For groundwater contamination source identification (GCSI), this study adopts a simulation–optimization framework (Mahar and Datta, 2001). There are various effective simulation techniques available for groundwater modelling. In this study, MODFLOW 6, including the Groundwater Flow and Groundwater Transport models (Hughes et al., 2017; Langevin et al., 2022), is adopted to simulate groundwater flow and pollutant transport, facilitated by the Python package FloPy (Bakker et al., 2016), which provides a convenient and flexible interface for model construction and execution. The governing partial differential equation for transient flow in a two-dimensional aquifer system can be given as follows: 1∂∂xiKi,j∂h∂xj+W=Ss∂h∂t where Ki,j denotes the hydraulic conductivity, md-1; h denotes the hydraulic head, m; xi and xj are the coordinates along the axis, m; Ss is the specific storage of the porous material; and W is the volumetric flux per unit area.

Solute transport can be described by the following advection-dispersion-reaction equation under known hydrogeological conditions: 2∂∂xiθDij∂Ck∂xj-∂∂xiθviCk+qsCks+∑Rn=∂(θCk)∂t where θ is effective porosity; Ck is the dissolved concentration of the species k, mgL-1; Di,j is the dispersion coefficient tensor, m2d-1; vi is the linear pore water velocity, md-1; qs is the volumetric flow rate per unit volume, representing sources or sinks; Cks is the source or sink concentration of species k, mgL-1; Rn is the chemical reaction term, mgL-1d-1.

UQPyL (http://www.uq-pyl.com, last access: 15 May 2026) is a Python package developed by our team to support uncertainty quantification and optimization in computational modelling. The package integrates a comprehensive set of tools, including sampling techniques, surrogate modelling, parameter analysis methods, and global as well as hybrid optimization algorithms. Its modular and extensible design enables users to flexibly combine different components, facilitating rapid prototyping and testing of new algorithms. Moreover, UQPyL includes a default interface to couple external numerical simulators, making it suitable for computationally intensive applications such as groundwater modelling. In this study, UQPyL provides the foundation for implementing the proposed SA-CSA-TS algorithm, conducting surrogate-model comparison experiments, and ensuring a consistent environment for benchmarking different optimization methods.

This study develops a surrogate-assisted hybrid optimization algorithm, SA-CSA-TS, built upon a multi-chain framework in which each chain iteratively performs a two-stage search. Global exploration is conducted using the Cooperative Search Algorithm (CSA), followed by local refinement using Tabu Search (TS). To reduce dependence on computationally expensive groundwater simulations, surrogate models with dynamic reconstruction are embedded into both stages. In addition, designed inter-chain communication enables the exchange of evaluated samples, enhancing data diversity and improving surrogate accuracy.

Figure 1 illustrates the overall workflow. The process begins with initial sampling, and the groundwater model is used to evaluate these samples to initialize the chain archive D. After that, the algorithm enters the multi-chain optimization phase.

During each iteration, surrogate models are at first constructed. The key feature is synergistic learning, where each chain builds its surrogates not only from its own history but also from the evaluated solutions shared by other chains (see the red arrows in Fig. 1). In the first stage, each chain independently performs CSA under the guidance of surrogates to explore the global search space. The best individual from each chain is then evaluated using the groundwater simulator and used to update D. Local refinement is performed in the second stage. Before activating TS, the surrogate models are reconstructed using all newly obtained evaluations. TS subsequently explores neighbourhood solutions through multiple-move operators, and cooperation among chains is realized via a shared tabu list, which prevents redundant searches and promotes effective diversification. Surrogates continue to pre-screen candidate solutions, and only the most promising candidate from each chain is evaluated with the groundwater model. This iterative process continues until the predefined maximum evaluations of groundwater model FEmax is reached.

