# Estimating Typhoon-Induced Maximum Flood for Spillway Safety Assessment—Case Studies in Taiwan

^{*}

## Abstract

**:**

## 1. Introduction

^{2}in New Zealand. Collier and Hardaker [7] estimated PMP in the U.K. using a convective storm model considering solar heating, orographic uplift, and mesoscale convergence. Svensson and Rakhecha [8] applied the storm transposition method to estimate PMP for the dams in the Hongru River basin of China to examine the safety of the dam design in comparison with earlier studies. Al-Mamun and Hashim [9] applied the storm transposition method to generate PMP isohyetal maps for Peninsular Malaysia. Fernando and Wickramasuriya [10] deducted 24-h point PMP values for seven meteorological stations of Sri Lanka using the hydrometeorological method and compared them with those generated using Hershfield’s method. Lee et al. [11] applied a regional climate model by adjusting initial and boundary conditions to reconstruct the severe storm event for PMP estimation in Korea. Afzali-Gorouh et al. [12] estimated 24-h PMP using statistical and hydrometeorological approaches in the humid climate of northern Iran. They concluded that the physical approach could provide the most reliable estimates for PMP. Liao et al. [13] developed a new storm separation technique using rainfall quantiles with rare return periods estimated via regional L-moments analysis to calculate the orographic intensification factors for PMP estimation. Considering typhoon is the primary reason for significant flooding in Taiwan, Wang et al. [14] developed a typhoon rainstorm model for PMP estimation. The complicated typhoon structure was simplified in this modeling, and only circulation and orographic rainfalls were considered. Wang [15] performed the typhoon rainstorm model to estimate the PMP to accomplish the spillway design. Liu et al. [16] evaluated the combined effect of typhoon cyclones and southwestern airflow to estimate PMP.

## 2. Storm Transposition Method for PMP Estimation

#### 2.1. Structure of Storm Transposition Method

_{1A}represents the precipitable water from the mean sea level (about 1000 mb) extending up to 10 km (about 200 mb). Considering the precipitable water due to the elevation variation between the meteorological station (the observation site) and the target site, an adjustment ratio is used to account for the moisture change in the atmosphere columns at different locations. The moisture adjustment ratio can be calculated as follows:

_{1A}and W

_{2A}are the precipitable water from the sea level to the top of the atmosphere at the observation and the target sites, respectively. W

_{1B}is the precipitable water from the sea level (1000 mb) to the elevation of the observation site; W

_{2B}is the precipitable water from the sea level to the representative elevation of the target site. The representative elevation of the target site is usually selected as the lowest elevation of the projected watershed (Z

_{2}in Figure 1), i.e., the watershed outlet, to maximize the moisture inflow to the target site.

_{1}in Figure 1. Determining the barrier height should consider the moisture inflow direction that is the typhoon track. As shown in Figure 2, the direction of the moisture current moving to the observation site is divided into eight azimuths. For example, azimuth VIII indicates that the observation site would receive typhoon moisture from the southeast-east. In this study, 45 straight lines are radially extended from the location of the observation site in this azimuth. The maximum elevation along a specified radial line for a distance of 50 km is recorded. As shown in this example figure for a typhoon moving from the azimuth VIII to the observation site, the average value of the 45 maximum elevations is recognized as the barrier height. Hence, the historically extreme rainfall transposing from the observation site to the target site can be emulated by multiplying the rainfall depth at the observation site by the moisture adjustment ratio.

