To manage flood disaster with an exceeding designed level, flood risk
control based on appropriate risk assessment is essential. To make an
integrated economic risk assessment by flood disaster, a flood risk curve,
which is a relation between flood inundation damage and its exceedance
probability, plays an important role. This research purposes a method to
develop a flood risk curve by utilizing a probability distribution function
of annual maximum rainfall through rainfall-runoff and inundation
simulations so that risk assessment can consider climate and socio-economic
changes. Among a variety of uncertainties, the method proposed in this study
considered spatio-temporal rainfall distributions that have high uncertainty
for damage estimation. The method was applied to the Yura-gawa river basin
(1882 km

A flood risk curve, a relation between the annual economic flood inundation damage and its exceedance probability, is essential information to establish an effective river planning to cope with flood disasters. In most basins, however, it is difficult to estimate a PDF (probability distribution function) of economic loss from disaster record because of few number of flood disaster data; thus estimating a PDF of the economic damage through rainfall-runoff and inundation simulations is an effective method.

One of the major approaches to estimate a PDF of the economic damage due to flood is to give a number of generated hydrographs on the main stream from its probability function and obtain a PDF of inundation depth or the economic damage through inundation simulations (Apel et al., 2006; Thieken et al., 2014). Another approach proposed here is to estimate a PDF of an annual maximum economic damage from that of annual maximum rainfall information through a rainfall-runoff model and an inundation model. A flood risk curve development from a PDF of rainfall has two advantages; one is to automatically consider the correlation among flood discharges on several tributaries through rainfall-runoff simulations, and the other is to consider the non-stationarity of rainfall characteristics due to climate change by giving a PDF of rainfall under climate change scenarios. For comprehensive flood risk assessment that considers climate change impact, this research presents a method to estimate a flood risk curve from a PDF of annual maximum rainfall information.

In converting a rainfall PDF into a PDF of the economic damage, it is important to consider the uncertainty of spatio-temporal rainfall distributions (hereafter referred to as rainfall patterns). This study treats the uncertainty of the rainfall patterns by incorporating typical rainfall patterns which happened in a study basin. The developed method was applied to the Yura-gawa river basin; and annual economic benefit of an existing dam in the basin was quantified by comparing flood risk curves with/without the dam operation.

In some river basins in Japan, a design flood is determined from a PDF of
the annual maximum rainfall through a rainfall-runoff model using the
Synthesis Probability Method (SPM). The method was developed to consider the
uncertainty of spatio-temporal rainfall distributions. The mathematical
background of this method is explained by Shiiba and Tachikawa (2013). In
this study, the SPM is extended to estimate a flood risk curve. The
assumptions to estimate a flood risk curve are as follows:

A total amount of rainfall in a pre-determined rainfall duration is independent from rainfall patterns.

For a fixed rainfall pattern, the economic damage monotonically increases as the total amount of rainfall increases.

The total number of the rainfall pattern

The rainfall events that cause the economic damage follow the Poisson
process with the occurrence ratio

The concrete procedure to obtain

to determine the rainfall duration

to obtain the distribution of the annual maximum basin rainfall

to prepare the typical

to obtain the normalized spatio-temporal rainfall pattern

for each rainfall pattern, to prepare a rainfall event with total
rainfall amount

to obtain a spatial distribution of runoff and river dishcarges over a
study basin caused by the

to obtain a spatial distribution of inundation depth using an inundation model with the runoff/discharges distribution as the boundary condition;

the economic damage is estimated using an estimated inundation depth and a spatial distribution of the asset of the study basin; and

to obtain a relation between total rainfall amount

To realize the above flood risk curve development, a rainfall-runoff model and an inundation model is necessary to estimate spatial distributions of flood inundation depth. This study applies a distributed rainfall-runoff model 1K-DHM (Tachikawa and Tanaka, 2013) and an inundation model coupled with the rainfall-runoff model developed by Tanaka et al. (2014).

1K-DHM is a distributed rainfall-runoff model based on a kinematic wave flow
approximation that considers surface-subsurface flow. The elevation and the
flow direction was determined using HydroSHEDS (Lehner et al., 2006), the
grid-based 1 km resolution DEM, and the model domain is composed of
river-channel cells and slope-runoff cells. A kinematic wave flow equation
is used in river-channel cells. Flow from slope-runoff for each cell is
modeled by the following two equations, which consider both saturated and
unsaturated subsurface flow components (Tanaka et al., 2014):

Relations between the non-exceedance probability of the annual
maximum rainfall and that of the annual maximum economic damage (the curve
bellow the horizontal axis is the CDF of the annual maximum rainfall

The inundation model coupled with the rainfall-runoff model (IMCR) was
developed to seamlessly give the simulated river discharge and lateral
inflow by 1K-DHM to the inundation model. The IMCR realizes the 1-D river
routing by the following local inertial equations (Bates et al., 2010);

For applying the method to develop a flood risk curve at the Yura-gawa river basin, the performance of 1K-DHM and the IMCR in the basin was validated. The Yura-gawa river basin has been suffered from a number of flood disasters such as the typhoon No. 13 in 1953, the typhoon No. 23 in 2004 and the typhoon 18 in 2013. This study focused on the typhoon 18, which damaged 3855 houses over the basin (Kinki Regional Development Bureau, MLIT, 2013).

