Enhancing a Real-Time Flash Flood Predictive Accuracy Approach for the Development of Early Warning Systems: Hydrological Ensemble Hindcasts and Parameterizations

Abstract

This study marks a significant step toward the future development of river discharges forecasted in real time for flash flood early warning system (EWS) disaster prevention frameworks in the Chugoku region of Japan, and presumably worldwide. To reduce the disaster impacts with EWSs, accurate integrated hydrometeorological real-time models for predicting extreme river water levels and discharges are needed, but they are not satisfactorily accurate due to large uncertainties. This study evaluates two calibration methods with 7 and 5 parameters using the hydrological Cell Distributed Runoff Model version 3.1.1 (CDRM), calibrated by the University of Arizona’s Shuffled Complex Evolution optimization method (SCE-UA). We hypothesize that the proposed ensemble hydrological parameter calibration approach can forecast similar future events in real time. This approach was applied to seven major rivers in the region to obtain hindcasts of the river discharges during the Heavy Rainfall Event of July 2018 (HRE18). This study introduces a new historical extreme rainfall event classification selection methodology that enables ensemble-averaged validation results of all river discharges. The reproducibility metrics obtained for all rivers cumulatively are extremely high, with Nash–Sutcliffe efficiency values of 0.98. This shows that the proposed approach enables accurate predictions of the river discharges for the HRE18 and, similarly, real-time forecasts for future extreme rainfall-induced events in the Japanese region. Although our methodology can be directly reapplied only in regions where observed rainfall data are readily available, we suggest that our approach can analogously be applied worldwide, which indicates a broad scientific contribution and multidisciplinary applications.

1. Introduction

As a result of the impacts of climate change, record-breaking disasters associated with unprecedented heavy rainfall, typhoon intensities and associated coastal sea disasters are occurring every year worldwide, causing significant societal and economic damage. Meanwhile, regional impact assessments of climate change on severe weather events, hazards and water resources are still limited [1]; therefore, connecting changes in atmospheric circulation and simulated floods is crucially needed to conduct robust impact assessments of regional climate change flood risk [2]. Rainfall-related disasters deliver much river water and associated nutrient pollution to river mouths and into coastal sea zones, where they greatly reduce sea water quality and can negatively impact marine environments for several days or weeks.

To cope with climate change impacts, mitigation of risks and adaptation for such disasters are needed. The development of real-time early warning systems (EWSs) for flash floods is one form of such disaster risk reduction mechanisms. Thus, it is important to develop real-time forecasting (also referred to as nowcasting) models to predict extreme river water levels and discharges before and during flood events. Ref. [3] comprehensively reviewed various existing solutions for disaster risk management, highlighting EWS nowcasting as one of the emerging technologies. The development of such nowcasting models is often associated with multiple levels of uncertainty in meteorological, hydrological and coastal ocean systems and corresponding real-time disaster prevention decision-making processes. Many recent studies [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] have used various hydrological or hydrometeorological approaches to develop such real-time forecasting or nowcasting models worldwide. In the aforementioned and other related flash flood forecasting studies, deterministic and ensemble approaches are applied regarding only meteorological modeling and hydrological ensemble cross-parameter calibration [26,27], where ensemble approaches are usually more accurate. However, to our knowledge, ensemble cross-event parameterization with such an analogical approach has not yet been applied regarding hydrological modeling, particularly to parameter calibration across various historical extreme rainfall-induced events. Due to their extensive meteorological data coverage, which can be easily extrapolated to other similar-scale events worldwide, case studies in Japan are used as integrals to address broader challenges posed by extreme events.

1.1. Heavy Rainfall Event in July 2018 in Western Japan

In July 2018, an unprecedented rainfall-induced disaster resulting from the interaction of Typhoon Prapiroon with the seasonal Baiu rain front occurred in western Japan, resulting in 224 casualties, 8 missing people and 459 injured people [28]. The disaster produced multiple catastrophic impacts, including floods, landslides and pollution transport through rivers into the coastal Seto Inland Sea, where this pollution severely damaged the local fishery and seashell industry. Since then, previous studies on this event from meteorological, geotechnical and hydrological perspectives have been performed with emphasis on possible forecasting methods for expected future occurrences. Geotechnical studies conducted on the HRE18 assessed the regional landslide hazard using geological and high-resolution hydrometeorological factors [29] and reported unprecedented types of geo-disasters in Hiroshima Prefecture due to the large amount of cumulative rain [30].

Several studies have described the meteorological context of the Heavy Rainfall Event of July 2018 (HRE18). Ref. [31] showed that the HRE18 was a combination of two extremely moist airflows of tropical origins along the stationary Baiu front, and [32] characterized the spatial and temporal pattern of the HRE18 as extreme by various criteria compared to other similar-scale events since 1982. In terms of future expectations, refs. [33,34] estimated that the intensity and frequency of extreme precipitation associated with the HRE18 and Baiu heavy rainfall will increase under a warming climate in the future. Furthermore, ref. [35] showed that precipitation and flood discharge responses to climate variability and climate change appear to be strong in relatively small and steep Japanese river basins. Therefore, developing models for real-time forecasting of such rainfall-induced disasters is becoming increasingly important in comprehensive meteorological, hydrological and coastal ocean contexts. Developing robust and accurate real-time meteorological forecasting tools is a first step toward conducting associated integrated hydrometeorological forecasts.

Several studies [34,36,37,38,39,40] have developed meteorological forecasts for the HRE18. Several hydrological studies have been conducted for the HRE18. Ref. [41] applied a 2D flood flow International River Interface Cooperative model using the solver Nays2DH (iRIC-Nays2DH) for the Sozu River and found that active sediment deposition caused severe inundation depths. Ref. [42] conducted field surveys along the Oda River Basin, a tributary of the Takahashi River, and found that the inundation depths exceeded 5 m along 1 km in the north-south direction and 3.5 km in the east-west direction on the north side of the river. Refs. [43,44] analyzed inundation in the Oda River Basin using a Rainfall-Runoff-Inundation (RRI) model with the default set of parameters (from [45]) but without calibrating them for application to the Oda River Basin. Ref. [46] evaluated the hydrological predictability and discussed a possible future nowcasting application of data on the HRE18 and Typhoon Hagibis in 2019 using an uncalibrated RRI model. Ref. [44] noted that a limitation of their methodology is that the model parameters also depend on the input scale of the topographic features and, therefore, noted the necessity of their calibration for different types of studies. However, to our knowledge, accurate hydrological models for nowcasting future extreme rainfall-induced flash flood disasters of similar scales as HRE18 have not yet been developed.

1.2. Recent Hydrological Studies on Heavy Rainfall Disasters

Many recent studies worldwide and in Japan have focused on simulating river water or inundation levels during heavy rainfall events using various approaches and methodologies. One frequently applied methodology is using physically distributed hydrological modeling, which can be further classified into modeling with kinematic wave-based methods [47,48,49,50,51,52], dynamic wave momentum equation methods [14,22,53], diffusive wave transfer methods [13,44,45,54,55,56,57] and multimodel comparisons from lumped to fully distributed [19]. Among them, refs. [14,19,22,48,51,52,56] recommended the possible application of their modeling to real-time forecasting flood events using various approaches and lead times. Refs. [51,52] proposed a methodology to calibrate the hydrological Cell Distributed Runoff Model version 3.1.1 (CDRM) with the Shuffled Complex Evolution optimization method, which was developed at the University of Arizona (SCE-UA), for nine first-class rivers in northeastern Japan. They showed that this calibrated model could accurately predict the extreme river discharges of two [51] and three [52] typhoons with similar trajectories and intensities and similarly be applied for nowcasting the river discharges associated with similar extreme events in the future. However, [52] noted that the application of their methodology to other types of spatial and temporal rainfall distributions that are different from typhoons must be further investigated.

1.3. Objectives, Hypothesis and New Contributions of This Study

Ref. [57] summarized that fundamental technical requirements for hydrometeorological flash flood forecasting include (1) a precipitation detection system; (2) a short-range numerical weather prediction model; and (3) a hydrological–hydraulic forecasting model capable of forecasting the stream response over a wide range of rainfall input scales. This study proposes an innovative methodology and approach for scientific advancement toward the third technical requirement. We develop a calibrated hydrological modeling system that can be used for real-time forecasting of river flash flood discharges caused by future extreme rainfall events in the Chugoku region of Japan and presumably be reapplied worldwide. In particular, the specific questions addressed in this study are summarized as follows:

(1) How can historical extreme rainfall events be classified into similar scales for calibrating a hydrological model?

(2) How accurately can the CDRM model parameters calibrated with the SCE-UA optimization method reproduce the river discharges that resulted from the HRE18 and similarly predict future peak river water discharges from similar-scale extreme rainfall events?

(3) How can our findings be used to predict the flood river discharges caused by future extreme rainfall events in real time?

We hypothesize that calibrated river basin parameters from ensemble historical extreme rainfall-induced extreme river discharge events can accurately predict river discharge hydrographs of extreme rainfall-induced future events of similar scale and reproduce similar past events and vice versa. This hypothesis intuitively arises from similarly using well-known ensemble forecasting technologies in meteorological flash flood forecasting. In meteorological applications, ensemble forecasting technologies often greatly reduce modeling uncertainty and increase accuracy; thus, we expect that similar effects should be valid for hydrological applications.

