Estimation of rainfall threshold for flood warning for small urban watersheds based on the 1D–2D drainage model simulation

Flood warning systems can provide an effective and low-cost solution to protect urban residents especially in developing countries suffering from severe flood damages caused by weak capacity of flood defense structures. Rainfall Threshold for Flood Warning (RTFW) is an essential key of accurate flood warning system. This study proposed an easy-to-use mathematical equation for RTFW for small urban watersheds based on computer simulations. First, a coupled 1D–2D dual-drainage model was used to simulate the flooded area in nine watersheds of Seoul, Korea corresponding to 540 scenarios of various synthetic rainfall events and watershed imperviousness. Then, the results of the 101 simulations that caused the critical flooded depth (0.25 m–0.35 m) were used to develop the equation that relates the value of RTFW to the rainfall event temporal variability (represented as coefficient of variation or CV) and the watershed imperviousness (represented as NRCS Curve Number or CN). The results suggest that (1) RTFW exponentially decreases as the rainfall CV increases; (2) RTFW linearly decreases as the watershed CN increases; and that (3) RTFW is dominated by CV when the rainfall has low temporal variability (e.g., CV < 0.2) while RTFW is dominated by CN when the rainfall has high temporal variability (e.g., CV > 0.4). For validation, the proposed equation was applied for the flood warning system with two storm events occurred in 2010 and 2011 over 239 watersheds in Seoul. The system showed the hit, false and missed alarm rates at 69% (48%), 31% (52%) and 6.7% (4.5%), respectively for the 2010 (2011) event.


