# The Revised Curve Number Rainfall–Runoff Methodology for an Improved Runoff Prediction

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## Abstract

**:**

_{a}and S), given by I

_{a}= 0.2S. However, instead of using the antilog equation in the power form (I

_{a}= S

^{0.2}) for simplification, the I

_{a}= 0.2S correlation was used to formulate the current SCS-CN rainfall–runoff model. To date, researchers have not challenged this potential oversight. This study reevaluated the CN model by testing its reliability and performance using data from Malaysia, China, and Greece. The results of this study showed that the CN runoff model can be formulated and improved by using a power correlation in the form of I

_{a}= S

^{λ}. Nash–Sutcliffe model efficiency (E) indexes ranged from 0.786 to 0.919, while Kling–Gupta Efficiency (KGE) indexes ranged from 0.739 to 0.956. The I

_{a}to S ratios (I

_{a}/S) from this study were in the range of [0.009, 0.171], which is in line with worldwide results that have reported that the ratio is mostly 5% or lower and nowhere near the value of 0.2 (20%) originally suggested by the SCS.

## 1. Introduction

_{a}= The initial abstraction amount (mm).

_{a}= λS, in which λ is the initial abstraction coefficient ratio to relate the I

_{a}and S values. As in 1954, there is a limited amount of rainfall data; thus, this equation was flimsily justified based on daily rainfall and runoff data. The only surviving evidence is documented in a different edition of the Natural Resources Conservation Services’s (NRCS) National Engineering Handbook, Section 4 (NEH-4) [8,9,10].

_{a}= λS. Thus, the linear correlation with the form of I

_{a}= 0.2S was introduced to simplify the SCS’s CN model and the calculations. In NRCS NEH-4, there is a note mentioning that the reason for proposing the λ as 0.2 is because 50% of the data points fall between the range of 0.095 < λ < 0.38 [10]. With the substitution of I

_{a}= 0.2S into the SCS’s CN model, Equation (1) will eventually be simplified into the conventional SCS runoff forecast model. The conventional (simplified) SCS rainfall–runoff prediction model is as follows:

_{a}= λS was used to formulate and simplify Equation (1) into (2). This can be a mathematical error or an oversight in the past which leads to the inaccuracy of the model’s runoff predictive ability. Since the SCS’s field data points were plotted on a log–log scale graph, the relation between the two key variables (S and I

_{a}) should be in the power correlation instead of a linear correlation.

_{a}and S and assessed the impact to its rainfall–runoff mechanism yet. Equation (2) exists in all hydrology textbooks; it also led to curve number derivation and has been applied in many designs while many types of software incorporated I

_{a}= 0.2S in their algorithms. If I

_{a}and S correlate otherwise, it will change the whole model and the way SCS curve number theory is taught. If the fundamental core theory of the SCS curve number rainfall–runoff model changed, the model and all its related downstream information will have to be re-derived.

## 2. Materials and Methods

#### 2.1. Validity of the Linear Correlation (I_{a} = 0.2S) and Introduction of I_{a} = S^{λ}

_{a}= 0.2S” based on the 1954 SCS original dataset and rejected the validity of the linear correlation model of the form I

_{a}= 0.2S, at alpha level = 0.01. Instead of a constant of λ = 0.2, the study proved that the best linear model for the 1954 SCS original dataset should be I

_{a}= 0.112S [16], which is also in line with the best regressed linear equation of I

_{a}= 0.111S reported in another study [17]. As the linear correlation of I

_{a}= 0.2S was reported as not statistically significant when reevaluated using the original 1954 SCS field data, the SCS rainfall–runoff predictive model Equation (1) should be recalibrated before it is used.

_{a}= 0.112S or I

_{a}= 0.111S in the past two studies [16,17], the best fitting model for the same datasets in a normal scale should be in the power model with the form of I

_{a}= S

^{0.112}or I

_{a}= S

^{0.111}to simplify Equation (1).

_{a}and S due to the SCS’s incomplete documentation and missing datasets. Thus, this study adopted the model calibration technique developed in our past studies and thoroughly assessed the use of the power correlation equation in the general form of I

_{a}= S

^{λ}to simplify the SCS’s CN base model, Equation (1), and recalibrate the runoff prediction model based on the regional rainfall–runoff dataset using inferential statistics.

#### 2.2. Study Site and Models’ Performance

_{a}= λS to determine the value of S using the values of P and Q data pairs with a closed-form expression for S [7,14,15,21,29]. However, with the introduction of the power model I

_{a}= S

^{λ}, the general formula for S does not have a closed-form expression in this study, so a numerical analysis technique will be applied to obtain the value of S. (The derivation and simplification of the general formula for S based on the power regressed model are summarized in Appendix A.) As a result, the simplified S general formula obtained was as follows:

_{λ}, according to the P and Q data pairs of the respective study sites. Besides the S

_{λ}values obtained from Equation (6) via the numerical analysis technique, the S

_{0.2}values for each of the P–Q data pairs for all the study sites were also determined based on the S general equation that was introduced in the previous study, where the linear correlation was applied, with the form of I

