Progress in Earth andPlanetary Science

Kobayashi et al.
Progress in Earth and Planetary Science(2023) 10:5
https://doi.org/10.1186/s40645-023-00537-3

Open Access

RESEARCH ARTICLE

Ensemble rainfall–runoff and inundation
simulations using 100 and 1000 member
rainfalls by 4D LETKF on the Kumagawa Riverflooding 2020
Kenichiro Kobayashi1* , Le Duc2, Takuya Kawabata3, Atsushi Tamura4, Tsutao Oizumi3, Kazuo Saito5,
Daisuke Nohara6 and Tetsuya Sumi7

Abstract
This paper presents the 1000 ensemble flood simulations using ensemble rainfalls simulated by 4D LETKF. The
number of ensemble rainfall members is large as 1000 compared to the operational rainfall products of two-digit
numbers to avoid sampling errors in the three-dimensional meteorological simulation based on chaotic theory. Using
the large data set, 1000 ensemble rainfall–runoff for dam catchments and high-resolution inundation simulations oflarge area are carried out focusing on the Kumagawa river catchment. Herewith, the comparisons were carried out
with 21-member ensemble rainfalls of an operational forecast by Japan Meteorological Agency and 100-member
4D-LETKF ensemble rainfalls simulated independent of 1000-member 4D-LETKF. At the same time, the accuracy of
selective 100-member ensembles out of 1000 members is investigated. As a result, although many previous researchworks show a large number of ensemble simulations are necessary for three-dimensional meteorological field, thenumber could be reduced in the catchment-average rainfall–runoff and 2.5-dimensional inundation simulations
given that the rainfall prediction has a certain level of accuracy since improving the discharge prediction accuracy
with lower dimension is sometimes possible by adjusting the horizontally/vertically integrated model parameters
determined by topography and soil characteristics in advance against the observed rainfall. Also, the 1000 ensembles
could be classified into several patterns in horizontally accumulated 2D rainfall field. Likewise, the flood flow moves
toward the low elevation area and river; thus, the resultant 2.5-dimensional flood field does not show much variety as
three-dimensional meteorological simulation. The paper summarizes these studies.
Keywords 1000 Ensemble, 4D LETKF, Rainfall–runoff, Storage function model, Inundation, Shallow water equation,
Kumagawa River*Correspondence:
Kenichiro Kobayashi
kkobayashi@phoenix.kobe-u.ac.jp
1
Research Center for Urban Safety and Security, Kobe University, Kobe,
Japan
2
Institute of Engineering Innovation, The University of Tokyo, Tokyo,
Japan
3
Meteorological Research Institute, Tsukuba, Japan
4
Department of Civil Engineering, Kobe University, Kobe, Japan
5
Atmosphere and Ocean Research Institute, The University of Tokyo,
Kashiwa, Chiba, Japan
6
Sustainable Society Laboratory, Kajima Technical Research Institute,
Chofu, Japan

7

Disaster Prevention Research Institute, Kyoto University, Kyoto, Japan© The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
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to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of thislicence, visit http://​creat​iveco​mmons.​org/​licen​ses/​by/4.​0/.

Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

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1 Introduction
Numerical weather predictions (NWPs) are the only way
for quantitative weather forecasting and the necessity ofpredicting future weather accurately is increasing due toincreasing weather-related disasters by realizing global
warming. However, NWPs have inherent errors, since(1) there are errors in the initial/boundary conditions
due to a lack of perfect observations representing theactual atmosphere; and (2) the governing equation of thenumerical weather model itself is not perfect due to lackof knowledge (Japan Meteorological Agency: JMA 2021).
Even if errors in the initial condition are very small, the
NWP in the long future would result in unrealistic predictions, which is known as the chaotic nature of theatmosphere.
To estimate the errors in NWPs, ensemble prediction
systems (EPS) are gaining more attention nowadays.
It is widely thought that spreads of EPSs, which consistof several model runs, represent errors in the predictions. To incorporate these spreads of prediction variations, global and mesoscale ensemble weather prediction
systems-GEPS (Yamaguchi et al. 2021) and MEPS (Onoet al. 2021), respectively, are in operation now in Japan.
The specifications of GEPS and MEPS are given in
Table 1 along with the global ensemble prediction system (ENS) by ECMWF. The usefulness of GEPS, MEPS as
well as ENS ECMWF is now shown by many stakeholders in Japan, not only for weather prediction but also forflood prediction including statistical post-processing andAI applications (Kobayashi et al. 2016a, b; Sayama et al.
2020; Nohara and Sumi 2020; Nohara et al. 2022; Hanasaki 2019).
On the other hand, the point we are paying attention
to in this paper is that the number of ensemble members in MEPS is 21, which is modest. However, the consideration is becoming more prevail that the number ofensemble members must be much greater to fulfill the
probability distribution of the ensemble simulations to
avoid sampling error. Especially in ensemble data assimilation, increasing the number of ensemble members isnecessary to be more precise, which helps the estimationof the errors owing to decreasing the sampling errors. In
this context, the research for mesoscale weather mega