#### 2.2. Procedure for PMP Estimation Using Storm Transposition Method

- (1)
- Collect historical severe typhoon rainstorm records to develop rainfall depth-area-duration (DAD) curves for each typhoon event (as the examples shown in Figure 3);
- (2)
- Determine the inflow barrier height (Z
_{2}) according to the topography of the target watershed. The inflow barrier height can be assigned as the elevation of the watershed outlet (as shown in Figure 1). - (3)
- Choose a meteorological station near the target site and find the 24-h highest dewpoint temperature (T
_{H}_{2}) in the historical records. Using T_{H}_{2}and the elevation of the meteorological station (H_{2}) to determine the sea-level dewpoint temperature (T_{2}). - (4)
- According to the sea-level dewpoint temperature (T
_{2}) to estimate the precipitable water, W_{2A}, in the atmosphere column between 1000 mb and 200 mb, and the precipitable water, W_{2B}, between 1000 mb and the elevation at the watershed outlet (Z_{2}). Then the precipitable water in the target watershed can be estimated by W_{2}= W_{2A}− W_{2B}; - (5)
- Select a meteorological station near the ith typhoon rainstorm center. Determine the inflow barrier height (Z
_{1})_{i}of the event according to the typhoon track and the surrounding topography of the meteorological station; - (6)
- Check the 24-h highest dewpoint temperature (T
_{H}_{1})_{i}in the ith typhoon event. Using (T_{H}_{1})_{i}and the elevation of the meteorological station (H_{1}) to determine the sea-level dewpoint temperature (T_{1})_{i}. - (7)
- According to the sea-level dewpoint temperature (T
_{1})_{i}, calculate the precipitable water (W_{1A})_{i}in the atmosphere column between 1000 mb and 200 mb and calculate the precipitable water of the atmosphere column from 1000 mb to the barrier height (Z_{1})_{i}in the ith typhoon event. Then the precipitable water at the meteorological station is ${({W}_{1})}_{i}={({W}_{1A})}_{i}-{({W}_{1B})}_{i}$; - (8)
- The moisture adjustment ratio of the ith typhoon event is ${r}_{i}={W}_{2}/{({W}_{1})}_{i}$;
- (9)
- Determine the T-hr rainfall depth ${({P}_{T}{}^{\prime})}_{i}$ from the DAD curve in the ith typhoon event corresponding to the area of the target watershed, and then the T-hr precipitable water at the target site can be estimated by ${({P}_{T})}_{i}={r}_{i}\cdot {({P}_{T}{}^{\prime})}_{i}$;
- (10)
- Repeat Steps (5) to (9) to calculate the precipitable water of different durations in the ith typhoon event;
- (11)
- Repeat Steps (5) to (10) to calculate the precipitable water for all the typhoon events;
- (12)
- Plot the precipitable water of different durations for all the selected severe typhoon events and develop an envelope curve at the target site (as the examples shown in Figure 4).
- (13)
- According to the envelope curve, the increment of the cumulated rainfall depth at each hour can be calculated. A design hyetograph can be developed by applying the alternative block method [19] using the calculated increments.
- (14)
- The PMP rainstorm hyetograph can be substituted into a rainfall-runoff model for runoff routing for the PMF estimation at the target watershed.

## 3. Typhoon Rainstorm Model for PMP Estimation

#### 3.1. Circulation Rainfall

_{c}is the circulation rainfall rate per unit area. ${V}_{1}$ and ${V}_{2}$ are the radial wind speeds towards the typhoon center at ${r}_{1}$ and ${r}_{2}$, respectively. H

_{s}is the specific humidity; g is the acceleration of gravity; ${p}_{1}$ is the atmospheric pressure at the top of the atmosphere column; ${p}_{0}$ is the pressure at the bottom of the atmosphere column. Wang [15] recommended that ${p}_{1}$ and ${p}_{0}$ can be set as 850 mb and 1000 mb in Taiwan, respectively.

#### 3.2. Orographic Rainfall

_{o}is the total amount of orographic rainfall for all the layers; V

_{z}is the vertical upward speed of the moisture current in the windward direction; $\Delta Z$ is the thickness of the atmospheric layer; R

_{d}is the dry gas constant; T is the temperature; $d{e}_{z}/dz$ is the variation rate of the vapor pressure due to elevation change; ${e}_{s}$ is the saturated vapor pressure. Wang et al. [14] suggested that the troposphere can be separated into two layers. The first layer is from 850 mb to 700 mb, and the second layer is from 700 mb to 550 mb. According to Equation (4), the upward speed in the windward direction is one of the main factors affecting the orographic rainfall, which is controlled by the wind speed and the terrain slope lifting along the windward path. The relationship between the effective terrain slope and the windward direction can be developed based on the DEM analysis to facilitate the orographic rainfall calculation.

#### 3.3. Design Typhoon Rainstorm

^{2}measured in Typhoon Galley in 1952 and the historical maximum typhoon radius of 777 km in Typhoon Ida in 1954. An empirical equation derived by [21] is adopted to estimate the wind speed of the design typhoon, which can be expressed as:

_{design}is the wind speed of the design typhoon; V

_{7}and R

_{7}are the maximum force 7 wind speed and radius, respectively; d is the distance from the basin’s center to the typhoon center.