In summary, SA-CSA-TS enhances GCSI efficiency through three integrated mechanisms. First, the multi-chain framework enables synergistic learning by sharing evaluated information across chains. Second, the sequential deployment of CSA and TS provides a strong balance between global exploration and local intensification. Finally, surrogate models with dynamic reconstruction reduce computational burden while preserving high-fidelity prediction accuracy to guide the search effectively.

For clarity, the pseudocode of SA-CSA-TS is also provided in Algorithm 1.

To alleviate the computational burden of repeated groundwater simulations, surrogate modelling is embedded into the proposed SA-CSA-TS framework. In GCSI, candidate parameters should be evaluated by the groundwater simulator to quantify the mismatch between simulated and observed concentrations at the monitoring wells (see the dashed line of Fig. 2). However, the entire optimization process typically requires thousands of forward simulations. To alleviate the computational demand, the SA-CSA-TS incorporates a surrogate modelling technique (see the solid line of Fig. 2). In this study, four commonly used surrogate models, namely Kriging (KRG), Gaussian Process (GP), Support Vector Regression (SVR), and Radial Basis Function (RBF), are considered as candidate approximators; detailed theoretical backgrounds of these models are available in Lophaven et al. (2002), Rasmussen and Williams (2006), Smola and Schölkopf (2004), and Buhmann (2003), respectively. Although KRG and GP are theoretically closely related, both are considered in this study because differences in practical implementation and hyperparameter estimation may still lead to different predictive performance. All four models are implemented in UQPyL, and their predictive performance is compared in Sect. 4. Rather than treating surrogate modelling as an independent component, the present study embeds it directly into the optimization workflow so that surrogate predictions can guide both the global exploration and the local refinement stage of SA-CSA-TS.

As illustrated in Fig. 2, a set of surrogate models is constructed to estimate the discrepancy (e.g., RMSE or R2) between simulated and observed concentrations at each monitoring well. Therefore, the number of surrogate models equals the number of observation wells. During optimization, these surrogates substitute for repeated groundwater simulations and provide rapid approximations of the error. The predicted discrepancies across all wells are then aggregated, and their sum is adopted as the overall objective function, guiding the evaluation of candidate parameters and the subsequent optimization.

In SA-CSA-TS, the first stage focuses on global exploration, where each chain independently executes the Cooperative Search Algorithm (CSA) with surrogate-based fitness evaluation. The CSA, proposed by Feng et al. (2021), is a population-based metaheuristic inspired by cooperative behaviours in social systems. Previous studies (Feng et al., 2022, 2024) have already shown its feasibility in related water-resources and hydrological applications, including cascade reservoir operation, discharge simulation, streamflow and flood forecasting. Here, CSA is adopted for GCSI because it emphasizes team communication, reflective learning and internal competition among individuals. These mechanisms are well suited to the high-dimensional, nonlinear, and potentially multimodal nature of the inverse problem, and are expected to identify promising regions for subsequent local refinement.

In CSA, a population of candidate solutions {xi}i=1N is initially generated. During the optimization process, individuals improve their positions by learning from others within the population. For example, at iteration t, the update of the ith individual typically follows a team communication rule: 3uit+1=xit+Ait+Bit+CitAit=log⁡(1/ϕ(0,1))⋅gindt-xitBit=α⋅ϕ(0,1)⋅(gmt-xit)Cit=β⋅ϕ(0,1)⋅pmt-xit where Ait, Bit and Cit denote the knowledge components from the chairman, board of directors, and board of supervisors, respectively. gindt is the indth global best individual at iteration t. The gmt represents the mean position of the top M global best individuals. The pmt is the mean position of the ith personal best individual. In addition to this team-communication update, CSA also employs reflective learning and internal competition to maintain population diversity and retain superior individuals; other detailed algorithmic formulations can be found in Feng et al. (2021).