_{a}= 0.2S to simplify and formulate the current SCS runoff predictive model [6,7,14,15,16,17,21,29]. (The S general Equation that was used to obtain the S

_{0.2}values was summarized in Appendix B.) The S general equation that was used to obtain the S

_{0.2}values was shown as follows:

_{λ}and S

_{0.2}values were obtained with Equations (6) and (7) according to the corresponding P and Q data pairs, the best correlation between S

_{λ}and S

_{0.2}and the corresponding best correlation equation can be mapped via the SPSS model regression module for a study site. When the optimal λ value is different from the conventional value of λ = 0.2, a correlation between the newly found λ value and 0.2 must be used to calculate the curve number again [7,11,14,16,21,29]. Therefore, it is important to identify the best correlation between S

_{λ}and S

_{0.2}and then convert the S

_{λ}back to the equivalent S

_{0.2}value to derive the conventional SCS curve number (CN

_{0.2}) value that SCS practitioners are familiar with for routine application. The best correlation between S

_{λ}and S

_{0.2}is determined by the highest adjusted R square value (R

^{2}

_{adj}), the lowest standard error of the estimate, and a model with a significant p value, which can be determined in SPSS through the regression module [31,32]. (The approach is summarized in Appendix C.)

_{λ}and S

_{0.2}for the Kerayong watershed, Malaysia, was identified in SPSS as follows:

_{0.2}= 3.223 S

_{0.199}

^{0.355}

_{0.2}= Conventional Curve Number, Curve number of λ = 0.2.

_{0.2}= Total abstraction amount of λ = 0.2 (mm).

_{λ}and S

_{0.2}will be very crucial as it will directly affect the overall performance of the rainfall–runoff model derived from the power regressed model. The correlation equation is also the key to convert S

_{λ}back to its equivalent S

_{0.2}value for the calculation of CN

_{0.2}for SCS practitioners [29,32].

## 3. Results

#### 3.1. Study Site and Models’ Performance

_{a}= S

^{λ}) within the 99% Confidence Interval of both S and λ values, accordingly [7,21,29,33,34]. Once the optimal S value and λ value were obtained for each corresponding dataset, the runoff prediction model can be formulated (refer to Appendix C section for model formulation guide) and the performance for each model can be analyzed. Table 1 shows the inferential statistics, which include the Nash Sutcliffe index (E), optimal S (S), optimal λ, initial absorption (I

_{a}), and the ratio of I

_{a}to S, for each corresponding dataset.

_{a}) for each corresponding dataset was determined using the newly proposed power correlation of I

_{a}= S

^{λ}. The ratios of I

_{a}to S values were calculated to benchmark against past study results. The I

_{a}to S ratios (I

_{a}/S) for each study site were in the range of [0.009, 0.171], which is in line with past results that reported that the ratio of I

_{a}to S was mostly 5% or lower and nowhere near the value of 0.2 (20%) as initially suggested by the SCS to simplify the SCS runoff predictive framework into Equation (2) [6,7,11,12,13,14,15,16,17,21,22,29,32,33,34].

_{λ}to S

_{0.2}for each dataset was modelled via SPSS. To measure the fitness of the correlation equation to the datasets, the adjusted coefficient of determination (R

^{2}

_{adj}) for each corresponding equation was also identified, together with the ρ-value. The SCS practitioner(s) was taught to choose the CN value from the NEH handbook according to the land use or land cover condition of the watershed and to calculate the S value from Equation (9) or (10) directly; however, it is no longer applicable as the linear correlation of I

_{a}= 0.2S was proven invalid [7,16,17,21,29,33,34]. Thus, the best-fitted correlation equation between S

_{λ}and S

_{0.2}proposed herewith should be used by the SCS practitioners, in which the optimal S value of each dataset must be converted back to the equivalent S

_{0.2}value via the mapped correlation equation (Table 2) prior to calculate the CN

_{0.2}value of a watershed [7,21,29,32,33,34] from Equation (9) or (10), as shown in the aforementioned example to derive Equations (8) and (11). SCS practitioners no longer have to decide and choose the CN value from any handbook; they can derive the CN value of a specific watershed according to the regional rainfall–runoff conditions.

#### 3.2. Derivation of Curve Number and Its Confidence Interval

_{λ}and S

_{0.2}were determined using SPSS in this study. The BCa 99% CI for the corresponding CN

_{0.2}value of each dataset can then be calculated once the respective correlation equation between S

_{λ}and S

_{0.2}from Table 2 is substituted into Equation (9) or (10) to derive the equivalent CN

_{0.2}range (refer to derivation steps in Appendix C), as shown in Table 3. Table 2 lists the best regressed correlation equation identified in SPSS while Table 3 summarizes the Bootstrapping BCa 99% CI for the S value and the calculated CN

_{0.2}value range of each dataset in this study.

