Estimation of Peak Flow in Ungauged Catchments Using the Relationship between Runoff Coefficient and Curve Number

Hourly flood flow estimation for gauged and ungauged catchments is a prerequisite for planning and water management. Various methods have been applied in a multitude of studies to calculate the peak flow for ungauged catchments. However, it is not simple for engineers to use the existing methods in practical applications. An easier method is suggested for this purpose in this study. The authors estimated the relationship between the runoff coefficient, intensity of rainfall, and curve number, and then utilized the relationship to calculated the peak flow using the rational method for ungauged catchments. Rainfall and flood time series for ungauged study catchments were generated by a simple data generation method and a distributed rainfall–runoff model. Results showed that the runoff coefficients simulated using the estimated relationship reasonably agree with the runoff coefficients in the studied ungauged catchments. In addition, the peak flow simulated using the rational method and the relationship highly agree with the peak flow in the ungauged catchments. Therefore, the peak flow in ungauged catchments can be easily calculated by this method, which is more pragmatic for engineers.


Introduction
Flood estimation with an hourly time step is a prerequisite for planning and water management [1]. However, many catchments are ungauged; therefore, there are limitations for flood calculation using rainfall-runoff models [1,2]. Flood estimation still can be conducted for ungauged catchments through regionalization methods either using (1) gauged catchment data with hydrologically similar areas [3][4][5] or (2) using catchment attributes of ungauged catchments [6][7][8][9][10]. However, the regionalization method has issues. It is difficult to determine the hydrological similarity of the catchment, and the parameters should be updated whenever land-use changes [11]. In addition, physically similar catchments cannot be easily assumed as being hydrologically similar for many catchments [12].
In order to overcome these shortcomings, geomorphological instantaneous unit hydrographs (GIUH) [13] have been applied [11]. Bhaskar et al. [2] used geomorphological characteristics of catchments to derive GIUH, which were then applied to Jira catchment in India to simulate 12 flood events. They found that the peak flows were properly simulated for ungauged catchments with a reasonable amount of errors in the peak flow and time to peak. Jain et al. [14] estimated the design The databases from three organizations were used for the climate input for the distributed rainfall-runoff model to simulate the flood flows (Korea Meteorological Administration, The databases from three organizations were used for the climate input for the distributed rainfall-runoff model to simulate the flood flows (Korea Meteorological Administration, http://www.kma.go.kr; Ministry of Land, Infrastructure, and Transport, www.molit.go.kr; and K-water, www.kwater.or.kr). The authors used hourly rainfall data from up to 27 rainfall gauge stations within and near the Andongdam catchment ( Figure 1). A total of 20 rainfall events, which have relatively large magnitudes, long durations, and reflect the monsoon climate, were selected for the period from 1998 to 2009 (Table A1). These events were chosen for quality purposes while a total of 31 years of records  is available with more missing data. Ordinary kriging was used to convert hourly rainfall records into the grid rainfall data (resolution 200 m × 200 m) for distributed rainfall-runoff model simulation. The ordinary kriging method was used because it can adequately represent the spatial distribution of rainfall.
A land use map, soil map, and digital elevation model (DEM; scale 1:25,000) were converted into grid form with a resolution of 200 m × 200 m for the physical inputs of the distributed rainfall-runoff model. Therefore, the distributed rainfall-runoff model utilizes the spatial variability of the physical catchment characteristics as well as the spatiotemporal variability of rainfall.

Objective Function and Performance Evaluation Statistics
The Nash-Sutcliffe Efficiency (NSE) [23], which has been widely used in rainfall-runoff modeling, was selected as the objective function for the parameter calibration: where Q obs,i is the observed flow at the time step i (hourly), Q obs is the mean of the observed flow, Q sim,i is the simulated flow, and n is the total number of time steps. The range of NSE is [−∞, 1], where unity represents the perfect match between the observed and simulated flow. The NSE emphasizes the high flow fit due to the squared differences between the observed and simulated values [24,25]; therefore, this objective function is appropriate for this study, which focuses on fitting hydrographs of flood. We select three model performance evaluation statistics: the NSE; percent bias (PBIAS); and modified correlation coefficient (r mod ; [26]). The use of NSE was recommended by the American Society of Civil Engineers (ASCE) [27] and provides the overall quality of flow simulations [28]. The use of PBIAS was recommended in Smith et al. [29] for the evaluation of the simulated flow and shows poor model performance in percentage terms with respect to the volume of the simulated flow [30]. The best PBIAS value is zero, because it represents zero difference between the simulated and observed flow. Positive PBIAS values represent overestimation of simulated flows.