In the proposed algorithm, CSA is embedded in a surrogate-assisted manner. As illustrated in Fig. 3, the objective values of candidate solutions are predicted by the trained surrogate models instead of being repeatedly evaluated by the computationally expensive groundwater simulator. This substitution substantially improves the efficiency of the global exploration stage. The superior solutions generated by CSA are then used to update the current position of each chain (Line 08 in Algorithm 1). For comparison purposes, this surrogate-assisted CSA module is also implemented as a standalone benchmark algorithm, denoted SA-CSA, so that the specific contributions of the multi-chain architecture and the subsequent Tabu Search can be explicitly assessed against the complete SA-CSA-TS framework.

Following the global exploration stage, SA-CSA-TS performs local refinement using Tabu Search (TS). TS is a local-search metaheuristic characterized by adaptive memory and strategic neighbourhood exploration. Its key feature is the tabu list, which records recently visited solutions or attributes and prevents their immediate reconsideration, thereby reducing cycling and encouraging exploration of new regions. To avoid excessive restriction, TS also incorporates an aspiration mechanism, under which a tabu status can be relaxed if the corresponding move leads to an improved solution. These characteristics make TS well suited for refining promising regions identified in the preceding global exploration stage. In the context of GCSI, TS is particularly useful for structured exploration of discrete source-location configurations, thereby helping the algorithm escape local traps and identify more competitive solutions.

Unlike the previous stage, where CSA operates independently in each chain, the Tabu Search (TS) stage is executed under a coordinated multi-chain framework. In this design, all chains share a common tabu list, which serves as a collective memory to prevent any chain from revisiting previously explored regions. Since the discrete variable corresponds to the index of a potential contamination-source area, the tabu list is defined over this finite set, and its maximum size is equal to the total number of candidate source areas. The corresponding search mechanism is illustrated in Fig. 4. Guided by the retrained surrogate model, each chain explores its neighbourhood to identify promising candidates. As shown, the search trajectories are strictly constrained by the shared history, enabling the algorithm to better navigate multi-modal landscapes. For example, moves that enter tabu-listed areas (highlighted by red arrows) are prohibited. After selecting the most promising solutions, the algorithm performs simulation-based evaluations and subsequently updates the shared tabu list, thereby allowing dynamic information exchange among all chains.

We describe the rule for updating the tabu list. Let xi and fi denote the current solution and its objective value of the ith chain, respectively, and let fbesti represent the historical best objective value recorded by that chain. The update mechanism consists of the following three cases:

If fi>fbesti, the discrete component of xi, denoted xid, is added to the tabu list T, preventing the algorithm from revisiting this configuration in subsequent iterations.

If fi<fbesti, and xid∈T, the tabu status of xid is removed, allowing the algorithm to reconsider this configuration since a better solution has been found.

If fi<fbesti and xid∉T, both the best solution xbest and the best objective fbesti are updated accordingly.

The study area is a two-dimensional, homogeneous confined aquifer (800m×1200m), as illustrated in Fig. 5. The left and right boundaries are assigned constant hydraulic heads, and the remaining boundaries are treated as no-flow. For simulation, the domain is discretized into a grid of 16×24 cells, with a uniform cell size of 50 m. The basic hydrogeological parameters used in this case are summarized in Table 1.

The potential contamination source zone, also shown in Fig. 5, represents an industrial area with intensive activities, where contaminants may be intermittently released into the aquifer. Within this zone, one or more contamination sources may exist. To capture solute transport behaviour and provide data for the inverse analysis, seven monitoring wells are distributed across the study area (the triangle in Fig. 5).

In Case 1, a single contaminant source is considered. The total simulation time is 40 months, divided into 20 stress periods (SPs), with the source releasing contaminants only during the first five SPs. The true source location and its release fluxes for these five SPs are listed in Table S1 in the Supplement. The contaminant plume distributions at the 5th and 10th SPs are shown in Fig. 6.

