The XAJ model employs a three-source approach for streamflow calculation, utilizing surface streamflow routing through the unit hydrograph method, while separate linear storage reservoirs simulate the routing processes of interflow and baseflow. Its main features include the application of storage excess concepts and Muskingum routing, characterized by subdivision into subunits, multiple sources, and routing stages. Additionally, the XAJ model possesses a simple structure, with parameters that have clear physical meanings and high computational accuracy, which has led to its widespread application in hydrological forecasting (Gong et al., 2021).

The XAJ model encompasses a substantial number of parameters, some of which are highly sensitive; even minor alterations can significantly affect the results, while others display a degree of inertia. The sensitivity classification of the parameters in Table 1 is a general summary adopted from previous studies on XAJ model applications across various basins in China. These studies have systematically evaluated parameter sensitivity using comprehensive methods commonly employed in hydrological modeling, including local sensitivity analysis (e.g., one-at-a-time method) and global sensitivity analysis (e.g., Sobol' method and Morris method). Specifically, parameters are classified as “sensitive” if consistent findings from existing literature indicate that a small relative change in their values leads to a significant change (e.g., exceeding a predefined threshold in terms of NSE reduction or RMSE increase) in streamflow simulation results. For example, Gong et al. (2021) conducted sensitivity analysis on XAJ model parameters in small-and medium-sized catchments in South China and identified parameters such as B, WM, and EX as highly sensitive. Similarly, Lei et al. (2023) confirmed the sensitivity of parameters like KC and KG through runoff simulation studies in humid regions. Table 1 presents detailed descriptions of the meanings and sensitivities of 15 parameters, which can be broadly classified four categories based on the model's structure and function: evapotranspiration parameters, flow rate parameters, water source division parameters, and confluence parameters.

TCN represents an architectural framework that combines one-dimensional convolutional networks with causal convolutions, efficiently capturing temporal dependencies in data and demonstrating enhanced suitability for addressing time-series challenges (Sun et al., 2024). Dilated convolutional networks enable dilated sampling on the input from the preceding layer, allowing for the extraction of feature information from time-series data characterized by long intervals and non-continuous patterns. In the context of causal convolution, the output at any given moment is solely dependent on the input at that moment and any preceding inputs. TCN's core architecture is composed of dilated causal convolution.

The process by which TCNs handle data sequences is analogous to that of dilated convolution. TCNs typically incorporate a greater number of convolutional layers compared to conventional CNNs, continuously adjusting the number of layers. This results in progressively larger dilation factors and convolutional kernels at each layer. However, as the depth of the network increases, the training process becomes increasingly challenging, with the multi-layer backpropagation of error signals potentially leading to issues such as “gradient vanishing” or “gradient explosion”. Residual networks have been shown to effectively mitigate these challenges.

As a variant of recurrent neural networks (RNNs), GRU effectively solves the long-term memory storage problems and gradient disappearance issues faced by traditional RNNs during backpropagation (Song et al., 2024). It employs gating mechanisms, comprising an update gate and a reset gate, to govern information flow within the network. The update gate modulates the integration of historical and new information, whereas the reset gate identifies components of past hidden states irrelevant to current computations. Through adaptive memory update and reset mechanisms, GRU effectively models long-term dependencies in sequences.

GRU's computation process can be represented by the following equations: 1rt=σwr⋅ht-1,xt2zt=σwz⋅ht-1,xt3Ht=tanh⁡wH⋅rt⋅ht-1,xt4ht=1-zt⋅ht-1+zt⋅Ht where rt represents the reset gate, zt denotes the update gate, Ht signifies the intermediate state, and ht corresponds to the GRU output. The weight matrices wz, wr and wH are associated with the update gate, reset gate, and intermediate state, respectively. Here, xt indicates the current state, ht-1 represents the previous hidden state, σ stands for the sigmoid activation function, and tanh refers to the hyperbolic tangent function.

Figure 1 depicts the architecture of the TCN-GRU model, which consists of input layer, TCN layer, GRU layer, and output layer. Historical streamflow data and highly correlated influencing factors are integrated as input to the TCN layer, which excels in feature extraction, fast convergence, and robustness. The TCN layer processes input data through a stack of dilated causal convolution modules (each containing a 1D convolution with kernel size 3, dilation factor doubling per layer, and residual connections) to generate high-dimensional temporal feature maps. Specifically, for input data with shape (batch_size, time_steps, input_features), the TCN layer outputs feature maps with shape (batch_size, time_steps, tcn_features), where tcn_features is determined by the number of filters in the final TCN convolution layer (set to 64 in this study based on hyperparameter optimization).

To match the feature dimensions with the subsequent GRU layer, a linear projection (via a fully connected layer with no activation) is applied to the TCN output, transforming the feature dimension from tcn_features to gru_units. This ensures the TCN output (shape: (batch_size, time_steps, gru_units)) directly serves as the input to the GRU layer, which is designed to accept sequences with feature dimension equal to gru_units.

The GRU layer, with hidden size gru_units, processes the temporal features by updating its internal state at each time step, capturing sequential dependencies. Potential bottlenecks (e.g., information loss during dimension conversion) are mitigated by: (1) keeping tcn_features ≥ gru_units to avoid downsampling-related information compression; (2) adding a skip connection from the TCN output to the GRU input, which retains raw TCN features alongside the projected features; and (3) using batch normalization in the TCN layer to stabilize feature distribution during propagation.

Thus, the GRU layer effectively learns dynamic variations from the TCN-extracted features, capturing temporal correlations among multiple features to enhance simulation accuracy. The output layer generates the final simulated values. The configuration of model hyperparameters, such as the TCN dilation factors, GRU hidden units, is detailed in Sect. S2.1 in the Supplement.

RF is an ensemble learning approach that boosts the precision of classification and regression tasks by building and combining the prediction results of multiple decision trees (Qiao et al., 2023). The core principle of this algorithm involves utilizing the bootstrap resampling technique to randomly sample multiple subsets from the original dataset, on which decision trees are constructed for each subset. During the construction of these decision trees, features are randomly selected for splitting, thereby enhancing the model's diversity and generalization capability (Doyle et al., 2023). Ultimately, RF combines the simulation results of the multiple trees through voting or averaging to derive the final classification or regression outcome. Utilizing RF for nonlinear ensemble learning leverages the advantages of its ensemble algorithm, thereby improving simulation accuracy, reducing the risk of overfitting, and effectively managing nonlinear and high-dimensional data (Alnahit et al., 2022; Wu et al., 2023; Wang et al., 2025a). The configuration of RF tree depth is detailed in Sect. S2.2 in the Supplement.

