https://orcid.org/0009-0002-1757-4222
ABSTRACT
Lumped hydrological models are widely used for hydrological simulations in basins due to their simple data requirement, ease of operation and reasonable performance. However, most lumped models are limited by their structure, making them unable to account for the impact of basin heterogeneity on runoff and soil moisture dynamics. To overcome these limitations, we proposed an extension approach that incorporates the Pareto distribution of soil water storage capacity to enhance lumped hydrological models, providing a statistical spatial description of runoff and soil moisture variability. This approach was applied to a structurally simple lumped time-variant gain model and tested over four basins in China to reproduce daily runoff and soil moisture dynamics. Additionally, numerical experiments were conducted to validate the effectiveness of this extension in terms of runoff mechanisms. The findings reveal that, compared with the original time-variant gain model, the enhanced model better captures hydrological dynamics, with a 7.9% improvement in total runoff accuracy, 11.2% in low-flow metrics and 30.8% in soil moisture simulation. This demonstrated that it is possible to enhance the ability of lumped models to respond to the basin spatial heterogeneity without significantly increasing the complexity, providing valuable insights for the further development of such models.
HIGHLIGHTS
Integrating the soil storage capacity curve into a lumped model enhances the model's hydrological simulation performance.
The extension method provides the lumped model with an implicit spatial representation.
Numerical experiments demonstrate that this method allows the model to possess a flexible runoff generation mechanism.
INTRODUCTION
Lumped conceptual hydrological models, favoured by hydrologists for their simple structures, computational efficiency and ease of data acquisition, are widely used for hydrological simulation studies in small- to medium-sized basins. However, many studies indicate that the spatial variability of basin underlying surfaces significantly affects the accuracy of hydrological model simulations, including runoff process (Jin et al. 2015; Athira & Sudheer 2021) and soil moisture dynamics (Jin et al. 2015; Yetbarek et al. 2020), while most lumped conceptual hydrological models, limited by their structure and input data, fail to account for the heterogeneity of basin geomorphic characteristics (Sivelle et al. 2022; Taheri et al. 2023). These limitations hinder the general applicability of lumped models in spatially heterogeneous basins and leave their theoretical structure incomplete (Chen et al. 2006; Bartlett et al. 2016a). Therefore, it is necessary to develop a simple and effective extension for lumped models to consider spatial heterogeneity. Such an approach should not significantly increase model complexity but should improve accuracy while refining the theory of the model with respect to heterogeneous representations.
Describing the spatial variability of runoff remains a significant challenge in hydrological modelling (Rigby & Porporato 2006). A common approach is to develop physically based distributed hydrological models, such as SWAT (Shah et al. 2021), GBHM (Yang et al. 2001) and DTVGM (Xia et al. 2005). These models divide the basin into various computational units, each with different parameters and model inputs to reflect spatial heterogeneity (Wang et al. 2009; Tan et al. 2020; Loritz et al. 2021). However, this approach significantly increases the complexity of the model, requiring detailed spatial input data and substantial computational resources. In small- to medium-sized basins where distributed data are scarcer, this modelling approach has not shown a clear advantage over simpler lumped models (De Niel et al. 2020; Vema & Sudheer 2020).
Another approach is to introduce soil water storage capacity probability distribution functions into lumped hydrological models, allowing them to account for the effects of basin spatial heterogeneity in a statistical sense. The Xinanjiang (hereafter, XAJ) model, proposed by Zhao et al. (1980), is one of the most representative models of this kind. It calculates the spatial heterogeneity of runoff generation through a Pareto distribution of the soil tension water curve and the free water reservoir distribution curve. Similarly, Moore (1985) developed the probability distributed model (PDM) based on the assumption of water exchange between spatial storage units, using different soil water storage capacity probability density functions (PDFs) to represent soil moisture spatial heterogeneity (Moore 2007). Liang et al. (1994) developed the semi-distributed VIC model using a saturation-excess principle with a Pareto distribution of water storage capacity as used in XAJ. Similar models using probability distribution functions also include the Arno (Todini 1996) and the HBV model (Bergström & Forsman 1973). The lumped models mentioned above bind the soil storage capacity curve (SSCC) with the saturation-excess runoff mechanism. This assumption makes the SSCC model more suitable for application in humid and easily saturated basins. However, in our view, this binding is not necessary. Moreover, since the SSCC is inherently part of the original model structure, it is not possible to isolate the effect of this statistical distribution curve on enhancing the spatial heterogeneity representation of lumped models. The time-variant gain model (TVGM) provides one example of a hydrological model that does not rely on the saturation-excess runoff mechanism (Xia 2002), instead employing an exponential runoff mechanism related to the degree of saturation (Cheng et al. 2020). However, effective methods have yet to be developed to enhance the TVGM's ability to capture the spatial heterogeneity of the basin while maintaining its simplicity.
While previous workers have explored the representation of spatial heterogeneity within lumped models, their focus has predominantly been on the development of hydrological models incorporating saturation-excess mechanisms, such as the XAJ model studied by Huang et al. (2016) and the SCS-CNx framework selected by Bartlett et al. (2016a, b). This study aims to investigate a simple and general approach to improve the lumped model's ability to capture the spatial dynamics of runoff and soil moisture without significantly increasing its complexity. Two conditions must be met: first, the lumped model applied should not rely on a specific runoff mechanism assumption; and second, the original model should have a simple structure with minimal interaction between modules, allowing for independent testing of the improvement's impact on model performance (Fenicia et al. 2011; Huang et al. 2016; Craig et al. 2020). The TVGM was chosen as an example for method development because its runoff calculation is solely dependent on net rainfall and relative soil saturation, making it sensitive to variations in basin soil moisture, and not reliant on a saturation-excess runoff mechanism. Furthermore, it has few parameters and a simple structure, which facilitates model development and comparison. In this study, we extend the SSCC with a Pareto distribution into the runoff structure, creating the TVGM-SSCC. The model was calibrated and validated using six consecutive years of daily streamflow data from four basins in China, with model performance evaluated based on streamflow processes and soil moisture dynamics. Subsequently, the stability of the model results was investigated through multiple simulations. Finally, scenario-based simulations were conducted to set up numerical experiments, elucidating how the extended model responds to basin heterogeneity in terms of runoff generation mechanisms.