## 4. Discussion

#### 4.1. Validity of the Linear Correlation (I_{a} = 0.2S) and Introduction of Power Regression (I_{a} = S^{λ})

_{a}= 0.2S was statistically invalid even to its own dataset [16,17]. In 1954, the SCS introduced the linear correlation (I

_{a}= 0.2S) to simplify the SCS rainfall–runoff model into the conventional SCS runoff prediction Equation (2) that is in use until today. However, worldwide studies reported runoff prediction accuracy and consistency issues with the model [6,7,11,12,13,14,15,16,17,21,22,29,30,31,32,33,34]. This simplification may have introduced an inherent risk of Type 2 error into the SCS CN model. (A Type 2 error, also known as a false negative, occurs when a statistical test fails to reject the null hypothesis when it is actually false.) In the context of the SCS CN model, this means that the model may incorrectly predict the runoff amount. The risk of Type 2 error in the SCS CN model has potential implications for downstream derivations and extended applications of the model in hydrology. Researchers and practitioners who use the SCS CN model should be aware of this risk and carefully evaluate the assumptions and limitations of the model in order to ensure the accuracy and reliability of their results.

_{a}directly to simplify its model framework without taking the antilog form [8,9,10]. In the authors’ opinion, it is possible that the antilog correlation form was not used in the past due to technical limitations in simplifying Equation (1) with I

_{a}= S

^{0.2}without the use of computers, or the fundamental mathematical mistake was overlooked. Two past studies reassessed the correlation between S and I

_{a}according to the 1954 SCS original dataset and reported that the best correlation equation should be I

_{a}= 0.111S or I

_{a}= 0.112S on the log–log scale graph [16,17]. As such, the antilog form or the power regression equations of I

_{a}= S

^{0.111}or I

_{a}= S

^{0.112}will be better correlation equations to the SCS original dataset as compared to what the SCS had proposed (I

_{a}= 0.2S). This finding necessitated the development of a revised form of the SCS rainfall–runoff model that incorporates the power correlation between I

_{a}and S in the general form of I

_{a}= S

^{λ}to simplify Equation 1 and rederive a new SCS runoff predictive model. The calibration of the SCS rainfall–runoff model (Equation (1)) can be performed based on the newly proposed power regressed model according to the regional or watershed specific rainfall–runoff dataset to improve the runoff prediction accuracy.

_{a}= λS equation according to watershed-specific rainfall–runoff datasets using inferential statistics and reported that λ = 0.2 was not statistically significant at the alpha = 0.05 level for modeling runoff [7,16,21,29,33,34], the novelty of this study is to calibrate the SCS CN model with I

_{a}= S

^{λ}and formulate new CN runoff prediction models. To date, no other researchers have attempted to do this.

_{a}= S

^{λ}to calibrate and formulate the runoff predictive model. It is noteworthy to highlight that although Equation (2) was able to achieve attractive E and KGE index values (Appendix D, Table A2) in certain datasets, the λ value of 0.2 was not significant even at α = 0.05 level for any dataset in this study. As such, the conventional SCS simplified model (Equation (2)) is not statistically significant for modelling any rainfall–runoff in this study. All SCS models have a tendency to underpredict runoff in this study. Additionally, the I

_{a}value of all SCS models violated a requirement from the SCS CN theory which states that there will be no runoff when the rainfall depth (P) is less than the I

_{a}value. Table A2 tabulates the total count of rainfall–runoff data pairs where the recorded P value is less than the SCS model’s I

_{a}value. In contrast, none of the newly formulated power models have this issue.

#### 4.2. Application of the Power Regression Model at Different Study Sites

_{a}to S (I

_{a}/S) for all the study sites were in the range of [0.009, 0.171] (Table 1) which is in line with past results whereby the ratios were mostly 5% or lower and nowhere near to the value of 0.2 (20%) as initially suggested by the SCS [6,7,11,12,13,14,15,16,17,21,22,29,32,33,34]. It is in line with the results of the largest scale of field tests on the SCS CN model, which analyzed more than half a million rainfall events across 24 states in the USA and reported that I

_{a}/S values were mostly below 5% to achieve better runoff modeling results in American watersheds [17,29,32]. Using the revised CN model may allow for more accurate and reliable predictions of runoff in different types of soils and hydrological conditions. This can be particularly useful for water resource management, as it allows for more informed decision making about the availability of water for various uses and the potential for flood hazards. It is important for researchers and practitioners to be aware of the revised form of the SCS rainfall–runoff model and to consider its use in their work.