The usefulness of r mod in the model evaluation was demonstrated in Smith et al. [17]. The r mod value complements the shortcomings of the correlation coefficient, namely, the significant influence from outliers and insensitivity to the differences in volumes [26]. The correlation coefficient is: The standard deviations of the simulated and observed flows are: Finally, the modified correlation coefficient is: The authors set the thresholds of the satisfactory model performance statistics as: NSE > 0.5 [28], PBIAS ± 30%, and r mod > 0.6. Smith et al. [17] reported with various rainfall-runoff simulations, including the distributed rainfall-runoff models, that the PBIAS values, except for outliers, were placed within these thresholds and r mod values were mostly greater than 0.6 for model calibrations.

Grid-Based Rainfall-Runoff Model
The grid-based rainfall-runoff model (GRM) [31], which is a physically based distributed rainfall-runoff model with hourly time steps, was employed in this study. This model is designed to handle the characteristics of Korean catchments. It simulates surface flow, infiltration, subsurface flow, baseflow, and the impact of flow control facilities ( Figure 2). The kinematic wave model is used to calculate surface flow, which is composed of overland flow and channel flow. The overland flow occurs by infiltration excess flow [32] and saturation excess flow [33]. Infiltration in the upper soil layer is calculated using the Green-Ampt model, and the infiltrated water becomes subsurface flow. If the downstream area of the subsurface flow is a stream, the subsurface flow becomes a lateral subsurface flow that flows into the lateral direction of the stream. If the downstream area of the subsurface flow is a surface saturated with water, the subsurface flow is returned as an overland flow [34]. The lateral baseflow in the lower soil layer is calculated using Darcy's law [35]. The total flow is the sum of overland flow, channel flow, lateral subsurface flow, and lateral base flow. The GRM has 12 parameters (Table A2). The parameter ranges were suggested by Choi [36]. The spatial resolution of the GRM simulation was set to 200 m × 200 m; the size of the grid meets the requirements of Bathurst [37], who argued that a single grid area should cover less than 1% of the catchment area. The governing equations of the GRM are as follows: ∂h ∂t ∂A ∂t where h is flow depth, q (q = uh) is flow rate per unit width, u is overland flow velocity in x direction, r is rainfall intensity, f is infiltration rate, q r is return flow into the overland flow, ∆y is the width of control volume, A is channel cross-sectional area, Q is discharge in the channel, q L is lateral flow from overland flow, q ss is subsurface flow, q b is baseflow, S 0 is surface slope, and S f is friction slope. A more detailed description of the GRM can be found in Choi and Kim [38].

Parameter Calibration and Flow Simulation in Ungauged Catchments
The SDE method, which is a simple data generation method for ungauged catchments, generates hourly flood hydrographs using the GRM by applying calibrated parameters of gauged catchments. This method is based on the hypothesis that the simulated flow at each upstream grid point is reasonable so long as the simulated model flow at the catchment outlet, which is the aggregation of the routed flows of all upstream grid points, is equal to the observed flow. The flood hydrographs for the ungauged catchments of interest can therefore be generated with this method, which is more efficient and pragmatic compared to the DMIP method [17]. Note that it is a special case of regionalization because this method is used to transfer information from downstream gauging stations to upstream ungauged catchments.