Introduction
Floods have posed serious challenges to urban areas, with high concentration of population and properties. Urban floods in history often caused widespread devastation, economic damages, and loss of human lives across the globe (Hammond et al. 2015;Miller et al. 2014;Miller and Hutchins 2017;Sarmah and Das 2018). Even worse, the magnitude of damages from the urban floods have been increasing at a rapid rate (O'Driscoll et al. 2010;Chen et al. 2015;Miller and Hutchins 2017) due to following reasons. First, the number of people living in the urban areas is increasing. In 2008, a research from UN showed that half of the world's population lived in urban areas, and it was projected to rise to 60% in 203060% in and 70% in 205060% in (UN 2008. This rapid urbanization rate is resulting in the increase of imperviousness and runoff in these areas, thereby leading to an increase in damage from the urban flood. Second, the climate change also led to the increase in magnitude and frequency of the rainfall events (Stott et al. 2004;Pall et al. 2011;Min et al. 2011). In some areas, 100-year flood event could become 2-5 year flood event leading to increase the flood risk and associated damages (Milly et al. 2002).
There have been various attempts to mitigate the urban flood damages, including structural and non-structural solutions. The structural solutions, such as levees, detention ponds, and underground conveyance tunnels, have been used to prevent the flooding itself. However, a typical structural solution is considered as expensive and ineffective for the urban areas because most of the ground surface and underground space are occupied by civil infrastructures. Conversely, early flood warning systems, which is the most popular and effective solution among non-structural solutions, do not require high costs that accompanies building the structural solutions. Thus, non-structural solutions are found to be more economic to reduce the flood damages in many urban places across the world compared with the structural measures (UNEP 2002;WMO 2011;Jha et al. 2012;Chen et al. 2015). The European Flood Awareness System, for instance, showed highly accurate performance in the early warnings for several recent heavy rainfall events, including the Balkan floods in 2014 (Pappenberger et al. 2015) and the Central European floods in 2013 (Haiden et al. 2014;Pappenberger et al. 2015). Paprotny et al. (2018) conducted study on the trend of losses due to flooding in the period of 150 years over Europe. The study found an increase in the flooded area and the number of affected people. However, there has been a significant decrease in the flood-related fatalities, reduced by 4.3% per year during the period from 1950 to 2016, due to the efficient non-structural method of incorporating the early flood warning system to the existing structural solutions.
Performances of these early flood warning systems significantly depend on the rainfall threshold to initiate the flood warning (also known as Rainfall Threshold for Flood Warning or RTFW). It is typically defined as the cumulative rainfall depth or rainfall intensity for a given duration and an initial soil moisture condition that is likely to cause the floods (Golian et al. 2010;Priest et al. 2011;Jang 2015;Thomas et al. 2018). The RTFW is derived based on either empirical or simulation-based approaches (Sene 2008;Zêzere et al. 2015). The empirical approaches of deriving RTFW values are primarily based on the observed data. These approaches identify the relationship between the various characteristics of floods such as timing, flood area, and the corresponding rainfall. The empirical approaches have been widely applied for its methodological simplicity (Glade et al. 2000;Duncan et al. 2011;Caine 1980;Hong et al. 2005;Dahal et al. 2009;Cannon et al. 2008;Guzzetti et al. 2007;Saito et al. 2010;Dao et al. 2020a, b). However, it is difficult to implement these methods for the urban flood analysis because of the challenges in detecting the urban flooding due to fast watershed response time (Emmanuel et al. 2012;Einfalt et al. 2004) and complex flooding mechanisms governed by drainage pipe network settings, terrain, and landcover Zhang and Pan 2014). Dao et al. (2020a) overcame this issue by using the flooded area data obtained from two massive field observation campaigns across the city of Seoul. A database containing every single flooded building during the two major flood events of 2010 and 2011 for 239 watersheds was used in the study. They discovered that the RTFW is strongly influenced by the temporal variability of the rainfall. A similar conclusion was found as Dao et al. (2020b) performed an analysis on Busan, Korea based on the flooded area data derived from Sentinel-1 synthetic aperture radar images.
Simulation-based approaches have been used to identify the rainfall threshold with hydrologic and hydraulic model simulations for various scenarios of rainfall and soil moisture conditions (Mogil et al. 1978). It is popular to define rainfall thresholds based on the numerical simulations for river and mountain basin (Reed et al. 2007;Miao et al. 2016;Golian et al. 2010;Kim et al. 2016a, b;Zêzere et al. 2015). However, the simulation-based approaches still have following challenges to be applied in the urbanized area. First, the approach requires extensive geospatial data, such as sewer network and high-resolution topography to develop a numerical urban flood model (Oh et al. 2016). Second, the urban flood simulations are computationally expensive, causing short warning lead time in operation Zhang and Pan 2014). Therefore, only few studies have been performed to determine the rainfall threshold for urban areas with the help of simulation-based method. Blanc et al. (2012) proposed the method to derive RTFW based hydrodynamic model. The InfoWorks CS model, a kind of 1D/1D model, was used simulate the flood for two urban areas in UK with various rainfall scenarios with rainfall duration and intensity ranging from 1 to 24 h and 1 to 10 mm/h, respectively. Based on the analysis on the relationship between the flooded area and the rainfall duration and intensity, authors derived an equation to estimate RTFW as a function of rainfall duration and intensity. Jang (2015) has developed a method to determine the rainfall thresholds for urban areas over Taiwan based on the hydrodynamic modeling. In this research, the flood simulation model was set up for 252 watersheds with 22 rainfall events occurred from 2004-2012. The initial rainfall threshold of each watershed was identified when the simulated water depth at a specific point is 50 cm or higher. The rainfall threshold was then stochastically adjusted based upon the observed data of 22 flood events. The results showed that the performance of warning system based on the rainfall threshold shows high accuracy ranging from 0.6 to 0.9 and the hit ratios outperforms false alarm ratio.
The contributions of this study compared to the previous ones are as follows: first, most previous studies on the RTFW of urban areas rely on the inundation areas simulated based on the 1D (pipe network)-1D (overland flow) models that often produce inaccurate overland inundation area: they do not have an algorithm to consider interaction between the overland flow and drainage inlets (Seyoum et al. 2011), which may lead to inaccurate flooded area and subsequently reduces the reliability of the RTFW (Chang et al. 2015). This study employed the 1D-2D model to resolve this issue. In addition, this study calibrated the model parameters based on the water depth time series observed at the watershed outlets. Even though the parameter regarding watershed imperviousness were later swapped with various values to derive its relationship to the RTFW, the remaining parameters (e.g. underground pipe roughness and overland surface roughness) remained identical, which helped to secure the realistic representation of the urban watershed flood process. Second, most previous studies suggested the RTFW in the format of rainfall depth-duration relationships with which the impact of rainfall temporal variability cannot be considered. Recent studies (Dao et al. 2020a, b) showed that rainfall temporal variability significantly influences the RTFW, which this study also explicitly addressed throughout the methodologies. Third, this study provided the RTFW in the format of the mathematical equation. The equation has only the two parameters (the coefficient of variation of the rainfall event time series and the watershed curve number) that can easily be obtained, which improves applicability of the equation.

