A Modified SCS Curve Number Method for Temporally Varying Rainfall Excess Simulation

Abstract

The SCS curve number (SCS-CN) method has gained widespread popularity for simulating rainfall excess in various rainfall events due to its simplicity and practicality. However, it possesses inherent structural issues that limit its performance in accurately simulating rainfall excess and infiltration over time. The objective of this study was to develop a modified CN method with temporally varying rainfall intensity (MCN-TVR) by combining a soil moisture accounting (SMA) based SCS-CN method with the SMA method in the Hydrologic Engineering Center’s Hydrologic Modeling System (HEC-HMS). In the MCN-TVR, the SMA-based SCS-CN method is utilized to simulate the cumulative rainfall excess and infiltration, while the SMA method in the HEC-HMS serves as an infiltration control function. A key advantage of the MCN-TVR is that it eliminates the need for additional input parameters by inherently linking the parameters in the two SMA-based methods. Sixteen hypothetical 24 h SCS Type II rainfall events with different soil types and five real rainfall events for the Rush River Watershed in North Dakota were used to assess the performances of the MCN-TVR method and the SMA-based SCS-CN method. In the hypothetical simulations, the rainfall excess simulated by the SMA-based SCS-CN and MCN-TVR models was compared to that simulated by a Green–Ampt model. Discrepancies were observed between the rainfall excess simulated by the SMA-based SCS-CN and Green–Ampt models, especially for coarse soils under relatively light rainfall. However, the MCN-TVR model, incorporating an infiltration control function, demonstrated its improved performance closer to the Green–Ampt model. For all the hypothetical events, the Nash–Sutcliffe efficiency (NSE) coefficient of the rainfall excess simulated by the MCN-TVR method compared to the Green–Ampt model was greater than 0.99, while the root mean standard deviation ratio (RSR) was less than 0.03. In the real applications, the SMA-based SCS-CN model failed to provide acceptable simulation of the direct runoff for rainfall events with durations of less than the time of concentration. In contrast, the MCN-TVR model successfully simulated the direct runoff for all the events with NSE values ranging from 0.65 to 0.91 and RSR values from 0.31 to 0.56.

1. Introduction

The Soil Conservation Service curve number (SCS-CN) method was developed for simulating rainfall excess from rainfall events [1,2,3]. Due to its simplicity, this method has been widely used in various regions of the world for flood forecasting, water resource management, and environmental impact assessment (e.g., [4,5,6]). However, there are many limitations to its application associated with the assumptions of this method [7,8,9,10,11,12]. Michel et al. (2005) [13] pointed out that the original SCS-CN method has severe structural inconsistencies. Based on the concept of the soil moisture accounting (SMA) procedure, they incorporated the initial water storage into the original SCS-CN method and developed an SMA-based SCS-CN method. Following that study, some researchers developed their modified SCS-CN methods and linked the initial water storage with the antecedent rainfall to overcome the sudden jump in the CN values (e.g., [14,15,16,17,18]). These revisions improved the performance and applicability of the SCS-CN method; however, their capabilities in terms of simulating the actual temporally varying rainfall excess and infiltration processes are still limited.

Compared with the original and modified SCS-CN methods, infiltration models, such as the Horton equation, Green–Ampt model, and the SMA model in the Hydrologic Engineering Center’s Hydrologic Modeling System (HEC-HMS), utilize infiltration capacity functions to regulate the entry of water into the soil [19,20,21]. The original Horton equation was designed to estimate the infiltration capacity under ponding conditions [19]. To expand its applicability to various real conditions, some studies integrated different drainage functions with the original Horton equation and established a relationship between the soil water storage and infiltration capacity [22,23,24,25]. These modified Horton methods overcome the ponding limitation of the original Horton equation, although the input parameters of these methods were assumed to be fixed for a specific location. In reality, the initial infiltration capacity and decay parameter can vary with the soil type, initial soil moisture, and surface coverage [26]. Therefore, additional efforts are needed to identify suitable input parameters for the modified Horton methods in real applications. The original Green–Ampt model was derived by applying Darcy’s law for unsaturated soil under ponding conditions [20]. Mein and Larson (1973) [27] developed a two-stage (i.e., pre-ponding and ponding stages) model to simulate the infiltration in homogeneous soil with a uniform initial water content during steady rainfall. Subsequent revisions extended the applicability of the Green–Ampt model to unsteady rainfall, homogeneous soil, and a non-uniform initial water content [28,29,30,31,32]. Although many studies have provided recommended parameters for different soil types in relation to the Green–Ampt model [33,34,35], it has been observed that the hydraulic conductivity and capillary pressure can vary under different rainfall events and soil conditions [36,37,38,39]. The HEC-HMS has been widely used for hydrologic modeling in highly urban and natural watersheds [40,41,42]. It offers the integration of diverse methods for both event and continuous simulations [21]. The user-friendly graphical interface of the HEC-HMS has further enhanced its applicability and usability [41,42]. The SMA model in the HEC-HMS utilizes a linear function to regulate the variation in the infiltration capacity based on the soil water storage [21]. However, due to the lack of guidance, the parameter estimation for the SMA model relies on the observed streamflow, limiting its application in data-poor watersheds [43].