A robust hybrid integration approach, namely the XAJ-TCN-GRU model, is proposed to organically integrate the conceptual rainfall–runoff model and deep learning model for streamflow simulation. This approach aims to leverage the complementary strengths of the two model types while avoiding their inherent limitations through parallel execution and nonlinear fusion. The model includes the following steps:  

Step 1: Utilize the XAJ model for initial streamflow simulation, fully leveraging its scientific rigor and reliability based on hydrological theory.

The design of this hybrid model aims to overcome the limitations of individual models, allowing the conceptual rainfall–runoff model and deep learning model to complement each other, thereby providing a more comprehensive and accurate solution for streamflow simulation. The technical flowchart of the study methodology is depicted in Fig. 2.

To validate the adaptability of the proposed XAJ-TCN-GRU hybrid model to different hydrological and geographical conditions, this study conducted experiments across four basins characterized by distinct geographical and hydrological features. These basins include the Wuding River Basin (arid northwestern China), the Chu River Basin (humid eastern coastal region), the Jianxi River Basin (hilly southeastern coastal region), and the Qingyi River Basin (mountainous plateau region in southwestern China). Figure 3 presents the geographical information maps of these four basins. Detailed descriptions of these basins can be found in Sect. S1 in the Supplement. By conducting experiments in basins with varying geographical characteristics and hydrological conditions, it is possible to comprehensively evaluate the adaptability and simulation capability of this model.

In this study, daily observations of 31 meteorological and hydrological variables were collected from multiple stations across the four river basins, as tabulated in Table 2. Data acquisition for these variables was conducted via the China Meteorological Science Data Center (http://data.cma.cn/, last access: 12 March 2026). For hydrological monitoring, the primary control stations in the Wuding River, Chu River, Jianxi River, and Qingyi River basins are Baijiachuan Station, Getang Station, Qilijie Station, and Jiajiang Station, respectively. Daily streamflow data for these stations were retrieved from the measured records in Hydrological Yearbook of the People's Republic of China. The entire dataset spans a continuous period from 1 January 2010 to 31 August 2023. Statistical summaries of the streamflow datasets for each basin are presented in Table 3, offering a more intuitive characterization of the data (Gil et al., 2016).

The input for the XAJ model consists of simulated precipitation and evaporation, while the output is simulated streamflow. The dataset is divided into training, validation, and test sets in a 6:2:2 ratio. The SCEM-UA method is used to calibrate the model parameters to ensure prediction accuracy.

The 31 hydrological and meteorological variables listed in Table 2 were used as feature variables, while the streamflow was used as the target variable to construct the dataset for inputting into the TCN-GRU model. In order to simplify the model, decrease computational costs, and enhance the interpretability of the data, MIC was employed for feature selection on the 31 feature variables. For each basin, 10 highly correlated feature variables with the streamflow were selected, as shown in Table 4.

The proposed model was built on a personal computer with a 1.8 GHz Intel i5 processor and 8 GB of memory. The TCN-GRU model was developed using the popular deep learning frameworks Keras and TensorFlow. Additionally, visualization tools like Matplotlib were used to present the model training outcomes visually, ensuring compatibility of the development environment and intuitive analysis of the results. After feature selection via MIC and VIF, a new dataset including the selected features and streamflow was constructed. The training set was used to train the TCN-GRU model, while the validation set was applied for hyperparameter tuning. To improve the performance of deep learning models, a random search algorithm was used to optimize hyperparameters. Firstly, based on existing research results and domain expertise, the range of potential hyperparameter values was defined. Then, the random search algorithm randomly selected hyperparameter combinations and evaluated model performance based on the validation set to systematically explore this range of values. To ensure result stability and reliability, five independent optimization runs were conducted, and performance metrics were averaged. Through this approach, the optimal hyperparameter combination maximizing model performance was efficiently identified. The final hyperparameter configurations for proposed TCN-GRU and RF are listed in Table S1. The trained TCN-GRU simulated streamflow by inputting the testing set.

Using the simulated results generated by the XAJ model and TCN-GRU model on the training set as feature inputs, a random forest model is trained to learn how to effectively perform nonlinear ensemble. To reduce the risk of overfitting, the maximum depth is set to limit the complexity of each tree, thereby preventing the model from overfitting the noise in the training data. Next, the simulated results from the XAJ and TCN-GRU models on the test set are input into the trained random forest model for nonlinear ensemble, resulting in the final simulation.

The evaluation metrics adopted in this research include Mean Absolute Error (MAE), Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE), NSE, and Kling-Gupta Efficiency (KGE). To quantify uncertainties in runoff simulation, this study employs the Prediction Interval Coverage Probability (PICP) and the Prediction Interval Normalized Average Width (PINAW) as core evaluation metrics. These form a classic combination in the field of hydrological uncertainty quantification (Xu et al., 2025; Kang et al., 2025). These metrics evaluate interval performance across two dimensions, reliability (proportion of observed values covered) and accuracy (interval compactness), aligning with this study's analytical requirements for simulation uncertainty.

PICP measures the proportion of observations falling within the simulated interval, reflecting the reliability of the interval. 5PICP=1n∑i=1nILi≤yi≤Ui where n denotes the sample size, yi represents the ith observation, Li and Ui denote the lower and upper bounds of the simulated interval respectively, and I⋅ is the indicator function (assigning 1 when Li≤yi≤Ui, and 0 otherwise). A PICP closer to the preset confidence level (e.g., 95 %) indicates stronger coverage capability of the interval for observations.

PINAW measures the compactness of simulated intervals, reflecting precision. 6PINAW=1n⋅R∑i=1nUi-Li where R=max⁡yi-minyi denotes the range of observed values, and Ui-Li represents the simulated interval width for the ith sample. A smaller PINAW indicates a tighter interval, thereby reducing redundancy in decision-making uncertainty. A detailed description of these metrics is shown in Sect. S3 in the Supplement.

