Estimation of river discharge using Monte Carlo simulations and a 1D hydraulic model based on the artificial multi-segmented rating curves at the confluence of two rivers

During extreme floods caused by climate change, reliable flow discharge data are essential for successful reservoir operation to mitigate downstream flood damage. Generally, the flow discharge is computed using the rating curve (RC) established from the relationship between the flow rate and water stage level. Determining the parameters of rating curves is subject to uncertainties related to the difficulties and limitations of flow monitoring in covering a wide range of flow variations. Especially at river confluences, the uncertainties are pronounced when floods occur owing to several factors such as roughness change, backwaters, and levee overflow. The Seomjin River Basin in Korea suffered from flood inundation that occurred at the tributary confluence during an extreme flood in 2020. To identify a reliable flow rate of the main stream and tributary, this study proposes an indirect flow assessment scheme using a 1D hydrodynamic simulation model to find the best simulated water level in an iterative manner based on Monte Carlo (MC) simulations. With a large amount of discharge data generated from random-number combinations, it is possible to obtain the best results automatically by specifying the reliability limitation considering the uncertainty of the predetermined RC parameters associated with the roughness coefficient. Nash Sutcliffe Efficiency (NSE) was incorporated to evaluate the reproduced water level to meet the threshold specified for NSE ≥ 0.75. The simulated flowrates computed from the revised RC and roughness coefficients revealed an error range of 8%–36.6% compared with the design flood. The approach proposed in this study is applicable for determining the valid parameters necessary to create a revised RC at an existing water level gauge station, where the uncertainties of the RC are pronounced, particularly in the vicinity of the channel confluence.


Introduction
Climate change has led to alterations in the frequency of droughts and floods [1].Danandeh et al [2], investigated the spatiotemporal variability of rainfall extremes using the outputs of various climate change scenarios.For successful water resource management, reliable flow discharge data are essential for establishing countermeasures to guarantee water supply security and flood damage mitigation.The time series of discharge data, usually in an open channel, is computed from the water stage level using the functional relationship between the water stage level and observed flow discharge.To convert the recorded water stage level into discharge from the stage-discharge relation, the rating curve (RC) estimates the flow discharge from a one-toone relation of the corresponding water stage level using the power-law equation [3].However, RC reflects many uncertainties and errors in the data production process because of the temporal and spatial limitations of flow measurements [4].The RC parameters are usually determined by regression analysis using measurement datasets.However, when the water level exceeds the measurement bound, extrapolation is necessary to determine the corresponding flow discharge.
The uncertainties in RC come from several factors: (1) monitoring errors involved in the flow discharge data estimated from the velocity-area method during field measurement, (2) different flow discharge due to loop characteristics of the flow curve, (3) the limited applicability of the flow data observed whether during the dry season or the flood event [5][6][7], and (4) missing data and outliers in the time-series flow data collected to estimate the RC parameters.For a rapid RC assessment, hydraulic modeling based on the Saint-Venant (SV) equations has been proved to be a good substitute for the traditional method because many difficulties are expected during flow observations [8].
To create a reliable RC, hydraulic modeling is straightforward when extrapolation of the RC is required in the absence of observational data.Manasanarez et al [9], presented a flow estimation method that considered the uncertainty of RC applying the Markov chain Monte Carlo (MCMC) sampling method for parameters of the roughness and slope.Le Coz et al [10], analyzed the uncertainty and flow rate of RC using a 1D Bayesian model and MCMC.Bessar et al [11], performed a flow uncertainty analysis using the Hydrologic Engineering Center's River Analysis System (HEC-RAS) and Monte Carlo (MC) simulation using the Latin Hypercube Sampling method for each stream section by specifying the parameter range for each segment.
The roughness coefficient is another important factor in estimating the flow rate because the Manning coefficient constantly changes depending on the river's hydrological and geomorphological processes.The roughness coefficient comprises a parameter set based on the basic river design plan and data [12].Shin et al [13], conducted a flood-level analysis using the HEC-RAS model, considering the uncertainty of the roughness coefficient and flow rate.Based on the observed water level and flow data, nonlinear regression analysis was performed on the RC parameters, and the generalized likelihood uncertainty estimation (GLUE) method was applied to estimate the flow uncertainty.
Various studies have been conducted to minimize the uncertainties of the flow discharge estimation related to the RC parameters.However, these studies mainly focused on a single channel without considering the impact of the lateral inflows, such as tributary confluences.When a flooding event occurs, the junction points of the rivers are very vulnerable to the flood inundation since flow discharge of the tributary and main stem increases simultaneously.During an extreme flood in August 2020, the Seomjin River Basin in Korea suffered from flood damages near the channel junction due to the confluence of the major tributary, the Yocheon River.Because the main stem of the Seomjin River is regulated by a multipurpose dam upstream, reservoir operation is very important to mitigate flood damage in downstream rivers.Accordingly, from the perspective of reservoir operations, determination of the discharge release is of great importance since the water level of the downstream river rapidly increases especially during the flood events.K-water, a government company responsible for multipurpose reservoir operation in South Korea, has developed a finite volume-based 1D hydrodynamic model, K-RIVER, to support the operation of various hydraulic structures such as reservoirs and multipurpose weirs.K-RIVER allows for stable hydrodynamic simulations without sacrificing the computational accuracy particularly rapidly varying flow during flood events and discontinuous flows related with the gate operation.
Comparing with the hydrodynamic simulation model such as HEC-RAS developed by USACE, one of the main advantages of K-RIVER is its applicability to various modeling purposes by customizing the source code of the program.In this study, the computational efficiency of the model was improved to cover functional variations to determine the flow characteristics, such as roughness coefficients and RC parameters, and to find the best simulation results following the observed water level.Regarding recent flood events in 2018, 2109, and 2020, the hydrodynamic simulation focused on flow estimation in the vicinity of the junction point where one of the major tributaries, the Yocheon River, joins the main stem of the Seomjin River.In the vicinity of the channel confluence during a flood, the reliability of RC is unsatisfactory because the water level increases significantly and exceeds the monitoring bounds combined with the backwater and irregular currents due to the flood flow released from the tributaries.
This study proposes an indirect flow estimation scheme that considers the uncertainties of two main influencing factors: RC parameters and roughness coefficient.Based on hydrodynamic simulations, focusing on the channel junctions, RC parameters were provided within certain ranges to minimize the difference between the observed and simulated water levels.Rather than using only specific data points to improve the existing RC, the virtual stage-discharge relations were proposed considering the uncertainties of the parameters combined with the roughness coefficient.The parameters of RC will be finally determined to meet the specified statistical criteria minimizing the error ranges between the simulated and observed water levels.The proposed flow discharge estimation scheme is useful to find a reliable flow discharge by reducing the uncertainties related to the extrapolation of RC when the water level exceeds the monitoring bounds and the highwater level due to the backwaters at the tributary junctions.