2. Materials and Methods

2.1. Study Sites

The extreme river discharges from the HRE18 in the Chugoku region of Japan were projected for all seven first-class river basins flowing into the Seto Inland Sea. From east to west, these rivers are the Saba, Oze, Ota, Ashida, Takahashi, Asahi and Yoshii Rivers. Arranged by the catchment area of the station farthest downstream with observed discharge data, these rivers are classified as follows: Takahashi (2644 km²), Yoshii (1996 km²), Asahi (1587 km²), Ota (1527 km²), Ashida (798.8 km²), Saba (423.1 km²) and Oze (323 km²). The total spatial distance from the eastern (Saba) to the western (Yoshii) river basin is approximately 300 km. Figure 1 shows the digital elevation model (DEM) of the seven targeted river basins, their associated mean centers, locations of the observed and simulated discharges near the river mouths, information on the position of the study domain within Japan, Typhoon Prapiroon’s track, and observed discharge near the Asahi River mouth, which had the largest cumulative 5-day rainfall values of 435 mm. The Figure is drawn using tools from [58].

2.2. Materials and Datasets

The observed hourly rainfall data were collected from the online database of the Japan Meteorological Agency [59] as point data from all rain gauges of the associated river basin with available published data (hereafter: observed rain gauge) for 168 h data inputs, with the aim of obtaining the hydrograph peak in the middle of the event on the 4th of 7 days. The observed discharge data were collected from the online database of the Ministry of Land, Infrastructure, Transport and Tourism [60]. Hydrological data and maps based on DEM, flow accumulation (ACC) and flow direction (DIR) data were obtained from the SHuttle Elevation Derivatives at multiple Scales datasets from the U.S. Geological Survey (HydroSHEDS) [61] with a resolution of 15 arc-sec (approximately 500 m resolution) for the DEM, ACC and DIR datasets.

2.3. Model Configurations

The CDRM model [51,52,62,63,64,65,66,67,68] calibrated by the SCE-UA [51,52,64,69,70,71,72] was applied to the seven first-class rivers in the Chugoku region of Japan at a spatial resolution of approximately 500 m (15 arc-sec) and a 1 h time step. River discharges were, in principle, simulated for the most downstream discharge stations with available observed data records. Simulating river discharges on the actual river mouth would need to include other very complex physical processes, such as tidal or backwater effects, which are not desirable in hydrological modeling, so the river mouth discharges were approximated from an observation station relatively close to the actual river mouth but sufficiently far away to not be significantly affected by tidal or backwater effects. In this way, tidal or backwater effects were omitted or greatly reduced from the hydrological modeling, and the study could therefore be focused solely on hydrological effects. However, if tidal effects were great enough to reduce the observed data reliability, then the closest upstream discharge station without a significant tidal effect was selected as the observation and simulation location (hereafter, river mouth). The particular river mouth location for each river is noted in the corresponding results section for that river.

The CDRM is a distributed rainfall-runoff model using the kinematic wave method for surface and subsurface runoff [68]. A very detailed description of another similar model with the same structure and accounting procedure as the CDRM was explained in [67]. Hereafter, we simplify the model description and explain novel additions to the model calibration and parameter optimization methodology applied in this study. Initial river discharge conditions for the hydrological model were obtained from [60], approximately 4 days before the peak of the HRE18. The model calculates subsurface flow in both capillary and non-capillary layers with a single set of storage–discharge equations. The watershed topography is presented with multiple land pixels with capillary and non-capillary pores above an impermeable bedrock. The slope gradient i is obtained from the DEM, while Manning’s roughness coefficient n is treated as a model parameter by setting lower and upper limits. The relationship between the discharge per unit width q and the water depth h with applied parameterization simplifications with the 5 calibrated parameters method (5-CPM) and without applied parameterization simplification with the 7 calibrated parameters method (7-CPM) is graphically represented in Figure 2, and the associated CDRM model structure is explained in great detail in the following. These optimal calibrated parameter sets were used throughout the entire simulations, with initial hydrological model conditions assumed to be identical and fixed for every subsequent step of each simulation.

When the volumetric water content (θ) is smaller than the maximum volumetric water content in the capillary pore (θm), the water flows in the capillary pore as unsaturated subsurface flow, which is modeled by the Darcy equation with a variable hydraulic conductivity k. For unsaturated capillary pore conditions (0 ≤ h ≤ d_m), lateral discharge from a pixel per unit width in the downslope direction in the capillary pore (q_m) is estimated as shown in Equation (1).

$${if\left(0\le h\le d_m\right):q}_m=d_m\times k_m\times {\left(\frac{h}{d_m}\right)}^{\beta }\times i$$

(1)

For 5-CPM: q_m = 0

where i is the slope gradient; km is the saturated hydraulic conductivity in the capillary pore; β is the permeability reduction degree; h is an equivalent water stage (h = D × θ), where D is the soil depth and d_m is the equivalent water stage to the maximum water content in the capillary pore (d_m = D × θm).

When the volumetric water content (θ) exceeds the maximum water content in the capillary pores (θm), the water flows in the non-capillary and capillary pores as saturated subsurface flow, which is modeled by the Darcy equation with saturated hydraulic conductivities. In the non-capillary pore (d_m < h ≤ d_a), the total discharge per unit width in the downslope direction (q_a) is estimated by adding subsurface flow in non-capillary pores and capillary pores, as shown in Equation (2).

$$if\left(d_m<h\le d_a\right):q_a={(h-d}_m)\times k_a\times i+d_m\times k_m\times i$$

(2)

$$\text{For 5-CPM:}if\left(dm<h\le da\right):q_a={(h-d}_m)\times k_a\times i$$

where ka is the saturated hydraulic conductivity in the non-capillary pore.

When the volumetric water content (θ) exceeds the effective porosity of the non-capillary subsurface layer (θa), saturation excess overland flow occurs (d_a < h), which is modeled by the Manning equation. The discharge per unit width (q_s) of overland flow in the downslope direction is estimated with Manning’s roughness coefficient (n), as shown in Equation (3).

$$if\left(d_a<h\right):q_s=\frac{\sqrt{i}}{n}\times {{(h-d}_a)}^m{+(h-d}_a)\times k_a\times i$$

(3)

$$\mathit{\text{For 5-CPM:}}if\left(d_a<h\right):q_s=\frac{\sqrt{i}}{n}\times {+(h-d}_a)\times k_a\times i$$

where m is the slope constant with a value of 5/3.

The total discharge (q) is estimated by adding overland flow and subsurface flow in non-capillary pores and capillary pores, as shown in Equation (4) and graphically in Figure 2.

$$q=\frac{\sqrt{i}}{n}\times {{(h-d}_a)}^m+{(h-d}_m)\times k_a\times i+d_m\times k_m\times i$$

(4)

$$\text{For 5-CPM:}q=\frac{\sqrt{i}}{n}\times {{(h-d}_a)}^m+{(h-d}_m)\times k_a\times i$$

The discharge-stage relationship shown in Equations (1), (2) and (4) and Figure 2, in combination with the continuity equation, which is calculated by the kinematic wave method [67,68], are used to simulate rainfall runoff in each individual cell of the CDRM model, as shown in Equation (5). This relationship enables us to track the rainwater propagation speed in combination with the continuity equation.

$$\left\{\begin{array}{c}\begin{array}{c}\begin{array}{c}q=f\left(h\right):Equations\_\left(1\right),\_\left(2\right)\_or\_\left(4\right)\_for\_various\_h\\ \frac{\delta h}{\delta t}+\frac{\delta q}{\delta x}=r_e\end{array}\end{array}\end{array}\right.$$

(5)

where r_e is the effective rainfall, which is equivalent to the difference between actual rainfall and canopy interception and evapotranspiration rates.

Dam effects on the major rivers were not included in the modeling, following the findings from [51], which note that their impact is negligible when they are built in the upstream parts of river basins. The land use and soil depth (D = 1000 mm) were assumed to be uniform to simplify the calibration process. River cells and nonriver cells were distinguished depending on the upstream contributing areas (ACC) as the threshold, where river cells were represented with a contributing threshold of 2.5 km² or larger, which is equivalent to 10 or more upstream contributing grid cells. The optimization method used in previous studies [51,52] was reused in this study with 5 calibrated parameters, referred to as the 5-CPM. With the five calibrated parameters, the modeling system calculates two levels of water flow: surface flow and non-capillary subsurface flow. The aforementioned studies assumed that all water starts flowing immediately after reaching the soil; thus, they simplified the original 7-CPM parameterization scheme to the 5-CPM. However, this study additionally evaluates the 7-CPM because the relative impact of the capillary subsurface flow layer is very important when heavy rainfall occurrence is prolonged for several days, which was the case for the HRE18. Therefore, the numbers of calibrated parameters in this study are both 5 and 7, where the 7-CPM additionally includes the capillary subsurface flow mechanism (Equation (1)). With the seven calibrated parameters, the modeling system calculates three levels of water flow: surface flow, non-capillary subsurface flow and capillary subsurface flow, as shown in Equation (4).

The five model parameters for the 5-CPM were the soil roughness coefficient (N_slo), river roughness coefficient (N_riv), effective porosity of the non-capillary subsurface layer (θa), saturated hydraulic conductivity (k_a) and the canopy interception and evaporation factor during the rainfall event (F1). The Manning roughness coefficient (n) of river cells is represented by the river roughness coefficient (N_riv), whereas that of nonriver cells is represented by the soil roughness coefficient (N_slo) to differently simulate the river runoff mechanism and surface runoff mechanism. Two additional parameters were used for the 7-CPM to represent the capillary subsurface flow: the effective porosity of the capillary subsurface layer (θm) and the coefficient that indicates the degree of reduction in permeability due to the decrease in the volumetric water rate (β). Since km and β have a simple one-to-one relationship, either of them can be selected for calibration when using the 7-CPM. β was selected for the calibration in this study. θa and θm were calibrated for each river and then used to calculate d_a (=D × θa) and d_m (=D × θm).