The parameters of 1K-DHM were calibrated by the SCE-UA algorithm (Duan et al., 1994) to optimize the hydrograph at the Ayabe station (see Fig. 2) for the typhoon No. 23, 2004, which caused the similar damage to the typhoon No. 18, 2013. In the calibration, rain gauge data at 17 stations and dishcarge at the Ayabe station shown in Fig. 2 was used. To consider the effect of dam control by the Ohno dam, the actual dam release flow was given to the model as an upper boundary condition of the rainfall-runoff model. The calibrated parameters are shown in Table 1.

Using the calibrated parameter values, the river discharge for the typhoon No. 18, 2013 was simulated for the model validation. Since rainfall data at some gauges (the black circles in Fig. 2) are deficient, other red gauges were used. Dam release from the Ohno dam was also given to the validation simulation. Figure 3 shows the observed and simulated hydrographs at the Ayabe station, which indicates that the simulated discharge well reproduced the observed one.

Inundation was simulated by the IMCR in 100 m spatial resolution for the
shaded area in Fig. 2, inputting the simulated runoff to river channel and
river discharge to the boundary of the target area. The Manning's roughness
coefficients for the river channel and the flood plain areas were set to
0.045 and 0.05, respectively. As the boundary condition at the downstream
end, the observed water level was given to the IMCR. The estimated spatial
distribution of the maximum inundation depth is shown in Fig. 4. In Fig. 4,
the thick black line represents the observed inundation area reported by
Kinki Regional Development Bureau, MLIT (2013), and the triangle marks show
the locations of the bank breaches. At the bank breach locations, the
heights of the river dikes were set to 0 m after overflow starts. We can see
that the observed inundation area is well overlapped with the simulated
inundation area. The simulated total inundation area is 20.21 km

The calibrated parameters of 1K-DHM.

The economic damage by the inundation caused by the typhoon No. 18, 2013 was estimated for each 100 m cell using the spatial distribution of the maximum water depth calculated by the IMCR. To estimate economic loss for each grid cell, the relation between the economic damage and the maximum water depth proposed by MLIT (2005) was used. It was developed for each land use type based on the field surveys for the several past flood disasters in Japan. The economic damage by the typhoon No. 18 for each item is shown in Table 2. The total economic damage was estimated at 58.2 billion Japanese yen, which is similar to the damage caused by the typhoon No. 23, 2004 officially reported as 61.7 billion Japanese yen by MLIT (2004).

The Yura-gawa river basin and the locations of discharge gauges and rainfall gauges (inundation is simulated in the shaded area; rectangulars are discharge gauges; small circles are rain gauges; and a triangle is the Ohno dam).

Observed and simulated discharge hydrographs at the Ayabe stations for the typhoon No. 18, 2013 (the solid line is the observed hydrograph; the broken line is the simulated one).

To develop a flood risk curve using Eq. (8), the CDF of the annual maximum
rainfall

Flood control benefit by the Ohno dam in the Yura-gawa river basin was
assessed by developing flood risk curves with/without the dam control.
Figure 5 shows the relation between the annual maximum basin rainfall in two
days and the economic damage for each rainfall pattern

The estimated economic damage of flood inundation caused by the typhoon 18, 2013.

The investigated inundation area (the area surrounded by thick black lines) and the spatial distribution of the maximum water depth for the typhoon No. 18, 2013.

The relation between the basin average total amount of rainfall in two days and the economic damage for five rainfall patterns (solid lines are the relations without dam control; broken lines are the relations with dam control; difference of thickness of the relations indicates difference of rainfall patterns; and the horizontal axis bellow the figure shows the corresponding non-exceedance probability of basin rainfall).

Flood risk curves in the Yura-gawa river basin with/without flood control by the Ohno dam (The solid line is a flood risk curve with dam control; and the broken line is a flood risk curve without dam control).

Finally, flood risk curves with/without dam control developed by Eq. (8) is
shown in Fig. 6. They are calculated by taking a weighted geometric mean of
the non-exceedance probability of the annual maximum rainfall

This research developed a method to estimate a flood risk curve from a PDF of the annual maximum rainfall utilizing a rainfall-runoff and inundation model considering the variety of rainfall patterns. The proposed method was successfully applied to the Yura-gawa river basin, Japan. As an example of utilization of a flood risk curve, the economic benefit of the existing dam in the basin was assessed by comparing flood risk curves with/without the dam control.

The future studies will focus on expanding the proposed method to consider a probabilistic bank break component and to use rainfall data set under various climate change scenarios.

We would like to express our gratitude to Kyoto Prefecture for providing the Ohno dam operation data.