# Assessing Future Rainfall Intensity–Duration–Frequency Characteristics across Taiwan Using the k-Nearest Neighbor Method

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of the Study Cases

_{short}), 24-h and 25-year events (event

_{long}), 1-h events (D1), 24-h events (D24), 2-year events (T2), and 25-year events (T25), as shown in the contingency table (Table 1). It should be noted that because the observed data has the periods of record of 20 years, only events with the 25-year return period and associated results are assessed in this study.

#### 2.2. Framework of Rainfall Projection

#### 2.3. Algorithm of the k-NN Method Used in the Study

_{p}between the observed and the generated consecutive n-day rainfall event is estimated using Equation (1). To determine the weights w

_{t}in Equation (1), while optimization methods can be used, this study adopts the reciprocal of the variance as the weights to estimate the distance with neighbors, as shown in Equation (2) [36]. The reciprocals of the variances are used as weights to allow more flexibility in selecting the most-alike historical events for the originally existed large variation of the n

_{th}day rainfall amount:

_{t}is the generated daily rainfall on the t

_{th}day of the consecutive n-day event; o

_{t,p}is the observed daily rainfall on the t

_{th}day of the p

_{th}historical event in the pool with a total of f events; w

_{t}are the weights. Notice that the events in the pool are extracted from the defined window period. In this study, the month of the generated daily rainfall is chosen as the window period.

_{p}with rank one (r = 1), is selected. The hourly rainfall h

_{q,t′}is determined by Equation (3), and the ratio of rainfall amount for each hour ($\frac{{h}_{q,t}}{{o}_{t,r=1}}$) is derived from the selected observed event:

_{q,t’}represents hourly rainfall simulated for the q

_{th}hour on the t

_{th}day; o

_{t,r = 1}is the total rainfall amount of the t

_{th}day of the observations; and h

_{q,t}is the hourly rainfall for the q

_{th}hour on the t

_{th}day from the observations. Here the observations refer to the nearest observed neighboring n-day rainfall event.

_{max}in the same month. Afterwards, either one of the following two possible conditions is determined for the generation of future hourly rainfall, as shown in Equations (4) and (5):

- (1)
- If D
_{max}is larger than half of m, then the m-day is divided into two sets, namely a and b days ($m=a+b$). All possible combinations of a and b are tried. After the m-day event is divided into new combinations, the distance d_{p}(Equation (1)) of historical and simulated a- and b-day events are summed up as Equation (4). The optimal a and b are determined when the minimal distance is obtained:$${d}_{p}{}^{\prime}={d}_{p,a}+{d}_{p,b},\forall \left(a,b\right)\mathrm{if}{D}_{max}\ge \frac{m}{2}$$ - (2)
- If D
_{max}is smaller than half of m, D_{max}is used to divide the m days as many times as possible. The m-day event becomes D_{max}+ … + D_{max}+ c, where c equals to m minus a multiple of D_{max}. With the new combination, the distance d_{p}(Equation (1)) of historical and simulated D_{max}-day events are calculated as well as c-day events (m = ${D}_{max}+\dots +{D}_{max}+c$). When the minimal d_{p}is obtained using Equation (5), the most alike multiple D_{max}-day and c-day events are determined and used as references to generate future hourly rainfall for the m-day event:$${d}_{p}{}^{\u2033}={d}_{p,{D}_{max}}+\cdots +{d}_{p,{D}_{max}}+{d}_{p,c},\mathrm{if}{D}_{max}\frac{m}{2}$$

## 3. Results and Discussion

_{short}) and 24-h events in the 25-yr return period (events

_{long}), to examine the variation among locations and scenarios. Next, the influences of return periods and durations on rainfall intensities are analyzed.

#### 3.1. Rainfall Variation among Locations and Scenarios

_{short}for the near- and far-future are depicted in Figure 4 and Figure 5, respectively. Both the rainfall intensities and the percentage changes compared to the observation in 1986–2005 are illustrated. The projected rainfall intensity of events

_{short}is about 30–65 mm/h in the near future (Figure 4), while a slight increase is found for the far future (Figure 5). It is found that the variation of rainfall intensities among the locations is larger than that among the scenarios. After converting to change percentages, the variations among the scenarios become equally prominent, indicating deviation of the future rainfall intensities from the past. The projected rainfall-intensity changes of −10~20% were found in the near and far future. Comparing the results of RCP 2.6 and 8.5, opposite signs which show obvious difference between the two scenarios are found at Kaohsiung (− to +), Hengchun (− to +), and Chenggong (+ to −). Changes are intensified under RCP 8.5 for most other stations, except for Zhuzihu, Taipei, Hsinchu and Alishan, although no obvious difference are found for the two scenarios.

_{short}. It is because the ventilation is good, and the height exceeds the zone where moisture is condensed. On the other hand, the altitudes and weather conditions of the stations are more even in southern and eastern Taiwan.

_{long}for the near and far-future are depicted in Figure 7 and Figure 8, respectively.

_{long}compared to events

_{short}. For future studies, events with longer return periods than 25-yr return periods should better be analyzed using record of the observed data longer than 20 years to disclose more information.

#### 3.2. Association between Return Periods/Durations and Rainfall Intensities

## 4. Conclusions

_{short}) and with 24-h duration and 25-yr return period (events

_{long}), were discussed. For events

_{short}, the projected rainfall-intensity changes of 10~20% were found under future scenarios compared to the observations. Considering the existing flood-warning thresholds, precautions of flooding are required for the frequently occurred short-duration storm events in Keelung, Danshui, Kaohsiung, and Suao, although rainfall intensities are projected to decrease from the near to the far future.