Methodology
The XP-SWWM model was used in this study to simulate inundation in the selected basins. First, the model was calibrated by comparing simulated and observed water level in the pipe located at the watershed outlet. The calibrated model was then used to simulate the flooded area based on various synthetic rainfall scenarios. For each rainfall scenario, the rainfall intensity and respective flood depth were collected to establish their relationship, which served as fundamental basis to determine RTFW. Through this analysis, it was identified that rainfall event temporal variability significantly influences RTFW. Then, the model simulation was repeated by modifying the originally calibrated parameters (namely, NRCS curve number) to further investigate the influence of watershed imperviousness on flooding. From this investigation, the equation to find RTFW was derived as a function of rainfall coefficient of variation and watershed curve number.

Study area
This study was applied to the metropolitan city of Seoul, the capital of South Korea, for the analysis. It covers an area of 605 km 2 ( Fig. 1c) with highly dense population. The city is featured by mountainous topography with steep slopes (average slope of 8.57%) and high percentage of impermeable areas (48.64%), which makes it prone to flooding when a heavy rain occurs (Yoon 2014). In addition, the majority of residential and commercial areas are located on floodplain, exacerbating damages caused by the flood (Yoon and Lee 2017). From 1980 to 2015, Seoul experienced seven extreme rainfall events (Dao et al. 2020a). In particular, the event of 2010 September 21st with a record accumulative rainfall of 480 mm caused flooding in 149 out of a total 239 watersheds across Seoul, causing financial loss of 42.9 million dollars and 17,905 flooded houses (Yoon 2014). The drainage system in Seoul is divided into 239 sub-watersheds (Fig. 1c). Due to Han River flowing through the city, the city's main drainage system is designed to discharge the excess water into the river.
In this study, nine watersheds (blue polygons in Fig. 1c) with high level of urbanization in Seoul were selected to perform flood simulation. These watersheds have area ranging from 0.82 to 4 km 2 and their average slope varies from 3 to 19.1%.

Data description
In this study, the data of rainfall, pipe network, and land cover were used as input for the XP-SWMM model to simulate inundation for the study areas. The data of the pipe network ( Fig. 2) are provided by the Metropolitan Government of Seoul (MGS 2019). Characteristics of the pipe network including pipe length, slope, shape, and diameter are used as input for the 1D model to simulate the hydraulic behavior of the flow in the pipe, while the manhole elevation was used to develop the 2D model to simulate the movement of flow over land surface. In addition, the flow data at the outlet of the study watersheds ( Fig. 2) were also collected to serve the model calibration process (SODF 2019).
Input data for flood simulation models including pipes and manhole data were provided by Seoul metropolitan government. The drainage network data was then simplified to reduce the computation time as well as uncertainty in the simulation results (Yoon and Lee 2017). Accordingly, pipes with diameters and lengths smaller than 0.5 m and 100 m, respectively, were removed. The characteristics parameters of drainage systems of watersheds used in the simulation model are listed in Table 1.
Observed rainfall data was used for model calibration. The in-situ rainfall data used in this study was collected from 31 ground-based stations located in and around Seoul with a measurement interval of 1 min (Fig. 1c). The rainfall event that occurred on July 23, 2013, which caused flooding in several areas in Seoul, were selected. To calibrate the simulation model, 1-min pipe flow depth data was acquired for each of the catchments recorded between 04:20 AM and 10:10 AM.