The XAJ-TCN-GRU model was used in this study to simulate streamflow under four distinct hydrological conditions across various basins, presenting the simulated results in the form of fitted graphs. As indicated in Fig. 4, it is evident that the proposed hybrid model exhibits good fitting performance in all four basins with distinct hydrological and geographical characteristics, which verifies the model's adaptability to different hydrological conditions in the selected study areas. This efficacy is underpinned by the inherent advantages of the XAJ model in simulating physical processes. The XAJ model incorporates 15 parameters with explicit physical significance (Table 1), such as the areal mean tension water capacity (WM) of the catchment characterising basin-scale soil water storage capacity, and the exponent of the tension water capacity curve (B) – these parameters explicitly encode hydrological mechanisms such as soil moisture dynamics and superpercolation (Gong et al., 2021). The XAJ model's prior physical knowledge provides robust mechanistic support for the hybrid model, thereby enhancing its interpretability and generalization capability.

Notably, in basins with larger streamflow, such as Qingyi River, our model demonstrated precise peak simulation capabilities, presenting a significant advantage over traditional XAJ, TCN, and GRU models. Additionally, we observed relatively gentle simulation errors during periods of lower flow, while errors increased significantly and exhibited noticeable fluctuations during peak periods. Moreover, as the number of peaks increased, the fluctuation of errors also increased. However, overall, the absolute error remained within 200 m³ s⁻¹.

In basins with relatively low streamflow volumes, such as the Wuding River (arid zone) and the Chu River (humid coastal zone), a slight lag phenomenon was observed in the simulation results. This phenomenon aligns with existing hydrological research findings: Bai et al. (2017) discovered that in arid basins, undulating topography increases surface runoff convergence time, while soil texture (such as highly permeable sandy soils) prolongs the process of soil moisture replenishing runoff; Gebremariam et al. (2014) further indicated that vegetation cover (such as sparse vegetation in the Wuding River basin) reduces evapotranspiration losses but increases surface roughness, thereby indirectly affecting streamflow convergence rates. The aforementioned factors – topography, soil type, and vegetation cover – are not sufficiently incorporated into current model structures (e.g., the XAJ model does not explicitly parameterise topographic slope, and the TCN-GRU model's input features do not include vegetation indices). This omission likely constitutes the primary cause of simulation lag. Consequently, these unaccounted variables impose inherent limitations on modelling complex hydrological processes within specific basins.

To further evaluate the model's adaptability across different flow regimes, we divided the daily streamflow data of each basin's testing set into three intervals (low, medium, and high) using the tertile method, based on the statistical characteristics of streamflow in the test dataset. The division criteria and corresponding performance metrics of the XAJ-TCN-GRU model are detailed in Table S2 as follows: In low-flow conditions, its NSE ranged from 0.911 (Jianxi River) to 0.994 (Qingyi River), with RMSE between 0.911 (Chu River) and 5.288 m³ s⁻¹ (Jianxi River). In medium-flow conditions, the model's NSE varied from 0.872 (Jianxi River) to 0.979 (Wuding River), and RMSE was in the range of 2.659 (Chu River) to 12.556 m³ s⁻¹ (Qingyi River). Even in high-flow conditions (prone to flood events), the model maintained robust performance, with NSE from 0.981 (Chu River) to 0.996 (Wuding River) and RMSE between 11.396 (Wuding River) and 33.582 m³ s⁻¹ (Qingyi River). Notably, the model's MAE in high-flow intervals showed moderate increases relative to medium-flow intervals across basins (e.g., Wuding River: 7.469 vs. 2.817; Qingyi River: 22.588 vs. 9.926), indicating its capability to capture high-flow dynamics effectively.

To better highlight the simulation capabilities of the proposed model, four baseline models (XAJ, LSTM, TCN, and GRU) and one hybrid model (TCN-GRU) were applied to simulate daily streamflow in the four basins as control models for the study. LSTM is a neural network architecture widely applied in time series forecasting and streamflow simulation. By including LSTM in our comparisons, we can conduct a more comprehensive evaluation of our XAJ-TCN-GRU model's performance, ensuring that our research findings have greater universality and robustness. In addition, to further highlight the advantages of nonlinear ensemble, we conducted two additional experiments. One group used the Linear Regression (LR) method to linearly combine XAJ and TCN-GRU models, resulting in a new model called XAJ-TCN-GRU&LR. The other group used the simulations based on XAJ as inputs for the TCN-GRU model, leading to the construction of the XAJ-Infused TCN-GRU model.

Table 5 provides a detailed overview of the streamflow simulation performance of the eight models in the four basins. Clearly, the XAJ-TCN-GRU model exhibits superior performance across all five metrics. The XAJ-TCN-GRU model exhibits relatively low RMSE and MAE values in each basin, particularly when compared to the traditional conceptual rainfall–runoff model XAJ, resulting in a significant reduction in simulation errors. For instance, in the Wuding River basin, the RMSE and MAE of the XAJ model are 69.212 and 64.320 m³ s⁻¹, respectively, while the XAJ-TCN-GRU model achieves only 7.032 and 3.876 m³ s⁻¹ for these metrics. Additionally, when compared with individual TCN and GRU models, the integrated TCN-GRU model exhibits higher NSE and KGE values across the four basins. This indicates the TCN-GRU has effectively combined complementary advantages of TCN and GRU, thus improving the model's simulation capability. The NSE values obtained using the TCN-GRU model in the Wuding River, Chu River, Jianxi River, and Qingyi River basins are 0.990, 0.947, 0.965, and 0.976, respectively. Upon nonlinear ensemble with the simulations of the XAJ model, the simulation accuracy significantly improves. The NSE values achieved using the XAJ-TCN-GRU model for streamflow simulation in the four basins are 0.991, 0.971, 0.984, and 0.986, respectively, representing enhancements of 0.10 %, 2.53 %, 1.97 %, and 1.02 % relative to the TCN-GRU model. Additionally, compared to the linearly integrated XAJ-TCN-GRU&LR model, the improvements are 1.33 %, 1.80 %, 1.66 %, and 0.62 %, respectively. The nonlinear ensemble approach is capable of better capturing intricate nonlinear dependencies within the data, thereby mitigating the risk of error propagation (Wang et al., 2024; Xu et al., 2025). The RF algorithm used for ensemble is less likely to amplify errors compared to linear combinations, as it captures the complex relationships between different model simulations through its decision tree-based structure. Each decision tree in RF is capable of capturing different patterns in the data, and the ensemble process averages out the errors and uncertainties, thereby reducing the impact of potential errors from individual models. In addition, the standard deviations of key validation metrics across the five runs were further quantified to reflect model stability: for NSE, the standard deviation ranged from 0.002 (Qingyi River Basin) to 0.005 (Chu River Basin); for Root Mean Square Error (RMSE), it varied between 0.32 (Wuding River Basin) and 0.87 m³ s⁻¹ (Jianxi River Basin); and for Mean Absolute Error (MAE), the standard deviation was between 0.18 (Wuding River Basin) and 0.54 m³ s⁻¹ (Jianxi River Basin). These small standard deviations indicate minimal fluctuations in model performance across independent optimization runs, confirming the robustness of the hyperparameter tuning process.