STUDY AREA AND DATA
Climatic and geographic characteristics of the four study basins
| No. | Basin | Area (km2) | P (mm/year) | PET (mm/year) |
|---|---|---|---|---|
| 1 | ZH | 1,510 | 598.5 | 1,085.3 |
| 2 | WG | 726 | 708.2 | 1,264.6 |
| 3 | GY | 1,831 | 1,037.8 | 989.8 |
| 4 | MDW | 1,599 | 812.1 | 1,055.6 |
| No. | Basin | Area (km2) | P (mm/year) | PET (mm/year) |
|---|---|---|---|---|
| 1 | ZH | 1,510 | 598.5 | 1,085.3 |
| 2 | WG | 726 | 708.2 | 1,264.6 |
| 3 | GY | 1,831 | 1,037.8 | 989.8 |
| 4 | MDW | 1,599 | 812.1 | 1,055.6 |
Note: The multi-year average annual values during the observation period are given. P, precipitation; PET, potential evapotranspiration. Calibration period: 2015–2018. Validation period: 2019–2020.
The spatial distribution of the study basins and the observation stations within these basins. According to the aridity index (AI) provided by Zomer et al. (2022), China is classified into different dry and wet regions as follows: AI < 0.2 for arid regions, 0.2 < AI < 0.5 for semi-arid regions, 0.5 < AI < 0.65 for semi-humid regions and AI > 0.65 for humid regions.
The spatial distribution of the study basins and the observation stations within these basins. According to the aridity index (AI) provided by Zomer et al. (2022), China is classified into different dry and wet regions as follows: AI < 0.2 for arid regions, 0.2 < AI < 0.5 for semi-arid regions, 0.5 < AI < 0.65 for semi-humid regions and AI > 0.65 for humid regions.
In this study, we collected precipitation, potential evapotranspiration, outlet streamflow and soil moisture for the four basins. The data span from 2015 to 2020, with a daily resolution. The first four years were used for calibration (with the first year as the warm-up period), and the last two years were used for validation. Precipitation data (P) were sourced from the National Meteorological Information Centre (http://data.cma.cn, last accessed on 24 May 2024) using data from various rain gauge stations (Figure 1). The basin-scale precipitation data were calculated using the Thiessen polygon method. Potential evapotranspiration data (PET) were obtained from the Global Land Evaporation Amsterdam Model (GLEAM) product (Miralles et al. 2011; Martens et al. 2017), with a spatial resolution of 0.25° (https://www.gleam.eu/#datasets, last accessed on 24 May 2024). The GLEAM provides data on various components of terrestrial evaporation (or ‘evapotranspiration’), including potential evaporation, plant transpiration and other variables. The GLEAM model uses the Penman equation to calculate potential evapotranspiration and is corrected based on global eddy-covariance and sapflow data. GLEAM data are widely used in regional hydrological simulation studies due to their broad spatiotemporal coverage and high accuracy. The basin-scale PET data were obtained by averaging the remote sensing values from multiple grids within the basin. Streamflow data (Q) were collected from the observation series at the outlet hydrological station. The observation data were complete without missing values, and there was no seasonal flow interruption across all basins. These data were sourced from the China Hydrological Yearbook. Soil moisture data (W) were obtained from the SMCI1.0 (Li et al. 2022) product (https://data.tpdc.ac.cn/en/data/, last accessed on 24 May 2024), with a spatial resolution of 0.1°. This dataset is based on in-situ observations of soil moisture at 10 depths (0–100 cm) from 1,648 stations provided by the China Meteorological Administration. It uses ERA5-Land meteorological forcing data, MODIS Leaf Area Index (LAI), USGS (United States Geological Survey) land cover type (Land types), the USGS DEM and soil properties from the CSDL (China Soil Dataset for Land surface modelling) as covariates. The dataset was derived using random forest machine learning techniques. Research indicates that within the Chinese region, the SMCI1.0 dataset outperforms other gridded soil moisture products (such as ERA5-Land and SMAP-L4) and is suitable for basin-scale hydrological modelling. In this study, the average soil moisture across these 10 layers was calculated for each grid, and multiple grid cells within the basin were averaged to obtain the basin-scale soil moisture data, matching the spatial input requirements of the lumped model.
METHODOLOGY
Soil storage capacity curve
The soil water storage capacity curve employs statistical distribution forms, such as the Pareto distribution (Moore 2007), power-law distribution (Bergström & Forsman 1973) or a distribution function similar to the SCS-CN function (Moore 1987; Bartlett et al. 2016a, b) to represent the spatial distribution of soil water storage capacity within the basin. Different distribution forms construct different lumped models. Several specific equations and corresponding models are detailed in Table 2.
The corresponding functions of SSCC in different saturation-excess models
| Model | VSA Function | Parameters | References |
|---|---|---|---|
| HBV |  |  | Bergström & Forsman (1973) |
| XAJ |  |  | Zhao et al. (1980) |
| VIC |  |  | Liang et al. (1994) |
| Arno |  |  | Todini (1996) |
| PDM |  |  | Moore (1985, 2007) |
| Model | VSA Function | Parameters | References |
|---|---|---|---|
| HBV | | | Bergström & Forsman (1973) |
| XAJ | | | Zhao et al. (1980) |
| VIC | | | Liang et al. (1994) |
| Arno | | | Todini (1996) |
| PDM | | | Moore (1985, 2007) |
Note: 
is the variable saturated (contributing) area fraction of the basin. 
is the shape parameter of the SSCC (determined through calibration), 
and 
denote the maximum and minimum point water storage capacity within the basin, and 
denotes the critical point storage capacity below which smaller stores are saturated.
and the shape parameter 
. Figure 2 represents the proportion of the variable saturated area, VSA, contributing to runoff under different β values when 
equals 0.5. Here, 
denotes the critical point soil water storage capacity below which smaller stores are saturated. It can be seen that when β ranges from 0 to 0.5, the curve is convex relative to the X-axis; when β equals 0.5, the distribution curve is a straight line; and when β is greater than 0.5, the curve is concave relative to the X-axis. This demonstrates the flexibility of SSCC in representing a wide variety of soil moisture distributions with distinctive characteristics. 
