# A Pragmatic Slope-Adjusted Curve Number Model to Reduce Uncertainty in Predicting Flood Runoff from Steep Watersheds

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## Abstract

**:**

_{IIα}) model used data from 1779 rainstorm–runoff events from 39 watersheds on the Korean Peninsula (1402 for calibration and 377 for validation), with an average slope varying between 7.50% and 53.53%. To capture the agreement between the observed and estimated runoff, the original CN model and its seven variants were evaluated using the root mean square error (RMSE), Nash–Sutcliffe efficiency (NSE), percent bias (PB), and 1:1 plot. The overall lower RMSE, higher NSE, better PB values, and encouraging 1:1 plot demonstrated good agreement between the observed and estimated runoff by one of the proposed variants of the CN model. This plausible goodness-of-fit was possibly due to setting λ = 0.01 instead of 0.2 or 0.05 and practically sound slope-adjusted CN values to our proposed modifications. For more realistic results, the effects of rainfall and other runoff-producing factors must be incorporated in CN value estimation to accurately reflect the watershed conditions.

## 1. Introduction

#### 1.1. The CN Model Framework

_{a}= λS = 0.2S; here, S (mm) is related to CN via

_{0.05}= 1.42S

_{0.2}, and leads to

#### 1.2. Effect of Slope on CN and Runoff Estimation

## 2. Materials and Methods

#### 2.1. Study Area Description and Climate

^{2}) to 911 m (879.10 km

^{2}) above mean sea level. The average slope of the watershed ranges between 7.50% and 53.53%. The majority of the land cover (about 70.50%) is upland forests, followed by 20.26% agricultural land, urban areas (5.22%), grassland (1.56%), and other land cover distribution (2.45%). The dominant soil types are loam and sandy loam, with some fractions of silt loam. The location of watersheds is shown in Figure 1, and other details can be found in [10].

#### 2.2. Data Collection and Interpretation

_{5}) was used to identify the watershed antecedent moisture [10,20,22,39]. The watershed weighted curve number (CN

_{II}) corresponding to the normal conditions were derived from the documented tables on the basis of land use/cover and soil types. The CN

_{I}(CN

_{III}) for dry (wet) conditions were adjusted as recommended by Mishra et al. [40].

#### 2.3. Slope-Adjusted Curve Number Considerations and Development

_{IIα}is the slope-adjusted CN for the antecedent runoff condition representing the watershed normal moisture (ARC-II), CN

_{II}and CN

_{III}are the handbook CN values obtained from watershed characteristics for ARC-II and ARC-III (wet condition), and α is the watershed average soil slope (m/m). The approach of Sharpley and Williams [17] has three empirical parameters—a, b, and c—with optimized values of 1/3, 2, and 13.86, respectively. Their adjusted relationship leads to

_{II}values applicable to a 5% average slope, another study [23] developed the following relationship to adjust CN values for other slopes:

_{II}and S

_{IIα}are the S values for normal moisture condition and slope-adjusted normal moisture conditions, respectively, and α is the watershed mean slope in percentage. The slope-adjusted CN can be obtained from the above equation using the general S and CN interrelationship as it is found in Equation (2). According to Huang et al. [19], the approach in Sharpley and Williams [17] has not been intensively verified in the field. Hence, they adopted a simplified approach for the CN

_{IIα}determination on the basis of their experiments for soil slopes ranging between 0.14 and 1.40, and proposed the following relationship:

_{IIα}using 1402 measured rainfall-runoff events from 31 watersheds and validated this with 377 rainfall–runoff events from the remaining eight watersheds. This is represented as

_{II}, CN

_{III}, and α as the mean slope of a watershed, the proposed slope-adjusted CN (CN

