*2.1. Analysis Procedure*

The method of this study is divided into four steps. First, the data for the annual maximum daily rainfall in the Kyushu region are collected. Next, the spatial correlation of rainfall data between the two stations is investigated. Following this, the annual maximum daily rainfall is normalized using quantiles, corresponding to the non-exceedance probability of 50% and 90% of the generalized extreme value (GEV) distribution. The second and third steps are performed to apply the station-year method. Then, the normalized rainfall data are combined and converted to the RP.

#### *2.2. Data Collection*

The annual maximum value for daily rainfall at 23 meteorological observatories operated by the Japan Meteorological Agency (JMA) in Kyushu until 2020 was used to estimate the RP of daily rainfall because long-term records are available (Figure 1 and Table 1). The longest record is 143 years at Nagasaki, the shortest is 59 years at Fukue, and the average for the 23 sites is 98 years. The highest elevation at the rainfall station is 677.5 m at Unzen, the lowest is 2.5 m at Fukuoka, and the average is 20.6 m. The distance between the stations is 25.0 km at the shortest and 441.8 km at the longest.

In addition, the annual maximum value for daily rainfall until 2020 from the Automated Meteorological Data Acquisition System (AMeDAS) Izumi, Morotsuka, AsoOtohime, and Asakura operated by the JMA was used to validate our method (Figure 1). These stations were selected because they have short-period records (approximately 40 years) and observed extreme rainfall events that triggered floods and landslides (Table 2) [12,32–34]. Specifically, a rainfall event that triggered a deep-seated landslide and debris flow was observed at AMeDAS Izumi in July 1997. At AMeDAS Morotsuka, a rainfall event that triggered deep-seated landslides was observed in September 2005. At AMeDAS Aso-Otohime, rainfall events that triggered floods and debris flow were observed in July 1990 and July 2012, respectively. At AMeDAS Asakura, an extreme rainfall event that triggered shallow landslides and debris flow including driftwood was observed in July 2017.

**Figure 1.** Location of rainfall stations. White circles and triangles indicate meteorological observatories and AMeDAS, respectively.


**Table 1.** Specifications of meteorological observatories in Kyushu.


**Table 1.** *Cont.*

**Table 2.** Specifications of AMeDAS.


#### *2.3. Investigation of Spatial Correlation of Annual Maximum Value for Daily Rainfall*

The Kendall rank correlation coefficient (Kendall's τ) [35] of the annual maximum daily rainfall between two stations was investigated because the station-year method assumes the spatial independence of stations [17]. In previous studies [17,28,30,36], spatial independence was investigated using Pearson's correlation coefficient, but Kendall's τ was used in this study since the annual maximum daily rainfall between two stations was not assumed to be Gaussian distribution.

Kendall's τ is a non-parametric method for testing the dependence between two variables based on an ordinal association between two measured quantities. Kendall's τ is given as:

$$\tau = \frac{\sum\_{i$$

where *sign* (·) is the sign function. To conduct this investigation, common period data from all stations between 1981 and 2010 were used.

#### *2.4. Normalization of Daily Rainfall Data*

Quantiles corresponding to the non-exceedance probability of 50% and 90% were used to normalize the daily rainfall data; in other words, the values corresponding to the RP of 2 and 10 years (2- and 10-year values). These indices were less variable since they were calculated by interpolation at each station and were considered suitable for normalization. Suzuki and Kikuchihara [24] also used 2- and 10-year values to normalize the daily rainfall data. The data for the annual maximum daily rainfall were normalized following Suzuki and Kikuchihara [24]:

$$y\_T = \frac{\mathbf{x}\_j - \mathbf{x}\_2}{\mathbf{x}\_{10} - \mathbf{x}\_2} \tag{2}$$

where *yT* is normalized daily rainfall and *xj* is the annual maximum daily rainfall. *x2* and *x10* are the 2- and 10-year values, respectively.