For this case, the parameters to be identified include: (a)

Hydrogeological parameters: The hydraulic conductivity (K) and the boundary head (H1 and H2). Their ranges are listed in Table 1;

Case 2 adopts the same hydrogeological setting and numerical configuration as Case 1, but involves a more complex contamination scenario. In this case, three independent contaminant sources are introduced within the potential source zone. Their true locations and time-varying release fluxes are summarized in Table S2. The contaminant plume distribution at the 5th and 10th SPs is illustrated in Fig. 7.

Compared with Case 1, Case 2 presents a significantly higher level of complexity for surrogate modelling and optimization. The number of discrete variables associated with source locations increases from 2 to 6, and the total number of unknown parameters rises from 10 to 24 due to the introduction of additional sources and their time-varying release fluxes.

This case study is designed as a realistic numerical experiment based on the hydrogeological conditions of a mining area in Henan Province, China. The study area covers approximately 2.67km×3km. According to exploration-stage geological archives and field investigations, the aquifer is conceptualized as a single-layer unconfined system composed mainly of weathered and fractured granite, with an average saturated thickness of about 30 m. The underlying fresh granite is considered impermeable and therefore forms the basal boundary of the model. The groundwater flow system is represented by a two-dimensional single-layer numerical model. In plan view, the model domain is discretized using a structured grid with a uniform cell size of 30m×30m, and the irregular outer boundary is represented by active and inactive cells, as shown in Fig. 8. The rivers along the western and eastern margins are treated as constant-head boundaries, whereas the northern and southern margins are specified as no-flow boundaries because they are bounded by relatively intact, low-permeability fresh granite. Groundwater recharge occurs primarily through vertical infiltration of precipitation and is represented using an average annual precipitation of 650 mm and a recharge coefficient of 0.12. To capture spatial heterogeneity, the aquifer is divided into four hydraulic-conductivity zones based on the exploration-stage geological archives: Zone I corresponds to alluvial sand and gravel near the riverbanks, Zones II and III represent highly weathered and moderately weathered granite, respectively, and Zone IV represents a localized tectonic fracture zone. The main hydrogeological parameters adopted in the model are summarized in Table 2.

A potential contaminant source region is delineated, as highlighted in pink in Fig. 8. Field investigations identify three waste-ore deposits (S1,S2,S3) within this region. These sources continuously release contaminants into the groundwater during the first five stress periods (out of a total of ten). Nine observation wells are distributed across the study area to monitor contaminant migration, and Fig. 9 illustrates the temporal concentration dataset used for the inverse analysis over the stress periods.

In summary, the parameters to be identified include: (a) Hydrogeological parameters: The hydraulic conductivity (K1, K2, K3, K4); (b) Source locations (SIi and SJi, i=1,2,3) and their release fluxes (SiPt, i=1,2,3 and t=1 to 5), with the value of each flux bounded between 0 and 100 kgd-1 Their reference values are listed in Table S3.

This study employs four commonly used surrogate models to investigate their performance in predicting the discrepancy between observed and simulated data for a given set of solutions: (a) Kriging (KRG); (b) Gaussian Process (GP); (c) Support Vector Regression (SVR); (d) Radial Basis Function (RBF).

To ensure a fair comparison, all surrogate models are constructed using UQPyL on a computer equipped with 12th Gen Intel(R) Core (TM) i5-12490F CPU, and 32.0 GB of RAM. Motivated by the cost–benefit perspective of surrogate tuning discussed by Ahrari and Verstraete (2023), only selected influential hyperparameters are tuned in this study using grid-search, whereas the remaining hyperparameters are retained at their default values in UQPyL. The tuned hyperparameters and their search ranges are summarized in Table S4.

For sample generation, Latin Hypercube Sampling (LHS) is used in Cases 1–3 to produce a set of parameter samples, which are subsequently input into the groundwater models to obtain contaminant concentrations. For each sample, the RMSE between the simulated and observed concentrations at all monitoring wells is calculated. RMSE is selected here because it provides a steeper and more informative gradient, which is advantageous for optimization. The generated parameter sets and their corresponding RMSE values constitute the full input–output datasets.