ensembles can be found in Kawabata and Ueno (2020),
Kunii (2014), Neckar et al. (2020), Duc and Saito (2017),
Duc et al. (2021) all with more than 1000 members.
Kawabata and Ueno (2020) investigated the nonGaussianity of the cumulonimbus prediction with 2 km
and 1000 members and found the non-Gaussianity was
initiated in the updraft at low troposphere. Kunii (2014)
dealt with 1000 members with a 15-km resolution
and illustrated that sampling errors in the backgroundcovariances significantly reduced in comparison with a
100-member ensemble, and concluded that an ensemble
size larger than 500 is enough large to approximate thecovariances. Necker et al. (2020) compared a 1000-member ensemble with a 15-km resolution by the Japanese
numerical weather model SCALE and a 40-member
ensemble by COSMO from the German Weather Service, and they found that sampling errors in backgrounderror covariances reduced in the 1000-member ensemble. The research aims of the above all experiments focus
mostly on sampling errors and Gaussianity in errors anddo not show their performances in precipitation predictions, even more in flood predictions.
On the other hand, the research also considering theensemble flood simulation driven by the 1600-member
ensemble weather predictions (Duc and Saito 2017) can
be found in Kobayashi et al. (2019, 2020). This was an initial work on probability flood forecasting using a physically based distributed rainfall–runoff model and a huge
number of ensemble rainfall forecasts obtained from afour-dimensional variational ensemble assimilation system. They showed that the probabilistic consideration in
the real sense with huge 1600-member ensembles was
possible along with the significant improvement in rainfall and flood simulations. Likewise, the result succeeded
in indicating the necessity of emergency flood operations
with enough lead times and probabilities. They also discussed the difficulty of selecting the best ensemble members due to the nonlinearity of the weather.
Recently, Duc et al. (2021) carried out 1000 ensemble
weather simulations targeting July 2020 Kyushu heavy
rain. Since the event brought extreme damage to flooded
areas, they designed the experiment for predicting heavy
rain, which is described in Duc et al. (2021), as well as the

Table 1 Specification of GEPS, MEPS and global ensemble (ENS) by ECWMF
GEPS

MEPS

ENS by ECWMF

Resolution

Approx. 40 km

5 km

18 km

Number of prediction per day

4 (00, 06, 12, 18UTC)

4 (00, 06, 12, 18UTC)

2 (00, 12UTC)

Duration of the prediction

132 (06, 18UTC)

39

360

21

51

264 (00, 12UTC)
Number of ensembles

27

Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

flood event. Thus, these 1000-member ensemble weather
simulations are used for the ensemble flood simulationsof this paper whose overview is explained later. Overall, this paper attempted to carry out 1000 ensemble
flood simulations driven by the 1000 ensemble weather
simulations using a lumped rainfall–runoff model with a
Kalman filter, and flood simulations with a shallow waterequation, which has different aspects from Kobayashi
et al. (2019) using a physically based distributed rainfall–
runoff model. The scientific question in the paper is thatin meteorology we have high uncertainty in rainfall forecasts based on chaotic nature, and this requires a large
number of ensemble members. However, in flood forecasting of hydrology, given that the rainfall prediction has
a certain level accuracy, it is not necessarily true because
certain factors like topography and river networks determine the flow movement very much, thus may reduce
uncertainties in flood forecasting compared to weather
forecasting. The large number may only be important forlong lead times. The paper attempted to answer this scientific question.
The basic attempts of this paper are: (1) fixing theparameters of the rainfall–runoff and inundation modelusing the radar–raingauge rainfall distribution data by
JMA (hereafter R/A) as observation and (2) investigating the variations/uncertainties of the flood predictions
mainly due to the uncertainties of ensemble weather
simulations only, then (3) estimating the rainfall–runoff
model state variables to minimize the effect of the runoff
model uncertainties using a Kalman filter. This is due tothe fact that, in general, state variables of runoff models
must be adjusted even against quasi-perfect rainfalls (i.e.,R/A) at each different rainfall event.
It is noted that Oizumi et al. (2022) also currently carryout 1000 ensemble flood simulation using Japan Meteorological Agency operational Runoff-Index Model, arainfall–runoff model different from this paper using thesame 1000 ensemble rainfall data set. However, the purpose and focus are different from this paper.
The structure of the rest of this paper is as follows. Section 2.1 presents a brief description of the 1000 ensemble rainfalls used in the paper. Section 2.2 explains the
Kumagawa river catchment, the target river basin in thispaper. Section 2.3 shows the flood simulation conditions
with a rainfall–runoff model and a shallow water equation model. Sections 3.1–3.2 show the results of 1000
ensemble rainfall–runoff simulation for the Ichifusa dam
catchment. Likewise, Sect. 3.3–3.4 shows the results
for the Kawabegawa dam catchment. Section 3.5 shows
the result of 1000 inundation simulations. Section 3.6
presents the application of a Kalman filter on the rainfall–runoff simulations. Finally, Chapter 4 states the concluding remarks and future aspects.

