*3.2. Spatial Distribution of 2- and 10-Year Values*

Table 3 shows the 2- and 10-year values and the SLSC for each station. As shown in Table 3, the SLSC values at all stations were below the JMA criterion (0.04) [39]; therefore, the observed rainfall and GEV distribution were a good fit.

Figures 3 and 4 show the spatial distribution of the 2- and 10-year values, respectively. As shown in Figures 3 and 4, the 2- and 10-year values tended to be lower in the northern plains and were larger in the south, especially in the east of the Kyushu Mountains. At the Fukuoka Regional Headquarters, the JMA [42] investigated the relationship between the spatial distribution of annual precipitation and the topographical conditions in northern Kyushu Island. They showed that the annual precipitation on the plain in the north was smaller than in the mountainous areas, the eastern hillsides near the Kyushu Mountains had more precipitation due to the influence of typhoons, and the annual precipitation at 32–33◦ N was higher than that of around 34◦ N because this area was affected by typhoons and Baiu precipitation [42]. Moreover, the JMA [39] examined the spatial distribution of

probable rainfall for 30, 50, 100, and 200 years using the annual maximum daily rainfall between 1901 and 2006 at 51 sites nationwide and showed that the probable rainfall in the Kyushu region (Fukuoka, Oita, Nagasaki, Kumamoto, Miyazaki, and Kagoshima) tended to be smaller in the north, but larger in the south. Thus, the 2- and 10-year values represented the regional rainfall characteristics and were good indexes for normalization.

**Figure 3.** Spatial distribution of the two-year value.

**Figure 4.** Spatial distribution of the 10-year value.


**Table 3.** The 2- and 10-year values at long-term record stations.

#### *3.3. Normalized Values of Annual Maximum Daily Rainfall at Each Station*

Figure 5 shows a boxplot for the normalized values of the annual maximum daily rainfall. As shown in Figure 5, there are wide variations in the data from all stations because the annual maximum value fluctuates by year. On the other hand, there are few differences in the maximum, minimum, interquartile range, and median between stations (Figure 5). As a result, the data exhibited the same probability distributions at all stations and were combined using the station-year method.

**Figure 5.** Boxplots for normalized values of annual maximum daily rainfall at 23 stations. Whiskers of the box show 25th (lower) and 75th (upper) percentile values. The gray line is the median value.

#### *3.4. Relationship between Normalized Daily Rainfall and the RP*

Figure 6 shows the quantile–quantile plot (Q-Q plot). As shown in Figure 6, the SLSC value was below the JMA criterion (0.04) [39]. Due to this, the standardized variate and the GEV distribution were a good fit.

**Figure 6.** Quantile–quantile plot.

Figure 7 shows the relationship between the RP and the normalized values of the annual maximum daily rainfall. As shown in Figure 7, the RP increased exponentially with the increase in the normalized values of the annual maximum daily rainfall when the RP was greater than 1.5 years. Furthermore, this relation indicates that an exponential approximation was obtained by the least-squares method.

$$RP = 2.156e^{1.525y\_T} \tag{12}$$

This equation accurately estimates the RP of daily rainfall up to 300 years (Figure 7) by using the 2- and 10-year values at the given stations in Kyushu.

**Figure 7.** Relationship between normalized values of annual maximum daily rainfall and the RP.

#### *3.5. Verification of Our Method at Short-Record Stations*

Figure 8 shows the RP of the annual maximum values for daily rainfall estimated by our method and previous methods. The black and white circle indicates the RP estimated by fitting the GEV distribution and by the plotting position formula, using data from each station, respectively. As shown in Figure 8, the RP estimated by our method was consistent with that calculated by other methods, excluding the AMeDAS Asakura. At the AMeDAS Asakura, the RP might be not consistent with the result of the plotting position formula because a record-breaking regional heavy rainfall event occurred in 2017 [14], and large values for daily rainfall occurred consecutively after periods when the GEV parameters were estimated (Figure 9).