To evaluate model performance, four training datasets, denoted as DS1–DS4 with sample sizes of 100, 200, 300, and 500, respectively, are constructed. An independent set of 50 samples is generated for testing.

As described earlier, SA-CSA-TS constructs individual surrogate models for each monitoring well, and the corresponding outputs are summed to derive an ensemble objective value for optimization. To evaluate the effectiveness of this approach, we first examine the ensemble prediction performance of four surrogate models across Cases 1–3, based on the coefficient of determination (R2) and Root Mean Square Error (RMSE). The results are summarized in Table 3.

Across all datasets (DS1–DS4) and all three cases, RBF clearly delivers the most stable and accurate ensemble predictions. KRG and GP achieve acceptable accuracy, whereas SVR consistently performs the weakest. All models benefit from increasing training data. In comparison, RBF demonstrates a superior sensitivity to data enrichment, aligning well with the iterative reconstruction strategy of SA-CSA-TS. In Case 3, the prediction task becomes significantly more challenging due to more complex hydrogeological conditions, leading to lower R2 values for all models. However, RBF still maintains robust predictive capability.

Based on the ensemble results, Cases 2 and 3 under dataset DS3 are selected for detailed surrogate evaluation at the individual monitoring wells. These two cases represent more challenging prediction scenarios. In addition, DS3 provides a sufficiently informative training set, yielding a clear performance improvement over DS2, while the additional gain from DS3 to DS4 is marginal. Figure 10a illustrates the prediction performance for Case 2 using DS3. Accuracy varies substantially across monitoring wells, primarily due to the spatial distribution of the contaminant plume. Wells 1, 2, and 5 are located within the main plume body, where steep and highly nonlinear concentration gradients dominate. Consequently, all surrogate models except RBF show marked reductions in R2 at these locations. In contrast, Wells 6 and 7 lie far from the plume centre, where concentration gradients are smooth, enabling all models to reach their highest performance. A similar trend is observed in Case 3 (see Fig. 10b). Wells situated in high-gradient zones (e.g., Wells 1, 2, 3, and 5) pose greater challenges, leading to noticeable performance declines for SVR, GP, and KRG. In contrast, RBF consistently maintains strong performance across all monitoring wells.

Figure 11a and b presents the sample-wise predicted values at representative locations: Well 1 for Case 2 and Well 5 for Case 3. In both scenarios, RBF achieves the highest R2 and lowest RMSE, followed by KRG, GP, and SVR. In Case 3, SVR fails to capture the nonlinearity of contaminant concentrations, with its predictions collapsing into a narrow range. For optimization applications, high fidelity in the low-value region of the response is particularly important, as deviations in this domain can significantly affect the quality of the optimal solution. RBF provides more stable and accurate predictions in these low-value zones, further reinforcing its reliability as a surrogate model for optimization.

In addition to prediction accuracy, the computational cost of training is a critical consideration for SA-CSA-TS, which involves iterative surrogate reconstruction. Theoretically, GP and KRG are computationally intensive with a complexity of O(k⋅N3), where k denotes the number of iterations required by the construction algorithm. In contrast, RBF and SVR offer higher computational efficiency, with complexities of O(N3) and O(N2)∼O(N3), respectively. This theoretical advantage is further supported by empirical results obtained using UQPyL on dataset DS4. In terms of actual training time, GP and KRG require approximately 1 and 4 s, whereas RBF and SVR significantly reduce the cost to 0.22 and 0.01 s.

In summary, RBF overcomes the precision limitations of SVR while avoiding the computational inefficiencies associated with KRG and GP. It thus provides the best balance between accuracy and efficiency, making it the most suitable surrogate model for the proposed SA-CSA-TS framework.