Flood simulation model
Recent studies have shown that combining 1D and 2D models is the most appropriate approach for urban flood simulation (Leandro et al. 2009;Chang et al. 2015;Fan et al. 2017;Jahanbazi and Egger 2014). Therefore, the XP-SWMM model was chosen to conduct the analysis in this study. In the XP-SWMM, the study area is divided into sub-watersheds based on the location of the manholes and the topographic data (Fig. 3). The rainfall-runoff process in each sub-watershed is simulated using RUNOFF package. Then, hydraulic behavior of the flow along the pipe network is modeled by 1D pipe network model, called EXTRAN package which uses a hydrodynamic equation and a continuous equation. Finally, 2D surface water model (TUFLOW package) simulates the movement of water routing onto the surface through manholes.
The model was simulated with rainfall input and the following parameters: curve number (CN), time of concentration of sub-watersheds, and roughness coefficient of pipe. Once the simulation was completed, each modeled water level was compared with the observed data at corresponding flow observation gauge. Based on the comparison results, a calibration process was conducted through adjusting the curve number and roughness coefficient of the pipe. This parameter adjustment follows the principle that the curve number affects amount of water and the roughness coefficient changes for the velocity of the flow in the pipe (USDA 1986). Once these parameters were adjusted, the model was run again and the water level was compared and evaluated by the coefficients of determination method (R 2 ). This procedure was repeated until the R 2 between observed and simulated water depth reaches the highest possible value. Figure 4 shows the flow depths (y-axis) of nine watersheds obtained from the simulation model (blue lines) and the observation data (dash-orange lines) corresponding to the analyzed rain event (x-axis). It indicates strong correlation between the simulated and the observed pipe water level height with R 2 value ranging from 0.68 to 0.96.

Inundation simulation with various scenarios
of rainfall to derive the RTFW.
After calibration, the model was run with various rainfall scenarios, focusing on the changes in intensity and rain distribution over the time of events. For generation of synthetic rainfall scenarios, a recent variant of the Poisson cluster rainfall model (Kim and Onof 2020) was used. Poisson cluster rainfall models generate rainfall time series as a series of rainstorms containing a series of rain cells.
Here, the arrival of rainstorms and rain cells are governed by two distinct Poisson processes, and each of the rain cells are assigned with random depth and duration drawn from  given probabilistic distributions. The models have five to eight parameters depending on the details of the assumptions made to mimic the complex rainfall process in reality. Unlike the traditional Poisson cluster models (Rodriguez-Iturbe et al. 1988;Onof and Wheater 1994;Kim et al. 2013;Kim et al. 2016a, b), the rainfall model employed in this study (Kim and Onof 2000) can accurately reproduce the extreme values. This is because (1) the model is based on the mathematical framework that can consider not only the first-and the second-order but also the third-order statistics of observed rainfall (Onof and Wang 2020); (2) the model can consider the inverse relationship between the rain cell duration and intensity that enables the reproduction of short-duration intense rainfall events (Kaczmarska et al. 2014);and (3) the model adopted the two-parameter gamma distribution to model depth of rain cells. For these reasons, the model of Kim and Onof (2020) shows dramatically improved performance in reproducing extreme values of observed rainfall at a wide range of timescales compared to its predecessors. The model parameters were calibrated based on the 20 years (2000-2019) of minutely rainfall data observed at Yongsan in-situ gauge (blue star mark in Fig. 1c). 500 years of 10-min rainfall time-series was generated using the calibrated rainfall model, from which 230 rainfall events were picked up. Here, 4 h of inter-storm dry period was used to differentiate one rainfall event from another. The relationship between accumulated rainfall and duration of the selected events are shown in Fig. 5. The color of the points represents coefficient of variation of rainfall event. Then, we further filtered out, for each of the watersheds, the events to obtain final 15 events of which average intensity ranges from 20 to 90 mm/h, which encapsulates what was identified by the previous studies (34 mm-62 mm by Dao et al. 2020a; 55 mm-73 mm by Dao et al. 2020b). Then, the flood simulation model was run for 135 rainfall events (15 rainfall events 9 9 watersheds) to derive flood map, and this entire process was repeated for four different levels of watershed imperviousness (CN = 80,85,90,95). As a result, this study performed a total of 540 simulations (15 rainfall events for each of the 9 watersheds with 4 different watershed imperviousness). For each simulation, we obtained the maximum flood depth and corresponding rainfall intensity to establish relationship. Note that the rainfall intensity may differ from a watershed for another even for the same rainfall event. This is because the intensity was calculated over the duration that is equal to the watershed time of concentration. The watershed imperviousness plays an important role in the flooding mechanism. Thus, it was assumed in this study that the watershed impermeability also has an impact on the relationship between rainfall intensity and inundation level in the study area. Therefore, instead of using the calibrated curve number (CN)   3 Results and discussions