The soil water storage capacity curves following a Pareto distribution for different β values are presented in the graph. The dimensionless Y-axis represents 
, the ratio of the critical soil water storage capacity below which saturation occurs to the maximum soil moisture capacity at a point in the basin. The value of 
ranges from 0 to 
. The black intersection point in the graph indicates the situation where the critical soil moisture 
equals to 0.5
, at which the proportion of variable saturated area in the basin, VSA, is marked for different 
values.
The soil water storage capacity curves following a Pareto distribution for different β values are presented in the graph. The dimensionless Y-axis represents , the ratio of the critical soil water storage capacity below which saturation occurs to the maximum soil moisture capacity at a point in the basin. The value of ranges from 0 to . The black intersection point in the graph indicates the situation where the critical soil moisture equals to 0.5, at which the proportion of variable saturated area in the basin, VSA, is marked for different values.
Hydrological model and integration method
The conceptual structure diagram of the TVGM and TVGM-SSCC model. Please refer to Equations (1)–(5) for the specific definition of the symbols.
) and potential evapotranspiration (
). In the actual evapotranspiration module, the actual evapotranspiration (Et) is expressed as a nonlinear function of soil moisture and potential evapotranspiration:
represents the soil moisture storage of the basin at the initial stage at time t, 
denotes the maximum soil moisture storage, both in millimetre (mm). 
(−) is the parameter that characterizes the nonlinear relationship between relative soil moisture, 
(mm) and 
(mm). The surface runoff 
and baseflow 
at time t are calculated as follows:
is the surface runoff coefficient, 
is the nonlinear exponent parameter related to soil moisture, 
is the nonlinear exponent parameter related to rainfall intensity, 
represents the net precipitation at time t, defined as the precipitation minus the actual evaporation.
is the core runoff mechanism of the TVGM, which assumes that surface runoff is related to the degree of saturation as a power-law function, without being constrained by a saturation-excess threshold. The calculation formula for the baseflow 
is given by 
. This formula represents an exponential storage-outflow model, initially proposed by Beven & Kirkby (1979) as part of the TOPMODEL concept, and is based on assumptions of quasi-steady state saturated throughflow in a soil in which hydraulic conductivity decreases exponentially with depth (Moore 1997). Subsequently, Peters and Aulenbach validated this exponential storage–discharge relationship for baseflow simulation in experimental basins (Peters & Aulenbach 2011; Aulenbach & Peters 2018). Cheng et al. (2020) then incorporated it to replace the linear storage–discharge relationship in hydrological models. The baseflow equation in Equation (2) adopts the baseflow structure from Cheng et al.'s study on the TVGM. Here, m(−) and b (−) are the fitted constants for the groundwater storage–discharge relationship. 
, in mm, is the updated soil moisture storage at time t after subtracting 
. When 
, the initial soil moisture storage 
is set to 50. Other symbols remain the same as mentioned above. For surface water routing, we use the instantaneous unit hydrograph method (Nash 1960), while groundwater routing is computed using a simple linear reservoir lag algorithm:
is the instantaneous unit hydrograph. N (−) is the shape parameter of the hydrograph, while K (−) denotes the parameter of time to peak flow. 
(−) is the linear reservoir lag coefficient, and U is the conversion factor between R and Q, given by the expression 
. In this study, the hydrological model is run on a daily time scale (
) for hydrological simulations. However, it is important to note that the model can also be extended to finer time scales (e.g., hourly), depending on the selection of the runoff parameters. For instance, if the calculation interval (
) for the IUH is set to 1 h and the conversion factor (U) for the groundwater reservoir is calculated over the same 
, the model can be applied to hourly flood forecasting. The update equation for soil water balance can be expressed as:
From the above description, the original TVGM considers the basin soil as a homogeneous bucket, neglecting the spatial heterogeneity of the underlying surface and its impact on actual hydrological processes. In this study, we introduce the concept of VSA and its corresponding distribution curve to represent the spatial heterogeneity of the basin's underlying surface, and based on this, we have developed the new hydrological model, TVGM-SSCC.
The 
concept is based on a hydrological assumption that soil water storage capacity varies spatially at different points within the basin, a variability that can be characterized by a statistical probability distribution. In this study, we utilize one of the most widely employed probability distributions, the Pareto distribution, which facilitates the derivation of a quantitative relationship between the basin's saturated area and its soil water storage capacity. Moore (2007) provided a detailed derivation of the relevant equations, which can be transformed into a quantitative equation linking the overall soil moisture storage of the basin with VSA. Similar equation structures are also present in models such as XAJ (Zhao et al. 1980) and VIC models (Liang et al. 1994). The specific derivation process is as follows.