In order to fully utilize the advantages of different models, several combined methods have been proposed. For example, Gabellani et al. (2008) [44] utilized the SCS-CN method to calibrate the parameters in a modified Horton model by assuming the equality between the cumulative runoff simulated by both methods. Chu and Steinman (2009) [45] related the parameters of the SCS-CN method and the SMA model in the HEC-HMS through formula derivation and utilized the simulation results of the SCS-CN method to derive the parameters for the continuous simulation using the SMA model. Grimaldi et al. (2013) [46] combined the SCS-CN and Green–Ampt methods. The former was used to calculate the total rainfall excess, while the latter was used to determine its temporal distribution. The effective hydraulic conductivity in the Green–Ampt method was adjusted to keep the total infiltration from the two methods consistent. Li et al. (2015) [47] utilized the initial abstraction in the SCS-CN method to calculate the cumulative infiltration and wetting front depth of the Green–Ampt method under ponding conditions. Although these combined methods expanded the applicability of the SCS-CN method, the combinations increased the number of input parameters or need for sufficient rainfall events to link the combined methods, which may limit their applications in data-poor regions.

The objective of this study is to develop a modified curve number method with temporally varying rainfall intensity (MCN-TVR) by combining the modified and SMA-based SCS-CN method proposed by Michel et al. (2005) [13] and the SMA method in the HEC-HMS [21]. As the two methods utilized for the combination were developed based on the concept of the SMA process, their parameters can be easily linked together, thereby reducing the need for extensive input parameters. The performance of the MCN-TVR method is assessed for both hypothetical and real rainfall events. Specifically, it is compared to the SMA-based SCS-CN method [13] and a Green–Ampt method [29] for 16 hypothetical 24 h SCS Type II rainfall events to emphasize its capability to simulate the temporal distribution of the rainfall excess under different rainfall depths and soil types. Additionally, both the MCN-TVR method and the SMA-based SCS-CN method are applied to the Rush River Watershed in North Dakota, USA to evaluate their performances under various rainfall types and initial soil moisture conditions.

2. Materials and Methods

2.1. SMA-Based SCS-CN Method

The original SCS-CN method was derived based on one mass balance equation and two fundamental hypotheses (Equations (1)–(3)). The combination of Equations (1) and (2) yields the general form of this method (Equation (4)).

$$P=I_a+Q+F$$

(1)

$$\frac{Q}{P-I_a}=\frac{F}{S}$$

(2)

$$I_a=\lambda S$$

(3)

$$Q=\frac{{\left(P-I_a\right)}^2}{P-I_a+S}$$

(4)

where $P$ is the cumulative rainfall of a rainfall event (L), $I_a$ is the initial abstraction (L), $Q$ is the cumulative rainfall excess (L), $F$ is the cumulative infiltration after P > I_a (L), $S$ is the potential retention that links to the curve number (L), and $\lambda$ is the initial abstraction coefficient.

By incorporating the mass balance equation for the soil water storage (Equation (5)) and two assumptions (Equations (6) and (7)) into the original SCS-CN method, Michel et al. (2005) [13] developed a modified SMA-based SCS-CN method (Equation (8)), which is referred to as the MCN-SMA in this study.

$$V=V_0+P-Q$$

(5)

$$S_a=\alpha S$$

(6)

$$V_{max}=S_a+S$$

(7)

$$Q=\left\{\begin{array}{cc}0& for V_0\le S_a-P\\ \frac{{\left(P+V_0-S_a\right)}^2}{P+V_0+V_{max}-2S_a}& for S_a-P<V_0\le S_a\\ P\left[1-\frac{{\left(V_{max}-V_0\right)}^2}{{\left(V_{max}-S_a\right)}^2+P\left(V_{max}-V_0\right)}\right]& for S_a<V_0\le V_{max}\end{array}\right.$$

(8)

where $V_0$ is the soil water storage before a rainfall event (L), $V$ is the soil water storage when the cumulative rainfall equals $P$ (L), $V_{max}$ is the maximum soil water storage (L), $S_a$ is a threshold that controls the generation of the surface runoff (L), and $\alpha$ is a coefficient that links $S_a$ to $S$ (If $V_0\le S_a$, $S_a=V_0+I_a$).

2.2. SMA Method in the HEC-HMS

In the SMA method of HEC-HMS 4.10, the infiltration capacity is determined based on a linear relationship with the soil water storage, which can be calculated using Equation (9). Comparing the MCN-SMA method (Equation (8)) to this SMA method, the soil water storage ($V$) and its maximum value ($V_{max}$) are identical for a given soil profile for both methods.

$$f_p\left(t\right)=f_{max}\left[1-\frac{V\left(t\right)}{V_{max}}\right]$$

(9)

where $f_p\left(t\right)$ is the potential infiltration capacity at time $t$ (L/T), $f_{max}$ is the maximum infiltration rate (L/T), and $V\left(t\right)$ is the soil water storage at time $t$ (L).

