Wang (2018) proposed a universal cumulative distribution function (CDF) to describe the spatial distribution of soil water storage capacity at the catchment scale: 1FC=1-1a+C+1-aSbaC+Sb2-2aSbC Here C is the point-scale soil water storage capacity (in meters) with C≥0. FC is the corresponding CDF; a is the dimensionless shape parameter with a range of 0–2; and Sb is the average soil storage capacity (in meters) in a catchment. Introducing the shape parameter allows for flexibility in representing various forms of spatial distributions of soil water storage capacity, ensuring that the CDF can represent a range of catchment types with different topographic organizations and associated soil storage capacity distributions.

Using this new CDF, Wang (2018) provided a theoretical explanation for the Soil Conservation Service Curve Number (SCS-CN) method (SCS, 1972) as a saturation excess runoff model, and revealed the theoretical common ground between the SCS-CN and VIC-type methods. Furthermore, Yao et al. (2020) developed a new PDM by using this CDF to quantify the spatial variability of soil water storage capacity and, hence, saturation excess.

In this study, we propose the Unified Runoff Scheme for Saturation and Infiltration Excess (URSSIE). In URSSIE, we keep the CDF in Eq. 1) for the spatial distribution of soil water storage. A byproduct of Eq. (1) is the spatial distribution of soil moisture deficits (portion of soil storage not occupied by water), which we utilize to derive the spatial distribution of infiltration capacities. This theoretical innovation enables us to incorporate both the spatial distributions of soil water storage and infiltration capacities within a single framework. Figure 1a depicts the spatial distribution of soil water storage capacity (C) with a uniform groundwater depth, similar to the previously discussed PDMs (e.g., HyMOD and VIC). This assumption is reasonable because groundwater table observations are often unavailable, and when the groundwater tables are relatively deep and laterally smooth, spatial differences tend to diminish (Yao and Wang, 2022). Correspondingly, there is a spatial distribution of soil water storage deficit (in meters), D. Usually, the larger the D value, the drier the soil, and the larger the infiltration capacity. Considering a one-to-one relationship between soil infiltration capacity and D, a spatial distribution of infiltration capacity can be yielded as a function of D, as shown in Fig. 1b.

We now conceptually divide the whole catchment area into four hydrologic subzones based on their hydrologic responses to rainfall events:

Antecedent saturation subzone (A0∼α0): Infiltration capacity is zero, and all the rainfall becomes saturation excess runoff. α0 represents the boundaries (–) of where soil is already saturated before the current event or time step.

Within the URSSIE scheme, the spatial distributions of infiltration and saturation excess generation areas evolve over time (Fig. 2a). The contour lines represent the conceptual boundaries between subzones experiencing different runoff generation mechanisms. The transition zones, depicted by the space between contours from time t to t+1, highlight the dynamic nature of these mechanisms across space within a catchment. At the core of this transformation is the interplay between four distinct subzones: A0∼α0, Aα0∼α1, Aα1∼α2, and Aα2∼1. As time progresses from t to t+1, the boundaries between these subzones evolve, reflecting the shifting dominance of saturation-excess and infiltration-excess runoff processes. When more catchment areas become saturated due to antecedent conditions or due to ongoing infiltration, the area contributing to saturation-excess runoff increases. This expansion occurs at the expense of the unsaturated subzones. The transition zones between the contours at time t and t+1 represent areas where infiltration mechanisms undergo significant changes. These zonal changes capture the gradual shift from infiltration-excess to saturation-excess runoff as the catchment becomes wetter (Fig. 2c).

