Hydrological Research Letters 12(4), 28–33 (2018)Published online in J-STAGE (www.jstage.jst.go.jp/browse/hrl). DOI: 10.3178/hrl.12.28

Flood risk curve development with probabilistic rainfall modelling and largeensemble climate simulation data: a case study for the Yodo River basin

Tomohiro Tanaka1, Yasuto Tachikawa2, Yutaka Ichikawa2 and Kazuaki Yorozu2

1Graduate School of Global Environmental Studies, Kyoto University, Japan2Graduate School of Engineering, Kyoto University, Japan

Abstract:

A flood risk curve is the relation between annual maxi-mum economic damage due to floods and its exceedanceprobability, which provides useful information for quantita-tive flood risk assessment. This study proposed to examinethe applicability of d4PDF, a large ensemble climate pro-jection dataset, to develop a probabilistic flood risk curvefor the Yodo River basin (8,240 km2), Japan. The d4PDF isa climate dataset under historical and 4 K increase condi-tions with tens of ensembles and provide a physically-basedand reliable estimation of ensemble flood risk curves andtheir future changes. We identified that d4PDF rainfall datahas bias for the spatial variability of rainfall probably dueto coarse spatial resolution, while not for basin-averagedrainfall. This typical type of bias was removed by incorpo-rating basin-averaged rainfall of d4PDF and observed spa-tial pattern of rainfall into analytically-based probabilisticrainfall modelling. Derived ensemble flood risk curves pro-vided a histogram of T-year flood damage. The histogramhad a long tail and showed that T-year flood damage maybe larger than its deterministic estimate located at themedian. Estimated ensemble flood risk curves at present/future climates showed a clear increase of flood risk and itsuncertainty at 4 K increase scenario.

KEYWORDS flood risk curve; ensemble projection;d4PDF; climate change

INTRODUCTION

In a changing climate, flood risk is expected to increasein many parts of the world. In river basins with high levelsof flood defenses, it is not feasible to cover all the incre-ments through reinforcement. For integrated flood riskmanagement including hardware/software countermeas-ures, a flood risk curve, which shows the relation betweenannual economic damage due to floods and its exceedanceprobability, is effective because it quantifies the currentflood risk and its reduction by both types of countermea-sures. Since it is hard to collect enough economic damagestatistics, several researchers have demonstrated the deriva-tion of a flood risk curve using the probability distributionof rainfall and/or flood peak discharge, considering their

Correspondence to: Tomohiro Tanaka, Graduate School of Global Envi-ronmental Studies, Kyoto University, Kyotodaigaku-Katsura, Nishikyo-ku, Kyoto, Kyoto 615-8540, Japan. E-mail: tanaka.tomohiro.7c@kyoto-u.ac.jp

spatiotemporal distribution (Apel et al., 2006; Neal et al.,2013; Tanaka et al., 2015).

The problem of small samples for this type of extremephenomena is also the case for future projection usingGlobal Climate Models (GCMs). Most GCMs provide cli-mate reproduction and projection for several decades undera few sea surface temperature scenarios and initial condi-tions. They are not enough to estimate quantiles corre-sponding to return periods over the simulation periods(such as 100 and/or 200 years). Furthermore, the smallnumber of ensemble members requires statistical resam-pling to identify the uncertainty range. To address thisissue, several inter-organizational research collaborationprojects have created a huge number of ensemble climatedata. A collaborative research project by several researchinstitutes in Japan has created a database for policy deci-sion making for future climate change (d4PDF), which haslarge ensembles under the historical (present) and a 4 Kincrease (future) scenario (and 2 K increase scenario avail-able since August, 2018). (d4PDF, http://www.miroc-gcm.jp/~pub/d4PDF/; Mizuta et al., 2017). Another inter-national research collaboration project, called Half a degreeAdditional warming, Prognosis and Projected Impacts(HAPPI), has made a multi-model ensemble climate data-set under 1.5 K increase scenario (HAPPI, http://www.happimip.org/about/). This makes it possible to produce asufficient number of extreme flood events causing over-flow even in well-protected areas to help assess flood riskand quantify its uncertainty (e.g. Endo et al., 2017;Barcikowska et al., 2018).