## *2.5. Extreme Value Analysis*

The 2- and 10-year values were calculated using parameters in the GEV distribution [22], estimated by the L-moment method [21]. The data for the annual maximum daily rainfall between 1981 and 2010 were used due to the need to unify the period for calculating the GEV parameters. The GEV cumulative distribution function and the quantile of the GEV corresponding to the non-exceedance probability are given as, respectively:

$$F(\mathbf{x}) = \exp\left\{-\left(1 - k\frac{\mathbf{x} - \mathbf{c}}{a}\right)^{\frac{1}{k}}\right\} \text{ for } k \neq 0 \tag{3}$$

$$\mathbf{x}\_p = \mathbf{c} + \frac{a}{k} [1 - \{-\ln(p)\}^k] \tag{4}$$

where *k* is the shape parameter, *c* is the scale parameter, *a* is the location parameter, and *p* is the non-exceedance probability. The parameters of the GEV are given as:

$$\begin{cases} \begin{array}{l} k = 7.8590d + 2.9554d^2 \\ a = \frac{k\lambda\_2}{\left(1 - 2^{-k}\right)\Gamma(1+k)} \\ c = \lambda\_1 - \frac{a}{k} \{1 - \Gamma(1+k)\} \end{array} \end{cases} \tag{5}$$

$$d = \frac{2\lambda\_2}{\lambda\_3 + 3\lambda\_2} - \frac{\ln(2)}{\ln(3)}\tag{6}$$

where *λ*1–3 are sample L-moments and Γ the gamma function. *λ*1–3 are given as:

$$\begin{cases} \lambda\_1 = \beta\_0 = \frac{1}{N} \sum\_{j=1}^{N} \mathbf{x}\_{(j)}\\ \lambda\_2 = \beta\_1 = \frac{1}{N(N-1)} \sum\_{j=1}^{N} (j-1)\mathbf{x}\_{(j)}\\ \lambda\_3 = \beta\_2 = \frac{1}{N(N-1)(N-2)} \sum\_{j=1}^{N} (j-1)(j-2)\mathbf{x}\_{(j)} \end{cases} \tag{7}$$

where *x*(*j*) is the *j*-th value from the smallest when the sample is arranged in increasing order.

The standard least-squares criterion (SLSC) [37,38] was used to evaluate the goodness of fit between the observed rainfall and the probability distribution. The SLSC compares the goodness of fit across distributions, and their smaller values imply better fits [37]. In this study, we judged a good fit when the SLSC value was below 0.04 in accordance with the JMA [39]. The SLSC value is given as:

$$SLSC = \frac{\sqrt{\frac{1}{N} \sum\_{j=1}^{N} \left\{ s \left( x\_{(j)} \right) - s^\* \left( p\_{(j)} \right) \right\}^2}}{|s\_{0.99} - s\_{0.01}|} \tag{8}$$

$$s(x\_{(j)}) = -\ln\left[\left\{ \left(1 - k \frac{x\_{(j)} - c}{a} \right)^{\frac{1}{k}} \right\} \right] \tag{9}$$

$$s^\*\left(p\_{(j)}\right) = -\ln\left[-\ln\left\{p\_{(j)}\right\}\right] \tag{10}$$

where *s*(*x*(*j*)) is the standardized variate by GEV parameters and *s\**(*p*(*j*)) is the standardized variate corresponding to the non-exceedance probability calculated by the plotting position formula [40]. *s*0.99 and *s*0.01 are the standardized variates corresponding to the

non-exceedance probability of 1% and 99%, respectively. The plotting position formula is given as:

$$p\_{(j)} = F\left(x\_{(j)}\right) = \frac{j - \alpha}{N + 1 - 2\alpha} \tag{11}$$

where *α* is a constant; we used Cunnane's formula (*α* = 0.4) [41] to give *pi* following the JMA [39].

#### **3. Results and Discussion**

#### *3.1. Spatial Correlation of Annual Maximum Value Daily Rainfall between 1981 and 2010*

Figure 2 shows Kendall's τ of the annual maximum daily rainfall between two stations plotted against distance. As shown in Figure 2, Kendall's τ of the annual maximum daily rainfall between two stations tended to decrease as the distance between the two stations increased. In general, the correlation of rainfall decreases [17,28,30,36]; for example, Kuzuha et al. [34] examined the spatial correlation structure of precipitation using rainfall data at AMeDAS in Japan and showed that Pearson's correlation coefficient of daily rainfall decreased exponentially as the distance between the two stations increased. Hence, our result agreed with previous studies.

Furthermore, Kendall's τ of the annual maximum daily rainfall between two stations was 0.57 at its maximum and was generally small (Figure 2). Overeem et al. [17] examined the spatial dependence to apply the station-year method by using Pearson's correlation coefficient of daily rainfall between the two stations. They assumed the spatial dependence even if the Pearson's correlation coefficient was around 0.60 [17]. Although the Kendall's τ and Pearson's correlation coefficient were not simply comparable, we believed that the data for the annual maximum daily rainfall at 23 stations were spatially independent considering a previous study [17].

**Figure 2.** Kendall's τ of annual maximum daily rainfall between stations plotted against distance.