Flooded depth versus flood inducing rainfall
Simulation results of 135 rainfall scenarios were used to establish the relationship between the rainfall intensity and the corresponding maximum flooding depth for the nine studied watersheds shown in Fig. 6. The non-flooded cases were excluded from the plot. A positive relationship exists between the rainfall intensity and the maximum inundation depth across all watersheds, indicating that higher rainfall intensity leads to higher the inundation level. However, there are significant uncertainties residing in this relationship as suggested by the R 2 value ranging from 0.64 to 0.98.

Impact of rainfall temporal variability on urban flood generation
We assumed that the uncertainty may stem from the temporal variability of the rainfall. To verify this assumption, coefficient of variation (CV) that represents the rainfall temporal variability was first calculated for 135 rainfall events. Then, the 135 events were classified into four categories based on four distinct intervals of the rainfall event CV. Here, boundaries of the four intervals were determined so that each category contains at least 20 events to secure the validity of further analysis including regression. Figure 7a-d shows the relationship between the rainfall intensity and the maximum flooded depth for the four intervals of the CV. The lines in these figures show best-fit exponential regression models (y ¼ ae bx ) of the scatters. Figure 7e shows the collection of the best-fit exponential regression lines shown in Fig. 7a-d. The regression line corresponding to greater CV range is located at the higher position, which suggests that, although significant uncertainty resides in the relationship, a storm with higher temporal variability generally results in greater inundation. For example, Fig. 8 compares the flood maps of the Hwagok watershed for the similar rainfall depths (63.56 mm/h vs 62.49 mm/h) but with different temporal variabilities (0.38 vs 0.13). Both the maximum flood depth and the flooded area were greater for the event with greater CV by 57 and 88%, respectively.

Impact of antecedent moisture condition on urban flood generation
Antecedent moisture condition (AMC) plays an important role in generation of runoff. Figure 11 compares the flood extents corresponding to the same rainfall event and the same imperviousness (CN = 90), but with different AMC.
To reflect the different AMC in simulation, the CNs were adjusted based on the following equations (USDA 1986): CNðdryÞ ¼ 4:2CNðnormalÞ 10 À 0:058CNðnormalÞ The CNs for dry and wet AMCs are 79 and 95, respectively when the CN under normal condition is 90. The comparison of the flood extents in the Figure clearly shows that the wetter AMC produces greater flooded area. Following this finding, this study also adjusted the CN using Eq. (1) while validating the methodology (Sect. 3.7). Note that no specific measures were taken to account for the influence of AMC while developing the RTFW. This is because (1) CN has a capacity to consider the AMC through Eq. (1) and because (2) reflecting the influence of the true AMC would require in-situ measurement and the use of more complex approach to simulate the AMC (e.g. Richard's equation and Green-Ampt equation), which subsequently would force the RTFW equation to include the terms that is hard to quantify.  Figure 12 shows the result of this contrasting simulations. The result suggests that the maximum flooded depth and the flooded area can be significantly greater for the case of temporally variable rainfall and wet AMC than for the case of temporally uniform rainfall and dry AMC. The relative difference was 180% and 160%, for the maximum flooded depth and the flooded area, respectively.