This study explores the applicability of d4PDF rainfalldataset to flood risk curve development as a case study ofthe Yodo River basin (8,240 km2), Japan. First, the reprodu-cibility of the present climate data was examined in termsof basin-averaged rainfall and its spatiotemporal character-istics (duration, temporal variance, and spatial concentra-tion). Then, we demonstrated the development of flood riskcurves for 50 ensemble members in d4PDF using probabil-istic rainfall modelling. Finally, future ensemble flood riskcurves were estimated assuming that spatiotemporal pat-terns of rainfall are consistent with those in the historicalstorm events.

Received 27 August, 2018Accepted 12 November, 2018

Published online 21 December, 2018

© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s)and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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FLOOD RISK CURVE DEVELOPMENT USINGPROBABILISTIC RAINFALL MODELLING

This study applied a probabilistic rainfall model pro-posed by Tanaka et al. (2017) which derives a flood riskcurve by modelling the occurrence of a storm event on thefollowing assumptions:1) The number of storm events per year follows the Poisson

distribution with the occurence ratio of μ (averagedannual event number is μΔt, where Δt is one year).

2) When a storm event occurs, its spatiotemporal pattern isgenerated from N patterns that occurred in historicalstorm events and i-th pattern ξi (i = 1, 2, ..., N) occurswith probability of pi = 1/N.

3) According to the rainfall duration di of the selected pat-tern ξi (i = 1, 2, ..., N), its basin-averaged rainfall r isgenerated from a conditional cumulative distributionfunction (CDF) GR|D (r|di).

4) For each storm pattern, the resulting flood damage mincreases as its basin-averaged rainfall r increases.i-th storm pattern ξi (x, y, t) is defined as the normalized

time series of rainfall intensity at time t and location (x, y).Denoting ri (x, y, t) as the rainfall intensity at time t andlocation (x, y) of i-th storm event, ξi (x, y, t) satisfies the fol-lowing equations:

ξ i x, y, t =ri x, y, t

Ri, and (1)

Ri = 1A∫0

d∫Bri x, y, t dxdy dt, (2)

where Ri is the basin-averaged rainfall depth of the i-thrainfall event, d is the rainfall duration, A and B are the areaand the domain of the target basin, respectively. Accordingto the above definition, non-exceedance probability ofannual maximum flood damage mmax is derived as

FM maxmmax = exp −μΔt 1

N ∑i = 1

N

1 − GR D ri mmax d i (3)

where ri (mmax) is the basin-averaged rainfall causing flooddamage mmax for the storm pattern ξi (i = 1, 2, ..., N). TheAssumption 2) is justified in this study by using a largenumber of storm patterns (N = 1,371 events for 35 years(1980 to 2014) in this study). The most unique feature ofthis method is that a storm event is seperated into basin-averaged rainfall and its spatiotemporal pattern. By prepar-ing these components and calculating the relation betweenbasin-averaged rainfall and flood damage for each stormpattern ξi through rainfall-runoff-inundation and damageloss simulation, a flood risk curve is derived from Equation(3) without Monte Carlo simulation.

BIAS IDENTIFICATION IN d4PDF

d4PDF rainfalld4PDF stands for database for Policy Decision making

for Future climate change and provides large ensembleclimate data, simulated by MRI-AGCM 3.2H in 60 km res-olution and downscaled by Non-Hydrostatic Regional Cli-mate Model (NHRCM) in 20 km resolution for Japan area

(d4PDF, http://www.miroc-gcm.jp/~pub/d4PDF/). The sim-ulation was performed for 50 and 90 ensembles for presentand future climates, respectively. Future climate is assumedto follow a 4 K increase scenario, mostly corresponding toa Representative Concentration Pathway (RCP), RCP 8.5,which is a scenario of rising radiative forcing pathwayleading to 8.5 W/m2 in 2100. Each ensemble has the lengthof 60 years, given different initial condition (sea surfacetemperature, sea ice, concentration of green house gasses,sulfuric acid aerosol, ozone, and volcanic aerosol) andsmall perturbation to the time series of boundary sea sur-face temperature and sea ice. Recent studies clarified thatthis dataset well reproduces frequencies of annual maxi-mum basin-averaged rainfall for some particular riverbasins of Japan (e.g. Tachikawa et al., 2017; Hoshino andYamada, 2018). This study first examined the bias ofd4PDF rainfall data in terms of basin-averaged rainfall,rainfall duration, and temporal and spatial concentration ofrainfall; then demonstrated the application of this hugeensemble rainfall dataset to flood risk curve development.Comparison of observed and d4PDF rainfall data