For the implementation of this method, the calibrated parameters related to the physical catchment characteristics, which are the parameters No. 2 to No. 5 (Table A2), should be reliable for all events, regardless of the storm characteristics, because the storm characteristics do not influence the physical characteristics of the catchments. Therefore, the authors optimized the physical parameters for all events simultaneously using the NSE objective function to obtain consistent physical parameter values. Theoretically, physical parameters are uniquely determined by the characteristics of a given catchment, therefore, they do not need calibration. However, usually, the parameters are still calibrated because of problems of measurement errors, scale difference between the model grid and the measurement data, and lack of data used to extract the parameter values [39][40][41][42][43]. After the calibration of the physical parameters, the initial soil saturation ratio (parameter No. 1 in Table A2) was optimized for each event because this parameter depends on the antecedent soil moisture condition for the respective event. The regional calibration method [44] has a key idea similar to the calibration method of this study. However, the regional calibration method optimizes the parameters for multiple catchments simultaneously. It is worth noting that the GRM simulation considers the physical spatial variability by using the physical input datasets (see Section 2.1). The consistent physical parameter values are used as the minimum thresholds. This simultaneous multievent calibration method uses all possible historical event data for a selected calibration catchment to generate data for ungauged catchments. Therefore, this method is different from the split-sample test [45] that uses distinct data series in the same catchment for validation. Instead, this method validates the calibrated parameters using historical events in the other dependent upstream gauged catchments (validation catchments), which are assumed to be ungauged catchments. Good simulated results in the validation catchments implies that the hypothesis of this method is validated, and these

Parameter Calibration and Flow Simulation in Ungauged Catchments
The SDE method, which is a simple data generation method for ungauged catchments, generates hourly flood hydrographs using the GRM by applying calibrated parameters of gauged catchments. This method is based on the hypothesis that the simulated flow at each upstream grid point is reasonable so long as the simulated model flow at the catchment outlet, which is the aggregation of the routed flows of all upstream grid points, is equal to the observed flow. The flood hydrographs for the ungauged catchments of interest can therefore be generated with this method, which is more efficient and pragmatic compared to the DMIP method [17]. Note that it is a special case of regionalization because this method is used to transfer information from downstream gauging stations to upstream ungauged catchments.
For the implementation of this method, the calibrated parameters related to the physical catchment characteristics, which are the parameters No. 2 to No. 5 (Table A2), should be reliable for all events, regardless of the storm characteristics, because the storm characteristics do not influence the physical characteristics of the catchments. Therefore, the authors optimized the physical parameters for all events simultaneously using the NSE objective function to obtain consistent physical parameter values. Theoretically, physical parameters are uniquely determined by the characteristics of a given catchment, therefore, they do not need calibration. However, usually, the parameters are still calibrated because of problems of measurement errors, scale difference between the model grid and the measurement data, and lack of data used to extract the parameter values [39][40][41][42][43]. After the calibration of the physical parameters, the initial soil saturation ratio (parameter No. 1 in Table A2) was optimized for each event because this parameter depends on the antecedent soil moisture condition for the respective event. The regional calibration method [44] has a key idea similar to the calibration method of this study. However, the regional calibration method optimizes the parameters for multiple catchments simultaneously. It is worth noting that the GRM simulation considers the physical spatial variability by using the physical input datasets (see Section 2.1). The consistent physical parameter values are used as the minimum thresholds. This simultaneous multi-event calibration method uses all possible historical event data for a selected calibration catchment to generate data for ungauged catchments. Therefore, this method is different from the split-sample test [45] that uses distinct data series in the same catchment for validation. Instead, this method validates the calibrated parameters using historical events in the other dependent upstream gauged catchments (validation catchments), which are assumed to be ungauged catchments. Good simulated results in the validation catchments implies that the hypothesis of this method is validated, and these calibrated parameter values can be applied to the actual upstream ungauged catchments for the generation of flood hydrographs. Therefore, this method is a special case of the parameter calibration and validation method for regionalization.
For the GRM parameter calibration in the Andongdam catchment (see Figure 1), two sub-catchments (Dosan and Socheon), which are assumed to be ungauged catchments, were used for the validation. Three performance evaluation statistics (NSE, PBIAS, and r mod ) were calculated for the model calibration and validation. The authors selected an additional 47 ungauged catchments (Table A3) and generated the hourly flood hydrographs using the calibrated parameter values. In addition, the areal rainfall time series for all 50 catchments were calculated using the grid rainfall data (see Section 2.1). The simulated flood hydrographs and areal rainfall time series for the 50 catchments are used to generate two intensity-duration curves in Section 3.2.