_{long}, the majority of the projected rainfall intensities exceeded the flood-warning threshold despite of the decreasing intensities compared to observation, which should be of concern. Compared to events

_{short}, the decreases in the rainfall intensities of events

_{long}are more obvious from near to far future. Among all stations, the land-subsidence regions in central and south, the landslide-sensitive mountainous region in north and central, the pluvial- and fluvial-flood prone region in north, and the regions with vulnerable infrastructures in east should be especially aware of possible long-duration extreme events. It should be noted that because the observed data has a period of record of 20 years, only the 25-yr return period are presented here; however, events with longer return periods than 25-yr return periods should better be analyzed to disclose more information of extreme events the cities face in the future.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Location of 21 weather stations across Taiwan for the northern, central, southern and eastern regions.

**Figure 2.**Relation between the IDF values in the historical period (1986–2005) used to verify the reliability of the K-NN method, derived based on (i) the observed daily rainfall, (ii) the projected daily rainfall, and (iii) the observed hourly rainfall.

**Figure 3.**Intensity-Duration-Frequency (IDF) curves from the k-NN method using both the simulated and observed daily precipitation of Tainan in the historical period (1986–2005).

**Figure 4.**Rainfall intensities (

**top**) and percentage changes (

**bottom**) of events with 1-h duration and 2-yr return period (events

_{short}) for the near future (2021–2040) under RCP 2.6 and RCP 8.5 for five GCMs (CCSM4, CESM1-CAM5, GISS-E2-R, HadGEM2-AO, MIROC5).

**Figure 5.**Rainfall intensities (

**top**) and percentage of changes (

**bottom**) of events with one-hour duration and two-year return period (events

_{short}) for far future (2081–2100) under RCP2.6 and RCP 8.5 for five GCMs (CCSM4, CESM1-CAM5, GISS-E2-R, HadGEM2-AO, MIROC5).

**Figure 6.**Difference of rainfall intensities (

**top**) and percentage changes (

**bottom**) of events

_{short}between the near future (2021–2040) and far future (2081–2100).

**Figure 7.**Rainfall intensities (

**top**) and percentage changes (

**bottom**) of events with 24-h duration and 25-yr return period (events

_{long}) for the near future (2021–2040) under RCP 2.6 and RCP 8.5 for five GCMs (CCSM4, CESM1-CAM5, GISS-E2-R, HadGEM2-AO, MIROC5).

**Figure 8.**Rainfall intensities (

**top**) and percentage of changes (

**bottom**) of events with twenty-four-hour du-ration and twenty-five-year return period (events

_{long}) for far future (2081–2100) under RCP2.6 and RCP 8.5 for five GCMs (CCSM4, CESM1-CAM5, GISS-E2-R, HadGEM2-AO, MIROC5).

**Figure 9.**Difference of rainfall intensities (

**top**) and percentage changes (

**bottom**) of events

_{long}between the near future (2021–2040) and far future (2081–2100).

**Figure 10.**Rainfall intensities of the 1-h events (D1) in different return periods for the four regions and four scenarios.

**Figure 11.**Rainfall intensities of events with twenty-four-hour duration (D24) changing with the return periods for the four regions and four scenarios.

**Figure 12.**Rainfall intensities of 2-yr events (T2) of different durations for the four regions and four scenarios.

**Figure 13.**Rainfall intensities of events with twenty-five-year return period (T25) changing with the durations for the four regions and four scenarios.

**Table 1.**Contingency table of the six types of events analyzed in this study (event

_{short}, event

_{long}, D1, D24, T2 and T25).

D1 | D24 | ||
---|---|---|---|

D (h) T (yr) | 1 | 24 | |

T2 | 2 | event_{short} | |

T25 | 25 | event_{long} |

Steps | Method Adopted in This Study |
---|---|

Generate daily rainfall | Richardson-type weather generator [14] |

Generate hourly rainfall | k-Nearest Neighbors (k-NN) method [36] |

Obtain rainfall intensities for events of various return periods | Weibull’s formula [54] |

Correct bias of historical simulation and future projection | Quantile-mapping-based bias correction [46] |

Parameter | Configuration |
---|---|

Window period of the neighbors | One month |

Weight for estimating the neighboring distance | Reciprocal of the variance |

Neighbor-selecting criterion | The nearest neighbors |

Number of realizations | Ten realizations of 20-yr data |

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**MDPI and ACS Style**

Chen, P.-Y.; Tung, C.-P.; Tsao, J.-H.; Chen, C.-J.
Assessing Future Rainfall Intensity–Duration–Frequency Characteristics across Taiwan Using the *k*-Nearest Neighbor Method. *Water* **2021**, *13*, 1521.
https://doi.org/10.3390/w13111521

**AMA Style**

Chen P-Y, Tung C-P, Tsao J-H, Chen C-J.
Assessing Future Rainfall Intensity–Duration–Frequency Characteristics across Taiwan Using the *k*-Nearest Neighbor Method. *Water*. 2021; 13(11):1521.
https://doi.org/10.3390/w13111521

**Chicago/Turabian Style**

Chen, Pei-Yuan, Ching-Pin Tung, Jung-Hsuan Tsao, and Chia-Jeng Chen.
2021. "Assessing Future Rainfall Intensity–Duration–Frequency Characteristics across Taiwan Using the *k*-Nearest Neighbor Method" *Water* 13, no. 11: 1521.
https://doi.org/10.3390/w13111521