The study area is shown in Figure 1. Red and green dotsshow the rain gauge stations and the center of d4PDF grids,respectively. Rainfall is observed for 35 years from 1980 to2014. Flood risk curves are developed for the confluencefloodplains (see red area in Figure 1) among the threemajor tributary rivers: Katsura, Uji, and Kizu Rivers, where

Figure 1. Map of the Yodo River basin, Japan. Red dots arethe rainfall gauge stations; green dots show d4PDF grid.The Katsura, Uji and Kizu River basins are shown as lightred, yellow, and light blue areas. The Lake Biwa basin(upstream of the Uji River basin shown as the orange area)was excluded from the catchment area because its outlet isclosed in flooding with the Seta Weir. Flood risk is assessedin the red floodplain area having several urban cities

ENSEMBLE FLOOD RISK CURVE DEVELOPMENT

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the lower parts of the historical Kyoto City area are located.This section examines the bias of d4PDF rainfall data at thepresent climate from the observed one in terms of basin-averaged rainfall and its spatiotemporal characteristics.Figure 2 shows the histograms of (a) basin-averaged eventrainfall, (b) rainfall duration and (c) temporal variancebetween d4PDF and observed rainfall. A storm event in thisstudy was defined as a rainfall event having dry periodsshorter than six hours, duration over 10 hours, and basin-averaged total rainfall over 10 mm. Temporal variance isdefined as

Ct = 1d∫0

dη t − η− 2dt (4)

where η(t) is the normalized basin-averaged rainfall at timet and η− is its mean value, and they are defined as

η t = 1A∫B

ξ x, y, t dxdy, η− = 1d∫0

dη t dt (5)

As in Figure 2, distributions of basin-average rainfall, rain-fall duration and temporal variance are in good agreementbetween d4PDF and the observed rainfall.

Spatial concentration of a storm pattern on a tributaryriver basin CΩ (denote its domain as Ω) is then defined bynormalized tributary rainfall on Ω as:

CΩ = 1A∫0

d∫Ωξ x, y, t dxdy dt 0 ≤ CΩ ≤ 1 (6)

Equation (6) indicates the ratio of basin-averaged rainfallover the tributary river basin to that over the whole riverbasin; thus, a rainfall pattern is evaluated as spatially con-centrated if CΩ is close to zero or 1. The histograms of thespatial concentration to the Katsura, Uji, and Kizu Riverbasins between the observed rainfall and d4PDF rainfall areshown in Figure 3. All the results indicated that normalizedtributary rainfall CΩ of d4PDF rainfall data are distributedaround 0.3 to 0.5, while that of the observerd rainfall con-

Figure 2. Histograms of (a) basin-averaged rainfall, (b)rainfall duration, and (c) temporal variance (see definitionin Equation (4)) between d4PDF (gray) and observed rain-fall (white) data of the Yodo River basin

tains more concentrated (0.1 to 0.3) patterns, indicating thatspatial distribution of d4PDF rainfal data is more spatiallyuniform. This bias often apprears in climate simulationresults in coarser spatial resolution and/or with hydrostaticassumption, and this is likely to result in smaller flooddamage than a storm event with actual spatial distribution.In summary, d4PDF data is comparable to observed datafor basin-averaged rainfall, rainfall duration and its tempo-ral variance while biased for spatial distribution.

ENSEMBLE FLOOD RISK CURVEDEVELOPMENT

Framework of deriving ensemble flood risk curveTo deal with the bias of d4PDF presented above, ensem-

ble flood risk curves for present and future climates werederived as follows (see Figure 4). The probabilistic rainfallmodelling in this study seperately defines the probability ofbasin-averaged rainfall (as the conditional CDF of rainfallon duration) and spatiotemporal rainfall patterns; therefore,d4PDF data was applied to estimate a conditional CDF ofbasin-averaged rainfall on duration GR|D (r|di). The rainfallpatterns were referred to ones of the observed 1,371 stormevents, and the basin rainfall-damage relation for each pat-tern ri (mmax) (i = 1, 2, ..., N) was simulated using rainfall-runoff and inundation models by inputting storm eventscomposed of the observed rainfall pattern and differentmagnitudes of basin rainfall. In addition to developingflood risk curves of each ensemble member, this study alsoapplied a set of all the ensemble members, i.e. 3,000 years(50 members for 60 years) at the present climate and 5,400years (90 members for 60 years) the future climate, to drawa single flood risk curve under each climatic condition ofd4PDF.