, then at any given time, when the critical storage capacity depth of the entire basin is 
, the proportion of the basin containing stores of capacity less than or equal to 
is:
is as follows:
represents the maximum soil water storage capacity at a point within the basin, while 
is the shape parameter of the distribution curve, reflecting the degree of heterogeneity within the basin. Substituting this into Equation (5) yields the probability distribution function 
, which is equivalent to the 
in this study.
concept can be found in Moore (2007) with its Corrigendum. In this study, we incorporated the SSCC into the TVGM. The model's runoff process is divided into two components: in the saturated area, the runoff coefficient is set to 1, meaning that all net precipitation is converted into runoff; in the unsaturated area, the original rainfall-runoff relationship, which follows a power-law relationship with relative soil moisture, is maintained. The integrated TVGM-SSCC maintains the original TVGM structure except for the runoff generation Equation (2), which is modified as follows:
In Equation (11), VSA (−) ranges from 0 to 1 and represents the portion of the saturated area. β (−) is the shape parameter controlling the distribution of the SSCC. Specifically, when β equals 0, the soil water storage capacity is constant across the entire basin. This indicates that when the critical soil storage capacity (
) is less than the maximum storage capacity 
, the VSA remains 0. Under this condition, the term 
on the right-hand side of Equation (11) will be zero, and the surface runoff equation becomes identical to that in Equation (2), corresponding to the original version of the TVGM. This indicates that the TVGM is a special case of the TVGM-SSCC model, where the soil water storage capacity distribution is constant.
The schematic depiction of the TVGM-SSCC is displayed in Figure 3(b). We integrated the SSCC into the TVGM using a parallel multiplication approach, a so-called flexible modelling method in Huang et al. (2016). This approach fully utilizes the original model's runoff generation and routing structure and variable settings. Only one additional parameter, 
, is introduced, maintaining the simplicity of the original model structure. The parameter settings and value ranges for both TVGM and TVGM-SSCC are presented in Table 3.
The parameters of the TVGM-SSCC and their prior range for calibration
| Parameters | Description | Prior range of parameters |
|---|---|---|
 ( − ) | Nonlinear parameter between soil moisture and  | (0.01, 3) |
 ( − ) | Shape parameter controlling the distribution of SSCC | (0.0.1, 5) |
 ( − ) | Surface runoff coefficient | (0.01, 1) |
 ( − ) | Nonlinear parameter related to soil moisture | (0.01, 3) |
 ( − ) | Nonlinear parameter related to rainfall intensity | (0.01, 3) |
 ( − ) | Nonlinear baseflow modelling structure index | (0.01, 200) |
 ( − ) | Magnitude index of baseflow | (0.01, 800) |
 (mm) | Saturated soil moisture storage | (50, 1,000) |
 ( − ) | Shape parameter of the unit hydrograph | (1, 40) |
 ( − ) | Delay time to peak flow | (1, 40) |
| KKG ( − ) | Subsurface linear reservoir lag coefficient | (0.7, 0.998) |
| Parameters | Description | Prior range of parameters |
|---|---|---|
| ( − ) | Nonlinear parameter between soil moisture and | (0.01, 3) |
| ( − ) | Shape parameter controlling the distribution of SSCC | (0.0.1, 5) |
| ( − ) | Surface runoff coefficient | (0.01, 1) |
| ( − ) | Nonlinear parameter related to soil moisture | (0.01, 3) |
| ( − ) | Nonlinear parameter related to rainfall intensity | (0.01, 3) |
| ( − ) | Nonlinear baseflow modelling structure index | (0.01, 200) |
| ( − ) | Magnitude index of baseflow | (0.01, 800) |
| (mm) | Saturated soil moisture storage | (50, 1,000) |
| ( − ) | Shape parameter of the unit hydrograph | (1, 40) |
| ( − ) | Delay time to peak flow | (1, 40) |
| KKG ( − ) | Subsurface linear reservoir lag coefficient | (0.7, 0.998) |
Note: Apart from the β, the number and value ranges of the parameters are the same for both models. In the TVGM, the value of β is fixed at 0.
Model calibration and evaluation
) (Nash & Sutcliffe 1970) as the objective function for the parameterization scheme (Equation (12)). According to previous studies on multi-objective function (MO) formulas (Nicolle et al. 2014; Santos et al. 2018), the chosen indices are complementary, ensuring a balanced representation of total runoff and low-flow characteristics. The parameterization scheme was calibrated based on observed total flow data Q at the outlet stations. The mathematical formulations are as follows:
and 
are the simulated and observed variables, respectively; the variables with overbar denote the mean value; and the subscript n means the 
value of the total N days. 
is a small constant equal to 
, thereby preventing the scenario wherein runoff equals zero during the dry season (Pushpalatha et al. 2012; Santos et al. 2018).Inspired by the study of Poulin et al. (2011), to avoid the lack of representativeness of results from a single calibration, each model was calibrated 20 times using different random seeds in each basin. The best result, based on the optimal muti-objective function (MO) value, was selected from all converged parameter sets as the final calibration result for each model in all basins.
The symbols are as previously described. The three indicators mentioned above are used to evaluate the model's performance in simulating total streamflow. To assess the model's performance in simulating low-flow, the logarithmic form of NSE 
) and the Pearson correlation coefficient of the baseflow (
) were calculated. To further assess the efficacy of the new model structure in representing basin spatial heterogeneity, this study compared the simulated soil moisture dynamics with regional soil moisture dynamics estimated from remote sensing products. Comparison was conducted by visual analysis of hydrograph and statistical analysis of various metrics.