Description of the study area 2.1. Backgrounds
The study area is part of the Seomjin River in Korea, in the vicinity of the channel junction where the river joins its major tributary, the Yocheon River.As shown in figure 1, the Seomjin multipurpose dam located very upstream of the river regulates the flow rate to meet the demands of downstream water use.Total length of the Seomjin River is 173.3 km with the basin area of 4,913 km 2 .The duration of flooding is relatively long to show a low flow concentration because the shape coefficient of the Seomjin River ranges from 0.08 to 0.18.The recorded average annual precipitation from the rainfall stations within the basin is 1,356 mm, which is comparatively higher than the average annual precipitation in Korea (1,278 mm, according to the water resources management plan of South Korea, Ministry of Land, Infrastructure and Transport, MOLIT).The Yocheon River, 60 km long and 486 km 2 basin area, joins the Seomjin River 50 km from the Seomjin multipurpose dam.
The square box in figure 1 displays the coverage of this study approximately 26.0 km of the Seomjin River starting from the Daegang gauge station to the Yeosung gauge station, including a 15.0 km stretch of the tributary.Six water level gauge stations are available, and their configurations are summarized in table 1.Two Geumgok and Godal stations were selected as target points to evaluate the uncertainties in the flow estimates.Because these two stations are located before and after the junction, it is possible to evaluate the impact of the tributary confluence.