This section aims to investigate the performance of SA-CSA-TS in GCSI. For comparison, three additional optimization algorithms are considered: Genetic Algorithm (GA), Cooperative Search Algorithm (CSA) and SA-CSA. GA is widely used as a benchmark, whereas CSA represents a state-of-the-art method in recent years. SA-CSA is included to isolate and assess the contributions of the multi-chain framework and the Tabu Search. All algorithms are implemented within UQPyL to ensure a consistent and fair computational environment.

For the standard evolutionary algorithms (GA and CSA), the maximum number of simulations (FEmax) and the population size Np are set to 20 000 and 100. For GA, the user-defined parameters pc, ηc, pm, ηm are set to 1, 20, 1/D, 20, respectively, where D denotes the dimensionality of the problem. For CSA, the parameters are set as α=0.10, β=0.15, and M=3.

For surrogate-assisted algorithms (SA-CSA-TS and SA-CSA), the RBF model is employed. The FEmax is reduced to 2000, as surrogate models enable efficient optimization with substantially fewer exact evaluations. Based on the results summarized in Table 3, the number of initial samples NI for surrogate construction is set to 300. For SA-CSA-TS, the number of chains is set to K=10.

For Cases 1–3, the optimization problem is formulated as: 4minimize:f=∑m=1M∑t=1TSmt-Omt2Tsubject to:LB≤{H,K,SI,SJ,SP}≤UB where Smt and Omt represent the simulated and observed concentrations at the mth monitoring well in stress period t, respectively. LB and UB are the lower and upper bounds of parameters to be estimated.

Surrogate models are incorporated into SA-CSA-TS to alleviate the computational burden of high-fidelity simulations. Figure 14 breaks down the runtime of all algorithms across the three cases into simulation time (blue) and algorithm time (red). It is evident that the simulation cost overwhelmingly dominates the total runtime. Although surrogate-assisted methods introduce a slight overhead for model construction and updating, this cost is negligible compared to the time savings achieved by reducing high-fidelity evaluations. Specifically, in three case studies, SA-CSA-TS reduces the total runtime by approximately 85 %–88 %, compared to the GA and CSA. This result confirms that the efficiency advantage of the surrogate-assisted framework becomes increasingly pronounced as the problem complexity grows.

Given the negligible overhead of surrogate modelling, the effect of the iterative reconstruction strategy is further examined. Figure 15 tracks the evolution of prediction accuracy (R2) of surrogate models on a validation set during optimization. In Case 1, the accuracy remains high and stable. In contrast, Cases 2 and 3 exhibit noticeable fluctuations. These oscillations are not indicators of failure but rather reflect the algorithm's active exploration of underrepresented regions. Driven by the Tabu Search mechanism, the optimizer periodically escapes local basins and enters unexplored areas where the surrogate model initially has lower accuracy. However, the subsequent recovery of R2 values confirms that the surrogate model successfully adapts to these new regions. Crucially, this dynamic updating process prevents the convergence stagnation observed in the other algorithms, ensuring that the search remains robust even in complex landscapes.

Groundwater contaminant source identification is an inherently multi-modal optimization problem, where inaccurate location estimates may easily trap algorithms in inferior local solutions. As observed in Fig. 12a–c, GA, CSA and SA-CSA frequently exhibited instability and stagnation. This failure is largely attributed to their reliance on a single search population or trajectory, which lacks the mechanism to escape local basins. In contrast, SA-CSA-TS successfully identified the source information in all three cases. To understand this mechanism, we examine the behaviour of the proposed multi-chain framework.

Figure 16 depicts the search-frequency maps of candidate source locations by SA-CSA-TS for Case 1 and Case 2. In both scenarios, the true source locations (marked by red bars) correspond to the highest visit frequencies (red circles), indicating that the majority of chains consistently converge toward the correct region. Notably, the surrounding cells also exhibit high visit frequencies. This phenomenon confirms the parameter-compensation effect, where spatial inaccuracies are temporarily balanced by adjustments in release fluxes or hydraulic conductivity. This “equifinality” trap explains why conventional algorithms often stagnate near, but not exactly at, the true source. Furthermore, Case 2 displays more dispersed secondary hotspots than Case 1, reflecting a more rugged landscape with stronger compensability. Despite this complexity, the proposed framework successfully concentrates the search effort on the true location, demonstrating robust global convergence. Further improvement may be achieved in future work by incorporating optimal monitoring well placement to provide stronger spatial constraints and further reduce the parameter-compensation effect.