The rainfall data input to the CDRM model was calculated as effective rainfall re [m/s] from Equation (5), where F1 reduces the value of actual rainfall (r) to the effective rainfall re (re [m/s] = F1 [/] × r [m/s]). For the actual rainfall (r) data input, we applied a spatial interpolation method based on the nearest neighborhood (Thiessen polygons) because of the high spatiotemporal density of the Japan Meteorological Agency’s rain gauge data, following the same methodology that was previously successfully applied in [51] and [52]. For instance, one Thiessen polygon covers an average of 203 km² of the Takahashi River Basin and 139 km² of the Ota River Basin with a temporal coverage of 1 h for all rain gauge stations. Figure 3 shows a more detailed DEM of the Ota and Takahashi River Basins with their associated mean centers, locations of the observed and simulated discharges near the river mouths, locations of the observed rain gauges and their associated Thiessen polygons. The exact number of observed rain gauges per river basin was 5 (Saba), 4 (Oze), 11 (Ota), 7 (Ashida), 13 (Takahashi), 9 (Asahi) and 9 (Yoshii).

2.4. Model Calibration and Validation

To compare the results of the two aforementioned methods, an identical setup for the CDRM model calibrations was applied to both the 5-CPM and 7-CPM. The SCE-UA control parameters were applied as follows: maximum number of trials allowed before optimization is terminated (MAXN = 10,000), number of shuffling loops where the criterion must improve (KSTOP = 5), percentage by which the criterion value must change in the specified number of shuffling loops (PCENTO = 0.001), number of complexes used for optimization search (NGS = 2), random seed used in optimization search (ISEED = −3) and flag for setting the control variables of the SCE-UA algorithm (IDEFLT = 0). The initial set of parameters and minimal and maximal searching ranges for the SCE-UA parameter calibrations were determined according to previous studies [51,52,64] and with consideration of physically realistic bounds for the parameters, and are shown in Supplementary Table S1.

The CDRM model parameter sets were calibrated against seven of the most extreme recorded historical events in terms of combined maximal 1-day, 2-day, 3-day and 5-day rainfall data [59] based on observations at the station located in the mean center of the associated river basin, for which observed river discharge data [60] from the associated observation station near the river mouth are available. Rainfall data were mostly available since 1976, whereas discharge data were mostly available since 1997 (Ota River) or 2002 (all other rivers), depending on the river basin.

Hereafter, a calibration event represents the case when the calibrated parameters and rainfall input are used from the same event, while a validation event represents the case when the calibrated parameters are used from one event and validated with the rainfall input from a different event. These seven historical events were selected for parameterization because we expected that the physical characteristics of river runoff from these events were similar due to their similar rainfall magnitudes. Hence, the calibrated river basin parameter sets from one event could be used to predict the river discharge values for the HRE18 in real time before the flood. Finally, we anticipate that our calibrated parameter sets can similarly be used to accurately real-time forecast or nowcast the river discharges for future extreme and unprecedented rainfall events for the seven first-class rivers in the Chugoku region of Japan. To validate our expectations, the ensemble averages (hereafter ensembles) of river discharges from six validation cases were compared. These ensembles were calculated as averaged values of the six validation cases of river discharges for every 168 hourly time steps (Equation (6)), which were computed using calibrated river basin parameter sets from the six designated historical events other than the HRE18 and the observed rainfall data from the HRE18.

$$P_t=\frac{1}{n}\sum _{i=1}^n{Q}_s^t\left(i\right)$$

(6)

where $P_t$ is the ensemble average at time t (t = 1, …, 168), $Q_s^t$(i) is the simulated discharge at t for the i-th validation case, and n is the total number of validation cases (n = 6).

Similarly, when the forecasted rainfall data for a future extreme event are available in real time before the flood, the real-time river discharge prediction can be easily obtained using the methodology from this study as ensembles of river discharges from seven historically calibrated river basin parameter sets. Six of these seven historical events are validated in this study, and the additional event is from the HRE18, all of which should be forced with the real-time forecast rainfall data from future extreme events. Figure 4 schematically shows the flowchart of this study.

The performances of the hydrological model for each individual simulated case and for the ensembles were evaluated by three measures, as in [46]: Nash–Sutcliffe efficiency (NSE) (Equation (7), [73]), Kling–Gupta efficiency (KGE) (Equation (8), composed of Equations (9)–(11), [74]) and the relative peak error (PE) (Equation (12)), which are widely used verification metrics in hydrological modeling.

$$NSE=1-\frac{\sum _{t=1}^T(Q_s^t-{Q_o^t)}^2}{\sum _{t=1}^T(Q_o^t{-\stackrel{\_\_\_}{Q_0})}^2}$$

(7)

$$KGE=1-\sqrt{{(r-1)}^2+{(\beta -1)}^2+{(\alpha -1)}^2}$$

(8)

$$r=\frac{\sum _{t=1}^T(Q_o^t-\overline{Q_o})(Q_s^t-\overline{Q_s})}{\sqrt{\left(\sum _{t=1}^T{(Q_o^t-\overline{Q_o})}^2\right)\left(\sum _{t=1}^T{(Q_s^t-\overline{Q_s})}^2\right)}}$$

(9)

$$\beta =\frac{\overline{Q_s}}{\overline{Q_o}}$$

(10)

$$\alpha =\frac{\sqrt{\frac{1}{T}\sum _{t=1}^T{(Q_s^t-\overline{Q_s})}^2}}{\sqrt{\frac{1}{T}\sum _{t=1}^T{(Q_o^t-\overline{Q_o})}^2}}$$

(11)

$$PE=\frac{Q_{p,s}}{Q_{p,o}}$$

(12)

where $Q_o^t$ is the observed discharge at time t, $Q_s^t$ is the simulated discharge at time t, $\overline{Q_o}$ is the mean observed discharge in an event, $\overline{Q_s}$ is the mean simulated discharge in an event, β is a measure of bias, α is a measure of the variability error, r is the correlation coefficient between $Q_o^t$ and $Q_s^t$, $Q_{p,s}$ is the simulated peak discharge, and $Q_{p,o}$ is the observed peak discharge.

3. Results

3.1. Classification of Historical Extreme Rainfall Events

In the following, we show the results of our classification of historical extreme rainfall events into categories of similar extreme rainfall scales. For each of the seven targeted rivers, we independently classified seven of the most extreme recorded historical events in terms of combined maximal 1-day, 2-day, 3-day and 5-day rainfall data observed at the station located in the mean center of the associated river basin (see Figure 1 dots). The mean centers are shown in

Table 1. Locations and elevations of selected mean centers of all seven simulated river basins.

River/Mean Center	Elevation (m)	LAT (◦N)	LON (◦E)
Saba/Wada	140	34.15	131.74
Oze/Hatsukaichi	317	34.37	132.19
Ota/Addition	210	34.61	132.32
Ashida/Fuchu	70	34.56	133.23
Takahashi/Jinjyama	529	34.83	133.52
Asahi/Kuze	144	35.07	133.75
Yoshii/Akaiwa	56	34.92	134.08

.

Table 2. Detailed classification of seven historical extreme rainfall events for each river.

