyeong Haman Weir −10.79. All sections achieved “Very Good” ratings ). The time-series reproducibility of the model improved with repeated runs (Fig. 3).
The quantitative improvements presented in Table 4 are visually substantiated in Fig. 3, which illustrates the time-series comparison of observed and simulated Chl-a concentrations for all eight weir sections. In each panel, the red dots represent the observed field data of Chl-a, the black line shows the initial model (default) simulation before optimization, and the progressively refined gray lines show the simulation results after each iteration of the optimization algorithm. The solid blue line represents the final simulation result after the optimization algorithm was completed.
The variability in the number of optimization iterations (represented by the number of gray lines in Fig. 3) across the different weir sections is primarily influenced by the initial parameter set (default). If the default parameters for a specific weir fall significantly outside the optimal application ranges determined by the sensitivity analysis, the algorithm requires more iterations to converge on a solution.
The effectiveness of the algorithm is particularly evident in capturing the temporal dominance periods of algae.
For instance, at the Dalseong Weir, as shown in Fig. 3(f), and the Changnyeong–Haman Weir, as shown in Fig. 3(h), the optimized model successfully captured the distinct bimodal peaks in late spring and autumn, a pattern the initial model (default) failed to represent. This marked improvement in reproducing the temporal variability of Chl-a concentrations was consistently observed across all other weir sections. These results demonstrate that the developed optimization algorithm not only minimizes the overall error but also significantly enhances the model’s capability to simulate the complex, dynamic behavior of algal blooms in the river system.
While the model’s overall efficiency was high, some periods could not be fully reproduced, highlighting its inherent limitations. The discrepancy at the Chilgok Weir, as shown in Fig. 3(d), between April and August, and the mismatch in March at the Dalseong Weir, as shown in Fig. 3(f), may be attributed to localized, short-term hydraulic events or un-monitored inflow changes that the model’s boundary conditions cannot fully capture. These localized discrepancies point to the challenges in modeling complex riverine systems, and future studies could focus on enhancing the model’s capability to reproduce such specific, event-driven phenomena.
3.3. Validation of Assumptions regarding the Temporal Dominance Periods of Algae
To assess the validity of the temporal dominant algal segments established based on the sensitivity analysis and previous studies, the %Difference between the optimized Chl-a simulation results and the observed data was evaluated for each period of algal dominance (Table 5). In Segments 1 (January–April) and 2 (May–September), all weir sections were rated “Fair” or better. However, in Segment 3 (October–December), some sections were rated “Poor” in terms of model efficiency. During this period of low Chl-a concentrations, even minor absolute deviations between simulated and observed values are magnified into large percentage errors, leading to an overestimation of the model’s discrepancy. As the algorithm was designed to prioritize accuracy during high-concentration bloom events, addressing this sensitivity for off-season performance remains a task for future research.
Since algal blooms in South Korea primarily occur during the spring and summer [38, 39], the optimization results reflected temporal trends appropriately. Although several studies have simulated algal blooms using the CE-QUAL-W2 model [4, 6, 19, 46], few have explicitly accounted for temporal characteristics. Therefore, the temporal dominant algal segments defined in this study provide a more systematic representation of seasonal variations in algal bloom dynamics.
3.4. Comparison with Previous Optimization Methods
A number of optimization algorithms, including Particle Swarm Optimization (PSO), Multi-Objective PSO (MOPSO), and Harmony Search, have been applied in previous studies to the calibration of hydrodynamic and water quality models [13, 20, 29]. These approaches generally emphasize multi-objective optimization schemes, in which several error functions for hydrodynamic and ecological variables (e.g., water surface elevation, temperature, dissolved oxygen, and chlorophyll-a) are minimized simultaneously through Pareto-based search strategies. For example, the MOPSO calibration of CE-QUAL-W2 in the Karkheh Reservoir demonstrated the ability of Pareto-optimal fronts to balance competing hydrodynamic and water quality objectives [20].
In contrast, the bagging-based optimization algorithm developed in this study was explicitly designed as a single-objective optimization framework, focusing exclusively on minimizing the relative error (%Difference) of chlorophyll-a. Temporal dynamics were incorporated by applying weighted error evaluations across seasonally segmented periods, thereby enhancing the representation of bloom peaks dominated by diatoms and cyanobacteria. This targeted formulation differs fundamentally from conventional optimization methods, offering a more interpretable outcome and producing parameter sets that can be directly integrated into algal bloom forecasting and management systems.
4. Conclusions
The optimization algorithm developed here is applicable to two-dimensional water quality models and performed better in terms of both the reliability of the optimization results and computational efficiency. Compared to the conventional bagging ensemble technique, the proposed method reduced the number of model runs required to generate an equivalent ensemble by more than 50 times. While optimization based on the overall %Difference has limitations in capturing temporal dynamics, our approach overcomes this by evaluating the %Difference for each observed event. Deriving parameter combinations at this granular level allows for the effective integration of temporal variability. Therefore, when applied to short-term forecasting systems or routine prediction programs, the algorithm should have enhanced predictive accuracy and reliability.
Although this study focused on the Nakdong River, the proposed approach is potentially transferable to other river systems with different hydrological and ecological settings. Future applications would require site-specific adaptations such as adjustments to parameter ranges, sensitivity characteristics, and the seasonal dominance of algal groups, ensuring reliable performance across diverse aquatic environments.
Sensitivity analysis was conducted during algorithm development to identify target parameters and define their applicable ranges. However, an inherent limitation remains, as values derived solely from model-based analysis may not fully reflect actual algal bloom patterns. Therefore, the parameters and ranges proposed here should be revised and adapted to reflect local field conditions when applied to other models.
Furthermore, as the parameters and ranges established in this study were derived from a two-dimensional model, their applicability to three-dimensional models is inherently limited. Nevertheless, with comprehensive sensitivity analysis and appropriate consideration of model-specific characteristics, the proposed optimization approach could be extended to three-dimensional models.
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