#### 4.3. Application of Machin-Learning Techniques to the Rainfall Runoff Model

## 5. Conclusions

- The linear correlation hypothesis (I
_{a}= 0.2S) proposed by the SCS was found to be statistically invalid, even in the 1954 original dataset. Therefore, it is important to revise the current conventional SCS runoff prediction model to better prepare for the challenges posed by climate change conditions. The use of I_{a}= 0.2S to simplify equation (1) into the conventional SCS runoff model (equation 2) may have been an oversight in 1954. The power correlation equation (I_{a}= S^{0.111}or I_{a}= S^{0.112}) should be used to simplify the 1954 SCS runoff model equation (1) and derive the runoff predictive model, as the original SCS dataset was plotted on a log–log graph. - The newly proposed power regression model (I
_{a}= S^{λ}) demonstrates good runoff prediction ability at different study sites, including those in Malaysia, Greece, and China. The calibrated SCS rainfall–runoff model based on the proposed power regression model is promising for modelling the rainfall–runoff characteristics of different watersheds in different countries. The ratio of I_{a}to S for all study sites is mostly 5% or lower, which is in line with past worldwide study results and much lower than the value of 0.2 (20%) as suggested by the SCS. - There is concern about the use of the original form of the SCS CN model in education, as it may teach students an oversimplified and potentially inaccurate model for predicting runoff, which could have serious consequences for fields such as water resource management, environmental science, and civil engineering. There is also concern about the widespread use of the original form of the model in educational materials, such as textbooks, software, and government agency handbooks and trainings, which may perpetuate the use of an oversimplified and potentially inaccurate model for predicting runoff.
- The proposed methodology has several limitations, including the need for a minimum sample size of at least 20 data pairs to obtain meaningful inferential results and the reliance on the bootstrap BCa method to produce confidence intervals for key variable optimization and the formulation of a new runoff predictive model. The statistical software used must also include the bootstrap BCa method as an option. There are several areas of research that have not been explored in this manuscript due to financial and time constraints. Future studies will examine the potential impacts of the proposed model on flood risk, financial losses caused by flooding, and its potential for downstream development and wider application.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{a}= The initial abstraction amount (mm).

_{a}and S was applied with the form of I

_{a}= 0.2S. The conventional SCS CN rainfall–runoff model was defined as followed:

_{a}= S

^{λ}) had substituted the linear regressed model (I

_{a}= λS); thus, the simplification of the SCS CN model will be carried out, as the attempt to obtain a closed form for the S general formula, similar to the previous study. In this derivation, the power regressed model is included in the SCS CN rainfall–runoff model and followed by the simplification of the model to obtain the closed-form S general formula for the new calibrated runoff model:

## Appendix B

_{a}= λS is still intact with the SCS CN rainfall–runoff model. The rearrangement of the SCS CN rainfall runoff model, eventually the S general formula, was as follows [7]:

_{0.2}values, the λ for Equation (A2) was set to be 0.2. When λ = 0.2, eventually the Equation (A2) can be simplified as follows:

_{0.2}values based on the P and Q values accordingly. Equation (A3) was also the same as the previous research study from Hawkins [6,7,17].

_{0.2}can be obtained for each P–Q data pair from all the study sites. Then, only the correlation between the S

_{0.2}and S

_{λ}for each study site can be determined via SPSS, and, later, the correlation equation will be substituted back into the SCS CN curve number model to derive the CN

_{0.2}values for each dataset. (Equation (A3) is also known as Equation (7) in this article.)

## Appendix C

_{0.2}value has to be adjusted as well as shown below:

_{0.2}value derivation and the calibration steps were summarized as follows:

- Given that the effective rainfall (P
_{e}) = P − I_{a}and I_{a}= λS, the SCS runoff model ($\mathrm{Q}=\frac{\mathrm{P}-{\mathrm{I}}_{\mathrm{a}}}{\mathrm{P}-{\mathrm{I}}_{\mathrm{a}}+\mathrm{S}}$) can be rearranged as follows:$$\mathrm{Q}=\frac{{\mathrm{P}}_{\mathrm{e}}{}^{2}}{{\mathrm{P}}_{\mathrm{e}}+\mathrm{S}}$$ - For each P–Q data pair (P
_{i}, Q_{i}), calculate corresponding λ_{i}and S_{i}values via numerical analysis techniques. - Perform bootstrap BCa procedure, Shapiro–Wilk, and Kolmogorov–Smirnov tests in SPSS (version 18.0 or an equivalent statistics software) for (λ
_{i}, S_{i}) to check on the normality of both λ_{i}and S_{i}.Check the Shapiro–Wilk and Kolmogorov–Smirnov test results of S_{i}and λ_{i}to see whether it is normally distributed or not:- (a)
- If yes, refer to the mean BCa confidence interval for S
_{i}and λ_{i}optimization. - (b)
- Otherwise, refer to the median BCa confidence interval for S
_{i}and λ_{i}optimization.

- Based on the normality of λ
_{i}and S_{i}, calculate the optimal value for λ_{i}and S_{i}, denoted by λ_{optimum}and S_{optimum}. - Substitute the λ
_{optimum}and S_{optimum}value into $\mathrm{Q}=\frac{\mathrm{P}-{\mathrm{I}}_{\mathrm{a}}}{\mathrm{P}-{\mathrm{I}}_{\mathrm{a}}+\mathrm{S}}$, where I_{a}= S^{λ}, to formulate the new SCS CN rainfall–runoff model and calibrate the model according to the given P–Q datasets. - Given (P
_{i}, Q_{i}) data pairs, compute S_{i}values and λ_{optimum}with ${(2{\mathrm{S}}_{\mathrm{i}}{}^{\lambda}+\mathrm{Q}-2\mathrm{P})}^{2}=4{\mathrm{S}}_{\mathrm{i}}\mathrm{Q}+{\mathrm{Q}}^{2}$ (Equation (A1) or (6)) via Excel’s numerical iteration. To date, there is no closed form for the S general equation formula. - Given (P
_{i}, Q_{i}) data pairs and λ = 0.2, compute S_{0.2i}values with Equation (A3) or (7). - Correlate S
_{0.2i}and S_{i}to obtain a correlation equation between S_{0.2i}and S_{i}via SPSS (or an equivalent statistics software). - Substitute the S correlation equation from step 9 into the SCS curve number formula ${\mathrm{CN}}_{0.2}=\frac{25,400}{{\mathrm{S}}_{0.2}+254}$ to derive CN
_{0.2}value for the watershed of interest.