Determination of RTFW
To determine RTFW, it is important to determine the threshold of flood depth over which the accumulated water is considered dangerous. This study adopted the flood depth threshold of 30 cm following Kramer (2016). Therefore, this study chose only the rainfall events corresponding to inundation depth ranging between 0.25 and 0.35 for the analysis to determine RTFW. There were 101 out of 540 events meeting this criterion. The scatters in Fig. 13 shows the relationship between rainfall intensity and coefficient of variation of these 101 events. Note that not all 101 events are clearly shown because many events share the same rainfall intensity and CV, so they overlap on the plot. The next step is to determine the RTFW formula. Sections 3.2 and 3.3 show that both CN and CV have clear positive relationship with flood depth, so high values of CN and CV would lower the RTFW value. Following this finding and the pattern of the scatters in the plot, we first assumed that RTFW would exponentially decrease as CV increases as Eq. (2). Then, we performed the least-square residual fitting on the pairs of CV (x) and intensity (y) corresponding to each of the CNs (80, 85, 90, 95). As a result, a set of parameters, namely b 1 , b 2 , and b 3 , was obtained for each of the CNs, which was shown in the subplots of Fig. 13. The subplot suggests that the parameters have the clear relationship with CN, so each of them were expressed as the quadratic equations (Eqs. 2a, 2b, and 2c).
where RTFW, CN, and CV represent rainfall threshold for flood warning (mm/hr), watershed curve number, and rainfall temporal variability. Solid color lines in Fig. 13 graphically represent Eq. (2). Each line represents different watershed imperviousness (CN). The relationship suggests that the RTFW is highly Fig. 12 Flood maps corresponding to the rainfall with similar rainfall intensity (63 mm/h) but with contrasting temporal variability (CV = 0.13 vs 0.38) and antecedent moisture conditions (wet vs dry) sensitive to the rainfall temporal variability at the range of CV less than 0.2. This is primarily because runoff amount corresponding to rainfall events with different temporal variability varies significantly at this low range of CV. Considering that the CV of frontal rainfall is generally low (Olivera et al. 2008;Kim et al. 2019), this result also suggests that the RTFW may be significantly different for the two similar frontal rainfall events even if they have slightly different temporal variability. Also note that, within this low range of the CV, the influence of the watershed imperviousness on the RTFW is not significant as suggested by little difference between the lines in the plot. This suggests that the influence of the rainfall temporal variability on rainfall-runoff process overwhelms the that of watershed imperviousness. For the range of large rainfall temporal variability (e.g. CV [ 0.4), the difference between the curves is significant and each of the curves has low slope with regard to the value of CV. This means that the watershed imperviousness rather than rainfall temporal variability is a dominating factor of the RTFW within this range of the CV. This is because the rainfall event with large CV is likely to have a hyetograph with a temporally intensified pattern, and the abstraction of rainwater is likely to take place in the intensified portion of the hyetograph, which subsequently makes the runoff amount sensitive to the CV.