RESULTS
Comparison of the total streamflow simulation between TVGM and TVGM-SSCC
) is 5.7% during the calibration period and 10% during the validation period. Figure 4 presents scatter plots comparing simulated and observed total runoff for both models across the entire period. It is evident that TVGM-SSCC consistently shows better agreement between observed and simulated values across all times, with an average 
value of 0.79, which is 0.07 higher than that of TVGM, representing an improvement of 9.7%. These results indicate that TVGM-SSCC outperforms TVGM under high-flow conditions and achieves greater advantages during the validation period compared with the calibration period, further demonstrating the consistency of TVGM-SSCC in hydrological simulations. The evaluation metrics for the total streamflow simulation performance of the two models during the calibration and validation periods
| Period | Watershed | NSE | WB | r | |||
|---|---|---|---|---|---|---|---|
| TVGM | TVGM-SSCC | TVGM | TVGM-SSCC | TVGM | TVGM-SSCC | ||
| Calibration | ZH | 0.77 | 0.83 | 0.85 | 0.86 | 0.90 | 0.88 |
| WG | 0.74 | 0.83 | 0.87 | 0.92 | 0.87 | 0.92 | |
| GY | 0.60 | 0.73 | 0.87 | 0.92 | 0.78 | 0.83 | |
| MDW | 0.67 | 0.75 | 0.95 | 0.94 | 0.86 | 0.88 | |
| Mean | 0.70 | 0.79 | 0.89 | 0.91 | 0.85 | 0.88 | |
| Validation | ZH | 0.66 | 0.76 | 0.75 | 0.78 | 0.82 | 0.90 |
| WG | 0.52 | 0.72 | 0.82 | 0.92 | 0.73 | 0.86 | |
| GY | 0.89 | 0.93 | 0.86 | 0.89 | 0.95 | 0.97 | |
| MDW | 0.60 | 0.69 | 0.64 | 0.77 | 0.92 | 0.91 | |
| Mean | 0.67 | 0.78 | 0.77 | 0.84 | 0.86 | 0.91 | |
| Period | Watershed | NSE | WB | r | |||
|---|---|---|---|---|---|---|---|
| TVGM | TVGM-SSCC | TVGM | TVGM-SSCC | TVGM | TVGM-SSCC | ||
| Calibration | ZH | 0.77 | 0.83 | 0.85 | 0.86 | 0.90 | 0.88 |
| WG | 0.74 | 0.83 | 0.87 | 0.92 | 0.87 | 0.92 | |
| GY | 0.60 | 0.73 | 0.87 | 0.92 | 0.78 | 0.83 | |
| MDW | 0.67 | 0.75 | 0.95 | 0.94 | 0.86 | 0.88 | |
| Mean | 0.70 | 0.79 | 0.89 | 0.91 | 0.85 | 0.88 | |
| Validation | ZH | 0.66 | 0.76 | 0.75 | 0.78 | 0.82 | 0.90 |
| WG | 0.52 | 0.72 | 0.82 | 0.92 | 0.73 | 0.86 | |
| GY | 0.89 | 0.93 | 0.86 | 0.89 | 0.95 | 0.97 | |
| MDW | 0.60 | 0.69 | 0.64 | 0.77 | 0.92 | 0.91 | |
| Mean | 0.67 | 0.78 | 0.77 | 0.84 | 0.86 | 0.91 | |
Scatter plots of observed versus simulated total streamflow over the entire simulation period, with red triangles representing TVGM and blue circles representing TVGM-SSCC. Subplots (a–d) correspond to the results for the ZH, WG, GY and MDW basins, respectively.
Scatter plots of observed versus simulated total streamflow over the entire simulation period, with red triangles representing TVGM and blue circles representing TVGM-SSCC. Subplots (a–d) correspond to the results for the ZH, WG, GY and MDW basins, respectively.
FDC of simulated runoff by TVGM and TVGM-SSCC compared with observed values under the 1% high flood condition for the entire period. Subplots (a–d) represent the ZH, WG, GY and MDW basins, respectively.
FDC of simulated runoff by TVGM and TVGM-SSCC compared with observed values under the 1% high flood condition for the entire period. Subplots (a–d) represent the ZH, WG, GY and MDW basins, respectively.
Comparison of soil moisture dynamics and low-flow simulation between TVGM and TVGM-SSCC
) for TVGM-SSCC is improved by 27.1% during the calibration period and by 34.4% during the validation period, compared with TVGM. This suggests that TVGM's simple soil structure lacks sufficient buffering, whereas the SSCC extension better captures soil moisture heterogeneity and improves dynamic simulation accuracy. The evaluation metrics for the low-flow simulation performance of the two models during the calibration and validation periods
| Period | Watershed | NSElog | rb | rw | |||
|---|---|---|---|---|---|---|---|
| TVGM | TVGM-SSCC | TVGM | TVGM-SSCC | TVGM | TVGM-SSCC | ||
| Calibration | ZH | 0.67 | 0.67 | 0.84 | 0.85 | 0.54 | 0.75 |
| WG | 0.60 | 0.72 | 0.80 | 0.86 | 0.56 | 0.68 | |
| GY | 0.61 | 0.70 | 0.79 | 0.85 | 0.76 | 0.91 | |
| MDW | 0.52 | 0.58 | 0.76 | 0.78 | 0.49 | 0.66 | |
| Mean | 0.60 | 0.67 | 0.80 | 0.84 | 0.59 | 0.75 | |
| Validation | ZH | 0.19 | 0.20 | 0.63 | 0.69 | 0.49 | 0.75 |
| WG | 0.52 | 0.72 | 0.73 | 0.86 | 0.56 | 0.80 | |
| GY | 0.68 | 0.84 | 0.84 | 0.92 | 0.67 | 0.87 | |
| MDW | 0.35 | 0.36 | 0.76 | 0.80 | 0.73 | 0.86 | |
| Mean | 0.44 | 0.53 | 0.74 | 0.82 | 0.61 | 0.82 | |
| Period | Watershed | NSElog | rb | rw | |||
|---|---|---|---|---|---|---|---|
| TVGM | TVGM-SSCC | TVGM | TVGM-SSCC | TVGM | TVGM-SSCC | ||
| Calibration | ZH | 0.67 | 0.67 | 0.84 | 0.85 | 0.54 | 0.75 |
| WG | 0.60 | 0.72 | 0.80 | 0.86 | 0.56 | 0.68 | |
| GY | 0.61 | 0.70 | 0.79 | 0.85 | 0.76 | 0.91 | |
| MDW | 0.52 | 0.58 | 0.76 | 0.78 | 0.49 | 0.66 | |
| Mean | 0.60 | 0.67 | 0.80 | 0.84 | 0.59 | 0.75 | |
| Validation | ZH | 0.19 | 0.20 | 0.63 | 0.69 | 0.49 | 0.75 |
| WG | 0.52 | 0.72 | 0.73 | 0.86 | 0.56 | 0.80 | |
| GY | 0.68 | 0.84 | 0.84 | 0.92 | 0.67 | 0.87 | |
| MDW | 0.35 | 0.36 | 0.76 | 0.80 | 0.73 | 0.86 | |
| Mean | 0.44 | 0.53 | 0.74 | 0.82 | 0.61 | 0.82 | |
represents the Pearson correlation coefficient between the baseflow simulated by the models and the observed baseflow sequence divided by the UKIH method, while 
denotes the Pearson correlation coefficient between the model-simulated soil normalized moisture and the remote sensing soil moisture product SMCI1.0.