2.3. Development of the MCN-TVR Method

2.3.1. Modification of the SMA Method in the HEC-HMS

Considering the similarities between the MCN-SMA method and the SMA method in the HEC-HMS, these two methods were chosen as the basis for developing the MCN-TVR method. In the MCN-SMA method, a rainfall excess occurs when the soil water storage surpasses a certain threshold value (i.e., $V>S_a$). Based on this assumption, the SMA method in the HEC-HMS was modified to align with the MCN-SMA method.

Assume that a rainfall event commences with a soil water storage $V_0$. If $V_0\le S_a-P$, no rainfall excess generates. In this case, the infiltration rate equals the rainfall intensity, as indicated by Equation (10).

$$f\left(t\right)=i\left(t\right)$$

(10)

where $i\left(t\right)$ is the rainfall intensity at time $t$ (L/T).

If $S_a-P<V_0\le S_a$, a rainfall excess generates after $V\left(t\right)>S_a$. According to Equation (9), the potential infiltration capacity can be expressed as Equation (11) when $V\left(t\right)>S_a$.

$$\begin{array}{cc}{f}_p\left(t\right)=f_a\frac{V_{max}-V\left(t\right)}{V_{max}-S_a}& for V\left(t\right)>S_a\end{array}$$

(11)

where $f_a$ is the potential infiltration capacity when the soil water storage equals $S_a$ (L/T).

Substituting Equation (7) into Equation (11) yields:

$$\begin{array}{cc}{f}_p\left(t\right)=f_a\left[1-\frac{V\left(t\right)-S_a}{S}\right]& for V\left(t\right)>S_a\end{array}$$

(12)

Considering the real rainfall intensity, if $S_a-P<V_0\le S_a$, the infiltration rate at any given time can be expressed as:

$$f\left(t\right)=\left\{\begin{array}{cc}i\left(t\right)& for V\left(t\right)\le S_a or i\le f_p when V\left(t\right)>S_a\\ f_a\left[1-\frac{V\left(t\right)-S_a}{S}\right]& for i>f_p when V\left(t\right)>S_a\end{array}\right.$$

(13)

If $S_a<V_0\le V_{max}$, the soil water storage (i.e., $V\left(t\right)$) is greater than the threshold storage for rainfall excess generation (i.e., $S_a$) at the beginning of the rainfall event. Therefore, according to Equation (9), the potential infiltration capacity can be calculated using Equation (14).

$$f_p\left(t\right)=f_0\frac{V_{max}-V\left(t\right)}{V_{max}-V_0}$$

(14)

where $f_0$ is the potential infiltration capacity when the soil water storage equals $V_0$ (L/T).

Similar to the condition of $S_a-P<V_0\le S_a$, for $S_a<V_0\le V_{max}$, the infiltration rate at any given time can be expressed as:

$$f\left(t\right)=\left\{\begin{array}{cc}i\left(t\right)& for i\le f_p\\ f_0\left[1-\frac{V\left(t\right)-V_0}{V_{max}-V_0}\right]& for i>f_p\end{array}\right.$$

(15)

By combining Equations (10), (13) and (15), the modification of the SMA method in the HEC-HMS can be expressed as the following equation:

$$f\left(t\right)=\left\{\begin{array}{cc}{f}_a\left[1-\frac{V\left(t\right)-S_a}{S}\right]& for i>f_p, when S_a-P<V_0\le S_a and V\left(t\right)>S_a\\ f_0\left[1-\frac{V\left(t\right)-V_0}{V_{max}-V_0}\right]& for i>f_p, when S_a<V_0\le V_{max}\\ i\left(t\right)& otherwise\end{array}\right.$$

(16)

2.3.2. Method Combination

In the MCN-TVR method, the MCN-SMA method is utilized to calculate the cumulative rainfall excess and infiltration, while the modified SMA method is employed to compute the temporal variation of the rainfall excess and infiltration and to update the soil water storage. The step-by-step calculation procedures are illustrated in Figure 1. Initially, the MCN-SMA method (Equation (8)) is applied to estimate the cumulative rainfall excess and infiltration for a given rainfall event. If $V_0\le S_a-P$, the infiltration rate equals the rainfall intensity at a time step. If $S_a-P<V_0\le S_a$, Equations (12) and (16) from the modified SMA method are utilized to calculate the rainfall excess and infiltration and to update the soil water storage for the time step. Otherwise, Equations (14) and (16) are used for the calculation. The infiltration capacity at the onset of the rainfall excess generation (i.e., $f_a$ and $f_0$) is initially set to zero and gradually increased until the cumulative rainfall excess estimated by the MCN-SMA method is greater than or equal to that calculated by the modified SMA method.