Further analysis of model performance differences across different basins reveals that their performance is closely related to basin hydrological characteristics: In the Wuding River basin (arid region), the XAJ model component contributes relatively more, as hydrological processes in arid regions (e.g., evaporation, surface runoff) are more significantly linearly influenced by meteorological factors (e.g., surface pressure), with stronger constraints from physical mechanisms; while in basins such as the Chu River (humid region) and Jianxi River (hilly region), hydrological processes are dominated by the nonlinear relationship between precipitation and runoff, resulting in a greater contribution from the TCN-GRU component, reflecting the ability of deep learning to capture complex nonlinear patterns. We also found that the simulation results of the XAJ-Infused TCN-GRU model are not satisfactory, with the KGE values for the Wuding River, Chu River, and Qingyi River basins even falling below those of TCN-GRU model. This result indicates that directly inputting the outputs from the physical-based model into the deep learning model may propagate the errors from the former to the latter, leading to a decrease in simulation accuracy. The XAJ-TCN-GRU model effectively leverages the strengths of both conceptual rainfall–runoff and deep learning models and compensates for their potential shortcomings in streamflow simulation through nonlinear ensemble.

Further analysis of the core drivers behind the performance differences in Table 5 reveals that XAJ's contribution lies not merely in enhancing model diversity, but in achieving rational corrections to the simulation process through its explicit physical parameter constraints. This conclusion is validated through controlled variable comparisons and mechanism decomposition: From a diversity perspective, the TCN-GRU model has already integrated the complementary strengths of TCN and GRU (NSE ranging from 0.947 to 0.990), indicating its diversity potential has been fully exploited. If performance gains were solely attributable to the addition of new model components, XAJ-TCN-GRU&LR (which similarly possesses the diversity combination of “XAJ+TCN-GRU”) should exhibit comparable performance. However, Table 5 data indicates that XAJ-TCN-GRU's RMSE (7.032–20.862 m³ s⁻¹) is reduced by 38.7 %–158.7 % compared to XAJ-TCN-GRU&LR, LR (12.613–29.370 m³ s⁻¹) by 38.7 %–158.7 %, while NSE improved by 0.62 %–1.80 %. This indicates that linear fusion utilises only the diversity of model outputs without leveraging XAJ's physical constraint value. In contrast, the XAJ-Infused TCN-GRU model treats XAJ outputs as inputs (transmitting data information without activating physical constraints), resulting in KGE (0.938–0.964) being lower than TCN-GRU (0.942–0.985). further demonstrating that XAJ's core function lies not in “adding a model member” but in its physical mechanism regulating simulation logic. This capability for correction based on physical mechanisms is absent in deep learning models that rely solely on data pattern learning. It is precisely this capability that constitutes the core reason for the XAJ-TCN-GRU model's significantly superior performance compared to other models.

Scatter plots provide an intuitive display of the model's explanatory power over data variations, as depicted in Fig. 5. Upon analysis, the XAJ model exhibits a tendency to overestimate streamflow during low-flow periods and underestimate it during high-flow periods. In contrast, deep learning models demonstrate superior streamflow simulation capabilities. Notably, the fitting line of the XAJ-TCN-GRU model is closest to the 1:1 line. The R2 values of this model in the four basins are 0.988, 0.981, 0.984, and 0.986, respectively, representing improvements of 3.56 %, 34.20 %, 20.59 %, and 20.10 % compared with the standalone XAJ model. In comparison with the TCN-GRU model, the improvements are 0.92 %, 2.51 %, 1.86 %, and 0.92 %, respectively. In the Chu River basin, the R2 value of the XAJ model is significantly low, indicating that traditional conceptual rainfall-runoff models are not applicable to this basin. Across the four basins, the simulation results of TCN and GRU are comparable, with R2 values both exceeding 0.94, highlighting the important role of these two models in each basin. The integrated TCN-GRU model effectively combines the advantages of both models, demonstrating good adaptability across all basins and the ability to handle complex hydrological conditions. These findings underscore the broad applicability and superiority of XAJ-TCN-GRU model in diverse basins.

The Taylor diagram is widely employed to evaluate the correlation and simulation accuracy between model predictions and observed data. Figure 6 illustrates that different symbols denote various simulation methods, with standard deviation on the horizontal axis and vertical axis, and the dotted line representing RMSE. The optimal simulation accuracy is represented by the center of the horizontal axis in the Taylor diagram. Notably, in Fig. 6, the XAJ-TCN-GRU model's predictions for the Wuding River basin are positioned at the optimal location in the Taylor diagram, and the correlation coefficient is 0.99. Likewise, for the Chu River, Jianxi River, and Qingyi River basins, the XAJ-TCN-GRU model's results outperform those of the comparative models and are closest to the observed values. By contrast, the XAJ model's simulated streamflow shows significant deviations from the observed data. This further highlights the proposed model's advantage in streamflow simulation.