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2 Methods/experimental
2.1 High‑resolution ensemble precipitation forecasts

In order to provide rainfall forecasts for the hydrologicalmodel, three ensemble prediction systems (EPS) are used.
The first EPS is the operational mesoscale ensemble prediction system (MEPS) running at JMA with 21 ensemble members. The other two EPSs are conducted by Duc
et al. (2021) using the JMA nonhydrostatic model (NHM)
and the local ensemble transform Kalman filter (LETKF)
to generate initial perturbations. The data assimilation
system NHM-LETKF is described in Duc et al. (2015). All
settings of these two EPSs are identical except the number of ensemble members, which are 100 and 1000. Thus,
by using the three EPSs, we can examine the impact of
the number of ensemble members on the uncertainty
quantification of rainfall and flood predictions.
NHM-LETKF run with dual resolutions: a low-resolution grid with 15-km grid spacing for analysis perturbations and a high-resolution grid with 5-km grid
spacing for analyses. The analysis domain is copied fromthe JMA’s operational domain, which is much larger
than the forecast domain [see Fig. 2 of Duc et al. (2021)].
Except for radiances and rain analyses, they assimilate
the same observations that are assimilated into the JMA’s
operational data assimilation system ASUCA-VAR. The
boundary conditions and boundary perturbations are
obtained from the JMA’s global forecasts. The assimilating run with 1000 ensemble members is relatively challenging because huge computational costs are involved.
In order to reduce the computational cost, they turn off
vertical localization in NHM-LETKF. Duc et al. (2021)
point out that removing vertical localization indeed has apositive impact on the heavy rain forecast in Kyushu.
The forecast domain is a 2-km grid-spacing domain
centered at Kyushu. This domain has 819 × 715 horizontal grid points and 60 vertical levels. The boundary conditions are interpolated from the JMA’s global forecasts.
The boundary perturbations are extracted from the JMA’s
1-week global ensemble forecasts. All ensemble members
start at 18 JST July 3rd 2020 and run for 24-h forecasts.
Note that the forecast domain of MEPS is the same as theanalysis domain of NHM-LETKF with the grid spacing of
5 km.
For illustration, Fig. 1 shows the deterministic
and probabilistic precipitation forecasts from the1000-member EPS in comparison with the observations
at the peak period 00–09 JST of the July 2020 Kyushu
heavy rain (July 4th). Clearly, these forecasts succeed
in predicting both the rainfall location and the rainfall
amount. In order to quantify this subjective evaluation,
Fig. 2 plots the Fractions Skill Score (FSS) (Roberts andLean 2008) calculated for the 3-h deterministic precipitation forecasts over Kyushu. This figure also displays

Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

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Fig. 1 a, b, c Observations of consecutive 3-h precipitations between 00 and 09 JST July 4th 2020. d, e, f The corresponding 2-km forecasts by
NHM-LETKF started at 18 JST July 4th 2020. g, h, i The corresponding 2-km probabilistic forecasts with respect to a rainfall threshold of 50 mm/3 h.
(Reproduced Duc et al. 2021)

Fig. 2 FSSs of the consecutive 3-h precipitations forecasted by ASUCAR-VAR and NHM-LETKF against the R/A observations between 00 and 09 JST
July 4th 2020 over the Kumagawa catchment. (Reproduced Duc et al. 2021)

the FSSs of the downscaling forecasts from the JMA’s
ASUCA-VAR analysis at 2-km grid spacing for comparison. It can be seen that the NHM-LETKF forecasts
even beat the operational forecasts in this case. Thus,
it can be expected that NHM-LETKF will yield goodprecipitation forecasts for the Kumagawa catchment.
This is indeed the case as verified in Fig. 3 in which theevolutions of the accumulated rainfall, averaged overthis area, are presented.

Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

2.2 Kumagawa river catchment

Receiving the three rainfall ensemble forecasts, this paper
presents its application to flood simulations of the Kumagawa river catchment. Figure 4 shows the Kumagawa
river catchment (catchment area: 1765 k­ m2). The river
experienced the largest-ever flooding on July 4, 2020. The
total accumulated rainfall at the Hitoyoshi rainfall observatory (Fig. 4) attained 420 mm between 00JST July 3 and24JST July 4. Note that the average rainfall of July in the
Hitoyoshi rainfall observatory was 471.4 mm. The hourly
rainfall over 30 mm continued for 8 h having a line shaperainfall band from the predawn of July 4. At Taragi and
Hitoyoshi rainfall observatories (Fig. 4), the 6-, 12- and24-h rainfalls exceeded the historical records of July in1965 and 1982 (Fukuoka district meteorological observatory 2020; The Kinki district of maintenance station,
Ministry of Land, Infrastructure, Transport and Tourisms 2020).
In Fig. 4, the catchment area of the already existing
Ichifusa dam is shown. The catchment area is 158 k­ m2.
The discharge into the Ichifusa dam was simulated using
a lumped rainfall–runoff model (hereafter RRM) or the
storage function model (hereafter SFM). The Kawabegawa dam (catchment area 470 k­ m2) is in the planningstage, though the catchment area is shown also in Fig. 4
since the ensemble discharges at the site were also estimated using the RRM. Figure 4 also shows the nested
computational area with a resolution of 5 m. The flood
simulation using a shallow water equation (hereafter SWE) was carried out in the nested computational
area. The outflows from Ichifusa and Kawabegawa dams

Page 5 of 22

were provided to the flood simulation by SWE; then, thepropagation of the river flows including overtopping was
simulated. At the same time, R/A was provided over theentire nested computational area. Thus, the inland flooding by the rainfall as well as the river flows by the two discharge inputs from the dams is simultaneously simulated.
The red color polygon in Fig. 4 shows the inundated
area (i.e., flood mark) by the Geospatial Information
Authority of Japan (hereafter GSI). The simulated flooding for the rainfall event of July 4, 2020, was comparedwith the flood marks and water levels at several observatories. Then, rainfall–runoff discharge simulations with
the SFM for the Ichifusa and Kumagawa dam catchments
are carried out using the three rainfall ensemble forecasts
as inputs. Finally, ensemble flood simulations with SWE
were carried out using the discharges from the two dam
catchments driven by the 1000 member (LETKF) ensemble forecasts.
2.3 Flood simulation conditions