Figure 3. Histograms of normalized tributary rainfall (seedefinition in Equation (6)) on (a) Katsura River basin, (b)Uji River basin, and (c) Kizu River basin between d4PDF(gray) and observed rainfall (white) data

T. TANAKA ET AL.

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Flood-inundation and damage modellingFlood damage is simulated through a rainfall-runoff, a

flood-inundation, and a damage simulation module.Rainfall-runoff (Tanaka and Tachikawa, 2015) and flood-inundation models of the target basin were constructed byTanaka et al. (2017). Flood damage is calculated from thesimulated maximum inundation depth and fragility curvesfor house, households, office stocks and depreciable assets,according to a manual of flood economic survey by theMinistry of Land, Infrastructure, Transportation and Tour-ism (2005). By simulating flood damage for each stormevent pattern extending basin-averaged total rainfall from50 mm to 1,000 mm in 50 mm increments, the relationbetween basin-average rainfall and flood damage for eachrainfall pattern ri (m) (i = 1, 2, ..., 1,371) is obtained asFigure S1.Estimation of rainfall probability distribution ford4PDF rainfall data

The conditional probability distribution of basin-averaged rainfall on rainfall duration GR|D(r|d) is derived bythe following equation:

GR D r d = 1gD d ∫0

rgRD x, d dx (7)

where gD(d) is the CDF of rainfall duration d. The jointprobability density function gRD(r, d) is derived from itsCDF GRD(r, d), which is estimated using the normal copula(Nelsen, 2006):

GRD r, d = ΦΣ Φ−1 GR r , Φ−1 GD d , Σ =1 θθ 1

(8)

where Φ is the one-dimensional CDF of the standardizednormal distribution; ΦΣ is the two-dimensional CDF of thestandardized normal distribution with the correlationmatrix; θ is the only paramter of this copula function, esti-mated by the maximum likelihood method. The generalizedpareto distribution and the gamma distribution were appliedto the marginal CDF of basin-average rainfall gR(r) andrainfall duration gD(d), respectively. The conditional CDF

Figure 4. Flowchart to derive ensemble flood risk curvesusing d4PDF data. d4PDF rainfall data for S membersunder present (S = 50) and future (S = 90) climates werereflected on the conditional CDF of rainfall on duration;basin-rainfall causing damage for different rainfall patternswere referred to rainfall-damage relations of observed rain-fall patterns of basin-averaged rainfall on duration of 24 and 96 hours

for 50 (present climate experiment) and 90 (4 K increaseexperiment) ensemble members of d4PDF are shown inFigure 5. The CDF of the observed rainfall data is shown asthe black line as well. The observed CDF is within therange of ensemble CDFs of d4PDF at the present climate.Therefore, this study utilizes d4PDF rainfall data for theconditional CDF of rainfall on duration. Comparing presentand future climates, it is clearly indicated that basin-averaged rainfall at the same percentile is higher at thefuture climate for both durations.Ensemble present/future flood risk curves

Flood risk curves for 50 ensemble members of thepresent d4PDF, derived by estimating the conditional CDFof basin-averaged rainfall GR|D(r|d) of each member in theabove process and applying Equation (3), are shown inFigure 6(a). The flood risk curve for each ensemble mem-ber largely differs from each other, especially for largerflood damage. The thick blue line shows a flood risk curvefor which all the ensemble members (3,000 years) wereused for estimating a conditional CDF GR|D(r|d). Figure 6(b)shows the histogram of exceedance probability for theflood damage of 800 billion Japanese yen among ensemblemembers. Non-exceedance probability of the flood riskcurve using all the ensemble members (shown as the thickblue vertical line) is located at the mode of the histogramand its shape is heavily distorted, which indicated that theestimated flood damage corresponding to a long returnperiod may become extreme, depending on the tail charac-teristics of basin-averaged rainfall. By combining the prob-abilistic rainfall model and large ensemble climate data,physically-based ensemble flood risk curves are derivedwithout assuming any parametric error distribution. Thevalidity of the probabilistic rainfall model used above wasevaluated in Text S1, by comparing the derived flood riskcurve with a probability plot of 3,000 flood damagescaused by annual maximum 24-hour rainfall of all the3,000 members at the present climate in d4PDF.