Figure 2b summarizes the dynamic evolution of the subzones in the form of the spatial distribution curves of infiltration capacity at the beginning of time step t and t+1, respectively. The rightward shift of the infiltration capacity curves indicates a progressive decline in the soil's capacity to infiltrate water. In other words, as water infiltrates, the water deficit D decreases, leading to a corresponding decrease in overall infiltration capacity. Figure 2c further illustrates the temporal evolution of the areas for infiltration and saturation excess runoff generation over time, assuming the rainfall duration is sufficiently long. The area for infiltration excess runoff gradually decreases as the groundwater table rises due to increased soil water storage and a decrease in unsaturated areas. On the contrary, the area for saturation excess runoff increases and eventually dominates over the whole catchment area. Overall, the dynamic nature of these subzones and their transitions in Fig. 2 highlights the spatiotemporal variability of runoff generation processes across the landscape owing to the complex interplay between antecedent conditions, rain event characteristics, and the spatial heterogeneity of landscape conditions.

To mathematically express the above conceptual framework, a formula directly linking D and soil infiltration capacity is necessary. There is a close causal relationship between them, i.e., the larger the D value, the drier the soil (particularly the topsoil layers), and thus the higher the infiltration capacity, as suggested by previous experimental and theoretical studies (Liu et al., 2011; Yao et al., 2018; Yang et al., 2020). However, classic infiltration equations do not provide such a linkage to represent the spatial distribution of initial soil moisture (Horton, 1933; Green and Ampt, 1911; Philip, 1957). We represent the point-scale infiltration capacity (fc) at any unsaturated location in a catchment as a power-law-type function of D: 2fc=mkDSbn Where D is the point-scale soil water deficit (in meters) at a given time. The coefficient, mk (m s⁻¹), sets the upper limit of infiltration capacity under extreme rainfall conditions. We recommend mk to be within a range of 0 to 2.315×10-5 m s⁻¹. The 2.315×10-5 m s⁻¹ value is slightly above the maximum historical daily rainfall intensity worldwide, 1825 mm d⁻¹ (Holland, 1993). The dimensionless exponent, n, reflects the spatial heterogeneity of landscape conditions within a catchment, such as vegetation cover and soil properties, which can be interpreted as a simplified representation of root-zone and ecosystem controls on infiltration behavior (Savenije, 2024; Gao et al., 2024). It adjusts the non-linearity of the relationship between D and fc and thus controls the response curve of the infiltration capacity to changes in soil moisture conditions. In this study, we recommend the range of n between 0.4 and 1.0 for the purpose of numerical modeling based on our numerical tests over the natural catchments in the United States (figure not shown). Water deficit ratio, D/Sb, indicates the overall soil moisture status: the higher the ratio, the drier the soil, and hence the larger the infiltration capacity. It thus acts as the dynamic scaling factor crucial to modeling how changes in soil moisture adjust the infiltration capacity. As noted by previous studies (Kollet and Maxwell, 2008; Soylu et al., 2011), a shallow groundwater table has a significant impact on surface soil moisture and infiltration capacity. As the groundwater table deepens, its influence diminishes and becomes decoupled from surface soil moisture.

Figure 3 illustrates how the infiltration capacity fc responds to changes in D/Sb, conditioned by varying values of the exponent n with an illustrative value of mk=500 mm d⁻¹. For low values of n (e.g., n=0.4 or 0.5), the infiltration capacity remains relatively high even at low water deficit ratios. This scenario is typical for clay-rich soils, where the infiltration rate is less sensitive to variations in soil moisture. In such cases, saturation excess may dominate, as the infiltration capacity remains higher than the incoming rainfall intensity. Conversely, for high values of n (e.g., n=0.9), the infiltration capacity increases rapidly with increasing water deficit ratios. This scenario is more representative of sandy or loamy soils, where the infiltration rate is susceptible to changes in soil moisture. In these cases, infiltration excess may dominate, as the infiltration capacity can quickly drop below the rainfall intensity, leading to the generation of surface runoff over unsaturated areas.