Data collection for simulation of the hydrodynamic model
The hydrodynamic simulation model was set up with two points for the upstream boundaries: one in the main stem and the other in the tributary.Except for one downstream boundary point, the remaining three stations near the junction were assigned as target points for flow assessment.Flow discharge and water level data corresponding to each flood event were collected and applied to the upstream and downstream boundary conditions.River topographic data were collected to reflect the hydraulic structure for river flow analysis.A total of 153 river cross-sections in the main channel and 97 river cross-sections in the tributary were compiled to build the model.The average distance between each river cross section was approximately 200 m.The roughness coefficients and river topographic data were found to be 0.32-0.35(MOLIT).Table 2 lists the flood events in chronological order selected to test the reliability of RC.Hourly water level ranges with total precipitation during three flood events were given at two gauging stations, one from the main stem and the other from the tributary.The recorded total precipitation of the Event 1 and Event 2 was 251 mm, and 229 mm, respectively, which increased up to the maximum of 607 mm in the year 2020 during the flood Event 3. The corresponding water level of the Event 3 exceeded the 100-year frequency flood, while the water level at the Donglim station remained 1.36-1.8m during Event 2, which was the lowest water level comparing with other two flood events.
Table 3 summarizes the RC of each gauge station applicable to the segmented water level ranges.The existing RCs of the Daegang station was not available until 2011, and the application range corresponds to 0.41-2.25 m, which needs extrapolation since the water level exceeds the monitoring bound 1.53 m.The RC equations of the Donglim station, assigned to the tributary upstream, were developed in 2020.Although the applicable water level ranges from 1.22 m to 6.00 m, the observed maximum water level during the flow monitoring was 2.97 m.As shown in table 2, the observed maximum water levels of the Daegang station during the flood events 1 to 3 exceeded the monitoring bounds.However, during the Event 2, the recorded water levels of the Donglim gauge station remained below the monitoring bounds.Although the RCs for the Geumgok and Godal stations were available, the reliability was not sufficiently tested since the equations were developed after the extreme flood in 2020.

Methods
Figure 2 shows the procedure for estimating the RC parameters.
Step 1: Collect hydrological data including water surface level, discharge, and RC equations.Additionally, cross-sectional channel geometry data are necessary to construct a hydrodynamic model.
Step 2: Establish a parameter set to incorporate uncertainties of the discharge data.These parameters are used to formulate the virtual RC and a set of roughness coefficients through random number combinations within a specified certain range.
Step 3: Prepare the hydrodynamic model input data using the generated RC and roughness coefficient.The output of this procedure is water level time-series data for the flood season.
Step 4: Select the optimal parameter set by evaluating the observed and simulated water levels using quantitative statistics.

Parameter selection 3.1.1. Parameters of the rating curve
The exponential equation presented by Lambie [14] is the most widely used form to express the relationship between water level and discharge, as shown in equation (1).An RC at the observation point is expressed through a single exponential formula; however, if restrictions of the water level are presented, the corresponding water level section is selected and the RC is applied.
where Q is the water discharge (m 3 /s), h is the water stage level of the local datum (m), and α is a scaling coefficient related to the control section or channel characteristics, β is an exponent describing the type of hydraulic characteristics, and h 0 is the water level at a point where the flow rate is zero.In this study, a scaling coefficient a and an exponent b are determined first; h 0 is determined separately through the observed water level and discharge data.Using these parameters, a range is selected based on an existing RC equation.These values reflect the mathematical uncertainties in the estimation process.Therefore, parameter values are randomly generated within a certain range and uncertainties are considered.The ranges of each parameter are as follows: Scaling coefficient a relates to the shape of the river and the regional characteristics of a constant value.Chang and Lee [15] reported that large values appeared in large rivers, and small values appeared in small rivers.varies depending on the shape and water level of the river.The exponent is a constant value that can provide a rough estimate of the shape of the river to reflect its control characteristics.Following Kennedy, E J [16], the suggested range of b is greater than 2.0 when cross-section control is dominant, and less than 2.0 if it is subjected to channel control.These ranges were used intermittently in previous studies.Hrafnkelsson et al [17], suggested a range of 1.0-2.67 to generalize the exponent of the RC equations through hydrodynamic proof.Petersen-Øverleir [6], presented a study that used the stage-fall discharge curve equation and considered various uncertainties such as equal variance, overflow effect, and multiple RC.Therefore, to include the ranges suggested in previous studies, b was set to be 1.0-4.0 in this study.

