# Assessing Spatial Flood Risk from Multiple Flood Sources in a Small River Basin: A Method Based on Multivariate Design Rainfall

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Design Rainfall and Flood Risk from Multiple Sources

#### 2.2. Copula Method for Rainfall Design

^{2}→(0,1) such that all is a copula [32]. This method separates joint distribution into a copula function and marginal distributions, and it has the advantage that the selection of an appropriate model for the dependence between varieties, represented by the copula, can then proceed independently of the choice of marginal distributions.

_{1},t

_{2}…t

_{n}, and r

_{t1}, r

_{t2}… r

_{tn}are the rainfall amounts for critical rainfall durations. Then, a joint distribution can be derived as follows:

_{1}, F

_{2},… F

_{n}are marginal distributions. Under the assumption of spatially identical rainfall, this joint distribution includes critical rainfall requirement information for multiple flooding sources. The steps for using the copula method are as follows:

_{1}, t

_{2}…t

_{n}are identified. The empirical methods discussed in Section 2.1 can be used.

#### 2.3. Spatial Flood Risk Assessment

## 3. Case Study

#### 3.1. Study Area

^{2}, and it includes three cities. The average annual temperature in the study area is 16 °C, and the average annual rainfall is about 1200 mm. Rainfall data were derived from four rain gauging stations with 49 years of hourly rainfall records.

#### 3.2. Rainfall Analysis and Rainfall Design

#### 3.2.1. Identification of Critical Rainfall Durations and Rainfall Data Pre-Processing

#### 3.2.2. Estimating the Correlation of Rainfall Amounts with Different Critical Rainfall Durations

#### 3.2.3. Fitting of Marginal Distributions

#### 3.2.4. Construction of Joint Distributions and Generation of Correlated Critical Rainfall

#### 3.2.5. Design and Generation of Rainfall Based on Joint Distribution

_{tp}represent the amount of peak rainfall, and r

_{1h}, r

_{2h}, r

_{3h}represent the 1 h, 2 h, and 3 h rainfall amounts simulated from the joint distribution of rainfall events, respectively. The rainfall amount for the critical duration of three hours can be determined as follows: r

_{tp}= r

_{1h}; r

_{tp-1}= r

_{2h}-r

_{1h}; r

_{tp+1}= r

_{3h}-r

_{2h}. Since the generated rainfall data contain not only information about 1 h, 2 h, and 3 h rainfall amounts, but also information about the joint probability of these events, the design rainfall event will share the same probability. In addition to the critical rainfall durations, the remaining duration of a rainfall event will also affect runoff but contribute less to the flood peak. Therefore, the remaining duration can simply be added using the statistical average of historical rainfall events. Using this strategy, rainfall was designed for the case study area. Because a joint probability function was adopted, more than one rainfall value can be expected for a certain return period. In Figure 8, five rainfall events with 50-year and 100-year return periods are shown. The generated rainfall can be interpreted as the design rainfall, which includes contributions from multiple flood sources such as, in our study, river flooding from the Ushitaki, Matsuo, and Makio Rivers, and local inundation from urban drainage. It is obvious that even for the same return period, variations in generated rainfall reflect different combinations of floods from different sources.

#### 3.3. Spatial Flood Risk Assessment

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Flood risk from flooding by two rivers and inundation, and the corresponding solution procedure.

**Figure 5.**3-D scatter plot of pseudo data and fitted copula densities for 1-h/2-h, 1-h/3-h, and conditional 2-h/3-h rainfall amounts.

**Figure 7.**Plots of 10,000 random rainfall values generated by the copula model. Black dots are random values; red dots represent observed real rainfall data.

**Figure 8.**Five rainfall cases for 50-year and 100-year return periods from copula-based rainfall analysis. The y-axis is rainfall amount, the x-axis is rainfall duration; 9-h rainfall events are designed.

**Figure 9.**Example of flood simulations (maximum inundation depth) under different rainfall scenarios using a 100-year return period.

**Figure 10.**Risk curve in terms of economic loss. (The exchange rate between U.S. dollar and Japanese yen is about 1:111.).

**Figure 11.**Spatial flood risk map for expected economic loss and three examples for risk curves at different locations. (The exchange rate between U.S. dollar and Japanese yen is about 1:111).

**Figure 12.**Comparison of 1-h, 2-h, and 3-h rainfall calculated using the copula-based design rainfalls and the IDF-curve-based ones for 1/50 and 1/20 events.

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**MDPI and ACS Style**

Jiang, X.; Yang, L.; Tatano, H.
Assessing Spatial Flood Risk from Multiple Flood Sources in a Small River Basin: A Method Based on Multivariate Design Rainfall. *Water* **2019**, *11*, 1031.
https://doi.org/10.3390/w11051031

**AMA Style**

Jiang X, Yang L, Tatano H.
Assessing Spatial Flood Risk from Multiple Flood Sources in a Small River Basin: A Method Based on Multivariate Design Rainfall. *Water*. 2019; 11(5):1031.
https://doi.org/10.3390/w11051031

**Chicago/Turabian Style**

Jiang, Xinyu, Lijiao Yang, and Hirokazu Tatano.
2019. "Assessing Spatial Flood Risk from Multiple Flood Sources in a Small River Basin: A Method Based on Multivariate Design Rainfall" *Water* 11, no. 5: 1031.
https://doi.org/10.3390/w11051031