Figure 5 shows the observed inflow and outflow of the Ichifusa dam. In the actual situation of 2020, the emergency
spillway gate release operation was considered in the dam,
but it was avoided in the end. The total storage volume ofthe dam is 40.2 million ­m3, and the effective volume is 35.1
million ­m3. On the other hand, Fig. 5 also shows the simulated inflow at Kawabegawa dam conducted by Sumi andNohara (2020) using a kinematic wave-type distributed
rainfall–runoff model. This value is used as the pseudoobservation in the following as it was simulated using aphysically based runoff model. The model parameter by

Fig. 3 Evolutions of the accumulated precipitation averaged over the Kumagawa catchment as forecasted by NHM-LETKF. The light gray zone
represents the intervals between the first and third quartiles of the probability density functions of the accumulated rainfall. The dark grayzone indicates the 95% confidence intervals, i.e., the intervals between the 2.5 and 97.5 percentiles, of the probability density functions of theaccumulated rainfall (Reproduced Duc et al. 2021)

Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

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Fig. 4 Overview of the Kumagawa river catchment

Fig. 5 Observed inflow/outflow of the Ichifusa dam and simulated inflow of assumed Kawabegawa dam by a distributed rainfall–runoff model(Sumi and Nohara 2020)

Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

Sumi and Nohara (2020) was adjusted so that the simulated
discharge was fitted to the observation at Yanase (Fig. 4) inthe downstream of Kawabegawa dam.
The observed outflow to Ichifusa dam after dam control and simulated inflow at the assumed Kawabegawa
dam was provided to the flood simulation with the SWE
at points 1 and 2 as shown in Fig. 4. As explained, the R/A
was also provided in the entire computational domain. The
flow can go out from the square boundary of the computational domain based on the weir formula. The governing
equations of SFM and SWE models applied are as follows.
The explanation of the Kalman filter used afterward is also
described.
2.3.1 SFM

The equations of the storage function model and Kalmanfilter applied are written by referring to JSCE (2002). For
the details see JSCE (2002).
A storage function model by Hoshi and Yamaoka (1982)(also in JSCE 2002) which is a function made up of two
terms was used.

s(t) = k1 q p1 (t) + k2

d p2
q (t)
dt

(1)

where q is the direct runoff in [mm/h], t the time in [h],k1 , k2 , p1 , p2 the model parameters. SFM is a lumped
model and driven by the basin-average rainfall of the
Ichifusa dam and Kawabegawa dam catchments. The
parameters identified are given in Table 2.
Then, the continuity equation of the storage height is:ds(t)= cr(t) − q(t)dt(2)

where c is the runoff ratio and r is the catchment-average
rainfall intensity in [mm/h].
Inserting Eq. (1) into (2) yields

dq(t)d2 q p2 (t)+ cr(t) − q(t)= −k1 p1 q p1 −1 (t)
k2
2
dt
dt

(3)
Equation (3) can be transformed into a system of ordinary differential equations (ODEs).
dx1 (t)= x2 (t)dt(4)

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p1/ −1dx2 (t)
k1 p1=−
x1 (t) p2 x2 (t)
dt
k2 p2
1/
c
1
− x1 (t) p2 + r(t)
k2
k2

(5)

p2

where x1 = q p2, x2 = dqdt .
By linearizing this ODE system, an update or system equation is obtained for an extended Kalman filter besides an observation equation derived from theobserved discharge and also linearized:

System Equationx(k + 1) = �(k)x(k) + α(k) + u(k)(6)

Observation Equationy(k) = Ŵ(k)x(k) + β(k) + w(k)(7)

where the variables are denoted as follows: k : time. x :system state variable (p-dimensional vector). : statetransition matrix (p × p matrix). α: system constant vector (p-dimensional vector). u: system noise, independent normal Gaussian white noise with zero mean and U
variance (p-dimensional vector). y : observation vector(m-dimensional vector (m ≤ p)). Ŵ: observation matrix(m × p matrix). β: observation constant vector (m-dimensional vector). w : observation noise, independent normal
Gaussian white noise with zero mean and W variance(p-dimensional vector).
The exact form of the two matrices �(k), Ŵ(k) and thetwo vectors α(k), β(k) is given in JSCE (2002). Referring
to Eqs. 4 and 5, state variables of Eq. (6) to be estimatedby the extend Kalman filter are defined as follows:

dq p2, x3 = k1 , x4 = 1/k2, ,dt
x5 =p1 , x6 = 1/p2 , x7 = c

x1 =q p2 , x2 =(8)

The initial estimation of the seven state variables
above must be given. Likewise, estimated error covariance matrix, the covariance matrix of the system noiseand observation noise must be also initially given for
the application. Then, the state variables are updated ateach time step following the algorithms of the extended
Kalman filter including the calculation of the Kalman
gain. See JSCE (2002) for further details.
2.3.2 SWE

Table 2 Parameter values for SFC
Parameters

k1

k2

p1

p2

Ichifusa

16.0

47.6

0.650

0.46

Kawabegawa

14.1

66.7

0.515

0.50

∂h ∂M
∂N
+
+
=R
∂t
∂x
∂y

(9)

Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

Page 8 of 22√∂M
u2 + v2
∂uM
∂vM
∂H
2
+
+
= −gh
− gn u
∂t
∂x
∂y
∂x
h1/ 3
(10)
√
∂N
u2 + v2
∂uN
∂vN
∂H
+
+
= −gh
− gn2 v
∂t
∂x
∂y
∂y
h1/ 3
(11)
where h is the surface flow water depth; M, N the discharge fluxes in the x- and y-directions, respectively(M = uh, N = vh); R the rainfall intensity; u, v the surface flow velocities in the x- and y-directions, respectively; H the water level (h + z); n the Manning’s
roughness coefficient; g the acceleration due to gravity. This SWE model was parallelized by MPI and OpenMP (Kobayashi et al. 2016b). The computational areaof the region is 26.545 km × 16.995 km with 5 m resolution; thus, the number of computational grids is5309 × 3399 = 18,045,291. The initial time of the simulation was 18JST July 3, 2020, and the simulation was carried out for 24 h. The Manning’s roughness coefficient n
in Eqs. (10–11) is 0.05.
Figure 6 shows the simulated water depth in the computational domain. The figure also shows the inundation
area record by GSI at downstream. The right-hand figure
is an enlarged view around Hitoyoshi city. Overall, it isclear that the simulated inundated area is in good agreement with the area where the record by GSI exists. The
simulated water levels at Yanase, Hitoyoshi and Taragi water level observatories were compared with theobserved water levels as shown in Fig. 7. Since there is nowater in the beginning of the simulation, the difference inthe beginning of the simulation is large but the levels at

the peaks were basically in good agreement for the prediction of the inundation level. The observation at Hitoyoshi was missing from the middle due to the flooding.

Fig. 6 Simulated water depth (2020/7/4, 0900)

3 Results and discussion
3.1 Ensemble rainfall–runoff simulations at Ichifusa dam

As mentioned, SFM by Hoshi and Yamaoka (1982) wasused for the simulation of dam inflow at Ichifusa andKawabegawa dams. The reason to use this simple conceptual rainfall–runoff model is that the computational
speed is fast. A 1000-member ensemble flood simulation
with 1000 rainfalls and SFM needs approx.15 s. using aworkstation with Xeon CPU E5-2670 (2.6 GHz).
In the followings, the comparisons with the 21-member ensemble by MEPS of JMA (hereafter 21 (MEPS)),the 100-member ensemble by 4D LETKF (hereafter 100(LETKF)) and the 1000-member ensemble by 4D LETKF(hereafter 1000 (LETKF)) are carried out. Likewise, ten
100-member ensembles selected from 1000 (LETKF)
are also compared. These ten 100-member ensembles
are simply selected as members 1–100, 101–200, 201–
300,…,900–1000 (hereafter expressed such as m1-100,m900-1000).
Figure 8a–c shows the catchment-average rainfalls of
the Ichifusa dam basin with 21(MEPS), 100 (LETKF) and1000 (LETKF), respectively. Both of 100 (LETKF) and1000 (LETKF) started at 18:00 on July 3 for 24 h, while
21 (MEPS) started at 21:00 on July 3 for 39 h. The figures
show that the ensemble means of the 21 and 100 members clearly underestimate R/A which is in the paper
regarded as the observed rainfall. Likewise, the grayclouds of ensembles by 21 and 100 ensemble members

Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

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Fig. 7 Observed and simulated water depth at water level observatories

can not necessarily cover the peak of R/A, while thoseby 1000 members cover the entire R/A except the very
beginning of the rainfall. Figure 8d–f shows the cumulative rainfall. The mean of the 1000-member ensemble
shows good agreement with that by R/A. Likewise, themean of the 100 ensemble rainfalls shows a relatively
good agreement with R/A although it is a little underestimation. The mean of the 21-member ensemble clearly
underestimates R/A.
Figure 9a–c shows the simulated 21 (MEPS), 100(LETKF) and 1000 (LETKF) ensemble discharges by
SFM. Same as rainfall, the gray cloud of 1000 member
discharges covers the observation, while that by 21 members could not encompass the observation though 1 outof 21 members having maximum peak discharges showsa good agreement with the observation. Several membersin the 100-member ensemble prediction also show a relatively good agreement with the observation, though thepeaks of many members appear earlier than the observation and the peak discharges are underestimated. The
ensemble mean of 1000 members relatively resembles theobservation, while those of 21 and 100 members clearly
underestimate the observation. Figure 9d–f shows cumulative discharges. The mean of 1000 members shows agood agreement with the observation, though it is slightly
underestimated. On the other hand, the final value of themean of 100 members also shows a good agreement withthe observation. The mean of 21 members underestimates the observation largely. Good prediction of cumulative inflow volume is also useful for reservoir managers

to know if the flood control capacity of the reservoir ismuch enough.
3.2 Selective 100‑member ensembles
from the 1000‑member weather and rainfall–runoff
simulation at Ichifusa dam

Figure 10a shows again 1000 (LETKF) ensemble discharge simulations, while figure b–f shows the 100-member ensembles selected from 1000 members. The
members are denoted by m1-100, m101-200,…, m401500. From the visual judgment only, each of the 100
selected members shows similar results. To quantify theagreement with the observation, Nash–Sutcliffe efficiency (hereinafter NSE: Nash and Sutcliffe 1970) is
calculated.