It indicates a discharge satisfying the condition of
To account for the uncertainties related to various factors involved in RC development, it is possible to generate a virtual RC by changing the coefficients and exponents of existing RCs.In this study, the coefficients and exponents of the existing RC were supposed to vary based on the uniform random distributions within the prescribed ranges to generate multiple virtual RCs.These RCs were applied to the upstream boundary conditions, particularly when the water level exceeded the monitoring bounds.The value of α depends on the existing RCs, if available, because the coefficient varies with channel size.Equations (2) and (3) specify the distribution functions for generating α and β where k is a constant set to 0.3 in this study, and U is a uniform distributed function.Based on the prescribed , a a¢ is generated using a uniform distribution in the range from

Roughness coefficients
The roughness coefficient constantly changes according to the hydrological and geometric processes in the river.The consistency of the roughness coefficient requires continuous observation and verification.However, in reality, it is challenging to obtain accurate time-relative values through field surveys.Thus, a constant value of the roughness coefficient is generally used for the calibration of hydraulic and hydrological models.Shin et al [13], considered the uncertainty in the roughness coefficient by multiplying a certain ratio with the previously established roughness coefficient, instead of applying the same roughness coefficient value to the entire river section.
The Seomjin River is mainly classified as a sandy river, with sand and gravel prevailing in the upstream reaches, and silt and clay in the downstream reaches.The roughness coefficient is set to 0.025-0.040 in this study.Based on the computed average of the roughness coefficients for all streams, random numbers were generated within a specified ratio that satisfied the range of roughness coefficients suggested by Chow [12].The roughness coefficient for each section has an average value of approximately 0.032.Based on a random number of combination, the range of k is specified to satisfy the prescribed roughness coefficient as follows: where M̅ is the average Manning coefficient corresponding to the total cross-sections of the target river.n min and n max are the minimum and maximum values, respectively, according to the type and shape of the channel.U denotes a uniform distribution function.

Description of the hydrodynamic simulation model
The hydrodynamic simulation of the study area focused on the stream gauge stations within the domain, involving particular upstream and downstream gauge stations in which the boundary conditions were applied.
To incorporate the uncertainties in the hydraulic data, water level data including the peak water level, RC, and roughness coefficient were applied during the simulation.Numerical simulations were based on K-RIVER, a public domain program (https://www.water.or.kr/) that can numerically compute the water level of an open channel based on the SV equations.Summarizing the characteristics and functions of K-RIVER, the following are noted: 1. Hydraulic analysis based on the Finite Volume Method (FVM) with Riemann Problem Solver.
2. Ability to consider generalized natural complex river cross-sections.
The governing equation of K-River is expressed as the SV equation in the form of integrating the river crosssection.
where vectors U, F, and S are the variables related to the flow, flow rate, and source term, respectively.A is the cross-sectional area of the river, Q is the flow rate, P is the average pressure at the section, ρ is the average density of the fluid, and S is the slope.S represents the friction slope, which is usually expressed using Manning's formula as follows: R is the hydraulic radius and n is Manning's roughness coefficient.This model can reduce mass-balance error and divergence by applying the FVM technique.The necessity of considering discontinuous flow in the analysis should be considered for the operation of various hydraulic structures in Korean rivers.Jeong et al [18], performed hydrodynamic simulations using K-RIVER and HEC-RAS models to estimate flood levels and flow discharge.Their results showed that K-RIVER could reproduce the discontinuous flow more accurately than the HEC-RAS model.

Performance evaluation of the proposed scheme
The Nash-Sutcliffe efficiency coefficient (NSE) was incorporated to evaluate the performance of the proposed scheme.NSE statistics avoid the disadvantage of the R-square error (R 2 ) method, which shows a tendency to increase at least when the observed and simulated data exhibit linear relationships despite poor modeling performance.Many groups, including the American Society of Civil Engineers (ASCE), Legates and McCabe [19].Moriasi et al [20], and Servat and Dezetter [21], have employed NSE metrics as objective functions to calibrate hydrological models.Nguyen et al [22], applied the NSE function to evaluate simulated water levels during flood events using various weather projection datasets.Nevo et al [23], applied NSE and persistent-NSE as evaluation criteria for a flood forecasting system in their study.Bruno et al [24], used NSE as an evaluation index to compare the water level calculated from HEC-RAS with the observed water level for a flood event.The NSE equation to evaluate the simulated hourly water level is as follows: where O i is the observed water level, O̅ is the mean observed water level, P i is the simulated water level at time i, and n is the total number of observations.In this study, the standard suggested by Moriasi et al [25], was used for NSE evaluation.As presented in table 4, the evaluation criteria 'Very Good' corresponds to NSE 0.75 to estimate the hourly discharge flow at the point of comparison based on the simulated water level.Root Mean Squared Error (RMSE, equation (9)) is an index that includes units of simulated variables and can be used to quantitatively identify errors.

RMSE
n O P 1 0 and 0inclusive 9 O i and P i are the observed and simulated values at time i, respectively, and n is the total number of observations.As the RMSE approaches zero, the modeling performance improves.The proposed scheme was evaluated based on NSE and RMSE.