Calculation of Important Factors Using the Rainfall-Intensity-Duration Curve (RIDC) and Flood-Intensity-Duration Curve (FIDC)
The flood-intensity-duration curve (FIDC; Figure 3) for 50 catchments was generated with the flood-duration-frequency (QDF) method [46,47] using the simulated flood hydrographs. The QDF method calculates the average flow with a moving average technique for different durations and calculates the frequency of the maximum flows among the averaged flows for different durations [48,49]. This method was employed in various studies [50][51][52][53][54]. The authors used the maximum flows for different durations (e.g., 1 h and 2 h) to generate the FIDC by plotting the maximum flows against the durations. Frequency analysis is not conducted because it is beyond the scope of this study. The rainfall-intensity-duration curve (RIDC) for the 50 catchments was generated with the traditional intensity-duration-frequency (IDF) method, which is similar to the QDF method but uses the areal rainfall time series as input. Note that the GRM utilizes the grid rainfall datasets as inputs to simulate the flood hydrographs, but we calculated the areal rainfall time series for each catchment to generate the RIDC. The unit for the FIDC and RIDC is mm/h. The effect of the point to grid rainfall conversion on the GRM simulation results is not studied because it is also beyond the scope of this study. calibrated parameter values can be applied to the actual upstream ungauged catchments for the generation of flood hydrographs. Therefore, this method is a special case of the parameter calibration and validation method for regionalization. For the GRM parameter calibration in the Andongdam catchment (see Figure 1), two subcatchments (Dosan and Socheon), which are assumed to be ungauged catchments, were used for the validation. Three performance evaluation statistics (NSE, PBIAS, and mod r ) were calculated for the model calibration and validation. The authors selected an additional 47 ungauged catchments (Table  A3) and generated the hourly flood hydrographs using the calibrated parameter values. In addition, the areal rainfall time series for all 50 catchments were calculated using the grid rainfall data (see Section 2.1). The simulated flood hydrographs and areal rainfall time series for the 50 catchments are used to generate two intensity-duration curves in Section 3.2.

Calculation of Important Factors Using the Rainfall-Intensity-Duration Curve (RIDC) and Flood-Intensity-Duration Curve (FIDC)
The flood-intensity-duration curve (FIDC; Figure 3) for 50 catchments was generated with the flood-duration-frequency (QDF) method [46,47] using the simulated flood hydrographs. The QDF method calculates the average flow with a moving average technique for different durations and calculates the frequency of the maximum flows among the averaged flows for different durations [48,49]. This method was employed in various studies [50][51][52][53][54]. The authors used the maximum flows for different durations (e.g., 1 h and 2 h) to generate the FIDC by plotting the maximum flows against the durations. Frequency analysis is not conducted because it is beyond the scope of this study. The rainfall-intensity-duration curve (RIDC) for the 50 catchments was generated with the traditional intensity-duration-frequency (IDF) method, which is similar to the QDF method but uses the areal rainfall time series as input. Note that the GRM utilizes the grid rainfall datasets as inputs to simulate the flood hydrographs, but we calculated the areal rainfall time series for each catchment to generate the RIDC. The unit for the FIDC and RIDC is mm/h. The effect of the point to grid rainfall conversion on the GRM simulation results is not studied because it is also beyond the scope of this study. R is the ratio between the rainfall peak ( peak P ) and Tc I ; and 2 R is the ratio between Tc I and the mean Figure 3. Schematic plot for the calculation of the runoff coefficient (C) and rainfall intensity ratios (R1 and R2); T c is the time of the concentration in hour; T c 2 is TC multiplied by two; C is the ratio between the flood peak (q peak ) and the mean rainfall intensity during TC (I Tc ); R1 is the ratio between the rainfall peak (P peak ) and I Tc ; and R2 is the ratio between I Tc and the mean rainfall intensity during two TCs (I Tc2 ). FIDC and RIDC are flood-intensity-duration curve and rainfall-intensity-duration, respectively.