Compared to the MCN-SMA method, the MCN-TVR method redistributes the rainfall excess and infiltration without requiring new input parameters. The cumulative rainfall excess and infiltration that are calculated using both methods are identical, and the two internal parameters (i.e., $f_a$ and $f_0$) in the MCN-TVR method can be determined through iterative procedures.

2.4. Test of the MCN-TVR Method

2.4.1. Hypothetical Rainfall Events

The performance of the MCN-TVR method in controlling the temporal distributions of the rainfall excess and infiltration was assessed by comparing its simulation results with those derived using the MCN-SMA model and a Green–Ampt model [29] for a series of 24 h SCS Type II rainfall events designed by Brevnova (2001) [48] (Columns 1–8 in

Table 1. 24 h SCS Type II rainfall events and model parameters.

Event No.	Rainfall Depth (cm)	Effective Hydraulic Conductivity (cm/h)	Capillary Suction Head (cm)	Saturated Water Content	Initial Water Content	Soil Texture	CN	S (cm)	Sa (cm)	V0 (cm)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)
1	8	2.450	7.773	0.4533	0.173	Sandy Loam	60.00	16.93	5.59	2.20
2	16	1.226	9.745
3	24	0.653	11.972
4	32	0.524	12.864
5	8	1.269	9.636	0.501	0.332	Silt Loam	70.00	10.89	3.59	1.42
6	16	0.394	14.119
7	24	0.306	15.342
8	32	0.292	15.578
9	8	0.201	17.602	0.464	0.316	Clay Loam	80.00	6.35	2.10	0.83
10	16	0.133	20.150
11	24	0.136	19.970
12	32	0.140	19.805
13	8	0.025	34.826	0.479	0.320	Silty Clay	90.00	2.82	0.93	0.37
14	16	0.027	33.699
15	24	0.029	33.258
16	32	0.029	33.971

). In this study, the rainfall excess and infiltration of the above events were simulated at 0.5 h intervals, with a total of 48 time steps for each event. For the Green–Ampt model, the input parameters consist of the effective hydraulic conductivity, capillary suction head, saturated water content, and initial moisture content. Brevnova (2001) [48] associated the input parameters of the Green–Ampt model with the curve numbers for the 24 h SCS Type II rainfall events (Columns 3–6, Table 1). The input parameters for the MCN-TVR and MCN-SMA models included $S$, $S_a$, and $V_0$. The values of S (Column 9, Table 1) were computed based on their corresponding curve numbers (Column 8, Table 1). According to Michel et al. (2005) [13], $S_a$ (Column 10, Table 1) was recommended to be 0.33$S$. In Brevnova (2001) [48], the initial abstraction coefficient (λ) was set to 0.2; therefore, $V_0$ (Column 11, Table 1) is calculated by:

$$V_0=S_a-I_a=0.13S$$

(17)

2.4.2. Real Applications

The MCN-TVR and MCN-SMA methods were applied to simulate the direct runoff of the Rush River Watershed in North Dakota (Figure 2). The watershed boundary was determined based on the ALOS Global Digital Surface Model [49]. The watershed covers an area of 254.87 km², with an outlet located at the USGS 05060500 Rush River gauging station in Cass County. According to the National Land Cover Database 2011 (NLCD 2011), the watershed is mainly covered by cultivated crops (89.35%), followed by developed area (4.35%) and herbaceous area (2.11%) [50]. The average curve number for the entire watershed was computed based on the 2011 NLCD and STATSGO2 (State Soil Geographic Dataset) soil-type data [50,51]. The time of concentration of the watershed was determined using the lag method [52].

Five rainfall events that occurred between 2019 and 2022 were chosen for evaluating the performances of the MCN-TVR and MCN-SMA models (

Table 2. Rainfall events used for calibration and validation (P₅ = 5-day antecedent rainfall depth; CR = cumulative rainfall in depth; DR = total direct runoff in depth).

Period	Event No.	Simulation Period	Rainfall Event Period	Event Duration (Hours)	P5 (cm)	CR (cm)	DR (cm)
Calibration	1	8 July 2019 19:00–16 July 2019 18:00	8 July 2019 19:00–9 July 2019 20:00	26	1.30	5.59	1.06
	2	14 August 2020 7:00–20 August 2020 21:00	14 August 2020 7:00 –14 August 2020 15:00	9	0.30	7.19	1.04
Validation	3	22 April 2022 4:00–29 April 2022 8:00	22 April 2022 4:00–24 April 2022 14:00	59	0.28	7.26	1.34
	4	29 April 2022 9:00–7 May 2022 20:00	29 April 2022 9:00–30 April 2022 20:00	36	1.32	5.23	0.94
	5	12 May 2022 19:00–18 May 2022 0:00	12 May 2022 19:00–12 May 2202 23:00	5	5.00	1.07	0.30