The proposed XAJ-TCN-GRU model demonstrates promising performance in streamflow simulation across the four representative Chinese basins with distinct hydrological and geographical characteristics (arid, humid, hilly, and plateau regions). Its performance advantages in these selected basins are mainly reflected in the effective integration of physical mechanism interpretability from the XAJ model and complex pattern capture capability from the TCN-GRU model, as well as the mitigation of error propagation through nonlinear ensemble. However, it is important to acknowledge that the relative standing of the proposed model compared with large-sample hydrological models developed based on extensive basin datasets (e.g., CAMELS) still needs further evaluation under standardized and comparable experimental frameworks.

To evaluate the robustness and resilience of the model to changes in the input data, we performed noise data injection testing and out-of-context validation.

In actual hydrological observations, due to equipment malfunctions, recording errors, and other reasons, observational data often contain erroneous data. In noise injection testing, to align with actual hydrological observation scenarios, the characteristics of the injected noise are first defined: Drawing upon common error sources in field hydrological monitoring (such as sensor drift, manual recording discrepancies, and data transmission losses), and considering the error background of Chinese hydrological observation data mentioned in the study, an initial noise proportion of 2 % is set as a conservative baseline (compliant with the lower limit requirement of 1 %–2 % equipment error specified in the Chinese Hydrological Observation Specification GB/T 50095-2014). To avoid the unrealistic assumption of independent noise, the injected noise follows a normal distribution (mean = 0, standard deviation = 5 %–10 % of the corresponding variable's observed value). This is combined with the correlation characteristics of variables across the four basins (e.g., Pearson's correlation coefficient between precipitation and relative humidity in the Wuding River basin is 0.72, and the correlation coefficient between average temperature and evaporation in the Qingyi River basin is 0.63), ensuring consistency with the patterns observed in the actual data (Wang et al., 2023; Zhao et al., 2024). Subsequently, we used this perturbed dataset to train the models and make streamflow simulations to test the models' responsiveness to noise data situations. Noise data injection helps evaluate how the models perform when facing unknown or noise data situations in the real world and aids in improving the models' performance and generalization capabilities. Additionally, we conducted 10 independent runs for each model and recorded the results of each run, including the model's outputs and performance metrics. By running the model multiple times, we could observe the variations and fluctuations in the models' performance across different runs, thereby enabling a more thorough assessment of its performance and improving the evaluation of its robustness and reliability.

The detailed outcomes of the noise data injection test are recorded in Sect. S4.3 in the Supplement. The boxplots in Fig. 7 provide a detailed representation of the normal distribution of indicators obtained from the 10 runs of each model during the noise data injection test. It is apparent that the deep learning models demonstrate good stability. Among them, the LSTM, GRU, and TCN-GRU models show relatively stable simulation performance across the four basins in the noise data injection test. However, it is noteworthy that the TCN model exhibits significant fluctuations in the tests conducted in the Chu and Jianxi basins. Overall, the XAJ-TCN-GRU model demonstrates the strongest stability and the best simulation performance in the noise data injection test. The XAJ-TCN-GRU model leverages the advantages of integrating conceptual rainfall - runoff and deep learning models, enabling it to capture the physical characteristics of streamflow data while also learning the deeper features of the data, thereby performing exceptionally well in handling noise data.

In the out-of-context validation, we excluded data from wet-year in the training sets of the four basins and used these data separately for model validation. A wet-year was defined as a hydrological year (June of the current year to May of the next year, consistent with China's hydrological year division) where the annual precipitation exceeded the 75th percentile of the long-term (2010–2023) annual precipitation series in the respective basin. This criterion aligns with the classification standard for wet years in the Technical Specifications for Hydrological Data Processing (General Administration of Quality Supervision, GB/T 50102-2014) in China. This approach aims to evaluate the model's performance under extreme hydrological conditions (i.e., wet-year) that it has not encountered before, providing a better understanding of its generalization ability and robustness. The out-of-context validation results for the four basins are shown in Table 6. By comparing Tables 5 and 6, we can see that the model's simulation accuracy declines slightly under wet-year conditions compared to regular years. However, it still maintains a high level of accuracy overall, indicating that the model has a certain degree of generalization ability and can make reasonable simulations under extreme flow conditions. In the Jianxi River and Qingyi River basins, the model's simulation accuracy decreases more significantly during wet-year, likely because these regions experience greater precipitation and flow fluctuations during wet-year. This variability may not be sufficiently captured in the training set, thereby affecting the ability of the model to learn and adapt under these extreme conditions.

To further refine the model robustness assessment framework, this study additionally designed the “drought year exclusion” approach and the “extreme event threshold segmentation” approach to comprehensively evaluate the model's stability under various exceptional conditions, such as extremely low flow and extremely high flow (Acuña Espinoza et al., 2025b).

“Dry-Year Exclusion” Approach: Years with annual precipitation < 25 % of the multi-year average were defined as “dry-year”. After excluding dry-year data from the training set, model training was conducted, followed by testing using the excluded dry-year data. Results are shown in Table S4. It indicates that the XAJ-TCN-GRU model achieved NSE values of 0.954–0.969 under extreme low-flow scenarios, with a performance decline of only 1.7 %–2.3 % compared to the full-data training scenario. This decline was lower than that observed for the TCN-GRU model (3.4 %–3.5 %) and the XAJ model (8.0 %–8.8 %), demonstrating its robustness under extreme low-flow conditions. Specifically, the Qingyi River basin, possessing a larger baseline runoff volume (mean daily streamflow 545.06 m³ s⁻¹), exhibited the smallest NSE reduction (1.7 %) after excluding dry years. Conversely, the Wuding River basin, characterised by arid conditions (mean daily streamflow 117.56 m³ s⁻¹), showed a relatively higher reduction (2.3 %), yet still maintained a low overall level.

“Extreme Event Threshold Segmentation” approach: Pearson Type III distributions were employed to calculate the 10-year return period flow thresholds for the four basins. Model training utilised only data below the thresholds, while data exceeding the thresholds tested the models' simulation capability for extreme high flows. Results are presented in Table S4. It indicates that the NSE of the XAJ-TCN-GRU model in extreme high-flow testing remained between 0.941 and 0.955, representing a 3.1 %–4.6 % reduction compared to the full-data training scenario. This decline was significantly lower than that observed for the TCN-GRU model (5.9 %–7.2 %) and the XAJ model (8.9 %–12.6 %). Notably, in the Qingyi River basin, where the 10-year return period threshold (2850.3 m³ s⁻¹) approaches its historical maximum discharge (3683.14 m³ s⁻¹), the model still achieves an NSE of 0.955 with a mere 3.1 % decline, demonstrating robust adaptability to high-intensity extreme flows. The Wuding River, characterised by arid basin conditions, exhibited a substantial disparity between extreme flow and mean flow (threshold 1120.5 m³ s⁻¹ being 9.5 times the mean 117.56 m³ s⁻¹), resulting in a relatively higher NSE reduction (4.6 %). Nevertheless, this reduction remained substantially lower than that of the comparison model.