Rearranging Eq. (2), we can express D as: 3D=fc1nmk-1nSb D can also be directly calculated as the difference between the point-scale soil storage capacity C (m) and the initial storage C0 (m): 4D=C-C0 By combining Eqs. (3) and (4) and solving for C, we obtain: 5C=C0+fc1nmk-1nSb, Eq. (5) allows us to link point-scale infiltration capacity with soil storage capacity and, importantly, dynamic soil moisture conditions. Next, we substitute Eq. (5) into Eq. (1), and yield the probabilistic distribution of infiltration capacity as a function of the dynamic soil water storage changes over unsaturated areas in a catchment: 6Hfc=1-1a+C0+fc1nmk-1nSb+1-aSbaC0+fc1nmk-1nSb+Sb2-2aSbC0+fc1nmk-1nSb Depending on the rainfall intensity i (in m s⁻¹), two cases are evident for the actual infiltration rate f (m s⁻¹) in unsaturated areas. The first case occurs when i≤fc at time t (in seconds). As such, the actual infiltration rate f=i. In the second case, i>fc, hence f=fc. For the second case, the decrease of f with increasing cumulative infiltration F (in meters), can be expressed by: 7f=fc=mkD0-FSbn Taking the derivative of Eq. (7) (see Supplement for details), we get 8dfdt=fD0-FSbn-1-mknSb Eq. (8) gives how the change rate of f depends on both f itself and F. As F increases, this derivative typically becomes more negative, indicating that the infiltration rate decreases more rapidly over time as the soil approaches saturation. Rearranging and substituting Eq. (7) into Eq. (8) and integrating both sides of the equation from t=0 to t for dt and from f=f0 to f for df (see Supplement for details), we obtain a new expression for the infiltration rate as a function of time: 9f=mk1-nnD0Sb1-n+n-1Sbmk1ntn1-n During a storm event (or a timestep), if the initial infiltration capacity of a spatial point is smaller than the rainfall intensity, infiltration excess begins immediately. The saturation time, ts1 (in seconds), is the time lapse from the beginning of the timestep to when the soil is saturated. The infiltration rate f at the saturation time is 0. Substituting f=0 and from t=ts1 into Eq. (9), we obtain the saturation time ts1 as: 10ts1=D01-nSbnmk1-n If the initial infiltration capacity is greater than the rainfall intensity i, infiltration excess does not start until the infiltration capacity decreases to i. There thus exists a critical soil water deficit value Di (in meters) satisfying the condition fc=i, which can be substituted into Eq. (3) to obtain: 11Di=imk1nSb Correspondingly, there is a critical areal fraction, α∗, which stands for the locations where D0=Di. Ponding time, tp (in seconds), is the time lapse from the beginning of the timestep to when infiltration excess starts, and can be obtained as tp=D0-Dii. Since the moment of tp, infiltration continues; after another duration of time, tsi, soil saturation is reached, and saturation excess takes over. tsi can be obtained by using D0=Di in Eq. (10). 12tsi=Di1-nSbnmk1-n The relationship between tsi and the timestep length, Δt (in seconds), determines the infiltration excess occurrence in different subzones of the catchment. We can consider two distinct conditions based on this relationship. i.

Condition-I, tsi>Δt: In this case, α∗ falls in α1∼α2 and infiltration excess occurs in both the Aα0∼α1 and Aα1∼α2 subzones. For the Aα0∼α1 subzone, infiltration excess occurs in the initial stage of the timestep, and then is taken over by saturation excess. For the Aα1∼α2 subzone, infiltration excess starts either at the beginning of the timestep (over some locations within the subzone) or after the ponding time (over the rest of the subzone), but saturation excess does not occur.

The applicability of the new approach is demonstrated through a semi-distributed model suitable for small, headwater catchments, where the effects of channel routing are negligible in terms of the magnitude and timing of streamflow, particularly at daily or longer time steps. A catchment is conceptually divided into four subzones, as outlined in Sect. 2.2, but the water balance is directly calculated at the catchment scale. Total surface runoff is further divided between direct runoff, which is directed into a quick storage tank, and groundwater recharge, which is directed into a slow storage tank. Note that these two storage tanks are intended solely for routing surface and subsurface runoff. Direct streamflow and baseflow are estimated as functions of quick and slow storage pools using a linear reservoir type method (Yao et al., 2020). Evaporation is estimated as a function of soil water storage. This model comprises a total of eight parameters, as listed in Table S1 in the Supplement.