). Hourly rainfall data for these events were obtained from a North Dakota Agricultural Weather Network station located at latitude 47°01′00″ N, longitude 97°12′50″ W [53]. Hourly discharge data at the outlet of the watershed were acquired from the USGS National Water Information System, situated at latitude 47°12′35″ N, longitude 97°25′50″ W [54]. To estimate the direct runoff, a recursive filter method (Equation (18)) was applied to the discharge data, with a recursive filter parameter of 0.925 [55,56]. For each of the selected rainfall events, the MCN-TVR and MCN-SMA methods were employed to calculate the rainfall excess, and the SCS unit hydrograph method was utilized for simulating the direct runoff.

$$q_d\left(t\right)=\gamma \times q_d\left(t-1\right)+\frac{1+\gamma }{2}\left[q\left(t\right)-q\left(t-1\right)\right]$$

(18)

where $q_d$ is the direct runoff at a time step (L³/T), $q$ is the streamflow at a time step (L³/T), and $\gamma$ is the recursive filter parameter.

Two rainfall events (i.e., Events 1 and 2 in Table 2) were selected to calibrate the MCN-SMA model, and the same parameters were used in the MCN-TVR model. The initial value of S was determined based on the average curve number for the watershed. S_a and V₀ were calculated using Equations (6) and (19). Initially, α and β were set to 0.33 and 0.1, respectively [13,15]. After the calibration process, the two models were applied to simulate the direct runoff for the remaining three rainfall events (i.e., Events 3 to 5) without changing the input parameters.

$$V_0=\beta \sqrt{P_5}S$$

(19)

where $\beta$ is a dimensionless coefficient and $P_5$ is the 5-day antecedent rainfall (L).

2.4.3. Model Performance Evaluation

For the hypothetical simulations, the results obtained from the MCN-SMA and MCN-TVR models for each event were compared to those of the Green–Ampt model. The differences between the MCN-SMA and MCN-TVR models and the Green–Ampt model were quantified using the Nash–Sutcliffe efficiency coefficient (NSE) (Equation (20)) and the root mean standard deviation ratio (RSR) (Equation (21)). For the real simulations, the same statistical metrics were used to evaluate the performances of the MCN-SMA and MCN-TVR models in simulating the direct runoff of all the real rainfall events.

$$\mathrm{N}\mathrm{S}\mathrm{E}=1-\frac{\sum _{i=1}^N{\left(R_i-S_i\right)}^2}{\sum _{i=1}^N{\left(R_i-R_{ave}\right)}^2}$$

(20)

$$\mathrm{R}\mathrm{S}\mathrm{R}=\frac{\sqrt{\sum _{i=1}^N{\left(R_i-S_i\right)}^2}}{\sqrt{\sum _{i=1}^N{\left(R_i-R_{ave}\right)}^2}}$$

(21)

where $R_i$ is the ith reference data (rainfall excess simulated by the Green–Ampt model in the hypothetical simulations or the observed direct runoff in the real simulations), $S_i$ is the ith rainfall excess or direct runoff simulated by the MCN-SMA or MCN-TVR models, $R_{ave}$ is the mean value of the reference data, and $N$ is the dataset size.

3. Results

3.1. Performance in the Hypothetical Event Simulations

3.1.1. Comparison of Rainfall Excess

Figure 3 shows the comparisons of the rainfall excess durations simulated by the MCN-SMA and MCN-TVR models with those simulated by the Green–Ampt model. Overall, the MCN-TVR model exhibited better agreement with the Green–Ampt model compared to the MCN-SMA model. This improvement can be attributed to the infiltration control function utilized in the MCN-TVR model. For most events, especially those involving coarse soils such as sandy loam and silt loam, the MCN-SMA model tended to overestimate the duration of the rainfall excess. This can be explained by the underlying mechanism of the MCN-SMA method, which triggers a rainfall excess once the cumulative rainfall surpasses a threshold value (i.e., S_a) regardless of the actual rainfall intensity. In contrast, the MCN-TVR method incorporates an SMA-based infiltration function to regulate the generation of the rainfall excess. As a result, the durations of the rainfall excess simulated by the MCN-TVR model closely aligned with the results of the Green–Ampt model. For silty clay, which has low permeability, the simulation results of the MCN-SMA model closely matched those of the Green–Ampt model.

Figure 4 illustrates the NSE and RSR values of the rainfall excess simulated by the MCN-SMA and MCN-TVR models compared to the Green–Ampt model. In Figure 4a, the NSE values exhibit an increasing trend when the rainfall depths increased from 8 cm to 32 cm for each soil type. Additionally, finer soils generally yield greater NSE values compared to coarser soils for the same rainfall depth. Conversely, the RSR values follow an opposite trend, as shown in Figure 4b. Therefore, the simulation results of the MCN-SMA model closely match those of the Green–Ampt model when the soil is finer or the rainfall depth is greater. In Figure 4c,d, the NSE and RSR values are consistently above 0.99 and below 0.03, respectively, for all the rainfall events. These results indicate that the MCN-TVR and Green–Ampt models exhibited similar performance in simulating the temporal variations of the rainfall excess. Furthermore, the variations observed in the NSE and RSR in Figure 4c,d highlight the differences in the infiltration control functions utilized by the MCN-TVR and Green–Ampt models.

