#### 6.1. Characteristics of Spatial and Temporal Structure of Extreme Rainfall Trends in South Korea

No trends are evident in the AM series of any of the durations examined at most of the employed stations. Additionally, no station consistently displays a trend for all of the durations examined. For the AM series of each month (May to October), there are statistically significant trends at a few stations. Additionally, no station consistently displays a trend for all of the durations examined. All of the detected trends in the annual and monthly AM series are an increasing trend. Thus, although the magnitude of extreme rainfall in South Korea is increasing over time, the changes are very small, and the temporal and spatial structures display inconsistent trends.

The temporal structure of extreme rainfall events can be represented by the scaling exponent. For example, when the scaling exponent derived from AM series is small, the differences in the location parameters of the AM series with different durations are small. Thus, at stations associated with small scaling exponents, the differences among the statistical properties of the AM series with different durations are small, and vice versa. The scaling exponent estimates derived from the AM series in South Korea differ strongly, based on the locations of the stations. For instance, the scaling exponents in the northeast are greater than 0.4, whereas the scaling exponents in the northwest are less than 0.3. The scaling exponents for the AM series of each month are spatially inhomogeneous. Moreover, their spatial distributions are dissimilar. Thus, the spatial distribution of the temporal structure associated with extreme rainfall in South Korea displays large variability. In addition, the interannual variability in the temporal structure of extreme rainfall in South Korea is very large.

The cause of spatial and temporal variation in various hydro-meteorological extremes is very complicated. Particularly, spatial and temporal variation of extreme rainfalls in South Korea are influenced by mountainous terrain, monsoon, and typhoon. Therefore, it is very difficult to analyze spatio-temporal variation in South Korea [

11,

71]. Chang and Kwon [

72] analyzed spatial variation of precipitation trends monthly during summer (rainy season) in South Korea using daily precipitation data between 1973 and 2005. They reported that spatial variations of the precipitation trends in summer was very different in each month. They discussed that it is likely to have some connection with large-scale atmospheric circulation, variation of sea surface temperature and geographic features. Consequently, amount and intensity of summer rainfall in South Korea have changed with time and space. They mentioned that these variations are the temporal redistribution of summer rainfall occurred in South Korea caused by climate change.

In this study, we also confirmed the temporal and spatial redistribution from the various contour maps of the scaling exponent estimate for each month. The scaling exponent estimates have changed with time and space during the rainy season. The variations of scaling exponents between June and July are most likely to be caused a seasonal rain front (i.e., monsoon) movement. In addition, spatial changes in the scaling exponent estimates from September and October are more likely to be due to typhoons. Spatial and temporal variations observed in this study may be due to hydro-meteorological phenomena such as monsoon and typhoon which are main contributors of extreme rainfalls in South Korea. Occurring time and intensity of these phenomena are different for each year. Therefore, the spatial distribution of the scaling exponents for each month are not consistent.

Scaling exponent estimates and scaling properties have been used to derive regional IDF curves [

70,

73,

74,

75]. Yu, Yang and Lin [

73] carried out a regionalization to identify homogeneous regions based on the scaling exponent. Additionally, since the scaling exponent represents the temporal structure of extreme rainfall events, it may be possible to use the scaling exponent to cluster stations that are homogeneous in terms of their extreme rainfall events for use in regional IDF analyses. Hence, the regionalization of the extreme rainfall events in South Korea should be performed with care, due to the large spatial and interannual variability in the temporal structure of the extreme rainfall events.

While the AM series at a small number of stations display statistically significant trends, significant trends are detected at many stations when the time series of scaling exponent estimates are used (see

Figure 4 and

Figure 7). Furthermore, both increasing and decreasing trends are detected in the time series of scaling exponent estimates, unlike the results of applying the trend analysis to the AM series. Even though the AM series of a small number of stations display significant trends, the scaling exponents present nonstationarity. These results indicate that applying trend analysis to extreme rainfall events is insufficient to check the nonstationarity associated with extreme events. The results of applying the BBS-MK test to the AM series of South Korea show that the AM series may contain very weak increasing trends; the AM series at most of the employed stations can be considered to be stationary, and the AM series at a few stations display increasing trends. However, the temporal structures of the AM series at many stations in South Korea can be considered to reflect nonstationary conditions, based on the results of applying the trend tests to time series of scaling exponent estimates. Hence, the detailed trends of the extreme rainfall events and their temporal structures should be assessed to investigate the nonstationarity in the extreme rainfall events.