Validation of RTFW
This study validated the RTFW values that were estimated from against 149 rainfall events that occurred at 112 subwatersheds in Seoul over the period of 2010 and 2011. During these two years, Seoul Metropolitan Government recorded every single flooded building (Open Data Platform, Seoul Metropolitan Government, 2020) in the city, which constitutes an accurate and precise spatial database for validation of the RTFW. The warning system first estimates the watershed parameters (i.e., Curve Number and time concentration) of the 239 watersheds of Seoul (Fig. 1c). Urban Storm Drainage Criteria Manual (Drainage 2016) was used to calculate time of concentration, and Curve Numbers of the watersheds were estimated as area-weighted averages of the CN values based on the land cover of the watershed (Fig. 1b, EGIS 2019). The CN value was then adjusted according to the accumulated rainfall over the prior of past 5 days to the event using Eq. (1) (USDA 1986). Second, rainfall that shed on each of the 239 watersheds was estimated based on the 10 min-1 km radar-gauge composite rainfall field (see Han et al. 2021 andDao et al. 2020a, b for radar-gauge rainfall data composition algorithm). This process is repeated every 10 min as the newly observed rainfall field is fed into the system. As a result of this process, each of the 239 watersheds is coupled with rainfall time series. Third, the most recent x-minute sub-time series is extracted from the time series for each of the watersheds. Here, the value x corresponds to the time of concentration of the watershed. Fourth, average rainfall intensity (I ave ) and the CV are computed based on the extracted sub-time series. Finally, the RTFW of each watershed was derived using Eq.
(2) based on the rainfall CV and the watershed CN, which is subsequently compared with the corresponding I ave . If I ave exceeds the RTFW, a flood warning is issued (Fig. 14). Figure 15 shows the validation result. In the figure, hits (orange columns in Fig. 15a) represent the number of watersheds that is flooded and warned; false alarms (orange columns in Fig. 15b) represent the number of watersheds that is not flooded but warned; and missed alarms (orange columns in Fig. 15c) represent the number of watersheds that is flooded but not warned. For the 2010 event, rate of hit, false alarm, and missed alarm were 69 (¼ 139 202 Â 100), 31 (¼ 63 202 Â 100) and 6.7 (¼ 10 149 Â 100)%, respectively. For the event of 2011, the rate of hit, false alarm, and missed alarm were 48 (¼ 107 223 Â 100), 52 (¼ 116 202 Â 100), and 4.5 (¼ 5 112 Â 100)%, respectively. The result of 2011 has lower rate of hit and higher rate of false alarm, which means that flood started to occur at lower rainfall intensity. This is primarily associated with the antecedent soil moisture condition of watershed, which plays a major role in flood generation. The 2011 flood was a frontal event that lasted several days while 2010 flood was a convective event that lasted only 6 h. One way to resolve this issue is to lower the RTFW by lowering the CN value based on the antecedent soil moisture condition, which may be inferred from the previous rainfall depths or remotely sensed data (Kim et al. 2016a, b).
The size of the watersheds tested in the warning system ranges from 0.5 to 6.8 km 2 , which means that the warnings can be given to a spatial scale of the smallest administrative unit in many countries. If the warning system runs on the merged watersheds with greater size, the rate of false alarm would significantly decrease and the rate of hit would also significantly increase (Dao et al. 2020b). Here, it is worth noting that the system failed to issue flood warning only for 6.7 and 4.5% of all flood events in 2010 and 2011, respectively, in spite of the high spatial precision.
Also note that the RTFW and the warning system is designed to work with the 10-min rainfall. Here, the RTFW is a function of the CV of the 10-min rainfall. However, rainfall is not often measured at this fine temporal resolution and measured mostly at hourly scale in many other regions of the world. This study further investigated the situation in which the rainfall data is provided to the system at only this coarse (e.g. hourly) temporal scale. If the measured hourly rainfall is uniformly distributed at 10-min Fig. 14 Operation procedure of the flood warning system level, the CV would be zero. This subsequently decreases the RTFW and significantly increases the miss rate of the system from 6.7 to 79.2%. One way to overcome this issue is to apply the average CV (0.167) of the rainfall events instead of using the value of zero. The blue-green column in Fig. 15a-c shows the performance of the warning system based on the average CV, which is similar to that of the original case with the improved hit rate at the expense of increased miss rate.