Figure 17 provides a deeper insight by analysing the distribution of search trajectories across ten independent chains in Case 3. The histograms for the source coordinates (SI and SJ) reveal a distinct multi-modal distribution, confirming the existence of multiple local optima. While the majority of chains converge to the primary peak (the true source), a few chains (e.g., Chains 1, 2, and 10) are entrapped in secondary peaks. Therefore, if a single-chain method (like standard GA or CSA) is used, and it happens to follow the trajectory of Chain 1, the identification would fail entirely. However, the multi-chain framework mitigates this risk by exploring multiple basins simultaneously. This collective intelligence allows the algorithm to filter out local optima and stabilize estimates around the true global solution, effectively overcoming the equifinality and multi-modality challenges that hinder conventional single-population methods.

To evaluate the robustness of SA-CSA-TS under data uncertainty, additional experiments are conducted based on three case studies. Random Gaussian noise with varying levels (0.5 %, 1 %, and 2 %) is superimposed on the noise-free observation data, following the equation: 5Cobs∗=Ctrue⋅(1+δ⋅ξ) where Cobs∗ and Ctrue denote the noisy and noise-free observations, respectively; δ denotes the noise level; and ξ is a random number following the standard normal distribution N(0,1).

Figure 18 illustrates the Average Relative Errors (ARE) for the three cases under these noise levels. A clear trend is observed where the identification error increases marginally with the noise intensity. Specifically, for Case 1, the ARE rises from 1.59 % (noise-free) to 3.09 % (2 % noise). For the more complex scenarios in Cases 2 and 3, the errors start at approximately 3.7 %–3.8 % and increase to roughly 4.5 % under the maximum noise level. Despite these increases, the average errors for all cases consistently remain below 5 %, indicating that the proposed method maintains high performance without significant degradation when observation data is subject to measurement noise.

Tables S5–S7 provide the specific identification results in three cases. Notably, the discrete source locations match the true values exactly across all noise levels. As for continuous variables, the hydrogeological parameters show only slight fluctuations. In comparison, the source release parameters exhibit relatively larger variations. This phenomenon is largely attributed to the complementary effects between different stress periods or among multiple sources, where slight deviations in one parameter may compensate for another. Despite this, the overall errors remain within an acceptable range, confirming the robustness of SA-CSA-TS against data uncertainty.

Despite the promising performance and robustness of SA-CSA-TS, some limitations should be further discussed. First, this study evaluates the proposed algorithm only using two-dimensional groundwater systems. The search strategy of SA-CSA-TS, however, is guided by the difference between simulated and observed responses at monitoring locations, rather than by the assumption tied to a specific groundwater model dimensionality. This gives the SA-CSA-TS potential for extension to three-dimensional groundwater models. Nevertheless, its applicability to three-dimensional flow and hydrodynamic dispersion systems has not yet been demonstrated in this study. Furthermore, an increase in vertical resolution (depth layers) not only raises the computational cost of groundwater simulation but also poses a challenge to the predictive fidelity of the surrogate model in complex cases, which should be confirmed through further investigation. Second, while the robustness analysis demonstrated resilience against Gaussian noise, real-world field conditions often involve more complex uncertainties. These include sparse monitoring networks, systematic measurement biases, and structural model errors. Therefore, future work should focus on testing SA-CSA-TS in three-dimensional systems and under combined uncertainty sources, in order to establish its robustness and reliability for complex groundwater inverse problems.