River	Date From	Date Until	Qmax (m³/s)	5 Day	3 Day	2 Day	1 Day	Max. Day
Saba	10 Jul 2010	16 Jul 2010	1060	444	376	349	337	06 Sep 2005
	21 Jul 2009	27 Jul 2009	1250	368	355	300	197.5	21 Jul 2009
	04 Sep 2005	10 Sep 2005	910	357	317	299	183.5	10 May 2011
	19 Jun 2016	25 Jun 2016	900	348.5	284	231.5	175	13 Jul 2010
	01 Jul 2005	07 Jul 2005	470	341	262.5	230	166	03 Jul 2005
	03 Jul 2018	09 Jul 2018	920	331	257	227.5	151.5	06 Jul 2018
	09 May 2011	15 May 2011	490	317	251	160.5	104	22 Jun 2016
Oze	03 Jul 2018	09 Jul 2018	1240	425	387	375	346	06 Sep 2005
	04 Sep 2005	10 Sep 2005	630	394	363	269	161	10 May 2006
	11 Jul 2010	17 Jul 2010	490	375	275	251.5	140	13 Jul 2010
	06 May 2006	12 May 2006	360	265	237	173	139	06 Jul 2018
	19 Jun 2016	25 Jun 2016	260	254.5	199	171	100	18 Jul 2003
	31 Jul 2004	06 Aug 2004	220	254	186	171	90	01 Aug 2004
	18 Jul 2003	24 Jul 2003	320	248	171	140.5	73.5	22 Jun 2016
Ota	11 Jul 2010	17 Jul 2010	4220	431.5	350.5	258	229	06 Sep 2005
	03 Jul 2018	09 Jul 2018	4530	341	303	256	145	06 Jul 2018
	04 Sep 2005	10 Sep 2005	7080	313	277.5	221	132	14 Jul 2010
	07 Jul 1997	13 Jul 1997	1620	311	233	205	131	24 Sep 1999
	21 Sep 1999	27 Sep 1999	3890	280	221	176	118	20 Jul 2009
	17 Jul 2009	23 Jul 2009	2250	238.5	212.5	147	105	08 Jul 1997
	31 Jul 2004	06 Aug 2004	1110	236	149	140	91	02 Aug 2004
Ashida	03 Jul 2018	09 Jul 2018	2090	394.5	373.5	285	181.5	06 Jul 2018
	31 Aug 2013	06 Sep 2013	920	232.5	189.5	162	110.5	04 Jul 2017
	19 Jun 2016	25 Jun 2016	990	211	185.5	134	90	20 Jun 2013
	01 Jul 2005	07 Jul 2005	300	198	163	132.5	83	02 Jul 2005
	19 Jun 2013	25 Jun 2013	460	190	161	120	78.5	04 Sep 2013
	11 Jul 2010	17 Jul 2010	1110	181	156	115.5	75.5	14 Jul 2010
	04 Jul 2017	10 Jul 2017	610	139	134	109	58	24 Jun 2016
Takahashi	03 Jul 2018	09 Jul 2018	7660	394.5	362	304.5	249.5	03 Sep 2011
	02 Sep 2011	08 Sep 2011	5110	306.5	305.5	290.5	174	06 Jul 2018
	31 Aug 2013	06 Sep 2013	3810	242.5	204.5	182.5	151.5	30 Sep 2018
	11 Jul 2010	17 Jul 2010	2520	231	188.5	164.5	109	14 Jul 2010
	17 Jul 2006	23 Jul 2006	3980	221	182.5	134	109	04 Sep 2013
	03 Jul 2012	09 Jul 2012	3050	194	163	128	84	07 Jul 2012
	29 Sep 2018	05 Oct 2018	4870	182.5	161	125.5	74	19 Jul 2006
Asahi	03 Jul 2018	09 Jul 2018	5330	435	415.5	325.5	178.5	06 Jul 2018
	16 Jul 2006	22 Jul 2006	2650	240	220.5	216.5	170	03 Sep 2011
	31 Aug 2013	06 Sep 2013	2030	230	206	161	123	09 Sep 2018
	01 Sep 2011	07 Sep 2011	3230	222	165	158	114	30 Sep 2018
	07 Sep 2018	13 Sep 2018	1070	177.5	163	146	109	20 Oct 2004
	19 Oct 2004	25 Oct 2004	2800	165	161	145	93	04 Sep 2013
	29 Sep 2018	05 Oct 2018	1870	161	145.5	118	76	17 Jul 2006
Yoshii	03 Jul 2018	09 Jul 2018	6700	314.5	280	213.5	170.5	03 Sep 2011
	31 Aug 2013	06 Sep 2013	4010	272.5	226.5	210	161	29 Sep 2004
	01 Sep 2011	07 Sep 2011	3750	218.5	216	183	146	06 Jul 2018
	17 Jul 2006	23 Jul 2006	4150	207	187	175	137	04 Sep 2013
	26 Sep 2004	02 Oct 2004	5740	191	183.5	175	136	09Aug 2009
	08 Aug 2009	14 Aug 2009	3170	183.5	176	165	120	20 Oct 2004
	19 Oct 2004	25 Oct 2004	3580	176	168	119	83	19 Jul 2006

Higher 7-CPM NSE Lower 7-CMP.

shows the detailed classification of the seven historical extreme rainfall events for each of the seven rivers. Columns are from left to right: river name; starting date; ending date; maximal observed discharge; maximal 5-day rainfall; maximal 3-day rainfall; maximal 2-day rainfall; maximal 1-day rainfall, and date of maximal 1-day rainfall. The colors of the events are ranked according to computed NSE values when using the 7-CPM calibrated parameter set from the particular event and HRE18 rainfall data for validation from the highest NSE (red) to the lowest NSE (gray), as indicated by the color bar in the caption of Table 2. This particular order of colors of each of the seven events defined in Table 2 follows the same color pattern in multiple Figures and Tables, as noted separately for each individual occurrence. The events are ranked and represented as red for the HRE18 calibration event and as orange, green, magenta, cyan, brown and gray for the validation events, where orange represents the highest computed NSE values and gray represents the lowest. In Table 2, the seven events for every river are ordered according to the 5-day rainfall (three leftmost columns) and separately according to each of the 3-day, 2-day and 1-day rainfall events. Finally, the exact day of the maximal 1-day rainfall was explicitly indicated.

With this presentation of Table 2, we wanted to emphasize the different features and nature of some events. For example, we can conclude that the 2013 event for the Takahashi River (brown color) featured relatively extreme rainfall from the 2nd to the 5th day, whereas the 2018-10 event for the Takahashi River (green color) featured relatively extreme 1-day rainfall.

Table 3. Selection criteria to classify the seven most extreme recorded historical events for the Ota and Takahashi Rivers. Blue color indicates and marks particular selection criteria (a–f) indicated in the first row.

River/Event	2010 (a)	2018	2005 (e)	1997 (f)	1999 (b)	2009 (d)	2004 (c)	Event/Day
Ota	81	63.5	10	90	131	26	3	3rd–5th days
	94.5	56.5	45	45	9	7.5	86	2nd–3rd days
	124	76	29	71	9	87	56	1st–2nd days
	132	145	229	105	131	118	91	1st day
River/Event	2018-7	2011 (a)	2013 (c)	2010 (d)	2006 (b)	2012 (f)	2018-10 (e)	Event/Day
Takahashi	32.5	1	54	26.5	60	31	0	3rd–5th days
	71.5	1	63	40	33	29	0	2nd–3rd days
	116.5	55	16.5	55.5	54	50	31	1st–2nd days
	174	249.5	109	109	74	84	151.5	1st day

explicitly explains the particular selection criteria used in this study to classify the seven most extreme recorded historical events. The colors that indicate particular events and their order are identical to those in Table 2. The rows indicate the observed rainfall of the particular events from the 3rd to the 5th day, the 2nd to the 3rd day, the 1st to the 2nd day and for the 1st day. Blue indicates the reason why that particular event was selected. In addition to the 2018 calibration event, six validation events were selected according to six criteria: the most extreme event in terms of the largest (a) maximum 5-day cumulative rainfall, (b) maximum 3rd–5th-day rainfall, (c) maximum 2nd–3rd-day rainfall, (d) maximum 1st–2nd-day rainfall, (e) maximum 1-day rainfall and (f) minimum 1-day rainfall among all four categories for a particular event, shown in Table 3. The selection criterion for each event is indicated in the first row.

3.2. Calibration and Validation of River Discharges

In the following, we show the calibration results of the CDRM model parameters with the SCE-UA optimization method and their validation using the six ensemble events. This calibration should enable the prediction of future peak river water discharges from similar-scale extreme rainfall events. In each of the following subsections, the exact river mouth location for every particular river is noted.

Table 4. Calibrated parameter set values for each of the seven targeted rivers. The events are ranked according to computed NSE values when using the 7-CPM calibrated parameter set from the particular event and the HRE18 rainfall data for validation (the same ranking criteria as in Table 2), from the highest NSE (red) to the lowest NSE (gray). “Cal” is the HRE18 calibration case, Val (1) is the validation case for the orange-colored event in Table 2, Val (2) is the green event, Val (3) is the magenta event, Val (4) is the cyan event, Val (5) is the brown event and Val (6) is the gray event.