_{IIα}) in its general form is presented as

#### 2.4. Steps of Slope-Adjusted CN Parameter Optimization

- Data pertaining to 39 watersheds in which 1779 rainstorms events occurred provided the known values of the rainstorm events, P; the observed runoff, Q
_{o}; and the optimized CNs for each watershed. The least squares nonlinear orthogonal distance regression objective function in Origin Pro 9.6 software produced the optimized CN values from the following equation.$${{\displaystyle \sum}}_{\mathrm{i}\text{}=1}^{\mathrm{n}}{\left({\mathrm{Q}}_{\mathrm{o}}-{\mathrm{Q}}_{\mathrm{e}}\right)}^{2}={\displaystyle \sum}{\left\{{\mathrm{Q}}_{\mathrm{o}}-\left[\frac{{\left(\mathrm{P}-0.2\text{}\times \left(\raisebox{1ex}{$25400$}\!\left/ \!\raisebox{-1ex}{$\mathrm{CN}$}\right.-254\right)\right)}^{2}}{\mathrm{P}+0.8\text{}\times \left(\raisebox{1ex}{$25400$}\!\left/ \!\raisebox{-1ex}{$\mathrm{CN}$}\right.-254\right)}\right]\right\}}^{2}=\text{}\mathrm{Minimum}$$ - To optimize parameter b in Equation (9), the CNs obtained for the 39 watersheds from Equation (10) were divided into two sets, those of 31 watersheds (1402 rainstorm–runoff events) for calibration and those of 8 watersheds (377 rainstorm-runoff events) for validation. For calibration, the optimized CNs in step 1 were set as the target values challenging the right side of Equation (9) using the nonlinear regression least squares Levenberg–Marquardt algorithm in SPSS v.25 software. To take into account the individual watersheds’ effects on parameter b optimization, the leave-one-out (LOOV) technique was adopted. The average of 31 calibrations repetitions was the value of b = 7.125. This led to recasting the proposed CN
_{IIα}as$${\mathrm{CN}}_{\mathrm{II}\mathsf{\alpha}}=\left(\frac{{\mathrm{CN}}_{\mathrm{III}}-{\mathrm{CN}}_{\mathrm{II}}}{2}\right)\left[1-{\mathrm{e}}^{-7.125\times (\mathsf{\alpha}-0.05)}\right]{+\mathrm{CN}}_{\mathrm{II}}$$

_{III}conversion from CN

_{II}after a suggestion in Mishra et al. [40] gives

_{IIα}relationship has twofold advantages over the previous three suggested relationships. The proposed model has only one parameter to be optimized compared to three in Sharpley and Williams [17] and Williams and Izaurralde [23], and two in Huang et al. [19], if the suggested parameter values are not applicable. Our proposed CN

_{IIα}works within the theoretical limits (i.e., 0 to 100), unlike that in Huang et al. [19], which loses its effectiveness after CN

_{II}= 94.27 using the highest average slope of their watersheds. Similarly, the adjustment in Williams and Izaurralde [23] and Ajmal et al. [10] also fails to follow the CN theoretical limits. The different variants of the CN model are shown in Table 1.

## 3. Statistical Analysis for Model Performance Evaluation

_{oi}and Q

_{ei}are the observed and estimated runoff values for rainstorm events 1 to n, and $\overline{{Q}_{O}}$ is the mean observed runoff in each watershed. The RMSE (0 to ∞) values closer to zero depict more appropriateness of the model to estimate runoff. The NSE (−∞ to 1) illustrates how well a plot of observed vs. estimated runoff fits a 1:1 line (i.e., a perfect fit) [39]. The PB (optimum = 0) describes the average tendency of estimated values to be larger or smaller than their observed ones. Positive (negative) values indicate underestimation (overestimation) bias [44]. It is notable that perfect agreement of the estimated vs. observed data does not essentially indicate a perfect model, because observed data could have uncertainties [39]. However, we are confident about the good quality of the data used in this study. Performance evaluation of different statistical indicators and their suggested ratings [44,45] are given Table 2.

## 4. Results and Discussion

#### 4.1. Models’ Analysis Based on Descriptive Statistics

_{IIα}and lower λ = 0.01 followed by the M6 and M5 models. However, the M6 model was preferred over the M5 due to its practically sound CN

_{IIα}to follow the CN theoretical bounds (0–100). In estimating runoff, the M2 model was not plausibly different from the M1 model. Therefore, lowering λ from 0.2 to 0.05, along with its corresponding CN adjustment using Equation (3), produced only modest changes in the estimated runoff values. Nonetheless, using λ = 0.05 and retaining handbook CN values without adjustment can improve the model’s runoff predictive capability, which is not shown in the assessment but is reflected in the comparison of the M6 and M7 models. The majority of the existing CN model variants underestimated the runoff in different watersheds. Nevertheless, it can be inferred that the watershed CN was not the only important parameter; selecting the proper λ also played a crucial role in estimating accurate runoff. Additionally, the prominent response of CNs to the rainstorm depth was vital in runoff depth estimation [1].