Figure 5 displays the simulated rainfall excess for four events using the MCN-SMA, MCN-TVR, and Green–Ampt models, providing insights into the variations observed in the NSE and RSR values. The results indicate that both the soil permeability and rainfall depth contributed to the discrepancies among the three models. For each soil type, the disparity between the MCN-SMA and Green–Ampt models decreased when the rainfall depth increased from 8 to 32 cm (e.g., Figure 5a,b). Consequently, the NSE values increased while the RSR values decreased with increasing the rainfall depth (Figure 4a,b) for each soil type. When the rainfall depth was the same, the disparity between the MCN-SMA and Green–Ampt models diminished as the soil permeability decreased (Figure 5). This trend also was found in the variations in the rainfall excess duration: the duration of the rainfall excess simulated by the MCN-SMA model aligned more closely with that simulated by the Green–Ampt model for soils with lower permeability (Figure 3). For all the soil types and rainfall depths, the MCN-TVR model consistently yielded similar simulation results to the Green–Ampt model (as shown in Figure 5). This agreement led to the higher NSE values and lower RSR values in Figure 4c,d, respectively.

3.1.2. Infiltration Control Function

To highlight the infiltration control function of the MCN-TVR model, the cumulative infiltration simulated by the MCN-SMA model was compared with that simulated by the MCN-TVR model for each soil. The NSE and RSR values in Figure 6 illustrate how the difference between the two models varies with the soil type and rainfall depth. For sandy loam, the difference between the models increases as the rainfall depth increases, indicating that the infiltration control function becomes more significant when there is sufficient rainfall. For silt loam and clay loam, the difference initially increases and then decreases with the increasing rainfall depth. In this case, the significance of the infiltration control function exhibits threshold behavior. For silty clay, the difference between the models decreases as the rainfall depth increases, indicating that the infiltration control function has a diminishing effect in this case.

Furthermore, the cumulative infiltrations simulated by the MCN-TVR and Green–Ampt models were compared to highlight the differences in their infiltration control functions. Figure 7a shows that the NSE values decrease with the increasing rainfall depth for each soil type, and the NSE values are generally greater for finer soils compared to coarser soils when the rainfall depth is the same. Conversely, the RSR values exhibit an opposite trend, as shown in Figure 7b. Overall, the difference between the MCN-TVR and Green–Ampt models increases with the increasing rainfall depth and decreasing soil permeability. Moreover, the difference tends to be more obvious for finer soils, such as sandy loam and silty clay.

Figure 8 shows the temporal variations in the normalized cumulative infiltration simulated by the MCN-SMA, MCN-TVR, and Green–Ampt models. The comparison between the MCN-SMA and MCN-TVR models reveals that the major difference occurred at peak rainfall intensity in each 24 h Type II rainfall event (i.e., simulation time = 12 h). For sandy loam, because most rainfall infiltrated into the soil, the infiltration control function of the MCN-TVR model was not significant when the rainfall depth was low (Figure 8a). However, as the rainfall depth increased, the infiltration control function became more significant due to the increase in the rainfall excess duration (Figure 3). For silt loam, the infiltration control function became significant when the rainfall depth increased from 8 cm to 24 cm (Figure 8d,e). However, the infiltration at peak rainfall intensity simulated by the MCN-SMA model decreased slightly from 2.83 cm to 2.71 cm when the rainfall depth increased from 24 cm to 32 cm. As a result, the significance of the infiltration control function decreased with further increases in the rainfall depth (Figure 8e,f). For silty clay, the infiltration at peak rainfall intensity simulated by the MCN-SMA model gradually decreased from 0.71 cm to 0.34 cm (Figure 8g–i). Consequently, the significance of the infiltration control function diminished as the rainfall depth increased.

As previously observed in Figure 7, the difference between the MCN-TVR and Green–Ampt models increased with the increasing rainfall depth for all the soil types. This observation is also evident in Figure 8 and can be attributed to the distinct methodologies employed by the MCN-TVR and Green–Ampt models. With the Green–Ampt method, the infiltration capacity initially decreased rapidly and then gradually stabilized. However, the MCN-TVR method specified an initial infiltration capacity, and the infiltration rate became zero when the soil was fully saturated. As a result, under ponding conditions, the infiltration simulated by the MCN-TVR model initially exceeded that of the Green–Ampt model and then gradually became less than that of the Green–Ampt model (e.g., Figure 8h,i).

3.2. Performance in Real Applications

The MCN-SMA and MCN-TVR models share the same input parameters, including the curve number, α in Equation (6), and β in Equation (19). These parameters were calibrated for Events 1 and 2 (Table 2) to align the simulated cumulative direct runoff with the observed data.