Normalized daily soil moisture series composed of remote sensing products SMCI1.0 (black line), TVGM (red dashed line) and TVGM-SSCC models (blue line). The left subplots (a–d) represent the calibration period, while the right subplots (e–h) represent the validation period. Each row corresponds to a basin, consistent with the order in Table 1. The calibrated parameter values for both models in the four basins can be found in Supplementary material, Table S1. Normalization was performed using the min–max scaling of the raw data.
Normalized daily soil moisture series composed of remote sensing products SMCI1.0 (black line), TVGM (red dashed line) and TVGM-SSCC models (blue line). The left subplots (a–d) represent the calibration period, while the right subplots (e–h) represent the validation period. Each row corresponds to a basin, consistent with the order in Table 1. The calibrated parameter values for both models in the four basins can be found in Supplementary material, Table S1. Normalization was performed using the min–max scaling of the raw data.
To evaluate the model's performance in low-flow conditions, 
and 
were used. The UKIH method (Gustard et al. 1992) was employed to separate the baseflow from total streamflow series. Table 5 presents the values of 
and 
during the calibration and validation periods. During the calibration period, the multi-basin averages for TVGM-SSCC's 
and 
are 0.67 and 0.84, respectively, which are 11.7 and 5% higher than those of TVGM. In the validation period, TVGM-SSCC's average 
and 
are 0.53 and 0.82, showing improvements of 20.5 and 10.8% over TVGM's. Although the simulation accuracy declines during the validation period for both models, TVGM-SSCC experiences a smaller decrease, and its advantage in low-flow simulation metrics remains stable compared with TVGM.
Comprehensive evaluation of the improvement and stability of the extended model
In this section, the performance stability of the two models across 80 calibrations (four basins × 20 calibrations) is examined. Table 6 presents the statistical indicators for these distributions, along with the exceedance percentage (PE) based on 80 simulations. PE (%) refers to the percentage of simulations in which the TVGM-SSCC outperforms the TVGM. During the calibration period, TVGM-SSCC shows the greatest improvement over TVGM in 
, with an average increase of 0.21, while its advantage in 
is the smallest, with an average increase of 0.02. Apart from 
, the PE for all other metrics exceeds 86.3%, with 
and 
reaching 100%. Despite TVGM having a lower standard deviation for 
during the validation period compared with TVGM-SSCC, the mean value is still 0.15 higher for the latter, resulting in a PE of 95% for this metric, indicating that TVGM still lags significantly behind TVGM-SSCC. All metrics in validation period have a PE exceeding 70%, with 
and 
once again reaching 100%.
Distribution of evaluation metrics between two models in 80 simulations (four basins × 20 calibration runs) under different calibration objective functions
| Metrics | Calibration | Validation | ||||
|---|---|---|---|---|---|---|
| TVGM | TVGM-SSCC | PE (%) | TVGM | TVGM-SSCC | PE (%) | |
| NSE | 0.60 ± 0.16 | 0.74 ± 0.07 | 86.3 | 0.61 ± 0.19 | 0.73 ± 0.13 | 80.0 |
| WB | 0.87 ± 0.06 | 0.89 ± 0.06 | 61.3 | 0.77 ± 0.13 | 0.81 ± 0.11 | 70.0 |
| r | 0.79 ± 0.10 | 0.87 ± 0.04 | 86.3 | 0.80 ± 0.11 | 0.91 ± 0.04 | 96.3 |
| NSElog | 0.50 ± 0.14 | 0.66 ± 0.06 | 95.0 | 0.37 ± 0.23 | 0.52 ± 0.26 | 95.0 |
| r_b | 0.77 ± 0.04 | 0.83 ± 0.03 | 100.0 | 0.71 ± 0.09 | 0.81 ± 0.08 | 100.0 |
| r_w | 0.50 ± 0.11 | 0.71 ± 0.11 | 100.0 | 0.42 ± 0.11 | 0.77 ± 0.10 | 100.0 |
| Metrics | Calibration | Validation | ||||
|---|---|---|---|---|---|---|
| TVGM | TVGM-SSCC | PE (%) | TVGM | TVGM-SSCC | PE (%) | |
| NSE | 0.60 ± 0.16 | 0.74 ± 0.07 | 86.3 | 0.61 ± 0.19 | 0.73 ± 0.13 | 80.0 |
| WB | 0.87 ± 0.06 | 0.89 ± 0.06 | 61.3 | 0.77 ± 0.13 | 0.81 ± 0.11 | 70.0 |
| r | 0.79 ± 0.10 | 0.87 ± 0.04 | 86.3 | 0.80 ± 0.11 | 0.91 ± 0.04 | 96.3 |
| NSElog | 0.50 ± 0.14 | 0.66 ± 0.06 | 95.0 | 0.37 ± 0.23 | 0.52 ± 0.26 | 95.0 |
| r_b | 0.77 ± 0.04 | 0.83 ± 0.03 | 100.0 | 0.71 ± 0.09 | 0.81 ± 0.08 | 100.0 |
| r_w | 0.50 ± 0.11 | 0.71 ± 0.11 | 100.0 | 0.42 ± 0.11 | 0.77 ± 0.10 | 100.0 |
The column for TVGM-SSCC represents the mean value ± standard deviation of each metric over 80 simulations, so on for the TVGM.