The factors important for the calculation of the peak flow in ungauged catchments are the runoff coefficient (C), rainfall intensity ratios (R1 and R2), and CN. The rational method's C, which is an indicator of the storm effect on the flood response in the catchments, can be calculated using the RIDC, FIDC, and the TC. Note that this study did not use the rational method to calculate the peak flow but used the concept of this method to calculate C. The rational method is given as: where Q peak is the peak flow, I Tc is the mean rainfall intensity for the TC (mm/h), and A is the size of the catchment. The authors used the TC equation of Kim [55], which is suitable for the Korean catchments: Boyd [56] and D'Odorico and Rigon [57] suggested a similar form of Equation (11) and demonstrated the suitability of the equation, even though it has only one variable, which is catchment area. C is calculated as: where the specific peak flow q peak (mm/h) is calculated as Q peak divided by A. Therefore, C reflects the effect of the mean rainfall intensity during the TC on the peak flow. Figure 3 illustrates this idea, where Tc represents the time of concentration. The temporal variation of the rainfall intensity could affect the peak flow. Therefore, the authors calculated the two rainfall intensity ratios R1 and R2: where P peak is the peak rainfall intensity; T c 2 represents two times the TC value; I Tc2 is the mean rainfall intensity during T c 2 (mm/h); R1 represents a variation of the rainfall intensity between P peak and I Tc ; R2 represents the variation of the rainfall intensity between I Tc and I Tc2 ; and R1 and R2 explain whether a storm is either intense over a short time or a longer time. For example, the gradient of the RIDC is steep when a storm is characterized by large intensity with a short duration; therefore, the value of R1 becomes large. The CN [58] can be calculated using the potential maximum retention (S) in mm: Hawkins [59] suggested Equation (16) to calculate the S if the initial loss is 20% of S, which was used in the U.S. Department of Agriculture, Natural Resources Conservation Service USDA-NRCS to develop the curve number method.
where P vol and Q vol are the volumes of rainfall and flow in mm, respectively. The rainfall and flow volumes can be calculated using the product of the intensities from RIDC and FIDC and the durations. For example, P vol is the product of I Tc and TC, whereas Q vol is the product of q Tc and TC, and q Tc is the mean flood intensity over TC (Figure 3). Given the Equations (15) and (16) for CN and S, respectively, the CN can therefore be calculated from the TC using the rainfall and flow volume.
3.3. Analysis of the Relationship between the Important Factors, and Simulation of the Peak Flow Using the Relationship Q vol , P vol , and S have the following relationship [58]: The runoff ratio (C r ) can be calculated using P vol and CN: The runoff coefficient C can be expressed as a function of C r , R1, R2, and CN in the analysis of the relationship between C, C r , R1, R2, and CN: The authors used stepwise multiple regression to estimate the relationship. Finally, the simulated q peak (q peak_sim ) can be calculated using the rational method, Equation (10), and the simulated C (C sim ), which is calculated by the Equation (19): Equation (20) shows that the peak flow in the ungauged catchments can be calculated using the rainfall time series, CN, and the relationship for C. It means that the peak flow of the ungauged catchment can be calculated conveniently using Equation (20) without repeatedly conducting hydrological simulation. Note that C r is a function of the P vol and CN as shown in Equation (18); therefore, P vol and CN are used for the calculation of C sim instead of using C r . The complete flow of the research method is summarized in Figure 4.   Figure 5 shows the performance of the GRM, and the acceptance boundaries for the three model performance statistics mentioned in Section 2.2 are shown in black dashed lines. The simulations of the Andongdam catchment shows satisfactory performance statistics for the 20 events ( Figure 5). Figure 6 shows the selected three simulated hydrographs of the Andongdam catchment that simulated the observed hydrographs relatively well. The simulations of the validation catchments Dosan and Socheon display reasonable performance results, except for some events. The PBIAS value of the 2000_Evt1 event in the Dosan catchment indicates that the NSE value, which is very close to the acceptance boundary, is acceptable while the simulated flow was slightly overestimated. The mod r is also satisfactory; therefore, the authors determined that the simulated flow for this event was acceptable. The model performance for Socheon catchment events was examined in the same way. All three performance evaluation statistics were unsatisfactory with the 2004_Evt1 event. The reason for this could be the water level measurement error, rather than the error in the GRM itself or climate input data, because the results from the other events for the Socheon catchment were acceptable with the same model structure. In addition, the results for the Andongdam and Dosan catchments for the 2004_Evt1 event were all reasonable with the same input rainfall data used for the Socheon catchment. It is worth noting that we investigated the observed flow of the Socheon catchment and found no outliers or anomalies. Therefore, the authors determined that the simulated flow for the 2004_Evt1 event in the Socheon catchment was also acceptable.  