The small, natural catchments used in this study are all from the Catchment Attributes and Meteorology for Large-sample Studies (CAMELS) dataset, which includes daily streamflow data from 1985 to 2014 for 671 catchments (Newman et al., 2015). We further select 181 out of 671 catchments where snowfall accounts for no more than 5 % of annual precipitation, as the focus of this study is on runoff generation only. The hourly climate forcing data used in this study are from Phase 2 of the North American Land Data Assimilation System (NLDAS-2) dataset, which includes precipitation, potential evapotranspiration, and air temperature (Xia et al., 2012). For a spatial map of these 181 catchments, see Fig. S2.

We determine the parameter values of the model following a strategy proposed by Abeshu et al. (2023), as outlined in Fig. S3. The period 1985–1986 was used for model warm-up, 1987–2005 for parameter determination, and 2006–2014 for model evaluation. For each of the 181 catchments, we first select samples from the entire parameter space to obtain 100 000 sets of parameter values and run the model using each parameter set. We then perform four rounds of filtering for each of the catchments by selecting the parameter sets leading to minimum normalized root mean square errors (NRMSE) between the simulated and observed annual mean streamflow, mean monthly streamflow, and annual maximum daily flows, respectively, and lastly, the highest modified Kling-Gupta efficiency (KGE, Kling et al., 2012) between the simulated and observed daily streamflow. We finally obtain one optimal parameter set for each catchment that ensures our modeling results reasonably well follow important hydrologic principles, such as preserving water balance at both the annual and seasonal scales. Note that we account for annual maximum daily flows here mainly due to the dominant role of surface runoff in flood peaks. This metric captures the catchment response during high flow conditions and serves as an important indicator for constraining parameters related to rapid hydrologic processes.

Despite its simplicity, the model performs well across various climate and landscape conditions, yielding KGE ≥0.5 at over 90 % (164 out of 181) of the catchments used in this study in the parameter determination period (see Fig. S4 for a spatial map of the KGE values). To demonstrate the improvement of the URSSIE model, we compared it with the SCS-PDM model (Yao et al., 2020) under consistent forcing data, calibration strategy, and study period. The two models show comparable performance in humid catchments, while URSSIE achieves noticeably higher KGEs (Δ KGE > 0.05) in 20 of the 66 arid basins for flood peaks, and 15 of the 66 arid basins for daily streamflow (see Fig. S8 and Supplement for details).

The satisfactory model performance is mainly due to the model reasonably well reproducing surface and subsurface runoff components at annual, seasonal, and event scales, particularly over arid catchments, as shown in Fig. 4.

Figure 4a depicts the relative dominance of infiltration and saturation excess runoff, expressed in the form of the ratio of annual mean infiltration excess runoff to annual mean total surface runoff, as a function of climate conditions. The climate aridity index (AI), which is the ratio of mean annual potential evapotranspiration (PET) to mean annual precipitation (P), is a classic signature of humid-to-arid climate regimes. The Phasing Index (PI), which is the correlation coefficient between monthly PET and P ranging from -1.0 to 1.0, serves as a signature for seasonal climate conditions (Ye et al., 2017). PI approaching -1.0, 0.0, and 1.0 means that seasonal PET and P cycles are strongly out of phase, not synchronized, and strongly in phase, respectively. Interestingly, despite various combinations of climate and landscape conditions, the hydrologic responses of the 181 catchments can be simplified into only three categories:  

Category I: For the catchments under humid climates (AI ≤ 1.0), saturation excess runoff dominates over infiltration excess runoff due to wet soil status before most storm events.

We note that although climate conditions govern surface runoff variations over most catchments, a small number of catchments do not fall within the categories above. These outlier catchments do not exhibit a clear and consistent hydroclimatic pattern, suggesting that additional local controls, such as vegetation dynamics or geological complexity, may influence their runoff behavior. Further investigation of these other factors is beyond the scope of this paper and is left for a future, full-scale study.