Table 3. Calibrated parameters for the real applications (CN = curve number; α = a dimensionless coefficient that controls the rainfall excess generation; β = a dimensionless coefficient for the calculation of the initial soil water storage).

Calibration Parameters	Range	Initial Value	Calibrated Value
CN	0–100 [11]	68.00	61.00
α	0–1 [15]	0.33	0.26
β	0–1 [15]	0.10	0.17

displays the range, initial values, and calibrated values of these input parameters.

Figure 9 depicts the comparisons of the direct runoff simulated by the MCN-SMA and MCN-TVR models with the observed data for both the calibration and validation events. The corresponding NSE and RSR values are presented in

Table 4. Simulation results for the calibration and validation events.

Period	Event No.	MCN-SMA		MCN-TVR
		NSE	RSR	NSE	RSR
Calibration	1	0.857	0.377	0.880	0.346
	2	0.854	0.380	0.838	0.402
Validation	3	0.109	0.941	0.651	0.589
	4	0.135	0.928	0.830	0.411
	5	0.899	0.317	0.906	0.305

Notes: Very Good: 0.75 < NSE ≤ 1.00, 0.00 < RSR ≤ 0.50; Good: 0.65 < NSE ≤ 0.75, 0.50 < RSR ≤ 0.60; Satisfactory: 0.5 < NSE ≤ 0.65, 0.60 < RSR ≤ 0.70; Unsatisfactory: NSE ≤ 0.50, RSR > 0.70 [57].

. During the calibration phase (i.e., Events 1 and 2), although the rainfall excess values simulated by the two models are different, the simulated direct runoff exhibits similarity. For Events 1 and 2, all the NSE values exceed 0.7 and all the RSR values are below 0.5 for both models, indicating a high level of agreement between the simulation results and the observed data [57]. This can be attributed to the relatively short durations of the rainfall events and the influence of the surface runoff routing process. For a rainfall event with a duration shorter than the time of concentration, the differences in the rainfall excess simulated by the MCN-SMA and MCN-TVR models can be mitigated by the surface runoff routing process. In the Rush River Watershed, with a time of concentration of 35 h, the durations of Events 1 and 2 are 26 and 9 h, respectively. As a result, both models exhibited similar performance for the two events. The discrepancies between the simulated and observed direct runoff are primarily associated with the peak flow characteristics (Figure 9a,b).

In the validation phase (Events 3 to 5), both the MCN-SMA and MCN-TVR models exhibited similar performance in simulating the direct runoff for Event 5 (Figure 9e). However, notable differences were observed in their simulated direct runoff for Events 3 and 4. For Event 5, the NSE and RSR of both models fell in the category of ‘very good’ [57] (Table 4). For Events 3 and 4, the performances of the two models varied significantly. The NSE value of the MCN-SMA model for Event 3 was less than 0.2, indicating poor agreement, while the RSR value was greater than 0.9, indicating substantial overestimation. In contrast, the performances of the MCN-TVR model were ‘satisfactory’ and ‘very good’ for Events 3 and 4, respectively (Table 4). Specifically, the MCN-TVR model captured the general variation trend of the direct runoff for Event 3, albeit with a delayed occurrence of the peak compared to the observed data. However, the MCN-SMA model failed to simulate the peak and the overall variation in the direct runoff (Figure 9c). For Event 4, the direct runoff simulated by the MCN-TVR model exceeded the observed data during the recession period, while the MCN-SMA model overestimated the peak flow and its timing (Figure 9d). The limitations of the MCN-SMA model in simulating Events 3 and 4 can be attributed to its overestimation of the rainfall excess duration. In summary, although both models demonstrated comparable performances for the rainfall events with durations shorter than the time of concentration, the MCN-TVR model outperformed the MCN-SMA model for the rainfall events that exceeded the time of concentration.

4. Discussion

4.1. Comparison with Existing Combined Methods

Integrating the SCS-CN method with other infiltration models to simulate the time-varying rainfall excess has been explored in previous studies (e.g., [44,45,46,47]). However, instead of simply combining the two methods, the MCN-TVR method proposed in this study accounts for the intrinsic linkage between the two SMA-based methods. Consequently, the integration in this study reduces the reliance on observed data and input parameters. For instance, previous studies, such as Gabellani et al. (2008) [44], collected a dataset of rainfall events from 125 stations spanning several decades to establish the relationship between the parameters in the SCS-CN method and a modified Horton method. Similarly, Grimaldi et al. (2013) [46] combined the SCS-CN method with the Green–Ampt method, which resulted in an increased number of input parameters from two to six. In contrast, the MCN-TVR method uses the same set of parameters as the MCN-SMA method, including the initial water storage (V₀), soil water storage before the generation of the rainfall excess (i.e., soil potential retention, S), and the threshold for the initiation of the rainfall excess (S_a). This enhances the applicability of the MCN-TVR method, particularly for ungauged watersheds.