Results and discussion
4.1.Determining the appropriate number of trials based on the monte carlo simulation MC simulation is a type of statistical analysis that relies on iterative random sampling for model validation.MC reliability increases with the number of trials; therefore, it is important to determine the number of samples for which the computational results consistently converge to a certain value.Bessar et al [11], applied approximately 10,000 trials using the MC method.More than 15,000 trials were performed to determine the posterior distribution of RC parameters using Bayesian and MCMCC methods.This study generated 10,000 samples using MC simulations of a flood event in 2018 to determine the appropriate number of trials that met the criteria and applied the results to other flood events.Figure 3 shows the number of trials versus the total number of simulations that satisfied NSE > 0.75 criteria.The best NSE results were obtained after 500 trials at the Geumgok and Godal gauge stations.The ratio of the final optimized results of the NSE appeared to converge after 2500 out of 10,000 trials.Thus, the minimum number of trials in this study was set to 2500.

Analysis of simulation results of the flood events
Multiple hypothetical RC equations and roughness coefficients were proposed and applied to compute the water levels of the target points.Figure 4 shows the NSE results to test whether the simulated results met the specified threshold by comparing the observed water level with the simulated results obtained after 2,500 MC simulation runs.
[A], [B], and [C] correspond to flood events 1 (2018), 2 (2019), and 3 (2020), respectively.The shaded part shows that the NSE of the two comparison points at the Geumgok and Godal stations corresponds to a performance rating of 'Very Good' (table 4).
The Y-axis of the figure 4 represents the Geumgok gauge station, located upstream of the junction where the direct assessment of the influence of main stem can be made.The X-axis represents the Godal gauge station located downstream of the junction, influenced by the inflow from the tributary.The insights obtained from the results are as follows: The NSE distribution in figures 4(a) to (c) exhibits an overall upward trend, indicating a proportional relationship between the NSE values at the two gauging stations.The high NSE values at the Geumgok gauge station, influenced only by the main stem discharge, correspond to high NSE values at the Godal observation station, suggesting that the main stem discharge has a significant impact on the NSE results at both stations.
Although these two gauging stations are closely located, it is observed that the shape of the NSE distribution differs depending on the scale of the flood.Larger floods result in increased influence from the tributary, leading to a broader dispersion of NSE distribution.In contrast, figure 4(b) corresponding to the 2019 flood shows a loop-shaped NSE distribution, indicating that the observed tributary water level was within the monitoring bounds of the rating curve (RC).Consequently, the flow was computed using the existing RC, and the loopshaped NSE distribution suggests that, in this region, the NSE distribution alone can provide an approximate assessment of the scale of the flood.

Analysis of the simulated results satisfying the NSE threshold
Figure 5 shows the simulated water levels at the Geumgok and Godal stations computed using the proposed scheme.The circle indicates the observed water level and the shaded part represents the simulated range that meets the specified statistical criterion (NSE > 0.75).
The solid line represents the best simulated mean water level displaying collections of arithmetic mean computed at both stations with respect to the NSE index.The results of MC simulations generated a number of various water levels within the ranges classified as 'Very Good.' RMSE and coefficient of determination (R 2 ) were incorporated as additional statistical indices to evaluate the performance of the proposed scheme.As presented in table 5, the RMSE and R 2 values varied with flood intensity, although the application scheme and locations were identical.
For Event 1, with the second-largest rainfall, RMSE were 0.068-0.441m and 0.105-0.433m at the Geumgok and Godal stations, respectively.For Event 2, with the least rainfall, the corresponding RMSE were 0.124-0.504m and 0.147-0.499m.The simulation results of Event 3, with the maximum rainfall, revealed that RMSE at the Geumgok and Godal stations varied by 0.33-1.13m.The R 2 estimates show that all simulations met the NSE > 0.9 threshold for each flood event.
The main factors to compute the water level are the virtual RC derived from a random number combination of a and , b and the roughness coefficients.Figure 6 shows the ranges of roughness coefficients corresponding to each flood event to meet the NSE > 0.75 condition.Median roughness coefficient was estimated to be 0.0363 and increased by 13% during the flood events compared with the predetermined values.However, although the roughness coefficients did not converge, it was identified that multiple combinations met the NSE > 0.75 condition, and the reproduced water levels show good agreement with the observed values.Figure 7 shows the ranges of α and β , which directly contribute to converting the reproduced water levels into the discharge flows corresponding to each flood event.The upper and lower edgest of the boxes represent  25% and 75% quantiles, respectively.Including the median (horizontal line) and average (cross) values, the parameters are indicated by the coefficients of RC as alpha and exponents as beta, separated by the main stem [M] and tributary [T] for each corresponding flood event.The RC of the tributary was divided into three sections based on the water level by assigning the low level as 1, high as 3, and middle as 2. The proposed scheme could not be applied to the parameters (T_α, T_β) of Event 2. This was because, as described previously, it was lower than the extrapolation level of RC in the tributaries during Event 2. The exponents of the main stream showed a tendency to converge to a particular value for each flood event, whereas the exponents at the tributary were widely scattered.