#### 4.2. Model Performance Evaluation in Watersheds Used for Calibration

_{IIα}values proposed by Williams and Izaurralde [23] and Sharpley and Williams [17], we compared only the latter with the other approaches. As mentioned earlier, the RMSE can vary from 0 to ∞, and a value close to zero indicates a nearly perfect fit [15,20,34]. On the basis of the RMSE (mean, median) values, the M2 (23.90, 21.91) and M3 (24.30, 21.90) models exhibited similar but improved runoff estimation compared to the M1 (26.49, 24.02) model. The mean value for all of the statistical indicators is shown on each box plot through connected lines. The M2 model’s enhanced runoff estimation could be attributed to the lower λ = 0.05 [2], whereas the M3 model’s improved predictability could be ascribed to CN

_{Iiα}, which was comparatively higher than the tabulated CN [17]. The M4 model (26.08, 23.78) showed almost no improvement compared to the M1 model. Comparatively better runoff prediction was found for the M5 model (23.53, 21.15), and that of the M6 model (23.23, 20.79) was almost equal in the calibration watersheds. However, the runoff predictive capabilities of the M7 model (21.06, 19.29) and M8 model (18.59, 16.87) were better, as was also evident from their overall RMSE values (Figure 2a). It can be inferred that setting a lower λ and a comparatively higher CN

_{Iiα}, as was the case in model M8, possibly reduces the infiltration and surface water retention capacity.

#### 4.3. Models’ Performance Evaluation in Watersheds Used for Validation

#### 4.4. Overall Performance of Models and Comparison Based on 1:1 Plot

_{IIα}and the handbook CN values (CN

_{IIα}–CN), which varied in the range of 0.73 to 1.46. The corresponding CN differences for the M3, M5, and M6 models were in the range of 1.37 to 6.52, 0.73 to 11.28, and 1.15 to 9.48, respectively. It is notable that the M6 and M8 models used the same CN

_{IIα}values. The M8 model’s outperformance in predicting runoff was probably because of its lower λ = 0.01, as suggested for Korean steep-slope watersheds [10], and its comparatively higher CN

_{IIα}values.

^{2}. The moderately high R

^{2}value supported better runoff prediction capability of the M2 model compared to the M1 model. However, deviation of the observed–estimated runoff best-fit-regression line from the 1:1 plot shows that both the M1 and M2 models underestimated the majority of the runoff events (Figure 4). Although the M2 model R

^{2}value was comparatively high, the runoff predictability of the M1, M2, and M4 models was almost indistinguishable. Nevertheless, the closeness of data points around the 1:1 plot and the higher R

^{2}values of the M5 through M8 models favored these models for comparatively better runoff prediction. The best agreement between the observed and estimated runoff was evidenced by applying the M8 model, as shown in Figure 4. It should be noted that the R

^{2}statistics used for model evaluation could mislead practitioners. These statistics are oversensitive to extremely high values and insensitive to additive and proportional differences between model predictions and measured data [44]. The overall promising results of the M8 model support its suitability for runoff prediction in the steep-slope watersheds. Therefore, the original CN model and the majority of its variants discussed here do not well represent complex watershed characteristics, and thus the abstraction coefficient, the CN values from watershed, and the CN model itself need to be revised for general application. A very recent and comprehensive review by the NRCS Task Group on Curve Number Hydrology [5] also suggested changes to update the handbook and its associated procedures on the basis of lessons learned from global experiences and additional data analyses. To avoid jumps in runoff estimation, the CN model could be made to be more robust by not fixing the initial abstraction coefficient and considering the effect of rainfall as well as the spatial and temporal variability while estimating the watershed CN values.