#### 3.4. Procedure for PMP Estimation Using Typhoon Rainstorm Method

- (1)
- Select historical typhoon events which had significant rainfall and caused large inflow in the target reservoir watershed;
- (2)
- Collect typhoon records from the nearby meteorological station, including typhoon track, wind direction, maximum force seven wind direction and speed, atmospheric pressure, temperature, rainfall depth, and radius of the typhoon circulation;
- (3)
- Analyze the effective slope for the target site according to the approach angle of the ith typhoon;
- (4)
- Estimate the total rainfall depth of the ith typhoon event by summing the circulation rainfall depth (Equation (3)) and the orographic rainfall depth (Equation (4));
- (5)
- Define a ratio to describe the deviation between the observed rainfall (P
_{obs}) and the estimated rainfall in the ith typhoon event as ${r}_{i}={\left[{P}_{obs}/({P}_{c}+{P}_{o})\right]}_{i}$; - (6)
- Repeat steps (3) to (5) to obtain the ratio for each typhoon event and to obtain an average adjustment ratio as $\overline{r}=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{r}_{i}}$;
- (7)
- Determine the design typhoon path based on the selected historical severe typhoon tracks (the bold red line, as shown in Figure 5);
- (8)
- Determine the parameters of the design typhoon and then calculate the circulation rainfall depth (P
_{c}) and orographic rainfall depth (P_{o}) along the design typhoon path at each time step; - (9)
- When the design typhoon circle covers the target watershed, use the average adjustment ratio ($\overline{r}$) to adjust the estimated rainfall depth at each time step;
- (10)
- The adjusted-rainfall depth in the progress of the design typhoon is regarded as the PMP of the target watershed.

## 4. Probable Maximum Flood Analysis

#### 4.1. Dimensionless Unit Hydrograph Model

_{lag}is the lag time; L is the maximum travel distance along the mainstream; L

_{ca}is the distance of the centroid of the watershed to the gauging site along the mainstream; S

_{c}is the average slope of the mainstream, and a and b are coefficients. The regression coefficients of a and b have been analyzed by the Taiwan Provincial Water Conservancy Bureau [23] for 23 major basins in Taiwan. Hence, the DUH can be applied to most watersheds in Taiwan for rainfall-runoff simulation.

#### 4.2. Kinematic-Wave-Based Geomorphologic IUH Model

_{j}with a mean value of ${T}_{{x}_{j}}$; $P(w)$ is the probability of a specified runoff path w; W is the runoff travel path space; and * denotes a convolution integral. Lee and Yen [22] suggested applying the kinematic-wave theory for runoff travel time estimations to alleviate the restriction of using empirical equations for runoff travel time estimation. The mean runoff travel time in the ith-order overland plane is [26]:

_{o}is the effective roughness coefficient for the overland planes; ${\overline{L}}_{{o}_{i}}$ is the mean overland-flow length of the ith-order subwatershed; ${\overline{S}}_{{o}_{i}}$ is the mean ith-order overland slope; i

_{e}is the rainfall excess intensity; and m is an exponent. The runoff travel time in the ith-order channel can be estimated by [22]:

_{i}is the width of the ith-order channel; ${\overline{L}}_{{c}_{i}}$ is the mean channel length of the ith-order subwatershed; n

_{c}is the roughness coefficient for channels; ${\overline{S}}_{{c}_{i}}$ is the mean ith-order channel slope; and ${h}_{c{o}_{i}}$ is the inflow depth of the ith-order channel due to water transported from upstream reaches, which can be determined from the watershed channel network structure. The merit of applying Equations (9) and (10) is that the hydrodynamic effect resulting from different rainfall intensities can be considered. Thus, the IUH can vary with different rainfall intensities and well reflect the overland and channel roughness conditions.

## 5. Case Studies of Shihmen and Feitsui Reservoir Watersheds

^{2}; the full water level area is about 8 km

^{2}, and the existing effective water storage capacity is $2.02\times {10}^{6}$ m

^{3}. There are six spillways on the dam crest controlled by radial gates; the total design discharge of the spillway and sluiceway is 13,800 m

^{3}/s. The Shihmen Reservoir was initially built mainly for irrigation and flood control, and now it also supplies public water in New Taipei City, Taoyuan City, and Hsinchu County.

^{2}; the full-water-level area is 1024 hectares, and the current water storage capacity is $3.35\times {10}^{6}$ m

^{3}. Radial gates control eight spillways on the dam crest; the total design discharge of the spillway and sluiceway is 9870 m

^{3}/s. It mainly provides domestic water in the Taipei area. Detailed geomorphologic factors of the Shihmen and Feitsui reservoir watersheds are shown in Table 1.