NSE = 1 −

N
i=1
N
i=1

Q0i − Qsi

Q0i − Qm

2
2

(12)

N

Qm =

1  i
Q0
N

(13)

i=1

where N is the total number of time steps (1 h interval),
Q0i is the observed dam inflow (discharge) at time i, Qsi is
the simulated dam inflow (discharge) at time i, Qm is theaverage of the observed dam inflows.
NSEs are shown in Fig. 11. Figure 11a, c shows all the
NSEs of 21 (MEPS), 100 (LETKF), 1000 (LETKF) and
10 selected 100-member ensembles from 1000 (LETKF)
with regard to rainfall and discharge, respectively.

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Fig. 8 Observed and simulated basin-average rainfalls for Ichifusa dam catchment. a–c Hourly rainfall intensity of (a) 21 members (MEPS), (b) 100
members (LETKF) and (c) 1000 members (LETKF) d–f Cumulative rainfall for (d) 21 members (MEPS), (e) 100 members (LETKF) and (f) 1000 members(LETKF)

Figure 11b, d shows the box plot of the NSEs also for
rainfall and discharge, respectively. From these figures, 1000 (LETKF) clearly outperforms 100 (LETKF)
and 21 (LETKF) for both rainfall and discharge. However, it is recognized that some NSEs of 21 (MEPS)
and 100 (LETKF) are close to one, thus those members
are in good agreement with the observations. On theother hand, each of the selected 100-member ensembles shows a similar tendency with 1000 (LETKF). In
other words, each of the 100-member ensembles is
randomly selected so that the set has similar statistics
to the 1000-member ensemble. Note that even though
the number of members is the same in the both 100(LETKF) and 100 (selective) cases, the two sets are
not equal in the sense that the selective 100-member ensembles inherit all statistics from the larger
1000-member ensemble, and therefore should beunderstood as representatives of this large ensemble. A
large number of ensemble members enable us to better capture uncertainties in atmospheric processes, andto better utilize information provided by observations,thus leading to significant improvement in rainfall
forecast. These 10 × 100 member sets also outperform
21 (MEPS) and 100 (LETKF) for both rainfall anddischarge.

Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

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Fig. 9 Observed and simulated discharge to Ichifusa dam catchment. a–c Hourly discharge of (a) 21 members (MEPS), (b) 100 members (LETKF)
and (c) 1000 members (LETKF). d–f Cumulative discharge for (d) 21 members (MEPS), (e) 100 members (LETKF) and (f) 1000 members (LETKF)

3.3 Ensemble rainfall–runoff simulations at the assumed
Kawabegawa dam

Figure 12a–c shows the ensemble basin-average rainfalls
for the Kawabegawa dam catchment. Again, each member of the 21-member (MEPS) ensemble and its mean
clearly underestimate R/A. Those 100 (LETKF) members
are better than the 21 members in terms of the amount,though the cloud of 100 members could not partiallycover R/A. The cloud of 1000 (LETKF) members entirely
covers R/A well, though the mean still underestimates
R/A. Figure 12d–f shows the cumulative rainfalls. The
cumulative rainfall of 1000 (LETKF) members resembles
that of R/A very well. That of 100 (LETKF) members also

shows a relatively good agreement with R/A while that of
21 (MEPS) members clearly underestimates R/A.
Figure 13a–c shows the simulated discharges by SFC
for 21 (MEPS), 100 (LETKF) and 1000 (LETKF) members. The 21 member discharges clearly underestimate
the discharge observation. On the other hand, the cloudsof 100 and 1000 members cover the observation well. It
should be noted that the 100-member ensemble also
shows good results with regard to the Kawabegawa dam
catchment. Some members of the 100-member ensemble
show a good agreement with the observation. Likewise,
the 1000-member ensemble well covers the observation.
Considerable members of the 1000-member ensemble

Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

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Fig. 10 Observed and simulated discharge to Ichifusa dam catchment by 1000 (LETKF). a Hourly discharge of 1000 members, b of m1-100, c
m101-200, d m201-300, e m301-400, f m401-500

show a good agreement with the observation. The meansof both 100 (LETKF) and 1000 (LETKF) members show
good agreement with the observation.
3.4 Selective 100‑member ensembles
from the 1000‑member ensemble weather
and rainfall–runoff simulations at Kawabegawa dam

Figure 14a shows again 1000 (LETKF) ensemble discharge simulations for Kawabegawa dam, while
Fig. 14b–f shows the 100-member ensembles selectedfrom the 1000-member ensemble. Same as Ichifusa,
the members are denoted by m1-100, m101-200,…,

m401-500. Each of the selected 100-member ensembles
shows similar results as the case of Ichifusa.
NSEs for Kawabegawa dam are shown in Fig. 15. Figure 15a, c shows all the NSEs of 21 (MEPS), 100 (LETKF),
1000 (LETKF) and 10 selected 100-member ensembles
from 1000 (LETKF) for rainfall and discharge, respectively. Figure 15b, d shows the box plots also for therainfall and discharge, respectively. From these figures,1000 (LETKF) clearly outperforms 100 (LETKF) and21 (LETKF) for rainfall. On the other hand, with regard
to discharge, 100 (LETKF) shows an equivalent level ofaccuracy with 1000 (LETKF). Likewise, it is recognizedthat some NSEs of 21 (MEPS) and a considerable number

Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

Fig. 11 a NSEs of rainfall, b NSE box plots of rainfall, c NSEs of discharge and d NSE box plots of discharge for Ichifusa dam

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Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