Figure 5 shows the performance of the GRM, and the acceptance boundaries for the three model performance statistics mentioned in Section 2.2 are shown in black dashed lines. The simulations of the Andongdam catchment shows satisfactory performance statistics for the 20 events ( Figure 5). Figure 6 shows the selected three simulated hydrographs of the Andongdam catchment that simulated the observed hydrographs relatively well. The simulations of the validation catchments Dosan and Socheon display reasonable performance results, except for some events. The PBIAS value of the 2000_Evt1 event in the Dosan catchment indicates that the NSE value, which is very close to the acceptance boundary, is acceptable while the simulated flow was slightly overestimated. The r mod is also satisfactory; therefore, the authors determined that the simulated flow for this event was acceptable. The model performance for Socheon catchment events was examined in the same way. All three performance evaluation statistics were unsatisfactory with the 2004_Evt1 event. The reason for this could be the water level measurement error, rather than the error in the GRM itself or climate input data, because the results from the other events for the Socheon catchment were acceptable with the same model structure. In addition, the results for the Andongdam and Dosan catchments for the 2004_Evt1 event were all reasonable with the same input rainfall data used for the Socheon catchment. It is worth noting that we investigated the observed flow of the Socheon catchment and found no outliers or anomalies. Therefore, the authors determined that the simulated flow for the 2004_Evt1 event in the Socheon catchment was also acceptable.

Estimation of the Relationship between the Runoff Coefficient, Runoff Ratio, Rainfall Intensity Ratios, and CN
The estimated relationship between the important factors C, r C , 1 R , 2 R , and CN using the stepwise multiple regression method is: Figure 7 shows the values of C and sim C for all 50 catchments ("All catchments") and the five selected catchments (Andongdam, Dosan, WP21, Socheon, and WP43) using 20 flood events. C was calculated using the outputs of GRM model as described in Section 3.2. sim C was calculated using Equation (21), where r C was calculated using Equation (18). C and sim C are very consistent in "All catchments", except for two outliers, and the r-squared value between them (Table 1) is high (0.9). The results in five selected catchments also show a good agreement with minor dispersion (Figure 7). Table 1 shows that most of the catchments' results have good r-squared values above 0.70, Figure 6. Observed hydrographs and simulated hydrographs by parameter calibration for three events in the Andongdam catchment.

Estimation of the Relationship between the Runoff Coefficient, Runoff Ratio, Rainfall Intensity Ratios, and CN
The estimated relationship between the important factors C, C r , R1, R2, and CN using the stepwise multiple regression method is: Figure 7 shows the values of C and C sim for all 50 catchments ("All catchments") and the five selected catchments (Andongdam, Dosan, WP21, Socheon, and WP43) using 20 flood events. C was calculated using the outputs of GRM model as described in Section 3.2. C sim was calculated using Equation (21), where C r was calculated using Equation (18). C and C sim are very consistent in "All catchments", except for two outliers, and the r-squared value between them (Table 1) is high (0.9). The results in five selected catchments also show a good agreement with minor dispersion (Figure 7). Table 1 shows that most of the catchments' results have good r-squared values above 0.70, except for four catchments (WP12, WP 28, WP42, and WP50). Therefore, Equation (21) can be used to calculate the C values for these catchments, which include the ungauged catchments. except for four catchments (WP12, WP 28, WP42, and WP50). Therefore, Equation (21) can be used to calculate the C values for these catchments, which include the ungauged catchments.   