Calibrated Parameter	Case	River (left—7 Parameters Method/Right—5 Parameters Method)
		Saba	Oze	Ota	Ashida	Takahashi	Asahi	Yoshii
Soil roughness coefficient N_slo [m−1/3s]	Cal	0.10/0.10	0.97/0.10	0.12/0.10	1.00/0.33	1.00/0.10	0.93/0.10	0.46/0.18
	Val (1)	0.75/0.10	0.59/0.10	0.56/0.10	1.00/0.70	0.68/0.10	0.97/0.10	0.62/0.15
	Val (2)	0.88/0.12	0.78/0.35	0.76/0.10	1.00/0.44	0.81/0.10	0.93/0.10	0.95/0.10
	Val (3)	1.00/1.00	0.83/0.39	0.51/0.10	0.95/0.10	0.94/0.10	0.99/0.14	0.51/0.10
	Val (4)	0.46/1.00	0.36/0.10	0.98/0.11	0.98/0.26	0.84/0.10	0.91/0.10	0.96/0.30
	Val (5)	1.00/1.00	0.29/0.99	0.10/0.53	0.57/0.10	0.80/0.10	0.32/0.10	0.73/0.10
	Val (6)	0.34/0.12	0.27/0.10	0.45/0.10	0.75/0.67	0.34/0.10	0.10/0.10	0.65/0.10
River roughness coefficient N_riv [m−1/3s]	Cal	0.03/0.04	0.03/0.04	0.06/0.04	0.03/0.04	0.06/0.07	0.05/0.06	0.05/0.05
	Val (1)	0.03/0.03	0.04/0.04	0.05/0.05	0.03/0.03	0.06/0.07	0.06/0.06	0.05/0.05
	Val (2)	0.04/0.04	0.03/0.04	0.04/0.04	0.04/0.04	0.05/0.06	0.03/0.03	0.03/0.04
	Val (3)	0.04/0.02	0.01/0.01	0.04/0.04	0.05/0.05	0.06/0.06	0.05/0.06	0.05/0.04
	Val (4)	0.02/0.02	0.04/0.01	0.02/0.02	0.04/0.05	0.07/0.07	0.04/0.01	0.03/0.03
	Val (5)	0.01/0.01	0.02/0.02	0.04/0.05	0.05/0.04	0.05/0.06	0.02/0.02	0.05/0.05
	Val (6)	0.04/0.01	0.02/0.03	0.05/0.03	0.05/0.04	0.06/0.05	0.02/0.02	0.02/0.03
Effective porosity of non-capillary subsurface layer θa [/]	Cal	0.47/0.31	0.27/0.32	0.37/0.23	0.22/0.22	0.10/0.20	0.23/0.26	0.18/0.16
	Val (1)	0.53/0.45	0.37/0.31	0.22/0.15	0.13/0.10	0.21/0.23	0.19/0.30	0.13/0.10
	Val (2)	0.46/0.70	0.15/0.10	0.58/0.57	0.23/0.25	0.14/0.19	0.24/0.23	0.15/0.27
	Val (3)	0.44/0.14	0.25/0.52	0.39/0.50	0.16/0.17	0.22/0.30	0.20/0.27	0.20/0.14
	Val (4)	0.32/0.11	0.22/0.41	0.26/0.44	0.45/0.32	0.14/0.22	0.49/0.47	0.10/0.13
	Val (5)	0.25/0.21	0.25/0.54	0.70/0.70	0.16/0.22	0.10/0.16	0.21/0.26	0.20/0.13
	Val (6)	0.62/0.43	0.34/0.18	0.40/0.48	0.38/0.34	0.13/0.14	0.33/0.42	0.13/0.20
Saturated hydraulic conductivity ka [ms—1]	Cal	0.50/0.19	0.41/0.18	0.07/0.50	0.31/0.24	0.48/0.40	0.35/0.50	0.37/0.43
	Val (1)	0.25/0.18	0.43/0.14	0.07/0.50	0.10/0.18	0.03/0.50	0.04/0.50	0.44/0.50
	Val (2)	0.39/0.44	0.18/0.04	0.04/0.50	0.35/0.37	0.10/0.50	0.21/0.42	0.49/0.50
	Val (3)	0.005/0.005	0.12/0.42	0.04/0.50	0.35/0.50	0.06/0.39	0.36/0.37	0.23/0.50
	Val (4)	0.44/0.005	0.12/0.50	0.43/0.18	0.006/0.005	0.04/0.50	0.005/0.26	0.09/0.50
	Val (5)	0.18/0.14	0.48/0.32	0.34/0.34	0.05/0.36	0.34/0.50	0.41/0.50	0.01/0.50
	Val (6)	0.09/0.50	0.14/0.09	0.07/0.50	0.007/0.005	0.28/0.49	0.50/0.50	0.37/0.50
Canopy	Cal	0.79/0.81	0.70/0.70	0.68/0.66	0.61/0.63	0.66/0.67	0.79/0.79	0.71/0.72
interception	Val (1)	0.77/0.76	0.67/0.68	0.69/0.60	0.60/0.60	0.64/0.60	0.75/0.71	0.71/0.69
and	Val (2)	0.94/0.90	0.60/0.60	0.74/0.71	0.69/0.71	0.60/0.60	0.70/0.60	0.67/0.65
evaporation	Val (3)	0.84/0.60	0.60/0.60	0.64/0.60	0.62/0.60	0.72/0.68	0.69/0.67	0.70/0.63
factor	Val (4)	0.60/0.60	0.62/0.60	0.60/0.60	0.79/1.00	0.72/0.72	0.97/0.60	0.60/0.60
F1 [/]	Val (5)	0.60/0.60	0.60/0.60	0.79/0.80	0.77/0.77	0.77/0.60	0.60/0.60	0.73/0.68
	Val (6)	0.64/0.62	0.74/0.75	0.83/0.82	0.88/1.00	0.65/0.60	0.60/0.60	0.69/0.67
Effective	Cal	0.27	0.28	0.37	0.10	0.11	0.24	0.10
porosity of	Val (1)	0.53	0.39	0.24	0.10	0.22	0.22	0.15
capillary	Val (2)	0.48	0.17	0.58	0.10	0.16	0.27	0.17
subsurface	Val (3)	0.44	0.10	0.39	0.18	0.25	0.22	0.24
layer	Val (4)	0.33	0.24	0.28	0.46	0.16	0.51	0.11
θm [/]	Val (5)	0.10	0.26	0.10	0.10	0.15	0.23	0.19
	Val (6)	0.62	0.32	0.42	0.39	0.16	0.30	0.15
	Cal	9.64	5.12	10.00	10.00	4.38	9.84	10.00
Permeability	Val (1)	10.00	9.43	6.74	10.00	7.26	4.94	9.52
reduction	Val (2)	9.23	4.92	9.97	10.00	7.08	9.09	5.83
degree	Val (3)	3.66	10.00	9.19	9.24	6.95	6.53	9.92
β [/]	Val (4)	9.66	6.39	8.22	6.81	5.13	2.48	9.43
	Val (5)	9.90	7.93	10.00	9.93	2.59	8.10	3.19
	Val (6)	10.00	8.05	9.93	4.70	6.68	10.00	6.07

shows the calibrated 7-CPM and 5-CPM parameter set values for each of seven targeted rivers, which were ranked using the same ranking criteria to classify extreme rainfall events as in Table 2.

Table 5. Summary of the computed calibration and ensemble average validation NSE, KGE and PE results for the 7-CPM and 5-CPM for each river and the cumulative and average results of all rivers. The green color indicates all the validation results with NSE ≥ 0.9 and 1.1 ≥ PE ≥ 0.9.

7 Parameters Method		Saba	Oze	Ota	Ashida	Takahashi	Asahi	Yoshii	All	Average
NSE	Calibration	0.98	0.98	0.98	0.95	0.98	0.98	0.98	0.99	0.98
	Ensemble average validation	0.95	0.95	0.97	0.84	0.96	0.90	0.95	0.98	0.93
KGE	Calibration	0.94	0.98	0.98	0.91	0.99	0.98	0.93	0.98	0.96
	Ensemble average validation	0.85	0.90	0.95	0.70	0.92	0.69	0.85	0.91	0.84
PE	Calibration	0.93	1.09	0.96	0.97	1.01	0.96	0.98	1.00	0.99
	Ensemble average validation	0.82	1.15	0.90	1.26	1.14	0.74	0.97	1.03	1.00
5 Parameters Method		Saba	Oze	Ota	Ashida	Takahashi	Asahi	Yoshii	All	Average
NSE	Calibration	0.98	0.98	0.96	0.96	0.98	0.97	0.98	0.99	0.97
	Ensemble average validation	0.91	0.95	0.93	0.94	0.98	0.85	0.96	0.98	0.93
KGE	Calibration	0.94	0.98	0.97	0.93	0.98	0.96	0.94	0.99	0.96
	Ensemble average validation	0.75	0.86	0.88	0.92	0.93	0.64	0.85	0.87	0.83
PE	Calibration	0.90	1.07	0.88	0.97	0.99	0.92	0.97	0.99	0.96
	Ensemble average validation	0.74	1.01	0.76	1.10	0.96	0.62	0.89	0.89	0.87

summarizes the computed calibration and ensemble average validation reproducibility metrics for the 7-CPM and 5-CPM for each river, a comparison of the two methods, and the range and average results for all rivers. These results are visualized in Figure 5 for the Ota River and Figure 6 for the Takahashi River in the following subsections. Ota and Takahashi are, hereafter, selected as representative rivers for detailed discussion because they are the largest in Hiroshima and Okayama Prefectures, respectively. Identical presentations and figures for the other five analyzed rivers are presented for reference in Supplementary Material 1.

3.2.1. Ota River

Figure 5 shows the Ota River calibration and validation discharge results for the HRE18 at Yaguchi Daiichi station, located 14.6 km upstream from the river mouth, using the 7-CPM and 5-CPM sets with the HRE18 rainfall data and ensemble average validation of all six calibrated parameter sets with the 7-CPM and 5-CPM sets compared with observed data. Headings “7” in panel (a) and “5” in panel (b) in Figure 5 and Figure 6 indicate the 7-CPM and 5-CPM parameter set plots, respectively. Hydrographs depicted in Figure 5 and Figure 6 show the flow of calibration and validation of seven calibrated parameter sets for HRE18 (calibration, red color) and six different events (validation) in panels (a) and (b), respectively, while panel (c) shows them as the calibration case and six averaged validation cases. In Figure 7, only panel (c) is shown because all rivers combined use parameter sets from six different validation events, as shown in Table 2. The NSE values in Figure 5a,b and Figure 6a,b were computed based on observed data and the respective calibration event timeline alone.

3.2.2. Takahashi River

Figure 6 shows the Takahashi River calibration and validation discharge results for the HRE18 at Sakazu station, located 10.2 km upstream from the river mouth, using the (a) 7-CPM and (b) the 5-CPM sets with the HRE18 rainfall data and ensemble average validation of all six calibrated parameter sets with the 7-CPM and 5-CPM sets compared with observed data.

3.2.3. All Rivers Cumulatively

Figure 7 shows cumulative calibration and validation discharge results for the HRE18 for all rivers at their river mouths, validated as cumulative ensemble average values of all six calibrated parameter sets for every river with the 7-CPM and 5-CPM sets compared with observed data.

3.3. Cross-Validation of River Discharges

In the following, we show the cross-validation results of all seven considered events using ensemble results, which were calculated as the calibrated parameters of six events different from the considered event and the rainfall input from the considered event. These cross-validation results should show that the prediction of future peak river water discharges from similar-scale extreme rainfall events can be used across multiple similar-scale events in general.

Table 6. Summary of the computed ensemble average cross-validation NSE, KGE and PE results for the 7-CPM and 5-CPM for each river, with the calculated average of all results and the best 94% of cases. The colors indicate particular events and are identical to those in Table 2.