4.2. Performance of the Proposed Method

According to the applications in relation to the hypothetical 24 h SCS Type II rainfall events, the MCN-SMA model failed to simulate the duration and temporal distribution of the rainfall excess, particularly in the case of coarse soils with a relatively light rainfall depth (e.g., sandy loam with a rainfall depth of 8 cm, Figure 3a and Figure 5a). These issues are commonly encountered in both original and revised SCS-CN models due to their basic empirical approaches and limitations [7,8,9,10]. However, these issues are successfully addressed in the MCN-TVR method by incorporating an SMA-based infiltration control function into the MCN-SMA method. Similar improvements have been reported in previous studies that combined the SCS-CN method with other infiltration models [58]. In the real applications, for events with durations shorter than the time of concentration, although there were differences in the simulated rainfall excess between the MCN-SMA and MCN-TVR models, the surface runoff routing process mitigated these differences. However, the performance of the MCN-SMA model was poor for the events with durations exceeding the time of concentration due to its tendency to overestimate the rainfall excess duration, which has been observed in both hypothetical tests and previous studies [46,47]. Although the MCN-TVR model effectively simulated the direct runoff for all the calibration and validation events, it exhibited discrepancies in accurately simulating the peak flow and the peak time for certain events. This can be attributed to the fact that the entire watershed was not subdivided into multiple subbasins during the delineation process [59,60]. In addition, the study area is located in the Prairie Pothole Region of North Dakota, which is characterized by many surface depressions. Previous studies have confirmed the retention [61,62,63] and acceleration [64] effects of surface depressions on the surface runoff. Therefore, the neglect of the dynamic influence of surface depressions could be another factor contributing to the discrepancies.

4.3. Limitations and Future Work

The MCN-TVR method inherits several assumptions from the SCS-CN method, which implies that some limitations associated with the SCS-CN method may also be applicable to the MCN-TVR method. For example, the SCS-CN method assumes that the surface runoff occurs only after the cumulative rainfall exceeds the initial abstraction, and that when the soil water storage reaches zero, the surface runoff is assumed to be equal to the rainfall. Therefore, there is still potential for further improvement in the MCN-TVR method. In addition, this study primarily concentrates on the development of the MCN-TVR method. Only 16 hypothetical rainfall events and 5 real rainfall events were used to evaluate the model. To enhance the reliability and generalizability of the method, it is necessary to conduct additional tests with diverse rainfall types and watershed conditions in the future. Moreover, by incorporating a soil water storage recovery function and utilizing continuous rainfall series from observed data or stochastic rainfall generators (e.g., [65]), the MCN-TVR method can be further applied for continuous rainfall-infiltration simulations in both gauged and ungauged watersheds.

5. Conclusions

In this study, a new MCN-TVR method was developed by combining an SMA-based SCS-CN method (i.e., MCN-SMA) and the SMA method in the HEC-HMS to simulate the rainfall excess and infiltration over time. In the MCN-TVR method, the MCN-SMA method is used to simulate the cumulative rainfall excess and infiltration, and the SMA method in the HEC-HMS is used to control the infiltration process. Unlike other combined methods, the approach used in this study avoids the need for additional input parameters by inherently linking the parameters in the two SMA-based methods. The performance of the MCN-TVR model was evaluated by utilizing 16 hypothetical modeling scenarios, which covered 4 soil types and 4 rainfall depths, and 5 real rainfall events in the Rush River Watershed in North Dakota.

In the hypothetical tests, the simulation results of the MCN-SMA and MCN-TVR models for the 16 24 h SCS Type II rainfall events were compared with those of the Green–Ampt model. Discrepancies in the rainfall excess simulated by the MCN-SMA and Green–Ampt models were observed, especially for coarse soils under relatively light rainfall. However, due to its infiltration control function, the MCN-TVR model exhibited a similar performance to the Green–Ampt model for all the hypothetical rainfall events. The significance of the infiltration control function in the MCN-TVR model varied with the soil permeability and rainfall depth. For sandy loam and silty clay, the significance of the infiltration function had a positive and negative relationship with the rainfall depth, respectively. For silt loam and clay loam, the significance exhibited threshold behavior.

In the real applications, the direct runoff simulated by the MCN-SMA and MCN-TVR models were compared with the observed data. In general, both models effectively simulated the direct runoff for the rainfall events with durations less than the time of concentration, and their performances were comparable due to the surface runoff routing process. However, for the rainfall events with durations longer than the time of concentration, the infiltration control function of the MCN-TVR model proved to be successful in regulating the occurrence of the rainfall excess.

Overall, the MCN-TVR method proposed in this study enhances the capability of the SCS-CN method to simulate the rainfall excess and infiltration over time. Compared to the MCN-SMA method, the simplicity of the MCN-TVR method is not compromised by the integration process. Thus, this method represents a viable alternative for simulating the rainfall excess in various hydrologic applications, particularly for ungauged watersheds.