, r and 
are relatively broad, ranging from −0.2 to 0.6, with mean values between 0.1 and 0.2. This indicates that the dynamic consistency of total runoff simulation is quite sensitive to parameter variations, and TVGM exhibits greater volatility, occasionally performing significantly worse than TVGM-SSCC (with differences exceeding 0.4). The distribution interval of the difference for 
is the smallest, entirely concentrated between 0 and 0.2, suggesting that baseflow simulation is less sensitive to parameter changes, allowing TVGM-SSCC to maintain a stable advantage. The 
has the second narrowest interval of differences, but has the largest average spread, with values exceeding 0.2 during the calibration period and reaching 0.35 during the validation period. This indicates that the structural advantages of TVGM-SSCC in capturing soil moisture dynamics are primarily due to the model structure itself, rather than parameter uncertainty, further supporting the rationality of the expanded model structure. Finally, WB exhibits the most distinct pattern, with its distribution evenly spread on both sides of the zero line and a mean close to zero. This indicates that TVGM-SSCC does not show a significant improvement in the accuracy of total water volume simulations. Two potential reasons for this are: (1) The WB value of TVGM is already relatively high, with a mean of 0.87 during the calibration period and 0.77 during the validation period, leaving limited room for improvement and making further gains more difficult. (2) The structural difference between TVGM and TVGM-SSCC lies primarily in the spatial distribution of soil moisture, which mainly controls the runoff coefficient and saturated area during rainfall events, affecting the redistribution of water within the basin but having less impact on the overall water inflow and evaporation. Consequently, the overall WB coefficient shows only minor differences between the models. 
Statistical distribution of the differences in evaluation metrics between TVGM-SSCC and TVGM over 80 simulations. Subplot (a) represents the calibration period, and subplot (b) represents the validation period. The upper section displays the kernel density distribution, with the black dashed line indicating the standard deviation and the black dot representing the mean of the distribution. The lower section shows the frequency of data points within each interval.
Statistical distribution of the differences in evaluation metrics between TVGM-SSCC and TVGM over 80 simulations. Subplot (a) represents the calibration period, and subplot (b) represents the validation period. The upper section displays the kernel density distribution, with the black dashed line indicating the standard deviation and the black dot representing the mean of the distribution. The lower section shows the frequency of data points within each interval.
DISCUSSION
The specific impact of the expansion module on the runoff process
In Section 4, we demonstrated that extending a lumped model by incorporating SSCC to account for spatially varying soil moisture can be successful through comparisons of observed and simulated data. However, a question arises: given that model calibration results are often region-specific, would the model hold up under different basin conditions or when evaluated using different metrics? To address this, we explore the fundamental reasons why this extended model structure improves performance by stripping away all external factors and focusing on the underlying runoff generation mechanisms.
A numerical experiment comparing the runoff generation mechanisms of the TVGM and TVGM-SSCC models was conducted using the Guanyin basin as the experimental site. The experiments employed the optimal parameter set for TVGM-SSCC, with an initial soil moisture set to 0.5 Wm. Three different precipitation strategies were applied: (a and d) feature continuous rainfall of 5 mm/day for 150 days; (b and e) involve continuous rainfall of 5 mm/day for the first 100 days and (c and f) include rainfall of 5 mm/day for the first 50 days and the last 50 days. To avoid influencing the runoff generation process, PET was set to 0.5 mm/day throughout the entire period. The top row shows the changes in total flow hydrographs, while the bottom row displays the changes in baseflow hydrographs.
A numerical experiment comparing the runoff generation mechanisms of the TVGM and TVGM-SSCC models was conducted using the Guanyin basin as the experimental site. The experiments employed the optimal parameter set for TVGM-SSCC, with an initial soil moisture set to 0.5 Wm. Three different precipitation strategies were applied: (a and d) feature continuous rainfall of 5 mm/day for 150 days; (b and e) involve continuous rainfall of 5 mm/day for the first 100 days and (c and f) include rainfall of 5 mm/day for the first 50 days and the last 50 days. To avoid influencing the runoff generation process, PET was set to 0.5 mm/day throughout the entire period. The top row shows the changes in total flow hydrographs, while the bottom row displays the changes in baseflow hydrographs.
Under the second rainfall condition, total runoff sharply declines as rainfall ceases because no surface runoff is generated, and excess water is discharged as baseflow from the soil (see Figure 8(b) and 8(e)). The total runoff decline peaks are nearly overlapping, and the baseflow processes are almost parallel. Combined with the above results, it can be seen that TVGM-SSCC significantly improves low-flow simulation, indicating that its simulation of the baseflow in the rising phase better matches the actual condition. This further demonstrates that the actual runoff process is spatially heterogeneous rather than uniformly replenishing soil moisture. Under the third rainfall condition, the basin experiences two rainfall events (see Figure 8(c)). The total runoff hydrograph of TVGM shows a slow initial rise during both rainfall events, without reaching saturation or exhibiting an inflection point. In contrast, TVGM-SSCC responds rapidly at the onset of both heavy rainfall events, with a steep increase in the total runoff hydrograph. During the recession phase without rainfall, the hydrographs of both models are very close, indicating that the main differences between the models lie in the runoff generation mechanisms during the rapid flood rise phase.