River	Event	Qmax (m³ /s)	NSE (7-CPM)	NSE (5-CPM)	KGE (7-CPM)	KGE (5-CPM)	PE (7-CPM)	PE (5-CPM)
Saba	2018	920	0.95	0.91	0.85	0.75	0.82	0.74
	2010	1060	0.97	0.95	0.94	0.87	0.95	0.87
	2016	900	0.75	0.75	0.55	0.57	0.64	0.63
	2005-9	910	0.69	0.16	0.35	−0.09	1.54	1.85
	2005-7	470	0.84	0.31	0.83	0.26	0.94	1.35
	2011	490	0.87	0.91	0.80	0.90	0.93	1.13
	2009	1250	0.68	0.89	0.50	0.80	1.40	1.02
Oze	2018	1240	0.95	0.95	0.90	0.86	1.15	1.01
	2010	490	0.94	0.92	0.93	0.92	0.97	0.87
	2016	260	0.87	0.78	0.84	0.74	1.11	0.98
	2005	630	0.87	0.94	0.83	0.85	1.21	1.06
	2006	360	0.93	0.80	0.83	0.63	0.89	0.76
	2004	220	0.91	0.68	0.86	0.58	0.85	0.91
	2003	320	0.84	0.75	0.77	0.67	0.82	0.67
Ota	2018	4530	0.97	0.93	0.95	0.88	0.90	0.76
	2005	7080	0.93	0.90	0.78	0.81	0.74	0.69
	2010	4220	0.98	0.94	0.97	0.91	0.96	0.77
	2009	2250	0.92	0.82	0.75	0.67	1.06	0.91
	2004	1110	0.78	0.42	0.72	0.32	1.30	1.35
	1997	1620	0.85	0.77	0.90	0.83	1.02	0.94
	1999	3890	0.75	0.73	0.56	0.61	0.54	0.48
Ashida	2018	2090	0.84	0.94	0.70	0.92	1.26	1.10
	2017	610	0.95	0.92	0.88	0.80	0.99	0.95
	2010	1110	0.93	0.90	0.88	0.73	1.14	0.88
	2013-9	920	0.93	0.93	0.79	0.82	1.13	0.86
	2013-6	460	−0.19	−0.44	−0.10	−0.28	1.40	1.59
	2016	990	0.90	0.79	0.76	0.59	0.77	0.63
	2005	300	−2.63	−4.28	−0.35	−0.78	1.10	1.50
Takahashi	2018-7	7660	0.96	0.98	0.92	0.93	1.14	0.96
	2011	5110	0.97	0.93	0.84	0.78	1.05	1.00
	2018-10	4870	0.97	0.96	0.84	0.81	1.13	1.01
	2010	2520	0.96	0.92	0.93	0.87	0.89	0.84
	2006	3980	0.93	0.89	0.81	0.75	0.80	0.71
	2013	3810	0.96	0.93	0.82	0.78	1.05	0.91
	2012	3050	0.96	0.90	0.92	0.82	0.92	0.81
Asahi	2018-7	5330	0.90	0.85	0.69	0.64	0.74	0.62
	2006	2650	0.88	0.83	0.77	0.75	0.77	0.70
	2013	2030	0.97	0.90	0.90	0.79	0.90	0.79
	2011	3230	0.94	0.94	0.92	0.92	0.97	0.91
	2018-9	1070	0.25	-0.05	0.25	0.01	1.28	1.49
	2004	2800	0.84	0.82	0.88	0.81	0.86	0.77
	2018-10	1870	0.84	0.84	0.71	0.65	1.17	1.09
Yoshii	2018	6700	0.95	0.96	0.85	0.85	0.97	0.89
	2004	5740	0.94	0.93	0.78	0.80	0.75	0.75
	2011	3750	0.94	0.88	0.88	0.81	1.05	1.03
	2013	4010	0.97	0.94	0.88	0.79	0.89	0.86
	2009	3170	0.94	0.97	0.92	0.78	0.95	1.09
	2006	4150	0.95	0.94	0.93	0.88	0.87	0.79
	2004	3580	0.91	0.91	0.88	0.89	0.84	0.87
Average (all)		2485	0.79	0.70	0.76	0.68	0.99	0.94
Average (top 94%)		2608	0.90	0.85	0.82	0.75	0.97	0.90

summarizes the computed ensemble average cross-validation NSE, KGE and PE results for the 7-CPM and 5-CPM for each river and the calculated average of all results and of the best 94% of cases. These results are visualized in Figure 8 for the Ota River and Figure 9 for the Takahashi River in the following subsections. The ensembles of six different calibrated parameter sets were calculated using the 7-CPM and 5-CPM and compared with the observed data. Identical presentations and figures for the other five analyzed rivers are presented for reference in Supplementary Material 2. In Figure 8 and Figure 9, similarly to Figure 7, only panel (c) is shown because all rivers combined use parameter sets from six different validation events, as shown in Table 2.

3.3.1. Ota River

Figure 8 shows the Ota River ensemble average cross-validation discharge results at Yaguchi Daiichi station, located 14.6 km upstream from the river mouth, using the 7-CPM and 5-CPM parameter sets and rainfall data input from the 2005, 2010, 2009, 2004, 1997 and 1999 events.

3.3.2. Takahashi River

Figure 9 shows the Takahashi River ensemble average cross-validation discharge results at Sakazu station, located 10.2 km upstream from the river mouth, using the 7-CPM and 5-CPM parameter sets and rainfall data input from the 2011, 2018-10, 2010, 2006, 2013 and 2012 events.

4. Discussion

4.1. Classification of Historical Extreme Rainfall Events

In the following, we discuss the method used in this study to classify historical extreme rainfall events into categories with similar extreme rainfall scales. From the proposed classification method, we expect that rainfall events with similar rainfall scales cause similar river discharge responses. Therefore, the cross-calibrated parameter sets from similar historical extreme rainfall events can then be used for validation and, ultimately, real-time river discharge forecasting of future extreme rainfall events.

The selected events represent each of the categories marked with a yellow color in Table 3: (a) 2010 for Ota and 2011 for Takahashi; (b) 1999 for Ota and 2006 for Takahashi; (c) 2004 for Ota and 2013 for Takahashi; (d) 2009 for Ota and 2010 for Takahashi; (e) 2005 for Ota and 2018-10 for Takahashi; and (f) 1997 for Ota and 2012 for Takahashi. The selection criteria of the same events were applied for all rivers. If it was not possible to find a representative event of some of the aforementioned categories due to the relatively short period of available rainfall data (e.g., approximately 20 years), then the most extreme event in terms of “(a) maximum 5-day cumulative rainfall” that had not been selected was used as its substitution. We expect and recommend that this proposed method of selecting a wide spectrum of the most extreme recorded historical events with relatively different rainfall distribution phenomena will increase the accuracy of the projection and prediction of other extreme events because their ensemble parameterization will consider a wide range of physical phenomena from extreme events that occurred in a particular river basin.

Among the seven extreme recorded historical events considered, the HRE18 was the only event common to all river basins, whereas the other six events were different in each river basin. Some events impacted several river basins, as expected. However, contrary to our expectations, no other event except HRE18 was common for all seven river basins. The next most widespread events occurred in July 2010, which impacted five rivers, excluding the most western ones of Asahi and Yoshii, and in September 2013, which impacted four rivers, excluding the most eastern ones of Saba, Oze and Ota. This finding emphasizes the importance of independently classifying the most extreme recorded historical events for every river basin for the calibration of river basin parameters instead of calibrating all river basins with the same historical events.

4.2. Calibration and Validation of River Discharges

We evaluate and compare reproducibility metrics in terms of NSE (Equation (7)), KGE (Equation (8)) and PE (Equation (12)) as the most applicable for our considered cases. NSE is widely used in the hydrological community; thus, most studies can be comparable with it. KGE combines the three components of NSE model errors (Equations (9)–(11)) in a more balanced way, whereas PE is a simple measure of the ratio between the simulated and observed peak river discharge hydrographs. Among them, the most pertinent and impactful reproducibility metric in our study is NSE, which is widely accepted to be satisfactorily large with a threshold of NSE > 0.7.

The results in Table 5 suggest that calibrated parameter sets for river basins are event-dependent, which is known as the equifinality issue in the calibration of hydrological models [75]. The quality of the calibration process of this study can only be properly evaluated by a simultaneous combination of seven parameters (for the 7-CPM) or five parameters (for the 5-CPM) and how they interact to predict the river mouth hydrographs, not by extracting a single parameter and evaluating its values in the seven calibrated parameter sets. For example, when the parameters are individually evaluated in the 5-CPM, the higher soil roughness coefficient, higher river roughness coefficient, lower effective porosity, higher saturated hydraulic conductivity and higher canopy interception and evaporation factor collectively contribute to a greater river mouth discharge. This indicates that we cannot objectively evaluate which parameter is the cause of the change in discharge because they act as an interconnection of 5 or 7 parameters calculating river discharge simultaneously at every cell. The SCE-UA optimization method suggests the best calibrated parameter set for the given calibration case. However, our expectation, which is not explicitly evaluated in this study, is that even the second, third and similarly successful calibrated parameter sets would obtain similar NSE, KGE and PE ensemble average validation results as those obtained with the best calibrated parameter set. The reason is that the best, second, third and similarly successful calibrated parameter sets can reproduce the river runoff mechanism for the given calibration case with similar accuracy, although they reproduce it with different combinations of 7 or 5 parameters, which do not necessarily have to be similar to one another. Therefore, we emphasize that the reproducibility metrics of our modeling results depend relatively less on the equifinality issue but relatively more on the accuracy of the applied parameter calibration methodology. If we evaluate only one parameter and conclude that the results are not reasonable based on the value of that particular parameter, it is misleading because we must evaluate all five parameters as a whole. Therefore, we evaluated these calibrated parameter sets by validating six historical extreme rainfall events of similar rainfall scales with respect to the river mouth hydrographs to demonstrate highly accurate results when we use our proposed methodology.