**Figure 8.** Comparison of RP estimated by the proposed regression equation, RP estimated by the GEV, and RP by the empirical method (Cunnane [39]) at AMeDAS Izumi (**a**), Morotsuka (**b**), Aso-Otohime (**c**), and Asakura (**d**).

#### *3.6. Usefulness and Limitation of Our Method*

Our method estimates the RP of daily rainfall up to 300 years in Kyushu using the 2 and 10-year values at a given station (Figure 7). Extreme daily precipitation is a trigger for floods, landslides, and debris flows (e.g., [43,44]), and estimating them is required to plan, design, and manage hydraulic structures against these disasters [15–18,45,46]. For example, the Sabo Planning Division, Sabo Department, NILIM, MLIT [46] summarized the methods of the Sabo master plan for preventing damage triggered by debris flows, including driftwood, in Japan. They decided that daily rainfall, corresponding to an exceedance probability of 100 years, can be used to design the scale of measures against debris flow and driftwood [46]. Hence, our method may be applied to estimate the extreme rainfall for developing measures against floods and landslides.

**Figure 9.** Annual maximum daily rainfall at the AMeDAS Asakura between 1976 and 2020. Gray circles indicate annual maximum value. Bold line indicates Sen's slope (Hipel and McLeod 1996 [47]; Sen 1968 [48]) (*p* < 0.05).

Extreme daily precipitation is generally estimated by extrapolation because rainfall records at individual stations are often short (e.g., [19]). By contrast, our technique only uses the 2- and 10-year values, which can be calculated by interpolation at many points. Kikuchihara and Suzuki [24] estimated the probability of daily precipitation using a similar approach. Meanwhile, they have not verified their applicability and limitations. In the current study, an empirical dependence (Figure 7) was verified using short-record data (approximately 40 years) at four stations where extreme rainfall was observed. As a result, the RP estimated by our method was consistent with that obtained by other means at most sites (Figure 8a–c); however, it was difficult to apply in stations with an increasing trend in daily rainfall (Figures 8d and 9). This problem may be solved to modify periods for calculating 2- and 10-year values, considering rainfall trends at each station. Overall, we concluded that our method estimates the probability of daily rainfall up to 300 years using data at given stations and contributes to developing strategies for measures against floods and landslides.

### **4. Conclusions**

This study estimated the probability of daily rainfall in the Kyushu region, Japan. The data for the annual maximum values of daily rainfall were obtained from 23 longrecord stations and were normalized by quantiles, corresponding to the non-exceedance probability of 50% and 90%. The normalized rainfall at all stations was combined by the station-year method, and the RP was calculated using GEV parameters estimated by the L-moment method. Then, a regression equation linking the normalized values of annual maximum daily rainfall and the RP was obtained; this accurately estimates the RP for up to 300 years. In addition, this relation was verified using the data at shortrecord stations that observed extreme rainfall events triggering floods and landslides. As

a result, the probability of daily precipitation estimated by our approach was consistent with the results of previous techniques. In contrast, our method reduced the uncertainty of extrapolation by using parameters estimated by interpolation. Nevertheless, our method was difficult to apply at sites overserving an increasing trend in daily rainfall; therefore, trends in daily rainfall should be examined to use them. In conclusion, our technique estimates the RP up to 300 years for daily precipitation in Kyushu using data from the given stations, considering outliers. Hence, our findings help to develop measures for floods and landslides.

**Author Contributions:** Conceptualization: T.S. and Y.S.; data curation: T.S.; formal analysis: T.S.; funding acquisition: Y.S.; methodology: T.S.; validation: T.S.; visualization: T.S.; writing—original draft: T.S.; writing—review and editing: T.S. and Y.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Grant-in-Aid for Scientific Research, JSPS (project number 21H01581).

**Data Availability Statement:** Daily rainfall data in Kyushu are available from the JMA website "https://www.data.jma.go.jp/gmd/risk/obsdl/index.php (accessed on 26 November 2022)".

**Conflicts of Interest:** The authors declare no conflict of interest.