Finally, ensemble flood risk curves for the present andfuture climates are shown in Figure 7. Similar to Figure 6,spatiotemporal rainfall patterns are applied from theobserved 1,371 patterns. Flood risk will clearly increaseand its uncertainty will become larger for the future cli-

Figure 5. Conditional CDF of basin-averaged rainfall onduration of (a) 24 hours and (b) 96 hours for the present(blue) and future (red) climate experiments (50 and 90ensemble members, respectively). The black line shows theobserved CDF

ENSEMBLE FLOOD RISK CURVE DEVELOPMENT

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mate. This increase is originated from the increased fre-quencies of heavy rainfall as shown in Figure 5. At thefuture climate (4 K increase scenario for d4PDF), theexceedance probability at flood damage zero correspondsto around a 50-year return period. The length of 60 yearsare not enough for plotting flood damage samples particu-

Figure 6. Results of the present experiment in d4PDF: (a)50 ensemble flood risk curves derived from Equation (3)with 60 years rainfall data of each member (thin lines) andwith the whole 3,000 years rainfall data (thick line); (b) his-togram of the exceedance probability of annual flood dam-age of 800 billion Japanese yen (the bar shows one for aflood risk curve derived with 3,000 years rainfall data)

Figure 7. Present (blue) and future (red) ensemble floodrisk curves. Thick lines show flood risk curve derived byEquation (3) using all the ensembles (50 members forpresent and 90 members for future climates)

larly in well-protected areas; thus, the proposed combinedapproach between large ensemble climate simulation dataand probabilistic rainfall modelling is effective to deriveensemble flood risk curves.

CONCLUSIONS

A flood risk curve is an effective criterion to quantify thepresent/future flood risk and its reduction by countermea-sures. As flooding rarely occurs especially in urbanizedareas, it is essential to show the uncertainty of flood riskcurve estimation. This study examined the applicability oflarge ensemble climate simulation data d4PDF to thepresent/future ensemble flood risk curve development as acase study of the Yodo River basin (8,240 km2). The bias ofd4PDF identified for spatial distribution of rainfall wasremoved by employing the observed patterns. In this com-bined approach, ensemble flood risk curves were success-fully demonstrated for the present and future climates.

The limitation of this approach is the assumption that thespatiotemporal rainfall pattern does not change in the futureclimate, which will be addressed by statistical/dynamicaldownscaling in future research.

ACKNOWLEDGMENTS

This research was supported by the grant-in-aid for sci-entific research (A) 17H01294 and the Integrated ResearchProgram for Advancing Climate Models (TOUGOU) of theMinistry of Education, Culture, Sports, Science and Tech-nology (MEXT), Japan. This study utilized the database forPolicy Decision making for Future climate change(d4PDF). Meteorological and hydrological data were pro-vided by the Yodo River Office, Kinki Regional Develop-ment Bureau, Ministry of Land, Infrastructure, Transporta-tion and Tourism.

SUPPLEMENTS

Text S1. Validation of probabilistic rainfall modelling usinghuge rainfall samples of d4PDF

Figure S1. Relation between basin-averaged rainfall andflood damage for 1,371 observed storm patterns

Figure S2. Histograms of normalized tributary rainfall (seedefinition in Equation (6)) on (a) Katsura River basin, (b)Uji River basin, and (c) Kizu River basin betweend4PDF (gray) and observed rainfall data without spa-tially concentrated patterns (white) (normalized tributaryrainfall is less than 0.35 for Katsura River, 0.33 to 0.5 forKizu River, and more than 0.35 for Uji River). A histo-gram of temporal variance between d4PDF and the selec-ted patterns is also shown in (d)

Figure S3. Validation of flood risk curve developmentusing Equation (3). Black dots show the Weibull plots ofannual maximum flood damage over 3,000 years causedby storm events of present climate with annual maximum24-hour rainfall, which is regarded as a reference floodrisk curve for validation. The black line shows thederived flood risk curve with all the members of present

T. TANAKA ET AL.

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climate, excluding spatially concentrated patterns to rep-resent rainfall pattern characteristics of d4PDF. The blueline shows the one with all the rainfall patterns (same asthe thick blue line in Figure 6)

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