Flood simulation is a core component of watershed management and disaster prevention planning. Especially during the flood season, accurate streamflow simulations are paramount for the prompt execution of flood control measures and the rational allocation of water resources. This section investigates streamflow simulations across four distinct basins during both flood and non-flood seasons, providing a comprehensive assessment of their capabilities.

Based on the climatic and geographical conditions, the flood seasons for the Wuding River and Chu River primarily occur from June to September, while for the Jianxi and Qingyi Rivers, they commence slightly earlier, from May to September. These flood seasons represent periods characterized by the most significant variations in streamflow, making them particularly challenging to simulate. According to the data presented in Table 7, the XAJ-TCN-GRU model demonstrates exceptional performance in simulating streamflow during the flood seasons across all four basins. Its NSE values for the Wuding River, Chu River, Jianxi River, and Qingyi River are 0.988, 0.968, 0.984, and 0.980, respectively, all of which are close to or exceed 0.98, indicating the model's high accuracy in flood season simulations. Furthermore, during non-flood seasons, the model maintains high NSE values of 0.987, 0.970, 0.980, and 0.987, respectively, further confirming its stability and reliability. It is important to note that although the accuracy of the XAJ-TCN-GRU in simulating streamflow during flood seasons for the Chu River and Qingyi River basins is slightly lower compared to non-flood seasons, this does not indicate poor performance. In fact, the complexities arising from factors such as climate change and topography make it challenging for any simulation model to achieve perfection during flood seasons. Nonetheless, compared with the other five comparative models, the XAJ-TCN-GRU model demonstrates the lowest RMSE, MAE, and MAPE in streamflow simulation during flood seasons across all four basins, highlighting its superiority in flood season simulation.

This study also placed a specific emphasis on the accurate simulation of extreme hydrological events. Flood events were identified using the Peak Over Threshold (POT) method, a widely accepted flood frequency analysis approach (Hosking and Wallis, 1997). For each basin, the threshold was determined as the 95th percentile of the daily streamflow series in the testing set, corresponding to a return period of approximately 2 years based on regional hydrological characteristics (as recommended by the China Hydrological Manual, 2017). Periods where daily streamflow exceeded this threshold and lasted for at least 2 consecutive days were classified as flood events. Through in-depth hydrological data analysis, a total of eight such flood events were identified across the four basins. Table 8 presents detailed simulation outcomes of these flood events using different models in the four basins. Clearly, the XAJ-TCN-GRU model exhibits the least simulation bias in all four basins when observing the data in Table 8, with remarkably lower RMSE and MAPE values than other models. In the four flood events of the Wuding River basin, the RMSE of XAJ-TCN-GRU was reduced by 30.60 %, 42.16 %, 66.47 %, and 43.01 % compared with TCN-GRU. This significant reduction not only highlights the importance of integrating the results of conceptual rainfall - runoff model to enhance the simulation accuracy of hybrid models but also further validates the high accuracy and dependability of the XAJ-TCN-GRU in simulating flood events. The XAJ-TCN-GRU model's in-depth understanding of hydrological processes provides an important foundation for flood peak simulation in areas with limited hydrological data, particularly in regions with sparse observation stations, where the model's portability is especially significant.

However, it is important to note that while this model provides insights into potential applications for flood peak simulation, our work focuses on streamflow simulation rather than direct flood simulation. Directly using streamflow simulation results for flood peak simulation has limitations. First, streamflow models often rely on coarse spatial and temporal scales, which may overlook critical local features and flood dynamics. Second, inaccuracies in model parameters and input data, particularly under extreme precipitation conditions, can amplify errors in flood peak estimation. These factors highlight the need for further development to improve the accuracy and reliability of flood peak simulations using streamflow models. The results suggest that, although the model's simulation can serve as an approximation of flood peak flows in areas with limited observation stations, further development is needed to improve the reliability of flood simulations. Future research may explore incorporating additional flood-related factors and advanced post-processing techniques to better translate streamflow simulations into accurate flood simulations.

Further quantitative analysis of the key characteristics of 8 extreme flood events shows that the XAJ-TCN-GRU model performs optimally across all core metrics (as detailed in Table S5):

Peak flow: The average peak error of the integrated model across the 8 events (6.1 %) was 62.3 % lower than that of XAJ (16.2 %) and 43 % lower than that of TCN-GRU (10.7 %). Among these, the peak simulation error for the September 2021 flood in the Qingyi River basin (peak 3,683.1 m³ s⁻¹) was only 4.3 %, while the TCN-GRU model, due to its excessive sensitivity to strong precipitation pulses, had an error of 12.6 %.

Traditional point simulations often prove insufficient to fully account for the inherent uncertainty in simulations, a challenge typically unavoidable in streamflow modeling. To obtain numerical estimates with their associated reliability, utilizing interval simulation methods represents a more reasonable approach. This approach not only provides an average simulation estimate but also captures the uncertainty level inherent in the simulation, thereby enabling more robust and scientifically grounded decision-making. In this section, interval simulation methods based on error modeling are investigated, building on the foundation of point simulations. The methodology adopted follows Song et al. (2015), namely, constructing simulation confidence interval at a specified significance level. The framework integrates the point simulation results from Sect. 4.2, using the logit distribution function to match the simulation error sequence and performing interval simulation at the designated significance level. Two evaluation metrics are used to assess interval simulation accuracy: PICP and PINAW. At a given significance level, a higher PICP value combined with a lower PINAW value indicates better model performance.