## Appendix D

Datasets | New SCS (Power) Model | Remark | ||
---|---|---|---|---|

E | BIAS | KGE | ||

DID HP 11, Malaysia | 0.823 | 2.347 | 0.805 | I_{a} < min rainfall data |

DID HP 27, Malaysia | 0.919 | 0.647 | 0.956 | I_{a} < min rainfall data |

DID HP 11+27, Malaysia | 0.875 | 2.531 | 0.863 | I_{a} < min rainfall data |

Kayu Ara, Malaysia | 0.810 | 0 | 0.865 | I_{a} < min rainfall data |

Kerayong, Malaysia | 0.888 | 0 | 0.818 | I_{a} < min rainfall data |

Attica, Greece | 0.786 | 0 | 0.798 | I_{a} < min rainfall data |

Wang Jia Qiao, China | 0.795 | 0 | 0.739 | I_{a} < min rainfall data |

Datasets | SCS Model (Equation (2)) | Remark | ||
---|---|---|---|---|

E | BIAS | KGE | ||

DID HP 11, Malaysia | 0.839 | −6.104 | 0.759 | I_{a} > 60 rainfall data (12.7%) |

DID HP 27, Malaysia | 0.910 | −4.085 | 0.895 | I_{a} > 6 rainfall data (2.6%) |

DID HP 11+27, Malaysia | 0.879 | −4.451 | 0.863 | I_{a} > 68 rainfall data (9.7%) |

Kayu Ara, Malaysia | 0.805 | −1.162 | 0.831 | I_{a} > 6 rainfall data (6.5%) |

Kerayong, Malaysia | 0.891 | −0.885 | 0.843 | I_{a} > 2 rainfall data (2.8%) |

Attica, Greece | 0.780 | −1.846 | 0.784 | I_{a} > 5 rainfall data (6.6%) |

Wang Jia Qiao, China | 0.482 | 1.586 | 0.418 | I_{a} > 4 rainfall data (13.8%) |

_{a}< rainfall depth (P).

_{a}value of all SCS models violated a requirement from the SCS CN theory that I

_{a}< the rainfall depth (P). Table A2 tabulates the total count of rainfall–runoff data pairs where the recorded P value is less than the SCS model’s I

_{a}value. In contrast, none of the newly formulated power models have this issue. The conventional SCS runoff model (Equation (2)) has a model bias ranging from -6.104 to 1.586 in this study, indicating inconsistency in its runoff predictions with an underprediction concern. In the context of climate change, the conventional SCS runoff predictive model is not reliable for accurately estimating runoff amounts.

## Appendix E

**Table A3.**The geographic coordinates of each catchment area that was included in the study sites from three different countries.

