*Article* **Estimation of Extreme Daily Rainfall Probabilities: A Case Study in Kyushu Region, Japan**

**Tadamichi Sato 1,\* and Yasuhiro Shuin <sup>2</sup>**


**\*** Correspondence: sato.tadamichi.343@s.kyushu-u.ac.jp

**Abstract:** Extreme rainfall causes floods and landslides, and so damages humans and socioeconomics; for instance, floods and landslides have been triggered by repeated torrential precipitation and have caused severe damage in the Kyushu region, Japan. Therefore, evaluating extreme rainfall in Kyushu is necessary to provide basic information for measures of rainfall-induced disasters. In this study, we estimated the probability of daily rainfall in Kyushu. The annual maximum values for daily rainfall at 23 long-record stations were normalized using return values at each station, corresponding to 2 and 10 years, and were combined by the station-year method. Additionally, the return period (RP) was calculated by fitting them to the generalized extreme value distribution. Based on the relationship between the normalized values of annual maximum daily rainfall and the RP, we obtained a regression equation to accurately estimate the RP up to 300 years by using data at given stations, considering outliers. In addition, we verified this equation using data from short-record stations where extreme rainfall events triggering floods and landslides were observed, and thereby elucidated that our method was consistent with previous techniques. Thus, this study develops strategies of measures for floods and landslides.

**Keywords:** extreme value analysis; daily rainfall; floods; rainfall-induced landslides; regional frequency analysis; station-year method; Kyushu region

**1. Introduction**

Extreme rainfall is an inducing factor for floods and landslides, which cause severe damage to humans and socioeconomics [1–3]. Hagon et al. [1] investigated six types of climate- and weather-related disasters (flood, storm, hydrometeorological landslide, wildfire, extreme temperature, and drought) worldwide between 1960 and 2020 and showed that floods affect more people globally than any other disaster. Ushiyama and Yokomaku [3] evaluated six types of disasters induced by heavy rainfall (storm surge, strong wind, flood, landslide, water accident in rivers excluding flood, and others) in Japan between 2004 and 2011. They showed that the number of deaths due to landslides was the highest and due to floods was the second highest [3]. In addition, increases in precipitation influenced by climate change are likely to increase floods and landslides [4–6].

Japan is within the East Asian monsoon region where torrential rainfall is frequent during the summer monsoon [7], and so floods and rainfall-induced landslides occur commonly and often cause damage [8]. In the Kyushu region, floods and landslides have occurred repeatedly [9–12] and may have been affected by climate change in recent years [13,14]; for example, an extreme rainfall event in the northern part of Kyushu in July 2017 induced severe damage from landslides, driftwood, and floods [12]. Thus, to mitigate the damage from floods and landslides, it is necessary to evaluate rainfall characteristics that may cause these disasters.

Rainfall frequency analysis statistically evaluates the magnitude of rainfall characteristics and provides basic information for the planning, design, and management of hydraulic

**Citation:** Sato, T.; Shuin, Y. Estimation of Extreme Daily Rainfall Probabilities: A Case Study in Kyushu Region, Japan. *Forests* **2023**, *14*, 147. https://doi.org/10.3390/ f14010147

Academic Editor: Filippo Giadrossich

Received: 9 December 2022 Revised: 10 January 2023 Accepted: 11 January 2023 Published: 12 January 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

structures (e.g., check dam and culvert) [15–18]. Similarly, rainfall frequency analysis is required in case studies of catastrophic disasters [15,19]. Hence, many researchers proposed theories for evaluating extreme rainfall (e.g., [20–23]). However, the rainfall record in the individual site is generally less than the return period (RP) required to design hydraulic structures [19]. Moreover, the record at one station may not include extreme rainfall because extreme rainfall is low frequency and has spatial heterogeneity [24]. Consequently, the results of rainfall frequency analysis may include uncertainty.

Regional frequency analysis (e.g., [25–28]) addresses these problems by using rainfall data observed at all stations in the region. The station-year method [15,16] combines the rescaled data from all stations into a single sample and fits a distribution by treating the combined sample as a single random sample [27]. Hosking and Wallis [27] mentioned that this method was rarely used because it is not appropriate to treat the rescaled data as a single random sample in many cases. Nevertheless, the station-year method has been used even in recent years (e.g., [29–31]) due to its simplicity.

In Japan, Suzuki and Kikuchihara [24] applied the station-year method to the annual maximum daily rainfall data from 137 stations and estimated the probability of extreme daily rainfall. They showed that daily rainfall for RP up to 1000 years was estimated successfully in the region using the return values of 2 and 10 years estimated by the plotting position formula (Hazen's formula) at a given station [24]. However, there was no comparison between the RPs estimated by their technique and by other processes at individual stations where extreme rainfall events occurred, that is, the usefulness and validation of their method were not sufficiently evaluated. In addition, rainfall data were not examined to the same standard because the periods for calculating the return values were different for each station.

The purpose of this study is to estimate regional daily rainfall in Kyushu by improving the previous method [24]. Additionally, we also examine the usefulness and limitations of our method in terms of development measures against floods and landslides. In this paper, we first describe the procedures for normalizing and combining daily rainfall at 23 stations in Kyushu, respectively (Section 2). Next, we examine the relationship between the RP and the normalized daily rainfall and propose an empirical dependence for estimating daily rainfall (Section 3.4). Then, we validate our method using daily rainfall data from shortrecord stations, including extreme events (Section 3.5). Lastly, we discuss the usefulness and limitations of our method (Section 3.6).

#### **2. Materials and Methods**