Three distinct significance levels were chosen for each of the four river basins, with corresponding PICP and PINAW results documented in Table 9. Interval simulation results of the XAJ-TCN-GRU model at these four sites are visualized in Figs. 8–10. Notably, at different levels of significance, the majority of actual observations are located within the shaded area of the simulated interval, indicating that the XAJ-TCN-GRU model exhibits good coverage probabilities across different significance levels. Furthermore, an apparent trend shows that as the significance level increases, the average width of the simulation intervals tends to decrease.

At a 0.05 significance level, the XAJ-TCN-GRU model realizes PICP rates of 94.29 %, 95.09 %, 95.29 %, and 95.09 % across the four river basins, respectively. These values show improvements over TCN-GRU model, highlighting the efficacy of integrating the conceptual rainfall-runoff model with the nonlinear ensemble. In terms of PINAW, the XAJ-TCN-GRU model yields values of 0.022, 0.044, 0.034, and 0.034, compared with the traditional conceptual rainfall-runoff model XAJ, which has PINAW values of 0.094, 0.244, 0.186, and 0.228. These results demonstrate a substantial reduction in the average width of simulation intervals for the XAJ-TCN-GRU model. The data in Table 9 further confirms that the XAJ-TCN-GRU model excels in generating more compact and narrower simulation windows, thereby reduces decision-making risk caused by inaccuracies and uncertainty. This effectively addresses uncertainties and better aligns with practical requirements.

Although the XAJ-TCN-GRU model demonstrates excellent performance across all basins, showcasing strong stability and robustness, the lack of transparency in deep learning models hinders the verification of their simulation logic. To improve the interpretability of the XAJ-TCN-GRU, this study employs three key indicators: mean absolute SHAP values (SHAPABS), Feature Importance (FI), and Permutation Feature Importance (PFI). These indicators can accurately calculate the specific contribute of each variable input to long-term trends on streamflow in the TCN-GRU model, thereby offering the clearer insight of the internal operation mechanism of TCN-GRU model.

Figure 11 presents density scatter plots of various input variables across the four basins, with the y-axis arranged in descending order according to their contributions to the simulated values per sample. Positive SHAP values indicate contributions that enhance the model's performance, with larger values reflecting more substantial contributions. In cases where multiple samples exhibit identical contribution values, these samples are vertically stacked within the plots. Figure 12 provides a comprehensive overview of the importance ranking of hydro-meteorological variables influencing streamflow, as determined by the TCN-GRU. The numerical values of three key indicators for the input variables of the TCN-GRU across the four basins, along with their normalization, are detailed in Sect. S4 in the Supplement.

Combining Figs. 11 and 12, it can be seen that dew point temperature is the most influential feature variable in the basins of Wuding River, Chu River, and Qingyi River. The phenomenon of dew point temperature as a key variable can be explained from the hydrological cycle mechanism: the difference between dew point temperature and ambient temperature directly reflects air moisture saturation; the smaller the difference, the closer the air is to saturation, and the more likely condensation and precipitation occur. Additionally, high humidity reduces evaporation rates, minimizing water loss in the basin. This influence varies across different climate zones: in humid regions (such as the Chu River), where water vapor is abundant, even minor changes in dew point temperature can significantly affect precipitation intensity; while in arid regions (such as the Wuding River), the synergistic effects of dew point temperature and surface pressure, among other factors, indirectly regulate limited precipitation processes by influencing convective activity. This, together with the basin's surface conditions (such as vegetation cover and soil permeability), ultimately determines the water balance outcome.

Furthermore, the second influential variables in each basin exhibit significant differences, reflecting the importance of varying hydrological characteristics and climatic conditions for flow simulation. In the Wuding River basin, surface pressure is identified as the second most important variable, which is closely related to the region's arid climate characteristics. In arid environments, surface pressure plays a critical role in the hydrological cycle, influencing the formation of precipitation and the evaporation of water. In contrast, in the Chu River basin, the daily maximum temperature is the second most important variable, indicating that temperature fluctuations in a humid environment may directly affect the evaporation and flow of water bodies. Meanwhile, in the Qingyi River basin, the influence of daily minimum temperature is noted, reflecting how variations in daily minimum temperatures may directly affect the cooling and condensation processes of water bodies, thereby impacting flow. We observed that evaporation also has a significant impact on streamflow in all four basins, with lower evaporation values having a greater impact on streamflow simulation. When evaporation is low, it implies that more water remains in liquid form on the surface or in water bodies, potentially leading to more streamflow.

Figure 11 shows that the importance rankings of different feature variables vary significantly across the three interpretation methods in the four basins. Dew point temperature generally ranks high in both SHAPABS and PFI across all basins, indicating its high importance in streamflow simulation. However, in FI, the importance of dew point temperature does not always rank at the top. For example, in the Chu River and Jianxi River basins, the FI method places more emphasis on other variables, such as surface pressure and evapotranspiration. This difference suggests that FI may focus more on the impact of local variables on model decisions, while PFI and SHAPABS take into account the overall contribution of variables to simulation performance.

In comparison, SHAPABS and PFI show relatively consistent rankings across most basins, with variables such as dew point temperature and ground temperature maintaining stable positions. This indicates that SHAPABS and PFI are more robust in identifying variable importance, while FI is more influenced by basin characteristics, leading to greater variation in variable rankings across basins.

These metrics offer meaningful insights by elucidating the individual contributions of each input variable to model performance. These techniques aid in identifying the most critical variables during the analysis, thereby deepening understanding of the factors driving model simulations and providing guidance for future research improvements.

Additionally, interpretability analysis for extreme events revealed significant stage differences:

Model component contributions: During the baseflow stage (flow < 0 % of peak), TCN-GRU contributed 65 %–70 %, while during the peak stage (flow > 70 % of peak), XAJ's contribution increased to 42 %–58 %. In particular, in torrential rain floods (such as Jianxi in August 2022), the XAJ streamflow generation module contributed 58 % to the physical description of the torrential rain–runoff conversion.

These results indicate that the advantage of the hybrid model in extreme events lies not only in improved overall accuracy but also in its ability to dynamically adapt to the dominant mechanisms of the different flood stages, providing more precise scientific basis for flood control.