The correlation of q peak and q peak_sim is shown in Figure 8 and Table 2, and the correlation was significant. q peak was calculated using the outputs of the GRM as described in Section 3.2 and q peak_sim was calculated by Equation (20) as described in Section 3.3. The agreement between q peak and q peak_sim is better than that of C and C sim in Figure 7 (compare the dispersion of the runoff coefficients and peak flows in Figures 7 and 8, respectively). Table 2 details the improved r-squared values compared to  Table 1, especially for WP42 and WP50. Although WP12 and WP28 have r-squared values below 0.90, the values were still reasonable (greater than 0.79). Table 3 lists the NSE values for q peak and q peak_sim for each of the 50 catchments and the aggregated 50 catchments ("All catchments"). The NSE value in "All catchments" is as high as 0.97. In most catchments, except for three (WP12, WP17, and WP28), NSE values are above 0.9 with a mean of 0.97, which represents a very good correlation. The three catchments WP12, WP17, and WP28 show acceptable NSE values above 0.75. Water 2018, 10, x FOR PEER REVIEW 15 of 22    Table 3. NSE values for the peak flow (q peak ) and simulated peak flow (q peak_sim ).  Figure 9 compares the simulated peak flow q peak_sim with the observed (q peak_obs ). The q peak_sim of the three catchments were highly correlated with the q peak_obs . Table 4 shows good r-squared values of the three catchments. While NSE values for the Andongdam and Dosan catchments were reasonable, the Socheon catchment had a lower value, which could be due to one large outlier shown in Figure 9. The authors inferred that the water level measurement error is the cause of the outlier, as described in Section 4.1. Therefore, it is determined that the peak flow in the ungauged catchments can be successfully calculated using the rational method, estimated relationship for C sim calculation, rainfall time series, and CN.

Catchment
were reasonable, the Socheon catchment had a lower value, which could be due to one large outlier shown in Figure 9. The authors inferred that the water level measurement error is the cause of the outlier, as described in Section 4.1. Therefore, it is determined that the peak flow in the ungauged catchments can be successfully calculated using the rational method, estimated relationship for sim C calculation, rainfall time series, and CN.    This study described a method for estimating the peak flow in ungauged catchments using the relationship between the runoff coefficient and curve number without the need for repetitive hydrological simulation. Users can use this method by referring to the steps shown in Figure 4.
However, this study has the following limitations. The range of error of the simulated peak flow for ungauged catchments may depend on the performance of the selected rainfall-runoff model. Therefore, further research is required for lower uncertainty in peak flow simulation via parametric and structural uncertainty analysis of the rainfall-runoff model (e.g., [60][61][62]). In addition, this study used one distributed rainfall-runoff model. As remarked by Onyutha [63], the influence from the selection of a hydrological model among others can lead to large under-and/or over-estimations of extreme events. Therefore it is recommended that application of the methodology in this study for related future research work should comprise several hydrological models. During future studies, it will be possible to investigate the degree of difference between simulation results by various models, which model can be relied on, and the confidence interval of prediction of peak flow. Finally, frequency analysis was not addressed in this study. Some previous studies, such as Onyutha and Willems [64], performed regionalization of peak flows based on frequency analyses to yield quantiles at ungauged catchments. However, in this study, frequency analyses were not conducted. It is envisaged that application of the methodology in this study based on results from frequency analyses will improve the results of the peak flows for ungauged catchments with respect to the applicability for design of water resources applications. However, reliable peak flow data should be used to obtain reliable frequency analysis results. Therefore, uncertainty analysis and model comparison analysis should precede frequency analysis.

Conclusions
The aim of this study is to analyze the relationship between runoff coefficients and curve numbers, which have different theoretical origins, using rainfall and flow data generated by the SDE method for many ungauged catchments. Previous studies did not find the relationship between runoff coefficient and curve number using data from ungauged catchments.
In summary, the performances of the GRM flow simulations for the gauged catchments were adequate; therefore, we used the SDE method to generate hydrographs of the ungauged catchments in the study area. Using the hydrographs and rainfall time series data, runoff coefficients and curve numbers were calculated and then their relationship was estimated. The presented results showed that the simulated runoff coefficient (C sim ) based on this relationship agrees well with C. In addition, the simulated peak flows (q peak_sim ) using the rational method and C sim show a good correlation with the peak flows in the gauged and ungauged catchments. Therefore, the proposed method presented in this study allows researchers and engineers to easily calculate the peak flow of an ungauged catchment in practical applications.