## 5. Conclusions and Practical Implications

_{IIα}) approach to improve the runoff prediction capability of the CN model in steep-slope watersheds in order to reduce possible uncertainties. The proposed CN

_{IIα}not only followed the theoretical limits (0, 100) [17], but in addition, unlike other existing CN

_{IIα}approaches [10,19,23], it provided a promising runoff prediction capability in the study area. The use of λ = 0.05 in place of λ = 0.2 and their adjusted CN

_{0.05}values modestly improved the CN model runoff predictability, but not well enough for runoff estimation from steep-slope watersheds. On the basis of different performance indicators, we found that the proposed CN

_{IIα}had a positive impact on the CN model runoff prediction. Users of the CN model should know the limitations in its procedures and assumptions because the model produces diverse responses when applied to different land types and watersheds [5]. Assuming a fixed λ value and its associated three fixed values of initial abstraction for dry, normal, and wet conditions are among the major limitations of the original CN model and variants used in this study. The model needs an overhaul for various compelling reasons to circumvent the fixed λ value, as well as unjustified sudden jumps in CN values and its associated estimated runoff. In this era of cutting-edge technology, researchers of different biomes have introduced new parameters in the model to improve its runoff prediction capability. However, inculcating new parameters has increased the model complexity and restricted its application in ungauged watersheds. The CN methodology must be overhauled using experiences from the modern hydrologic engineering without losing the simplicity rule.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Location of watersheds in the study area. The watersheds in italics were used for validation.

**Figure 2.**(

**a**) Root mean square error (RMSE), (

**b**) Nash–Sutcliffe efficiency (NSE), and (

**c**) percent bias (PB) for eight variants of the CN model using data of 30 out of 31 calibration watersheds.

**Figure 3.**(

**a**) RMSE, (

**b**) NSE, and (

**c**) PB for eight variants of the CN model using data of eight validation watersheds.

**Figure 4.**Observed and estimated runoff comparison for eight variants of the CN model using cumulative data of all 39 watersheds.

Parameters | |||
---|---|---|---|

Model Identity | λ | CN (CN_{IIα}) | Model Expression |

M1 | 0.20 | *NEH-4 Tables | Equations (1) and (2) |

M2 | 0.05 | NEH-4 Tables | Equations (1)–(3) |

M3 | 0.20 | Sharpley and Williams [17] | Equations (1), (2) and (5) |

M4 | 0.20 | Huang et al. [19] | Equations (1), (2) and (7) |

M5 | 0.20 | Ajmal et al. [10] | Equations (1), (2) and (8) |

M6 | 0.20 | Proposed | Equations (1), (2) and (12) |

M7 | 0.05 | Proposed | Equations (1)–(3) and (12) |

M8 | 0.01 | Proposed | Equations (1), (2) and (12) |

Performance Rating | NSE [44] | NSE [45] | PB (%) |
---|---|---|---|

Very good | 0.75 < NSE ≤ 1.00 | 0.90 < NSE ≤ 1.00 | −10 < PB < +10 |

Good | 0.65 < NSE ≤ 0.75 | 0.80 ≤ NSE ≤ 0.90 | ±10 ≤ PB < ±15 |

Satisfactory | 0.50 < NSE ≤ 0.65 | 0.65 ≤ NSE < 0.80 | ±15 ≤ PB < ±25 |

Unsatisfactory | NSE ≤ 0.50 | NSE ≤ 0.65 | PB ≥ ±25 |

**Table 3.**Summary statistic of rainfall (P), observed runoff (Q

_{o}), and modeled runoff (M1–M8) in the calibration and validation watersheds.

Calibration Watersheds (1402 Rainstorm–Runoff Events) | ||||||
---|---|---|---|---|---|---|

Parameter/Model | Mean | Minimum | First Quartile (Q1) | Median | Third Quartile (Q3) | Maximum |