A further analysis of the core role of the XAJ model in the ensemble reveals that its value lies not merely in improving NSE, but in providing three key supports through physical mechanisms that deep learning models cannot replace:

First, constraint-based corrections of physical processes. Although deep learning models such as TCN-GRU can capture the relationship between meteorological variables such as dew point temperature and streamflow, they cannot distinguish between the physical path of “high humidity–precipitation–streamflow” and the false correlation of data noise. For example, during a high dew point temperature event in the Wuding River basin (arid area) as showed in Table S4, TCN-GRU overestimated streamflow (128.6 m³ s⁻¹ vs. observed 104.3 m³ s⁻¹) due to coincidental high cloud cover data (LCC = 0.8) during the same period, while the XAJ model identified through its physical calculation of tension water capacity curve parameter B (= 0.3) and free water capacity SM (= 15 mm) to identify that the humidity had not been converted into effective precipitation (actual precipitation was only 2.1 mm). After correction, the simulated value was 107.5 m³ s⁻¹, and the hybrid model's final output was further optimized to 105.2 m³ s⁻¹, demonstrating the screening effect of physical mechanisms on data correlations.

Second, process assurance in extreme scenarios. During the flood confluence stage, the Nash unit line parameters (k= 6–10 h, n= 2–3) of XAJ are set based on the topographical characteristics of the watershed to ensure the physical rationality of the streamflow propagation speed. For example, in the September 2021 flood in the Qingyi River watershed, the TCN-GRU model, due to the limited number of large flood samples in the training set (only 5 instances), simulated a confluence time 4 h shorter than the actual value, which showed in Table S5. In contrast, the XAJ model, based on k= 8 h calculations, accurately matched the confluence process, and after integration, the confluence time error was reduced to 1 h, with the peak simulation error decreasing from 21.3 % to 7.8 %. During the dry season, the groundwater drawdown coefficient CG (= 0.95) was used to stabilize the baseflow simulation, resulting in the hybrid model achieving an RMSE of 8.179 m³ s⁻¹ in the Jianxi River basin during the dry season, a 49.0 % reduction compared to TCN-GRU (15.947 m³ s⁻¹).

Third, the noise data interference resistance benchmark. XAJ model parameters (such as evaporation coefficient KC = 0.85 and deep evaporation coefficient C= 0.1) have clear physical significance and are significantly less sensitive to observational errors than the weight parameters of deep learning models. In a 2 % noise data test of Table S3, the increase in simulation error for the XAJ model (12.6 %) was only 44.5 % of that for TCN-GRU (28.3 %), providing a stable physical benchmark for RF integration. This resulted in a noise-induced NSE decline (0.009) for the hybrid model that was far lower than that for TCN-GRU (0.021).

These analyses indicate that the integration of the XAJ model is not “unnecessary complexity”, but rather addresses the shortcomings of deep learning models in process constraints, extreme adaptability, and interference resistance through physical mechanisms, serving as the core guarantee for hybrid models to achieve high accuracy, high reliability, and interpretability.

Additionally, a further comparison of the core differences between the single GRU model and the hybrid model reveals that the value of this study lies not only in the improvement of NSE from 0.942–0.987 to 0.971–0.991 (Table 5), but also in three breakthroughs. First, a significant enhancement in the simulation capability of extreme events: in eight flood events, the RMSE of the hybrid model was on average 52.3 % lower than that of the GRU (e.g., in the August 2022 flood in the Jianxi River basin, the GRU's RMSE was 92.535 m³ s⁻¹, while the hybrid model reduced it to 50.825 m³ s⁻¹), which is critical for flood disaster prevention and control; Second, enhanced robustness: in noise data tests, the NSE fluctuation range (±0.012) of the hybrid model was significantly smaller than that of GRU (±0.022), indicating greater reliability when actual observational data contains errors; Third, integration of physical interpretability: by constraining the physical framework of the XAJ model, the hybrid model avoided GRU's overestimation of low flows during drought periods (e.g., during the dry season of the Wuding River, the GRU simulation values were on average 18.7 % higher, while the hybrid model deviation was reduced to 3.2 %). Combined with methods such as SHAP, it reveals the mechanism of action of key variables, which provides a scientific basis for understanding the hydrological processes in the basin, far surpassing the “black box” simulation of a single model. These advantages demonstrate that the complexity of the hybrid model is a necessary means to address the multidimensional challenges of hydrological simulation, possessing clear scientific value and practical significance.

To further contextualize the performance of the proposed XAJ-TCN-GRU model, we provide a qualitative discussion regarding several representative studies in the field. Kratzert et al. (2021) demonstrated the capability of LSTM networks to synergistically utilize multiple meteorological datasets through multi-basin training across over 500 basins in the CAMELS dataset, reporting a median NSE of 0.82. Li et al. (2025) explored the ensemble of a differentiable physical model (δHBV) with LSTM under diverse meteorological forcings, focusing on enhancing spatial generalization (e.g., in the PUB and PUR tests) within a large-sample framework. Solanki et al. (2025) employed machine learning methods (RF and XGB) to post-process ensembles of multiple physical hydrological models at the basin scale. It is important to note that a direct numerical comparison of NSE values between these studies and ours is not scientifically robust, as the experimental settings are fundamentally different. The aforementioned studies primarily adopted large-sample, multi-basin training and evaluation protocols, with reported metrics (often median values) reflecting performance across hundreds of heterogeneous basins (Kratzert et al., 2024). In contrast, the present study focuses on a hybrid modeling strategy that deeply integrates a physical hydrological model (XAJ) with a deep learning model (TCN-GRU) via nonlinear ensemble, and it is evaluated on four individual basins with separate training, a setting more relevant to scenarios where building a large, unified training set is challenging. Therefore, the performance of the XAJ-TCN-GRU model (achieving high NSE values in these four basins) should be interpreted within its specific context. It demonstrates the effectiveness of the proposed parallel integration and nonlinear fusion framework for daily streamflow simulation in individual basins, particularly in leveraging physical mechanisms to complement data-driven patterns. However, the methodological approaches and insights from these previous large-scale studies provide valuable perspectives for future work. In other words, whilst this study has validated the performance of the XAJ-TCN-GRU model across four Chinese river basins with distinct hydrological characteristics, the model's generalisability to a wider range of river basins and cross-regional scenarios still requires further validation through large-scale experiments in subsequent research.