Since the trends in the time series of scaling exponent estimates may influence the regionalization, the homogeneous regions clustered using current data will change in the future. These trends should be taken into consideration in deriving a regional IDF curve for South Korea. These trends should also be considered for the nonstationary IDF curves of the extreme rainfall events in South Korea. Conventional nonstationary IDF curves assume that the temporal structure of extreme rainfall events is constant over time [

76,

77]. For example, Cheng and AghaKouchak [

78] employed a nonstationary GEV distribution with a time-varying location parameter to model a nonstationary IDF curve. They fitted the nonstationary GEV distribution for each duration examined, and their study provided an IDF curve for extreme rainfall events in the U.S. Sarhadi and Soulis [

79] modeled a nonstationary IDF curve using the nonstationary GEV distribution with time-varying location and scale parameters. However, the temporal structure of the extreme rainfall events of South Korea change, although there is no trend in the mean of extreme rainfall events. Thus, the conventional approach used in modeling nonstationary IDF curves, which involves the use of the nonstationary GEV distribution with a covariate, may not properly represent extreme rainfall in South Korea.

The model used to derive the nonstationary IDF curve for South Korea should account for trends in temporal structure. For example, the scaling exponent in Equation (5) can be allowed to vary with time, instead of the location or scale parameters of the GEV distribution. Employing the scaling exponent as a time-varying parameter in the GEV simple scaling framework may be an adequate parameterization for the representative model of the nonstationary IDF curve in South Korea, since significant trends are not detected in the AM series at many stations. In cases where a trend exists in the AM series at a given station, the location parameter and the scaling exponent can be permitted to vary with time.

#### 6.2. Limitation of the Employed Methodology

The scaling exponent series have the serial correlations. Due to the serial correlation, the trend tests employed in the current study consider the serial correlation in the data. In the current study, results of two trend tests are very similar but not the same. The methods to derive the distribution of Mann-Kendal statistics are different in two employed tests. The modified version of Mann-Kendall test considered that the distribution of Mann-Kendal statistics follows the normal distribution. This test takes consideration of the influence of the serial correlation in the trend detection by modifying the variance of the mean. In the BBS-MK test, the distribution of Mann-Kendall statistics may not be the same as the normal distribution since the distribution is obtained by the block bootstrapping method. The distribution of Mann-Kendall statistics in the BBS-MK test may be different depending on the data.

Both tests have pros and cons. The subjective decisions are not required in the modified version of the Mann-Kendall test. For the large sample size, the distribution of the Mann-Kendall statistics follows the normal distribution [

39]. However, in small sample size, the distribution of the Mann-Kendall statistics may not exactly follow the normal distribution [

80]. In the case of small sample size, the modified version of Mann-Kendall test can lead to low power to detect trends due to the normal distribution assumption. The BBS-MK test derives the distribution the distribution of the Mann-Kendall statistics by the block bootstrapping method. The shape of the Mann-Kendall statistic distribution for various sample size and data may be close to the distribution of the Mann-Kendall statistic in population. However, in the BBS-MK test, obtaining an accurate distribution shape from bootstrapping is difficult due to difficulty in the selection of an optimal block size.

According to the results of the Mann-Kendall test, some stations where the BBS-MK test detected trends did not trend. Some stations where the modified version of the Mann-Kendall test detected trends did not have trends based on the results of the BBS-MK test. These results may come from the issue of small sample size. Basically, the results of two tests should be similar because both tests attempt to detect a trend in the data. If the sample size is very large, two different tests may provide the same results. The small sample size is the limitation of the employed methodology in the current study. Thus, to attenuate and avoid the probability to obtain inappropriate results of trend detection from the small sample size, a number of trend tests should be employed. The application of a number of trend tests leads to more reliable results for trend detection in the serially correlated data with the small sample size. Although the results of the current study may not be the perfect results due to the limitation, the results can provide the glance to the trends in the spatial and temporal structure of extreme rainfall in South Korea.