In this numerical experiment, the value of β in TVGM-SSCC was varied while keeping other parameters constant. Therefore, the baseflow generation formula remains identical across the four scenarios, with differences arising from variations in soil moisture during the same period. As shown in Figure 8(d), the four curves exhibit differences in the rising phase of baseflow (before soil moisture saturation), with TVGM showing the fastest increase, indicating that most precipitation is converted into soil moisture and baseflow. As the β value increases, the rise in soil moisture becomes slower, suggesting that the priority of surface runoff gradually increases. When the soil becomes saturated and high-intensity rainfall persists, the baseflow curves for all four scenarios run parallel and remain at their maximum values. Figure 8(e) and 8(f) primarily illustrate the baseflow simulation during the no-rain recession period. According to Equations (2) and (11), it can be inferred that the baseflow recession rate increases with higher soil moisture. In the early stages of the recession, TVGM shows the highest soil moisture, resulting in greater baseflow and a faster corresponding soil moisture decline. It can also be observed that when two curves have similar soil moisture levels at the beginning of the recession, their recession processes overlap (e.g., the blue and yellow curves in Figure 8(e)). However, since the baseflow magnitude is relatively small and PET is set low (0.5), the recession processes of the four curves are approximately parallel and do not dominate the total runoff composition.
Based on the above analysis, we can draw the following views. Firstly, TVGM-SSCC is more responsive to precipitation, allowing for a quicker reaction to storm runoff events. This is particularly advantageous in small- to medium-sized basin runoff simulations, where flash floods are common. In semi-arid small basins, storm events often have short durations and high intensities, leading to sharp and narrow flood peaks. TVGM-SSCC can capture this feature. Secondly, the runoff generation pattern of TVGM is of a single shape and lacks flexibility, which may pose challenges for simulations of broader basins. On the other hand, TVGM-SSCC can generate a variety of runoff patterns as the β value changes, better accommodating the spatial heterogeneity of runoff across different basins.
Significance of the study and future work
The analysis above highlights the theoretical and practical value of the lumped model extension proposed in this study. Previously, some researchers have explored expanding the spatial representation of lumped models. For example, the most renowned is the XAJ model proposed by Zhao et al. (1980), in which the SSCC with a Pareto distribution was applied to lumped hydrological models. Subsequent studies, such as Bartlett et al. (2016a, b) and Wang (2018), explored improvements by combining soil water storage capacity distribution curves with the SCS-CN method. Huang et al. (2016) enhanced runoff simulation by flexibly combining multiple SSCC based on the XAJ model. However, all these studies are based on the assumption of a saturation-excess runoff mechanism. In contrast, the TVGM used in this study does not rely on saturated-excess or infiltration-excess runoff mechanisms. Its runoff equation is a power-law function of the degree of soil saturation in the basin, without a ‘runoff threshold’ constraint. In this study, SSCC is used to represent the spatial heterogeneity of soil water storage capacity within the basin, and it is not confined to the saturation-excess mechanism. Runoff can still occur in unsaturated areas, which aligns with the spatial characteristics of basin runoff. Theoretically, the model extension approach proposed in this study should be universal, replacing the homogeneous soil moisture structure in lumped models and providing an implicit spatial representation, thereby improving the model's ability to capture real hydrological dynamics. Moreover, the numerical experiments that validate the theoretical foundation for the success of the improvements and offer a comparative framework and methodology have been rarely explored in previous research: thus, the study provides a valuable reference for future researchers in related fields.
It is worth noting that further research is needed to improve this study, such as integrating this extension scheme into more lumped models to assess its reliability and conducting comparative studies with distributed models to understand to what extent it enhances the representation of spatial heterogeneity. Additionally, the role of this extension across different time scales in hydrological models (e.g., in hourly flood forecasting) should be investigated. It is believed that integrating local spatial runoff models into future lumped model development can enhance the realism of basin-scale hydrological simulations with minimal computational cost.
CONCLUSION
This study extends the soil water storage capacity, which was originally treated as a constant distribution in the TVGM, by using a statistical probability distribution SSCC. This extension enhances the runoff and soil moisture simulation performance without significantly increasing the model complexity. This is a novel attempt to apply SSCC to a nonsaturation-excess mechanism hydrological model. Moreover, we have demonstrated the theoretical foundation of the proposed extension approach through numerical experiments. The main conclusions are as follows:
(1) The proposed approach of extending the lumped model with SSCC effectively enhances the lumped model's ability to represent spatial heterogeneity, improving its performance in runoff simulation and soil moisture dynamics capture.
(2) Compared with TVGM, the TVGM-SSCC model achieved comprehensive improvements. Overall, during the whole periods, the enhanced model improved total runoff simulation metrics (
) by 7.9%, low-flow simulation metrics (
, 
) by 11.2%, and soil moisture dynamics fit metrics (
) by a significant improvement of 30.8%. Across 80 simulations, TVGM-SSCC outperformed TVGM in 88.2% of cases during the calibration period and 90.2% during the validation period, indicating greater stability in the model results.(3) In the numerical simulation experiments, the model effectively reflects the spatial heterogeneity of the basins, and the flexible
parameter enables it to adapt to the runoff generation needs of different basins. This makes TVGM-SSCC a more comprehensive and refined lumped model.
As this extension introduces only one additional parameter without altering other hydrological processes, it is easy to implement and promote, providing a simple and effective approach for improving the simulation performance of lumped hydrological models.
ACKNOWLEDGEMENTS
This work was supported by the National Natural Science Foundation of China under Grant (No. U2340213). Apart from the financial support, the foundation did not participate in the specific research of this paper. This work was performed as part of the IAHS HELPING Working Group on ‘Development & application of river basin simulators’. We sincerely thank the reviewers for their careful, meticulous, and professional comments and suggestions, which greatly improved the quality of this manuscript. We also thank Professor Chong-Yu Xu from the University of Oslo for his valuable contribution to the writing and improvement of this work.
AUTHOR CONTRIBUTIONS
Y.Z.: Conceptualization, Methodology, Software, Writing – original draft. J.X.: Supervision, Methodology, Resources, Writing – review & editing. A.Y.: Methodology, Validation. L.Z.: Investigation, Validation.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The authors declare there is no conflict.