Comparison of the flow discharges at reference points based on the reproduced water levels
A hydrodynamic simulation successfully reproduced the water levels of the three flood events.Based on the computation results, it is possible to convert the reproduced water levels into flow discharge using the predetermined virtual RCs. Figure 8 shows the flow discharge with a variational range in accordance with the reproduced water levels.Because the flow discharge of the target points was not available, model verification was performed indirectly.From the frequency analysis, the rainfall in August 2020 exceeded the 100-year levels which corresponds to 7,470 m 3 /s and 5,590 m 3 /s at the Godal and Geumgok gauge stations, respectively.The flow estimation scheme proposed in this study projected 8,065-9,080 m 3 /s and 6,057-7,638 m 3 /s, at Godal and  Geumgok respectively, with variation ranges of approximately 8%-36.6% differences.Although the proposed scheme generally overestimated the flood discharge, the flow variation was within a reasonable range compared with the frequency analysis of the flow discharge.

Conclusions
This study proposed an indirect flow discharge estimation scheme using a hydrodynamic simulation model focused on channel junctions, where flow uncertainties occur due to channel confluences.Targeting water level gauge stations, by specifying the multiple segmentations, the application range of RC varies with water level changes.Although the coefficients and exponents of the RC differ according to each segmentation, the predefined RC parameters are subject to uncertainties when the water level exceeds the monitoring bounds.
Particularly at the channel junctions, the backwater and irregular currents produce uncertainties in the flow discharge estimates.This study presents an efficient flow estimation scheme that combines a hydrodynamic simulation model with virtual RCs generated by random number combinations.The utility of the proposed scheme was verified using the extreme flood events in the Seomjin River Basin in Korea.The uncertainties associated with the roughness coefficient were considered.Consequently, it was possible to estimate the RC parameters and roughness coefficients at the upstream boundary points for each flood event.
Based on the estimated parameters, the ranges of the time-series flow discharge were successfully identified for flow estimation at the target points in the vicinity of the channel confluences.The proposed scheme is also applicable for estimating the flow discharge during a flood event when the relationship between the water level and the corresponding flow discharge shows a loop curve pattern.The dynamic relationships between the discharge data computed using the proposed scheme and the observed water levels should be further verified.Such considerations are essential for a comprehensive understanding of the interactions between water levels and flow discharge in rivers.

Figure 1 .
Figure 1.The Seomjin River Basin in Korea focusing on the confluence of the Seomjin River and its tributary, the Yocheon River.

Figure 2 .
Figure 2. Flow chart for the proposed scheme.
value of .b The exponent b

Figure 3 .
Figure 3. NSE percentage (0.75 or greater) for the Geumgok and Godal gauge stations versus the total number of simulations.After 2500 trials, no significant change is observed.

Figure 4 .
Figure 4. NSE of the hourly water level reproduced at the Geumgok and the Godal gauge stations.

Figure 5 .
Figure 5.The reproduced hourly water levels (The left is for the Godal, while the right for the Geumgok gauge station).

Figure 6 .
Figure 6.The quantiles of the roughness coefficient.

Figure 7 .
Figure 7.The quantiles of the RC parameters (During Event 2, the observed water levels in the tributary (T) were within the confidence interval; hence, we applied the existing RC.Consequently, the range of the variable was represented by a single value.).

Figure 8 .
Figure 8. Estimations of hourly based flow discharge (The left is for the Godal, while the right for the Geumgok gauge station).

Table 1 .
Configuration of the available water level gauge stations.

Table 2 .
Total precipitation at the Donglim station and the water level of each gauge station for three flood events.

Table 3 .
RC equations and water level segmentations at two upstream boundary points.

Table 5 .
Range of the RMSE and R 2 for the reproduced water levels of each event.