P | 80.96 | 12.10 | 39.92 | 59.09 | 98.27 | 519.68 |

Q_{o} | 38.60 | 0.17 | 8.23 | 19.61 | 49.04 | 348.46 |

M1 | 25.57 | 0.00 | 1.49 | 6.13 | 27.03 | 415.63 |

M2 | 23.56 | 0.00 | 1.14 | 7.26 | 25.79 | 383.27 |

M3 | 28.79 | 0.00 | 1.30 | 7.95 | 32.94 | 436.28 |

M4 | 26.06 | 0.00 | 1.52 | 6.31 | 28.33 | 419.65 |

M5 | 30.06 | 0.00 | 1.35 | 8.83 | 35.39 | 443.28 |

M6 | 30.26 | 0.00 | 1.23 | 9.38 | 35.34 | 445.73 |

M7 | 28.98 | 0.00 | 2.54 | 10.77 | 34.57 | 417.11 |

M8 | 39.67 | 0.53 | 7.93 | 20.13 | 49.30 | 458.55 |

Validation Watersheds (377 Rainstorm–Runoff Events) | ||||||

P | 75.22 | 20.52 | 40.97 | 57.05 | 86.95 | 376.86 |

Q_{o} | 35.03 | 0.24 | 8.30 | 19.10 | 43.20 | 364.38 |

M1 | 22.04 | 0.00 | 1.48 | 6.35 | 20.35 | 294.27 |

M2 | 19.85 | 0.00 | 0.85 | 5.55 | 19.93 | 265.59 |

M3 | 24.75 | 0.00 | 1.52 | 6.27 | 25.99 | 309.31 |

M4 | 22.49 | 0.00 | 1.39 | 6.63 | 21.48 | 296.26 |

M5 | 26.48 | 0.00 | 2.03 | 7.87 | 30.12 | 309.72 |

M6 | 26.07 | 0.00 | 1.71 | 6.66 | 29.04 | 314.48 |

M7 | 24.98 | 0.00 | 2.10 | 9.43 | 26.71 | 293.91 |

M8 | 34.77 | 0.87 | 7.70 | 17.91 | 40.12 | 325.07 |

**Note:**The highlighted values show the good agreement between the observed and the estimated runoff.

M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | |
---|---|---|---|---|---|---|---|---|

Performance Criteria | NSE [44] | |||||||

0.75 < NSE ≤ 1.00 | 14 | 15 | 14 | 14 | 14 | 14 | 20 | 30 |

0.65 < NSE ≤ 0.75 | 3 | 10 | 7 | 3 | 10 | 12 | 13 | 6 |

0.50 < NSE ≤ 0.65 | 10 | 9 | 13 | 13 | 11 | 9 | 4 | 2 |

NSE ≤ 0.50 | 12 | 5 | 5 | 9 | 4 | 4 | 2 | 1 |

NSE [45] | ||||||||

0.90 < NSE ≤ 1.00 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 5 |

0.80 ≤ NSE ≤ 0.90 | 6 | 12 | 12 | 8 | 11 | 11 | 11 | 20 |

0.65 ≤ NSE < 0.80 | 10 | 12 | 8 | 8 | 11 | 13 | 19 | 11 |

NSE ≤ 0.65 | 22 | 14 | 18 | 22 | 15 | 13 | 6 | 3 |

PB (%) | ||||||||

−10 < PB < +10 | 1 | 1 | 5 | 1 | 5 | 6 | 6 | 19 |

±10 ≤ PB < ±15 | 0 | 0 | 3 | 0 | 6 | 5 | 8 | 9 |

±15 ≤ PB < ±25 | 10 | 11 | 12 | 10 | 13 | 12 | 12 | 7 |

PB ≥ ±25 | 28 | 27 | 19 | 28 | 15 | 16 | 13 | 4 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ajmal, M.; Waseem, M.; Kim, D.; Kim, T.-W.
A Pragmatic Slope-Adjusted Curve Number Model to Reduce Uncertainty in Predicting Flood Runoff from Steep Watersheds. *Water* **2020**, *12*, 1469.
https://doi.org/10.3390/w12051469

**AMA Style**

Ajmal M, Waseem M, Kim D, Kim T-W.
A Pragmatic Slope-Adjusted Curve Number Model to Reduce Uncertainty in Predicting Flood Runoff from Steep Watersheds. *Water*. 2020; 12(5):1469.
https://doi.org/10.3390/w12051469

**Chicago/Turabian Style**

Ajmal, Muhammad, Muhammad Waseem, Dongwook Kim, and Tae-Woong Kim.
2020. "A Pragmatic Slope-Adjusted Curve Number Model to Reduce Uncertainty in Predicting Flood Runoff from Steep Watersheds" *Water* 12, no. 5: 1469.
https://doi.org/10.3390/w12051469