#### 5.1. Probable Maximum Precipitation Analysis

_{m}was set as 10.01 in Shihmen and 10.23 in Feitsui in performing Hershfiled’s method. The PMPs generated by the storm transposition method and typhoon rainstorm model for the Shihmen watershed are pretty close, and the Hershfield method provides the largest estimation. In the Feitsui watershed, the typhoon rainstorm model and the Hershfield method provide very close estimations; the PMP estimated by the storm transposition method gives the largest one. The large value of PMP by the storm transposition method may result from a low elevation at the outlet of the Feitsui watershed, which causes a high precipitable water in the atmosphere column. Previous studies showed that the ranges of PMP were 1375~2922 mm for Shihmen and 1476~1794 mm for Feitsui [27,28,29]. The difference in the PMP values between the previous studies and this research may be due to the selection of historical typhoon events and the meteorological stations. Moreover, the ways to determine the barrier height in the storm transposition method and to calculate the effective terrain slope in the typhoon rainstorm model have also affected the results.

#### 5.2. Spillway Safety Assessment Based on Probable Maximum Flood Analysis

_{ca}, and S

_{c}were substituted into Equation (7) to obtain the DUH (as shown in Figure 9). More detailed geomorphologic factors were derived using DEM and substituted into the KW-GIUH model. The applicability of using the KW-GIUH model for runoff simulation was verified through the flow records. In performing the model, the effective roughness coefficient for overland flow, No, and the channel-flow roughness coefficient, n

_{c}, were set as 6.0 and 0.05 in the Shihmen watershed, respectively; and they were 2.0 and 0.02 used in the Feitsui watershed. As shown in Equations (9) and (10), the runoff travel time depends on the rainfall intensity, which in turn changes the shape of the IUH (Equation (8)). A higher rainfall intensity will result in a larger peak discharge and a shorter time to peak discharge. On the contrary, a lower rainfall intensity will generate a hydrograph with a smaller peak discharge and a longer time to peak discharge. The IUHs corresponding to three different rainfall intensities (i = 10 mm/h, 50 mm/h, and 100 mm/h as examples) for the Shihmen and Feitsui watersheds are shown in Figure 10. As shown in Figure 10, the unit of the IUH is hr

^{−1}because the IUH indicates the probability distribution of the hydrological response of the watershed receiving an instantaneous unit rainfall input (as shown in Equation (8)). The IUHs are then convoluting the PMP hyetographs to generate flow hydrographs for the PMF estimations.

^{3}/s in the Shihmen watershed observed on 25 August 2004 and 4450 m

^{3}/s in the Feitsui watershed on 28 September 1974. As shown in Table 4, the discharges corresponding to the return periods of 100-yr and 1000-yr following the Pearson type-III distribution in the Shihmen watershed are 8678 m

^{3}/s and 11,988 m

^{3}/s, respectively. In the Feitsui watershed, 5586 m

^{3}/s and 7667 m

^{3}/s for the 100-yr and 1000-yr return periods follow the extreme value type-I distribution. The PMF values estimated by the three methods in the two watersheds are larger than the discharge corresponding to 1000-yr return periods.

^{3}/s for Shihmen in previous research [26,27,28], but no related report can be found for Feitsui. In the Shihmen Reservoir, all the PMF estimations are larger than the spillway design discharge (=14,100 m

^{3}/s) constructed in 1963. The same situation is found in the Feitsui Reservoir; the spillway design discharge (=9870 m

^{3}/s) is all smaller than the PMF estimations except for that generated by Hershfield’s method and routing with the DUH case.

^{3}/s. The tunnel is expected to reduce sediment deposition by about 640,000 m

^{3}per year, and the design flood discharge can increase from 14,100 m

^{3}/s to 14,700 m

^{3}/s. A second phase for sediment sluicing is an extra sluicing tunnel that will be constructed at Da-wan-ping, in which the design discharge is 1600 m

^{3}/s. Hence, the drainage capacity can be increased to 16,300 m

^{3}/s in the Shihmen Reservoir. However, no extra floodway is planned for the Feitsui Reservoir. Currently, the two reservoirs have established real-time flood warning systems to compensate for the insufficient spillway capacity. The early warning system can provide the incoming 6-h upstream inflow information for reservoir operation based on the next 6-h forecast rainfall provided by the Quantitative Precipitation Forecasts (QPF) system. It would allow timely action to prioritize releasing floodwater to mitigate the risk of excessive flooding in the spillways.