No. | Catchment Locations | Latitude | Longitude |
---|---|---|---|

1 | Attica, Greece | 38° 4′ N | 23° 50′ E |

2 | Wangjiaqiao, China | 31° 8′ N | 111° 41′ E |

Peninsula Malaysian Catchments | |||

1 | Kayu Ara, Malaysia | 3° 8′ N | 101° 37′ E |

2 | Kerayong, Malaysia | 3° 6′ N | 101° 42′ E |

3 | Parit Madirono di Weir | 1° 41′ N | 103° 16′ E |

4 | Sg. Johor di Rantau Panjang | 1° 46′ N | 103° 44′ E |

5 | Sg. Sayong di Jam. Johor Tenggara | 1° 48′ N | 103° 40′ E |

6 | Sg. Kahang di Bt.26 Jln. Kluang | 2° 15′ N | 103° 35′ E |

7 | Sg. Lenggor di Bt.42 Kluang/Mersing | 2° 15′ N | 103° 44′ E |

8 | Sg. Muar di Buloh Kasap | 2° 33′ N | 102° 45′ E |

9 | Sg. Serting di Jam.Padang Gudang | 3° 06′ N | 102° 28′ E |

10 | Sg. Triang di Jam. Keretapi | 3° 14′ N | 102° 24′ E |

11 | Sg. Bentong di Kuala Marong | 3° 30′ N | 101° 54′ E |

12 | Sg. Lepar di Jam. Gelugor | 3° 41′ N | 102° 58′ E |

13 | Sg. Kuantan di Bukit Kenau | 3° 55′ N | 103° 03′ E |

14 | Sg. Lipis di Benta | 4° 01′ N | 101° 57′ E |

15 | Sg. Cherul di Ban Ho | 4° 08′ N | 103° 10′ E |

16 | Sg. Kemaman di Rantau Panjang | 4° 16′ N | 103° 15′ E |

17 | Sg. Dungun di Jam. Jerangau | 4° 50′ N | 103° 12′ E |

18 | Sg. Berang di Menerong | 4° 56′ N | 103° 03′ E |

19 | Sg. Telemong di Paya Rapat | 5° 10′ N | 102° 54′ E |

20 | Sg. Lebir di Kg. Tualang | 5° 16′ N | 102° 16′ E |

21 | Sg. Nerus di Kg. Bukit | 5° 17′ N | 102° 55′ E |

22 | Sg. Chalok di Jam. Chalok | 5° 26′ N | 102° 50′ E |

23 | Sg. Lanas di Air Lanas | 5° 47′ N | 101° 53′ E |

24 | Sg. Besut di Jambatan Jerteh | 5° 44′ N | 102° 29′ E |

25 | Sg. Pelarit di Titi Baru | 6° 35′ N | 100° 12′ E |

26 | Sg. Buloh di Kg. Batu Tangkup | 6° 33′ N | 100° 17′ E |

27 | Sg. Kulim di Ara Kuda | 5° 26′ N | 100° 30′ E |

28 | Sg. Kerian di Selama | 5° 13′ N | 100° 41′ E |

29 | Sg. Plus di Kg. Lintang | 4° 56′ N | 101° 06′ E |

30 | Sg. Raia di Keramat Pulai | 4° 32′ N | 101° 08′ E |

31 | Sg. Kampar di Kg. Lanjut | 4° 20′ N | 101° 06′ E |

32 | Sg. Bidor di Bidor Malayan Tin Bhd. | 4° 04′ N | 101° 14′ E |

33 | Sg. Sungkai di Sungkai | 3° 59′ N | 101° 18′ E |

34 | Sg. Slim At Slim River | 3° 49′ N | 101° 24′ E |

35 | Sg. Bernam di Tanjung Malim | 3° 40′ N | 101° 31′ E |

36 | Sg. Selangor di Rasa | 3° 30′ N | 101° 38′ E |

37 | Sg. Gombak di Damsite | 3° 14′ N | 101° 42′ E |

38 | Sg. Batu di Kg. Sg. Tua | 3° 16′ N | 101° 41′ E |

39 | Sg. Lui di Kg. Lui | 3° 10′ N | 101° 52′ E |

40 | Sg. Langat di Dengkil | 2° 59′ N | 101° 47′ E |

41 | Sg. Linggi di Sua Betong | 2° 40′ N | 101° 55′ E |

42 | Sg. Melaka di Pantai Belimbing | 2° 20′ N | 102° 15′ E |

43 | Sg. Kesang di Chin Chin | 2° 17′ N | 102° 29′ E |

44 | Sg Sembrong di Bt 2 Air Hitam, Yong Peng | 2° 4′ N | 103° 22′ E |

45 | Sg Segamat di Segamat | 2° 31′ N | 102° 51′ E |

46 | Sg Sayong di Johor Tenggara | 1° 48′ N | 103° 35′ E |

47 | Sg Muar di Bt 57 Jln GemasRompin | 2° 25′ N | 102° 30′ E |

48 | Sg Lepar di Jam Gelugor | 3° 43′ N | 102° 56′ E |

49 | Sg Lenggor di Bt 42 KluangMersing | 2° 12′ N | 103° 41′ E |

50 | Sg Kuantan di Bkt Kenau | 3° 53′ N | 103° 8′ E |

51 | Sg Kepis di Jam Kayu Lama | 2° 41′ N | 102° 20′ E |

52 | Sg Kemaman di Rantau Panjang | 4° 15′ N | 103° 16′ E |

53 | Sg Kecau di Kg Dusun | 4° 22′ N | 102° 6′ E |

54 | Sg Kahang di Bt 26 Jln Kluang | 2° 10′ N | 103° 31′ E |

55 | Sg Johor di Rantau Panjang | 1° 37′ N | 103° 54′ E |

56 | Sg Cherul di Ban Ho | 4° 10′ N | 103° 8′ E |

57 | Sg Berang di Menerong | 4° 57′ N | 103° 0′ E |

58 | Sg Bentong di Jam K Marong | 3° 31′ N | 101° 55′ E |

59 | Sg Bekok di Bt 77 Jln Yong Peng/Labis | 2° 7′ N | 103° 6′ E |

60 | Sg Sungkai di Sungkai | 4° 2′ N | 101° 18′ E |

61 | Sg Raia di Keramat Pulai | 4° 35′ N | 101° 14′ E |