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Fig. 12 Observed and simulated basin-average rainfalls for Kawabegawa dam catchment. a–c Hourly rainfall intensity of (a) 21 members (MEPS),(b) 100 members (LETKF) and (c) 1000 members (LETKF) d–f Cumulative rainfall for (d) 21 members (MEPS), (e) 100 members (LETKF) and (f) 1000
members (LETKF)

of 100 (LETKF) are close to one, or in good agreementwith the observations. On the other hand, each of theselected 100-member ensembles shows a similar levelof accuracy with 1000 (LETKF) and resembles them instatistics.
3.5 Ensemble flood simulation with a shallow water model

Then, ensemble inundation simulations are carried out
using 1000 (LETKF) ensemble discharges at Ichifusa/
Kawabegwa dams and the corresponding ensemble rainfalls in the previous chapters as inputs. The results areshown in Fig. 16. The figure shows the probabilities of the
maximum inudation depth of the selected ensemble members beyond the maximum inundation depth simulated by

R/A without the discharge cut of dams as shown on theright-hand side. Herein, 1–100, 1–200, 1–500 and 1–1000
members are selected. As in the figures, the probability becomes even more than 50% in the mountain area
whose maximum inundation depth is directly simulatedby the ensemble rainfalls. However, in the main channelsof the Kumagawa and Kawabegawa rivers, the probability is below 50%. One of the interesting features is that,as the case of discharges at the two dams in the previous
chapters, the distribution of the probability does not differ much according to the number of selected ensembles.
This may indicate that the place of inundation is governedmostly by the topography or ground elevation in addition to the rainfall distribution, thus the place where theKobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

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Fig. 13 Observed and simulated discharge to Kawabegawa dam catchment. a–c Hourly discharge of (a) 21 members (MEPS), (b) 100 members(LETKF) and (c) 1000 members (LETKF). d–f Cumulative discharge for (d) 21 members MEPS, (e) 100 members and (f) 1000 members

maximum inundation occurs does not differ much as theflood water flows into the lower elevation area. In otherwords, a reasonable number of ensemble members could
be smaller when the inundation depth is predicted, unlike
the three-dimensional weather ensembles.
3.6 Kalman filter application on the rainfall–runoff
simulation

Finally, a Kalman filter (Kawamura 1993; JSCE 2002,
Ueda 1984; Athans 1968) was applied to the first 11, 12,
13, 14, 15 and 16 h of the SFC ensemble simulations.
Herein 21 (MEPS), 100 (LETKF) or 1000 (LETKF) statevariable sets of SFC were adjusted so that simulatedhydrographs become similar to the observed discharge
while the Kalman filter was applied. By doing this, the 21,
100 or 1000 simulated hydrographs become similar to theobservation for the first 11–17 h, then the hydrographs
afterward show the spread of the discharges according to
the variations of ensemble rainfalls. This attempt was carried out since there are uncertainties in the runoff model
state variables even against quasi-perfect rainfall (i.e.,
R/A). Thus, those uncertainties are minimized against
the 1000 (LETKF) rainfalls. Note that the predicted rainfalls and not the observed rainfall are used in the Kalmanfilter. The challenge herein is to make a successful prediction beyond the accuracy of the rainfall predictions.

Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

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Fig. 14 Observed and simulated discharge to Kawabegawa dam catchment by 1000 (LETKF) a Hourly discharge of 1000 member, b of m1-100, c
m101-200, d m201-300, e m301-400, f m401-500

Figure 17 shows the application of the Kalman filter on
the discharge simulations of 21 (MEPS), 100 (LETKF)
and 1000 (LETKF) ensembles for the Ichifusa dam. Oneof the interesting features of this Kalman filter application is that the ensemble mean becomes quite close to
the observation especially as it approaches the peak discharge. The difference between the ensemble mean andobservation is smaller in the case of the 1000 (LETKF)member ensemble than the 21 (MEPS) or 100 (LETKF)member ensembles with regard to Ichifusa dam. This
indicates that 1000 ensemble simulations are superior to

100 or 21 ensemble simulations in this aspect because of
fewer sampling errors. However, 1 of 21 ensemble members shows always a good agreement with the observation in this case which is preferable for the application of
MEPS currently operated by JMA.
On the other hand, Fig. 18 shows the 21 (MEPS), 100(LETKF) and 1000 (LETKF)-member ensembles for the
Kawabegawa dam. In this case, the mean and ensemblesof 21 members basically underestimate the observation.
However, those of 100 and 1000 members show both
good agreements. Although there are less large discharge

Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

Fig. 15 a NSEs of rainfall, b NSE box plots of rainfall, c NSEs of discharge and d NSE box plots of discharge for Ichifusa dam

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Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

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Fig. 16 The probabilities (%) of selected ensemble members beyond the inundation depth simulated by R/Amembers in the 100-member ensemble compared withthe 1000-member ensemble, the accuracy of the mean of
100 members compared with the observed discharge is
not inferior to the 1000 members.