## 6. Conclusions

^{3}/s) and in Feitsui (Q = 9870 m

^{3}/s), respectively. Since the biggest metropolitan of Taiwan is located downstream of the reservoirs, extending the capacity of the spillways and/or constructing upstream flood bypass channels should be urgent to avoid catastrophic flooding downstream.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Historically severe typhoon tracks and the design of typhoon paths in the Shihmen and Feitsui watersheds.

Watershed | Shihmen | Feitsui |
---|---|---|

Watershed area (km^{2}) | 758.9 | 302.38 |

Mainstream length (km) | 101.61 | 53.29 |

Mainstream slope (m/m) | 0.02689 | 0.00969 |

Watershed mean slope (m/m) | 0.4161 | 0.3022 |

Watershed mean elevation (m) | 1408.1 | 477.5 |

Watershed outlet elevation (m) | 224.2 | 69.9 |

Dam height (m) | 133.1 | 122.5 |

Reservoir storage (106 m^{3}) | 2.02 | 3.35 |

Design flood discharge (m^{3}/s) | 13,800 | 9870 |

Typhoon Rainstorm | Observation Stations | Target Basin | |||
---|---|---|---|---|---|

Station | Elevation (m) | Dewpoint (°C) | Sea-Level Dewpoint (°C) | Barrier Height/ Wind Azimuth | |

1959/08 T.D. | Tainan | 40.8 | 24.7 | 24.9 | 20.4 m (NNW) |

1960/07 Shirley | Taichung | 84.0 | 24.1 | 24.4 | 223.3 m (NNW) |

1963/09 Gloria | Taipei | 6.3 | 25.2 | 25.2 | 396.7 m (ENE) |

1996/07 Herb | Alishan | 2413.4 | 15.6 | 25.2 | 1206.7 m (NNW) |

2001/09 Nari | Chiayi | 26.9 | 24.1 | 24.2 | 1335.4 m (NNE) |

2004/06 Mindulle | Taichung | 84.0 | 23.5 | 23.8 | 499.3 m (SSW) |

2004/08 Aere | Hsinchu | 26.9 | 23.6 | 23.7 | 29.4 m (WNW) |

2008/09 Sinlaku | Taichung | 84.0 | 25.0 | 25.3 | 817.7 m (NNE) |

2009/08 Morakot | Alishan | 2413.4 | 16.5 | 26.1 | 1206.7 m (NNW) |

Watershed | ${\mathit{R}}_{\mathit{T}}$ (mm) | PMP (mm) | |||
---|---|---|---|---|---|

100-yr | 1000-yr | Storm Transposition Method | Typhoon Rainstorm Model | Hershfield’s Method | |

Shihmen | 1140 | 1592 | 1359 | 1386 | 1934 |

Feitsui | 772 | 1036 | 1823 | 1526 | 1576 |

Watershed | Q_{T} (m^{3}/s) | PMF (m^{3}/s) | ||||||
---|---|---|---|---|---|---|---|---|

Storm Transposition Method | Typhoon Rainstorm Model | Hershfield’s Method | ||||||

100-yr | 1000-yr | KW-GIUH | DUH | KW-GIUH | DUH | KW-GIUH | DUH | |

Shihmen | 8678 | 11,988 | 17,107 | 16,582 | 25,023 | 23,235 | 26,962 | 24,163 |

Feitsui | 5586 | 7667 | 14,168 | 10,950 | 15,949 | 14,630 | 11,435 | 9350 |

^{3}/s (Shihmen), 9870 m

^{3}/s (Feitsui).

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**MDPI and ACS Style**

Lee, K.T.; Hsu, Y.-H.; Yang, J.Z.
Estimating Typhoon-Induced Maximum Flood for Spillway Safety Assessment—Case Studies in Taiwan. *Water* **2023**, *15*, 3040.
https://doi.org/10.3390/w15173040

**AMA Style**

Lee KT, Hsu Y-H, Yang JZ.
Estimating Typhoon-Induced Maximum Flood for Spillway Safety Assessment—Case Studies in Taiwan. *Water*. 2023; 15(17):3040.
https://doi.org/10.3390/w15173040

**Chicago/Turabian Style**

Lee, Kwan Tun, Yu-Han Hsu, and Jing Zong Yang.
2023. "Estimating Typhoon-Induced Maximum Flood for Spillway Safety Assessment—Case Studies in Taiwan" *Water* 15, no. 17: 3040.
https://doi.org/10.3390/w15173040