Impact of the other factors on urban flood generation
This study estimated the RTFW based on two factors affecting the flood mechanism which are temporal variability of rainfall precipitation and watershed imperviousness. However, flood generation is also affected by other factors such as spatial variability of rainfall and drainage network. Various studies analyzed the influence of rainfall spatial variability on the generation of runoff (Anquetin et al. 2010;Bedient et al. 2000;Brath et al. 2004;Smith et al. 2004), however, with contrasting conclusions: some concluded that rainfall spatial variability has a strong influence on runoff (Anquetin et al. 2010;Bedient et al. 2000), others showed the insignificant influence (Nicótina et al. 2008;Tarolli et al. 2013;Brath et al. 2004). In addition, Weijian et al. (2015) confirmed that the effect of rainfall spatial variability on runoff strongly depends on watershed area. They found that the rainfall spatial variability just contributes less than 10% on total runoff if the watershed has the size less than 10 km 2 . Dao et al. (2020a) analyzed the spatial variability of rainfall for 239 small watersheds (area ranges from 0.5 to 6.8 km 2 ) in Seoul city based on the radar-gauge composite rainfall data with spatial resolution of 1 km. The result shows that the CV of the pixel rainfall values ranges from 0 to 0.35 and that 83% of the 239 watersheds have the CV values less than 0.05. These results indicate that the temporal variability of rainfall over small watersheds is either not significant or cannot be accurately measured even with the most recent technology of spatial rainfall measurement. Therefore, we chose not to consider the influence of the rainfall spatial variability on RTFW. Drainage network configuration also significantly influences the flooding mechanism. Seo et al. (2015) showed that the sinuosity of drainage network has great impact on magnitude and the timing of peak flow and the degree of overland flow. This finding suggests that the accuracy of the RTFW would increase if it can consider the influence of the drainage network configuration. The challenge to overcome in this regard is to develop a method to accurately parameterize the network configuration in the perspective of watershed flooding. The centrality, betweenness, and closeness are typically used to parameterize networks, but they have not been extensively analyzed in the perspective of watershed flooding.

Conclusions
This study proposed an equation for rainfall threshold for flood warning that can be applied to small (e.g. less than 10 km 2 ) based on 540 dual drainage hydrodynamic model simulations. The major findings are as follows: (1) The RTFW ranged from 20 to 90 mm/h. The RTFW decreased as the rainfall temporal variability increases and as the watershed imperviousness increases. The RTFW was provided as mathematical formula with two parameters (CV and CN) for enhanced applicability. Unlike the previous studies, this study suggested a way to explicitly consider the influence of rainfall temporal variability on the RTFW. (2) The RTFW was validated over 239 watersheds over Seoul based on 1 km radar-gauge composite rainfall data. The ratio of hit, false alarm, and miss ranged at 48-69%, 31-52%, and 4.5-6.7%, respectively. (3) The RTFW exponentially decreased as the CV increased, and the RTFW decreased as the CN increased following a linear trend. Two distinct regimes of rainfall temporal variability were identified over which RTFW behaves differently. When the CV is less than 0.2, the RTFW was more sensitive to the rainfall temporal variability than it is to the watershed imperviousness. The opposite was observed when the CV is greater than 0.4. This result suggests that the rainfall temporal variability should be carefully considered when giving the flood warning during the frontal rainfall event with low temporal variability. Conversely, watershed imperviousness should be carefully considered when giving the flood warning during the convective events that exhibit large high variability.
The accuracy of the RTFW of this study would increase by considering influences of the additional factors such as drainage pipe networks and terrain characteristics. Development of a method to parameterize these factors in the perspective of urban flooding is a challenging yet intriguing tasks, which this study considers as a future work. Availability of data and material Rainfall data that was used in this study is available over the internet website of Korea Meteorological Administration (www.kma.go.kr). The drainage pipe network data that was used in this study is available upon request to Seoul Metropolitan Government (www.seoul.go.kr). The land use and the DEM data will be available upon request to National Geographic Information Institute of Korean Government (www.ngii.go.kr).

Declarations
Conflict of interest All authors declare that they have no conflict of interest.