Figure 5 and Figure 6 show the (a) 7-CPM and (b) 5-CPM calibrated and validated results for each of the seven historical events, so the intuitive question is which of these seven calibrated parameter sets should be used in future nowcasting applications. As we could not know the accuracy of the calibrated parameters of each of the six validated cases, we did not assign higher or lower importance to any calibrated parameter set. Instead, we emphasized their ensemble values. This is because we believe that the methodology used in this study should be very accurate in general when used over many calibrated events rather than being accurate for each of the individual events separately. Every single event separately had different spatiotemporal rainfall characteristics, which were not all similar to those of the HRE18, but all events, in general, had relatively more similar rainfall values to HRE18 in terms of combined maximal 1-day, 2-day, 3-day and 5-day rainfall. Therefore, the results of this study should be evaluated in terms of Figure 5 and Figure 6c, which present six validation case ensembles of the 7-CPM and 5-CPM results. These results and the HRE18 calibration reproducibility metrics are systematically summarized in Table 5. The lowest NSE reproducibility metric for these calibration results is 0.95 for the Ashida River when we used the 7-CPM, which shows a very high reproducibility of the applied calibration process. These validation ensemble results have highly accurate reproducibility metrics as follows: an average NSE of 0.93 for both the 7-CPM and 5-CPM (0.84–0.97 for the 7-CPM and 0.85–0.98 for the 5-CPM), an average KGE of 0.84 for the 7-CPM (0.70–0.95) and 0.83 for the 5-CPM (0.64–0.93), and an average PE of 1.00 for the 7-CPM (0.74–1.26) and 0.87 for the 5-CPM (0.62–1.10). These highly accurate reproducibility metrics indicate that our methodology can accurately predict the river discharge hydrographs for the HRE18 and accurately forecast or nowcast the river discharge hydrographs for future extreme and unprecedented rainfall events for the considered rivers in real time.

When we separately compare the ensemble validation results when using the 7-CPM and 5-CPM, we find that the 7-CPM is generally better for the Saba, Ota and Asahi Rivers, the 5-CPM is generally better for the Ashida and Takahashi Rivers, and both methods are generally equivalent for the Oze and Yoshii Rivers. We assume that these differences may be due to the geology of each river basin. We expect that the 5-CPM performs better for river basins where flash flood river discharge hydrographs more readily occur because the 5-CPM does not calculate the effect of the capillary subsurface layer, which is of relatively lower importance for flash flood river discharge hydrographs. Therefore, when the methodologies from this study are used for future forecasting, the 7-CPM is recommended when a high predictive accuracy of the hydrograph is desirable. A combination of both the 7-CMP and the 5-CPM is recommended when a wider and safer range is preferred for the predictive accuracy of the hydrograph. An alternative might be to use the 7-CPM for rivers where it performs better (Saba, Ota and Asahi) and the 5-CPM for those where it is more accurate (Asahi and Takahashi).

Furthermore, Figure 7 shows the validation of cumulative river mouth discharge ensembles from all rivers, with an NSE of 0.98 for both the 7-CPM and 5-CPM, a KGE of 0.91 for the 7-CPM and 0.87 for the 5-CPM and a PE of 1.03 for the 7-CPM and 0.89 for the 5-CPM. NSE and KGE values larger than 0.7 are generally considered satisfactory in terms of reproducibility performance. Therefore, these results indicate extremely high reproducibility metrics for the two methods used in this study, with the 7-CPM showing slightly better performance than the 5-CPM, as expected, because it considers two more parameters. These results demonstrate better validation performance of the CDRM model when considered cumulatively for all seven rivers (with discharge values of up to 25.000 m³/s) than when each river was considered separately. To put these results and reproducibility metrics into context, we can compare them with the only other hydrological study on the HRE18 that used the same reproducibility metrics as in [56]. The computed values and standard deviations from that study were 0.78 ± 0.11 for NSE, 0.75 ± 0.10 for KGE and 0.87 ± 0.15 for PE. However, in their study, the RRI model used was not calibrated, whereas in this study, we calibrated the CDRM model, so this difference in accuracy values can be attributed to the calibration method used, which includes the proposed historical rainfall event classification. Moreover, [51] and [52] similarly applied the CDRM model with the SCE-UA optimization method to the major northeastern Japanese rivers, but only for the 5-CPM, as their studies assumed that all water starts flowing immediately after reaching the soil. Refs. [51,52] jointly obtained values of 0.97 ≤ NSE ≤ 0.99 for three calibrated events and 0.83 ≤ NSE ≤ 0.95 for three cross-validated typhoon events of similar rainfall amount scales and spatial trajectories. Their results are similar to those obtained in our study for calibration, but our ensemble parameter calibration approach greatly improved the validation results compared to theirs. Furthermore, the applicability of their methodology is limited to similar scales of typhoon events with similar trajectories, which is not a limitation of our study.

Several other studies have reported reasonable model performances of hydrological parameter calibration and validation for joint application with hydrometeorological flood forecasting input. A process-oriented semi-distributed hydrological model [24,25] was used for operational flood forecasting with a cross-parameter ensemble calibration approach. Ref. [28] obtained a hydrological model performance of 0.71 ≤ NSE ≤ 0.84. Ref. [19] obtained KGE values of 0.64 ≤ KGE ≤ 0.85 with NSE only for peak flows (PNSE) of 0.32 ≤ PNSE ≤ 0.62 during the calibration period and 0.65 ≤ KGE ≤ 0.82 with 0.32 ≤ PNSE ≤ 0.44 during the validation period of five semi-distributed rainfall-runoff hydrological models of varying complexity. Ref. [22] obtained 0.52 ≤ NSE ≤ 0.79 during the calibration period and 0.59 ≤ NSE ≤ 0.73 during the validation period of event-type flash flood river discharge hydrographs across China. All of these reproducibility metrics are far below the metrics obtained in our study, but direct comparison is not fairly applicable due to some objective reasons discussed later in this section.

A brief comparison with hydrometeorological forecast studies shows that [19] obtained 0.42 ≤ KGE ≤ 0.75 and 0.32 ≤ PNSE ≤ 0.67 during the calibration period, 0.65 ≤ KGE ≤ 0.82 with 0.32 ≤ PNSE ≤ 0.44 for 1-day lead times and 0.22 ≤ KGE ≤ 0.46 with 0.07 ≤ PNSE ≤ 0.28 for 2-day lead times by applying deterministic meteorological forecasts. Ref. [27] obtained Nash–Sutcliffe efficiency values of 0.14 ≤ NSE ≤ 0.71 when forcing two deterministic forecasts and 0.29 ≤ NSE ≤ 0.83 when forcing three ensemble precipitation nowcasts on two event scales, while [5] obtained 0.32 ≤ NSE ≤ 0.48 for the mean of ensemble discharge forecasts. These findings confirm expectations that the reproducibility of river discharge hydrograph results decreases when the hydrological model forced with observed data shows the best performance, followed by the ensemble and then by deterministic meteorological forcing data, because of increasing rainfall uncertainty levels.

For unbiased and objective comparison of our findings with other studies using proper context, several important similarity conditions should be satisfied, in particular: rainfall data availability to calibrate models should be in comparable range; rainfall data input should be based on observed data, not on predicted data; duration of time series and input–output data of river discharge results should be in comparable range; watershed sizes should be in comparable ranges; reproducibility metrics for results comparison should be identical. To our knowledge, no other previous studies satisfy all of these conditions. Thus, in the following, we propose context to establish a benchmark that may be used for comparing our proposed methodology and approach outcomes with other studies. We highlight that the most important new contribution of our study is that the validation reproducibility metrics of our study are largely improved with our ensemble calibration approach, sometimes even more than the calibration reproducibility metrics for the same event. Therefore, we propose that only validation cases from one or more parameter sets should be used for the comparison of reproducibility metrics. Ultimately, the objective number of calibrated parameter sets likely converges toward some number. In our study, sensitivity analysis showed that the six calibrated parameter sets combined often show better validation results than the calibrated parameter set, but this number might be different in various contexts. In addition to cumulative river discharge validation ensembles (Figure 7), the union of all validation reproducibility metrics for all seven considered river validation ensembles for both the 5-CPM and 7-CPM (14 cases, systematically shown in Table 5) shows obtained values of 0.84 ≤ NSE ≤ 0.98, 0.64 ≤ KGE ≤ 0.95 and 0.62 ≤ PE ≤ 1.26. These large reproducibility metrics were obtained for the very wide range of modeled river basin areas, ranging from 323 to 2644 km², showing that our approach is very robust and can be used across a wide range of rivers. Of the aforementioned studies, ref. [56] had study sites and an event scale common to our study, but their RRI model was uncalibrated. Refs. [19,28] had much smaller watershed areas (70 and 186 km2, respectively), and their calibration (6 years for both studies) and validation periods were much longer than those in our study. Ref. [22] used an event-type temporal scale but a different parametrization scheme and overall approach. Another different approach related to emphasizing the importance of estimating initial hydrologic conditions was shown in [76]. Only [51,52] shared a directly comparable approach with our study. Therefore, any direct comparison of results with other studies that used different approaches would not be objective. We recommend evaluating our computed NSE, KGE and PE reproducibility metrics with an absolute scale, not relative to other studies.

4.3. Approach to Developing Flash Flood River Discharge Hydrograph Nowcasting Applications

In the following, we discuss, give recommendations regarding and note the limitations of how the CDRM model parameters calibrated with the SCE-UA optimization method can be used to reproduce peak river water discharge hydrographs from future extreme rainfall events with similar scales.