4 Conclusions
This paper dealt with the 1000 ensemble rainfall–runoff and inundation simulations driven by 1000 ensemble weather simulations. To better sample uncertainties
in ensemble weather simulations, it is now prevailingto increase the number of ensembles even up to 1000–
10,000 in the academic field of meteorology. Receiving
these ensemble weather simulations, this paper carriedout ensemble rainfall–runoff and inundation simulations.
The results of discharge simulations at the Ichifusa andKawabegawa dams using ensembles show that the overall trend of the rainfall–runoff simulation is similar to
the weather simulation. In other words, the 1000 ensemble discharge simulations show basically better coverageof observed discharge as the case of ensemble rainfalls.
Likewise, the ensemble mean of 1000 discharge simulations becomes closer to the observation, which is considered due to better representation in rainfall simulations.
However, there were cases in the discharge simulation that the 100 (LETKF)-member ensemble was also

not bad. This is considered because the discharge by therainfall–runoff simulation is determined not only by therainfall but also by the topography and subsurface characters. The rainwater converged to the catchment outlet
in case of rainfall–runoff simulation; thus, some differences of the rainfall pattern in the catchment could becanceled out. This means that the accuracy of discharge
simulation is not necessarily the same to the accuracyof weather simulation if the soil parameter is appropriately set up. This change of accuracy is more confirmed
when the state variables of the rainfall–runoff model
are adjusted by the Kalman filter in real time using
observations. If the real-time discharge observation
exists 1–3 h ahead of the peak, the discharge prediction around peak time can be better. Nevertheless, 1000
ensemble simulations are preferable when the prediction long time before such as half to one day is necessary since the adjustment of rainfall–runoff model statevariables has almost no meaning at the early stage when
the rainfall prediction indicates small values. In thissense, the research on mega ensemble rainfall simulations with enough lead time is worth pursuing aiming
at citizens an early evacuation before the flood incident occurs. Likewise, when 1000 ensemble inundation
simulations were conducted, the results also indicatedthat the area where the maximum inundation occurs
was determined again not only by the rainfall but also
by the topography. Thus, the probability distribution ofKobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

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Fig. 17 Application of the Kalman filter on the discharge simulations of 21, 100 and 1000 ensembles for Ichifusa dam (a–g): The Kalman filter wasapplied until: a 11, b 12, c 13, d 14, e 15 and f 16 h

the inundation beyond the observation (i.e., simulated
by R/A without dams) did not differ much between
selected 1–100, 1–200, 1–500 and 1–1000 member

ensembles. In other words, the number of ensemble
members could be reduced in case of 2.5-dimensional
inundation predictions unlike the three-dimensional

Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

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Fig. 18 Application of the Kalman filter on the discharge simulations of 21, 100 and 1000 ensembles for Kawabegawa dam (a–g): The Kalman filter
was applied until: a 11, b 12, c 13, d 14, e 15 and f 16 h

Kobayashi et al. Progress in Earth and Planetary Science(2023) 10:5

weather/rainfall simulations, although finding out theproper number of ensemble members remained as the
future work.
Abbreviations
LETKF	Local Ensemble Transform Kalman Filter
NWPs	Numerical Weather Predictions
JMA	Japan Meteorological Agency
EPS	Ensemble Prediction System
GEPS	JMA’s Global Ensemble Prediction System
MEPS	JMA’s Mesoscale Ensemble Prediction System
ECMWF	European Centre for Medium-Range Weather Forecasts
SCALE	Scalable Computing for Advanced Library and Environment
COSMO	Consortium for Small-scale Modeling numerical weather forecasting model
NHM	JMA nonhydrostatic model
FSS	Fractions Skill Score
RRM	Rainfall–runoff model
SFM	Storage function model
SWE	Shallow water equation
R/A	JMA’s radar–raingauge rainfall
GSI	Geospatial Information Authority of Japan
Acknowledgements
The first author joined the investigation committee for the Kumagawa Riverflooding on July 4, 2020, by the Japan Society of Civil Engineers headedby Prof. Omoto from Kumamoto University. Through the activity, we havereceived useful comments and hydrological data from the Kumamoto Prefecture, Ministry of Land, Infrastructure, Transport and Tourisms (MLIT) and JMA.
The authors would like to thank them.
Author contributions
KK conceptualized the entire research work in hydrology, carried out flood
simulations, made the figures/tables and wrote many parts of the paper. LD
carried out the 1000 ensemble weather simulations and wrote the meteorological summary in the paper, TK took the leading role in the researchof the meteorological part, TA worked together with KK for many parts ofhydrological research, TO supported 1000 ensemble flood simulations using
a supercomputer, KS advised the research, especially the meteorological part,
DN carried out physically based rainfall–runoff simulation which was used as abenchmark for the rainfall–runoff simulation by SFM, and TS advised in many
aspects of the hydrological research. All authors read and approved the final
manuscript.
Funding
This research used computational resources of the supercomputer Fugaku
and other computers of the High Performance Computing Infrastructure(HPCI) system provided by the RIKEN R-CCS (ID:hp200128, hp210166,
hp220167) supported by MEXT (JPMXP1020200305) as “Program for Promoting Researches on the Supercomputer Fugaku” (Large Ensemble Atmosphericand Environmental Prediction for Disaster Prevention and Mitigation). Likewise, this work was supported by JSPS KAKENHI Grant Number 19K04618.
Availability of data and materials
JMA-NHM is available under the collaborative framework between the Meteorological Research Institute (MRI) of Japan and related institutes or universities. Likewise, the DRR and SWE models are available under the collaborative
framework between Kobe University and related institutes or universities. The
meteorological data and hydrological data were provided by MLIT, Kumamoto Prefecture and JMA. We will consider making other data available uponrequest. Research cooperation is preferable for data provision.

Declarations
Competing interests
The authors declare that they have no competing interest.

Page 21 of 22

Received: 24 July 2022 Accepted: 21 January 2023

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