62 | Sg Pari di Jln Silibin, Ipoh | 4° 36′ N | 101° 4′ E |

63 | Sg Semenyih di Sg Rinching | 2° 56′ N | 101° 50′ E |

64 | Sg Selangor di Rasa | 3° 27′ N | 101° 27′ E |

65 | Sg Plus di Kg Lintang | 4° 56′ N | 101° 9′ E |

66 | Sg Pelarit di Wang Mu | 6° 34′ N | 100° 13′ E |

67 | Sg Melaka di Pantai Belimbing | 2° 20′ N | 102° 14′ E |

68 | Sg Lui di Kg Lui | 3° 9′ N | 101° 54′ E |

69 | Sg Linggi di Sua Betong | 2° 37′ N | 101° 60′ E |

70 | Sg Langat di Dengkil | 2° 58′ N | 101° 38′ E |

71 | Sg Kurau di Pondok Tg | 4° 59′ N | 100° 32′ E |

72 | Sg Kulim di Ara Kuda | 5° 23′ N | 100° 32′ E |

73 | Sg Kerian di Selama | 5° 12′ N | 100° 38′ E |

74 | Sg Kinta di Weir G, Tg Tualang | 4° 21′ N | 101° 3′ E |

75 | Sg Kesang di Chin Chin | 2° 17′ N | 102° 31′ E |

76 | Sg Durian Tunggal di Bt 11 Air Resam | 2° 19′ N | 102° 17′ E |

77 | Sg Cenderiang di Bt 32 Jln Tapah | 4° 15′ N | 101° 10′ E |

78 | Sg Bidor di Bidor Malayan Tin Bhd | 4° 2′ N | 101° 11′ E |

79 | Sg Bernam di Tg Malim | 3° 46′ N | 101° 3′ E |

80 | Sg Selangor di Rantau Panjan | 3° 27′ N | 101° 27′ E |

Sabah State, Malaysian Catchments | |||

1 | Sg Tawau di Kuhara | 4° 16′ N | 117° 53′ E |

2 | Sg Kalabakan di Kalabakan | 4° 27′ N | 117° 23′ E |

3 | Sg Kalumpang di Mostyn Bridge | 4° 38′ N | 118° 9′ E |

4 | Sg Talangkai di Lotong | 4° 43′ N | 116° 26′ E |

5 | Sg Mengalong di Sindumin | 4° 59′ N | 115° 34′ E |

6 | Sg Kuamut di Ulu Kuamut | 5° 4′ N | 117° 26′ E |

7 | Sg Lakutan di Mesapol Quarry | 5° 7′ N | 115° 37′ E |

8 | Sg Segama di Limkabong | 5° 7′ N | 118° 7′ E |

9 | Sg Sook di Biah | 5° 15′ N | 116° 8′ E |

10 | Sg Baiayo di Bandukan | 5° 26′ N | 116° 8′ E |

11 | Sg Apin-Apin di Waterworks | 5° 29′ N | 116° 15′ E |

12 | Sg Kegibangan di Tampias P.H. | 5° 41′ N | 116° 22′ E |

13 | Sg Papar di Kaiduan | 5° 46′ N | 116° 5′ E |

14 | Sg Papar di Kogopon | 5° 42′ N | 116° 2′ E |

15 | Sg Labuk di Tampias | 5° 43′ N | 116° 51′ E |

16 | Sg Moyog di Penampang | 5° 54′ N | 116° 6′ E |

17 | Sg Tungud di Basai | 6° 3′ N | 117° 18′ E |

18 | Sg Tuaran di Pump House 1 | 6° 9′ N | 116° 14′ E |

19 | Sg Sugut di Bukit Mondou | 6° 11′ N | 117° 14′ E |

20 | Sg Kadamaian di Tamu Darat | 6° 15′ N | 116° 27′ E |

21 | Sg Wariu di Bridge No.2 | 6° 19′ N | 116° 29′ E |

22 | Sg Bongan di Timbang Batu Sabah | 6° 26′ N | 116° 48′ E |

23 | Sg Bengkoka di Kobon | 6° 37′ N | 117° 2′ E |

Sarawak State, Malaysian Catchments | |||

1 | Sg Kayan di Krusen | 1° 4′ N | 110° 29′ E |

2 | Sg Kedup di New Meringgu | 1° 3′ N | 110° 33′ E |

3 | Sg Entebar di Entebar | 1° 0′ N | 111° 32′ E |

4 | Sg Ai di Lubok Antu | 1° 2′ N | 111° 49′ E |

5 | Sg Sabal Kruin di Sabal Kruin | 1° 8′ N | 110° 53′ E |

6 | Sg Sarawak Kanan di Pk Buan Bidi | 1° 23′ N | 110° 6′ E |

7 | Sg Sarawak Kiri di Kg Git | 1° 21′ N | 110° 15′ E |

8 | Sg Tuang di Kg Batu Gong | 1° 20′ N | 110° 26′ E |

9 | Sg Sekerang di Entaban | 1° 19′ N | 111° 37′ E |

10 | Sg Layar di Ng Lubau | 1° 29′ N | 111° 35′ E |

11 | Sg Sebatan di Sebatan | 1° 48′ N | 111° 20′ E |

12 | Sg Katibas di Ng Mukeh | 1° 50′ N | 112° 37′ E |

13 | Sg Sarikei di Ambas | 1° 58′ N | 111° 30′ E |

14 | Btg Rajang di Ng Ayam | 1° 56′ N | 111° 53′ E |

15 | Sg Oya di Setapang | 3° 1′ N | 112° 35′ E |

16 | Btg Mukah di Selangau | 2° 54′ N | 112° 5′ E |

17 | Sg Sibiu di Sibiu (Atc) | 3° 13′ N | 113° 9′ E |

18 | Sg Limbang di Insungai | 4° 44′ N | 114° 59′ E |

19 | Sg Trusan di Long Tengoa D | 4° 35′ N | 115° 20′ E |

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**Figure 2.**Three country locations (seven rainfall–runoff datasets) of this study. (The information of all the catchment locations is summarized in Appendix E.)