Although this study evaluated the river mouth location for observation and validation to represent the entire river basin processes, the CDRM model calculates a river discharge hydrograph at any desired point within a river basin, which is a very useful feature for the development of future real-time forecasting or nowcasting applications.

From Table 6 as well as Figure 8 and Figure 9, we emphasize that the cross-validation analysis in this study has very high reproducibility metrics. For the discussion hereafter, we will consider 94% of the best results from Table 6 and discard 3 of 49 analyzed events where at least one method obtained negative NSE results; these discarded events are outliers with notably low reproducibility metrics. These discarded events are Ashida (2013-6), Ashida (2005) and Asahi (2018-9). When considering 94% of the best results, the average cross-validation reproducibility metrics were 0.90 (NSE), 0.82 (KGE) and 0.97 (PE) for the 7-CPM and 0.85 (NSE), 0.75 (KGE) and 0.90 (PE) for the 5-CPM. From the 49 considered cases, the NSE was larger than 0.9 in 30 cases for the 7-CPM and 10 cases for the 5-CPM and larger than 0.7 in 44 cases for the 7-CPM and 40 cases for the 5-CPM. This result may indicate that the 7-CPM is much more accurate in terms of excellent hydrograph reproducibility (NSE ≥ 0.9), while both methods are similarly accurate in terms of satisfactory hydrograph reproducibility (NSE ≥ 0.7). These high reproducibility metrics imply that our proposed methodology is very accurate and robust for predicting similar-scale extreme events; therefore, they can be used to nowcast a wide range of extreme river discharge events. The three discarded events with the worst reproducibility metrics all represent the validated cases with the lowest maximum observed discharge for the particular river. Therefore, the proposed methodology may not always be reasonably accurate when using calibration cases from more-extreme events to predict less-extreme events. However, the proposed methodology is, on average, very accurate when calibration parameter sets from less-extreme events are used to predict more-extreme events.

We have shown that our methodology could have accurately predicted the HRE18 river mouth discharge hydrographs when using ensemble values from the six validation cases for each river and the total freshwater discharge into the Seto Inland Sea, and vice versa for the past events, by showing a high accuracy of cross-validation results. In their extensive evaluation of crucial scientific gaps for the development of flash flood operational nowcasting, [77] highlighted that understanding and predicting drastic changes in hydrological and geomorphic functions are among the greatest scientific challenges because they require extrapolations far beyond the available observed data range. Our proposed approach and methodology greatly contribute to improving this highlighted scientific gap, as large reproducibility metrics were obtained through cross-validation of multiple river discharges with wide ranges of watershed areas. When using our proposed ensemble hydrological calibration approach, both unprecedented HRE18 river discharge hydrographs were accurately predicted from the calibrated parameters of six past events, and most of the past events were accurately predicted from the calibrated parameters of the other six events, including the HRE18 (Figure 8 and Figure 9). Therefore, these results strongly support our established hypothesis that calibrated river basin parameters from ensemble historical rainfall-induced extreme river discharge events can accurately predict river discharge hydrographs of extreme rainfall-induced future events of similar scale and vice versa.

This study used observed historical rain gauge input data with high spatiotemporal resolutions to calibrate parameter sets and for validation. Ref. [26] also used an ensemble calibration approach, but a cross-parameter ensemble, whereas our study used a cross-event ensemble approach. They obtained a superior performance of pluviometer-based nowcasts compared to radar-based nowcasts when dealing with flash flood simulations in a gauged catchment and discussed that it is important that parameter calibration is conducted using the same rainfall methodology input as the nowcast rainfall input. An interesting alternative to the Thiessen polygon rainfall input method to be considered for future studies is using weather radar rainfall input data for parameter optimization instead of data from rain gauge stations, which would likely improve the reproducibility of spatial rainfall variability. However, in this study, we applied the Thiessen polygon method mainly for the two reasons discussed below. An important advantage of our approach and methodology is that it can presumably be applicable anywhere in the world where any rainfall gauge data are recorded because no region-specific data for Japan were used in our study, whereas radar data are less available worldwide. Second, despite its lower spatial rainfall variability compared to radar data, the Thiessen polygon method in this study produces quite high reproducibility metrics, as shown in Section 4.2. If radar or some other rainfall input data method (i.e., satellite remote sensing) would indeed produce more accurate results than the Thiessen polygon method, that methodological comparison might be a topic of follow-up studies. Presumably, our proposed calibration and parameter optimization methodology needs to be adjusted and modified for predictions in ungauged basins using different rainfall estimation methods, such as in [78], but the proposed hydrological ensemble calibration approach can still be analogously applied regardless of the rainfall data input method.

For future nowcasting applications, we also calibrated the parameter set for the HRE18, which makes these seven-parameter sets a basis for the calculation of ensemble values. However, we additionally need to integrate this study’s calibrated parameter sets with robust real-time forecast rainfall from future events to extend the hydrological tools developed in this study into integrated nowcasting hydrometeorological tools of future flood events. The forecast rainfall can be obtained either from available real-time meteorological forecasts or from some existing meteorological models that can reproduce or forecast spatiotemporal rainfall data for the HRE18 in real time [34,36,37,38,39,40]. In follow-up studies, real-time meteorological forecasts will be applied as rainfall forcing instead of the observed data to develop a real-time forecasting application to be implemented in flash flood EWSs. When the forecast rainfall for a future event is readily available, the ensemble-calibrated CDRM model can very quickly, within a few minutes of computational time for both the 7-CPM and 5-CPM, produce real-time predictions of river discharges and contribute to the active development of EWSs for the reduction of flash flood disaster impacts.

Modeled river discharge hydrographs at river mouths, such as the ensemble values in Figure 7, are useful input data to force coastal ocean models that can simulate the effect of extreme river water levels on coastal sea dynamics and associated storm surge and pollution transport disasters [51,79,80,81,82,83]. Therefore, these accurate real-time river discharge predictions improve comprehensive land–river–ocean disaster prevention systems.

Developing the ensemble parameter calibration approach and methodology proposed in this study also has potentially large applicability potential in climatological studies. Ref. [2] reported that the reliability of future climatological river discharge hydrograph results depends on the reproducibility of extreme rainfall against historical events. Therefore, robust hydrological ensemble-calibrated models are a significant step forward toward calculating river discharge hydrographs of future climatological events. The extended application possibilities of our approach and methodology to climatological studies will be evaluated in follow-up studies.

5. Conclusions

While [51,52] established the initial tools for developing real-time forecasting or nowcasting models, this study proposes the next step toward implementing these models. Adding data on a new study site and improving the methodology provided more reproducible and more accurate extreme river mouth discharge hydrographs than previous studies.

We introduced a new extreme historical rainfall event classification methodology, where the calibration of parameter sets based on similar extreme rainfall patterns for seven historical events was used to accurately reproduce the river mouth discharge hydrographs associated with the HRE18. We expect that our fundamental proposed hydrological ensemble approach of using several calibration parameter sets for predicting river discharge ensembles for future events shows promising potential to be analogously applied anywhere, regardless of rainfall data availability, and we recommend its use. However, when rainfall data are readily available, our methodology can even be directly reapplied by using the same or different hydrological models and the parameter optimization method.

We showed that the CDRM model with seven historically calibrated parameter sets using the SCE-UA method could have accurately predicted the river discharge hydrographs for the HRE18 and, similarly, can also accurately predict river discharge hydrographs for future extreme and unprecedented rainfall events in the Chugoku region of Japan in real time before a flood. Our cumulative ensemble validation results of river mouth discharge hydrographs from all rivers (Figure 7), with an NSE of 0.98 for both the 7-CPM and 5-CPM, have large reproducibility metrics, thus showing the high accuracy of our proposed approach and methodology. Therefore, we recommend that these results be used as an important step toward the future development of real-time forecasts of river water levels and discharges for EWSs and the prevention of associated flash floods. This is the most important scientific contribution of this study.

To further demonstrate the robustness and applicability of the proposed method to a wide range of study sites and events, we cross-validated the results for each of the seven considered rainfall events for every river basin and obtained large reproducibility metrics, except for 6% of the cases. This is an important takeaway from this study because the accurate prediction of unprecedented extreme rainfall-induced river discharge events, which occur increasingly often due to ongoing climate change impacts, is currently one of the most challenging topics for hydrologists. This robustness strongly supports the expected applicability potential of our established hypothesis.

A limitation of this study is the occurrence of the equifinality issue in the calibration of hydrological models, when the values of the calibrated parameter sets are sometimes very different across the seven calibrated events. Therefore, the unanswered question in this study is which of the calibrated parameter sets represents the associated river basin the most accurately. However, we note that the quality of this study’s calibration process can only be properly evaluated by the simultaneous combination of seven parameters (for the 7-CPM) or five parameters (for the 5-CPM) and how they interact to predict river mouth hydrographs, and not by extracting a single parameter and evaluating its values in the seven calibrated parameter sets. Another unanswered technical challenge that we introduced is the optimal number of validation events to form ensembles. We showed that highly accurate hydrograph reproducibility metrics can be obtained by applying our methodology with six validation events to form ensembles, but different numbers of selected validation events may provide even more accurate results.

In follow-up studies, the calibrated CDRM hydrological model parameter sets, which highly accurately reproduce ensemble hindcasts of modeled river mouth discharge hydrographs, should be integrated with real-time rainfall forecasting meteorological models to develop integrated hydrometeorological forecasting tools to predict future flash flood river discharge hydrographs in the Chugoku region in real time before floods. The further extension of the proposed approach to predictions of future climatological events is also expected. We expect our approach and methodology to be replicated and applied not only for the Chugoku region, but also more broadly, with an accuracy similar to that in this study.