**Table 1.**Inferential Statistics of all the datasets, including DID-HP 27, DID-HP 11, the combination of DID-HP 27 and DID-HP 11, Kerayong, Kayu Ara, Attica, and Wang Jia Qiao.

Dataset & Location | E | N | Bias (mm) | KGE | S_{λ} (mm) | I_{a} = S^{λ} (mm) | I_{a}/S |
---|---|---|---|---|---|---|---|

DID HP 11, Malaysia | 0.823 | 474 | 2.35 | 0.805 | 139.48 | 3.71 | 0.027 |

DID HP 27, Malaysia | 0.919 | 227 | 0.65 | 0.956 | 165.94 | 2.33 | 0.014 |

DID HP 11+27, Malaysia | 0.875 | 701 | 2.535 | 0.863 | 143.99 | 3.19 | 0.022 |

Kayu Ara, Malaysia | 0.810 | 94 | 0 | 0.865 | 24.79 | 2.20 | 0.089 |

Kerayong, Malaysia | 0.888 | 73 | 0 | 0.818 | 9.08 | 1.55 | 0.171 |

Attica, Greece | 0.786 | 77 | 0 | 0.798 | 23.73 | 1.85 | 0.078 |

Wang Jia Qiao, China | 0.795 | 29 | 0 | 0.739 | 386.57 | 3.57 | 0.009 |

**Table 2.**Adjusted coefficient of determination (R

^{2}

_{adj}) and ρ-value of all the datasets’ correlation equations, including DID-HP 27, DID-HP 11, the combination of DID-HP 27 and DID-HP 11, Kerayong, Kayu Ara in Malaysia, Attica in Greece, and Wang Jia Qiao in China. Power correlation models were identified as the best correlation model in SPSS with the lowest standard error of estimate.

Dataset & Location | Correlation Equation | R^{2}_{adj} | p-Value |
---|---|---|---|

DID HP 11, Malaysia | S_{0.2} = S_{0.265}^{0.878} | 0.977 | <0.001 |

DID HP 27, Malaysia | S_{0.2} = S_{0.166}^{0.874} | 0.997 | <0.001 |

DID HP 11+27, Malaysia | S_{0.2} = S_{0.234}^{0.880} | 0.998 | <0.001 |

Kayu Ara, Malaysia | S_{0.2} = 4.263 S_{0.246}^{0.438} | 0.893 | <0.001 |

Kerayong, Malaysia | S_{0.2} = 3.223 S_{0.199}^{0.355} | 0.849 | <0.001 |

Attica, Greece | S_{0.2} = 5.057 S_{0.194}^{0.357} | 0.846 | <0.001 |

Wang Jia Qiao, China | S_{0.2} = S_{0.214}^{0.702} | 0.975 | <0.001 |

**Table 3.**Bootstrapping BCa 99% CI for the S value and CN

_{0.2}value of all the datasets’ correlation equations, including DID-HP 27, DID-HP 11, the combination of DID-HP 27 and DID-HP 11, Kayu Ara, Kerayong, Attica, and Wang Jia Qiao.

Datasets | BCa 99% Confidence Interval | |||
---|---|---|---|---|

S & Curve Numbers | S_{0.2} (mm) | CN_{0.2} | ||

Dataset & Location | Lower | Upper | Lower | Upper |

DID HP 11, Malaysia | 100.99 | 139.48 | 76.88 | 81.54 |

DID HP 27, Malaysia | 118.65 | 165.94 | 74.46 | 79.62 |

DID HP 11+27, Malaysia | 115.01 | 143.98 | 76.21 | 79.60 |

Kayu Ara, Malaysia | 20.75 | 331.07 | 82.43 | 94.04 |

Kerayong, Malaysia | 8.77 | 397.55 | 90.40 | 97.33 |

Attica, Greece | 23.73 | 314.15 | 86.57 | 94.19 |

Wang Jia Qiao, China | 289.86 | 553.69 | 75.08 | 82.60 |

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## Share and Cite

**MDPI and ACS Style**

Lee, K.K.F.; Ling, L.; Yusop, Z. The Revised Curve Number Rainfall–Runoff Methodology for an Improved Runoff Prediction. *Water* **2023**, *15*, 491.
https://doi.org/10.3390/w15030491

**AMA Style**

Lee KKF, Ling L, Yusop Z. The Revised Curve Number Rainfall–Runoff Methodology for an Improved Runoff Prediction. *Water*. 2023; 15(3):491.
https://doi.org/10.3390/w15030491

**Chicago/Turabian Style**

Lee, Kenneth Kai Fong, Lloyd Ling, and Zulkifli Yusop. 2023. "The Revised Curve Number Rainfall–Runoff Methodology for an Improved Runoff Prediction" *Water* 15, no. 3: 491.
https://doi.org/10.3390/w15030491