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Extreme precipitation events are a major cause of severe floods worldwide. The statistical modelling of extreme rainfall is essential in designing water-related structures, for use in agriculture and in weather modification, conducting risk assessment and monitoring climate changes, among others. Many researchers have analysed extreme daily rainfall, including Manton et al. (2001), Katz et al. (2002), Feng et al. (2007), Choi et al. (2009), Costa and Soares (2009), Re and Barros (2009) and Rahimzadeh et al. (2009).
Heavy rainfall occurs frequently in Korea during the warm season (from late June through early September) because of synoptic disturbances, typhoons or convective systems within an air mass (Lee et al., 1998). Because of the highly concentrated population and property in Korea, the country has been vulnerable to water disasters in terms of considerable amounts of the affected population and damaged property. In this regard, heavy rainfall in Korea has been an important research topic (Park et al., 2001; Park and Jung, 2002; Nadarajah and Choi, 2007).
Statistical distributions have long been applied to time series of climate extremes. This study uses a non-stationary generalized extreme value (GEV) distribution on the summer extreme daily and 2-day rainfall data observed by 28 stations in South Korea (herein ‘Korea’) to obtain reliable quantile estimates for several return periods. We performed statistical significance tests on the extreme rainfall data to explore linear temporal trends in the course of climate change. Of the 28 stations considered, we found evidence of non-stationarity in the form of increasing trends for six stations from AMP1 and for five stations from AMP2. This trend is consistent with the results from a regional climate model derived by the A1B emission forcing of IPCC AR4. We quantified the changes in extreme rainfall for each station; the return levels and their 95% confidence intervals for various return periods are provided.
Nadarajah and Choi (2007) analysed the annual maxima of daily rainfall in Korea from 1961 to 2001 using a technique similar to that used in this paper. They also looked for trends in the data, but found no evidence. The results of the present study indicate the Gumbel distribution as the most reasonable model for 18 stations for AMP1 and 15 for AMP2. Further, we found evidence of increasing trends for six stations for AMP1 and five for AMP2. These discrepancies may be due to differences in the data period and the number of stations considered.
Section 2 describes the climatology and descriptive statistics of the annual maximum daily precipitation in Korea. Section 3 presents the study methodology, and Section 4 presents the results and the isopluvial maps of the quantiles at the selected return periods. Section 5 discusses the results and concludes.
2. Data and climatology
The original data consisted of daily rainfall records up to 2007, which were observed by 28 stations in Korea (Korea Meteorological Administration: http://www.kma.go.kr). Figure 1 identifies these stations. Two time series—annual maxima of daily precipitation (AMP1) and annual maxima of 2-day precipitation (AMP2)—were constructed. Table I shows the descriptive statistics of the annual daily maximum precipitation for the locations where N is the sample size and IQR stands for the inter-quartile range (sample upper quartile minus sample lower quartile). Table I lists only those stations with more than 45 annual maximum observations. The sample sizes of the stations not listed in Table I were between 35 and 42. Park et al. (2001) provided similar statistics. The lengths of the time series (until 2007) varied because the stations were established at different times. The modern meteorological observation has been in operation since 1904 and is managed by the Korea Meteorological Administration (KMA). Between 1904 and 2007, because of the Korean War, daily precipitation data for three years from June 1950 to July 1953 were unavailable for some stations (Gangneung, Seoul, Incheon, Pohang and Chupungryong). To ensure the validity and integrity of the data for the intended applications, both the real-time and delayed-time quality control (QC) procedures were applied for synoptic weather stations, upper-air observing stations, marine and automated weather systems (AWS), among others; the procedures, which included format, completeness, tolerance and consistency tests, reflected the guidelines and supplementary materials of the World Meteorological Organization (WMO) on climate practices (WMO, 1983). In addition, to maintain consistency between recent 1-min interval precipitation records by AWS and historical hourly precipitation data obtained through the weighting rain gauge, the KMA digitized the database for the entire period by using the same format and temporal resolution of modern observations, compiling 2.5 million analogue-recorded sheets from 1904 to 1999 for 102 stations. Another factor that may raise a homogeneity issue in precipitation data is changes in the location of weather stations. The KMA (1995) stated that only one station (Gwangju) out of the 28 stations was relocated during the observation period, which was not a highlighted station in this study.
Table I. Descriptive statistics of the annual daily maximum precipitation (AMP1) data (unit:mm)
Climatologically, the high intensity of precipitation in Korea is mainly due to torrential rain events and typhoons. Another reason for heavy rainfall may be that the convergence zone of the monsoon front (called Changma) is centrally located between China, Japan and the Korean Peninsula, where the condition manifests for approximately 3–4 weeks because of the stationary planetary wave over East Asia (Svensson and Berndtsson, 1996). Another factor is the strong influence of the subtropical Pacific high pressure over the period from mid-August through early September, known as the fall rainy season. A feature of the synoptic-scale environment, which is favourable for heavy rainfall, is the relative locations of the upper- and lower-level jets, which are located to the north and south of the heavy rainfall area, respectively. The annual precipitation in Korea thus depends largely on precipitation during the summer rainy season (from late June through July). This season is strongly controlled by the Far East Asian summer monsoon system. Noteworthy is that the horizontal distribution of heavy rainfall amounts is roughly similar to that of the total annual precipitation (Lee et al., 1998; Park and Jung, 2002). Jung et al. (2002) identified an increasing trend in the frequency of extreme precipitation events in Korea; the highest rate was observed in the 1990s. The mean frequency of extreme precipitation events in the 1990s increased by 53% with respect to that of the 1950s (1954–1960).
This study is motivated by Nadarajah (2005). We used Coles' R program (Coles, 2001) to compute the maximum likelihood estimates of the GEV distribution for this work.
This study employs the GEV distribution, which has all of the flexibility of three extreme value distributions: Gumbel, Frechet and negative Weibull. The cumulative distribution function (cdf) of the GEV distribution is:
where µ, σ> 0 and ξ are the location, scale and shape parameters, respectively. The case of (1) for ξ = 0,
is a Gumbel distribution, whereas the cases ξ> 0 and ξ< 0 of Equation (1) are known as the Frechet and negative Weibull distributions, respectively (Coles, 2001; Kotz and Nadarajah, 1999). The reason behind the widespread use of the GEV distribution is that the three types of extreme value distributions are the only possible limits (as the sample size becomes larger) for the distribution of the extreme statistic regardless of the distribution of the population.
The maximum likelihood method was used to fit Model GEV0 and related models to the data. We follow the same statistical methods as Nadarajah (2005) and Feng et al. (2007). Consequently, some descriptions throughout this paper are the same as theirs.
Assuming the independence of the data, the likelihood function is the product of the assumed densities for the observations x1, x2, …, xn. For Model GEV0, we have
The estimates of µ, σ and ξ, denoted as , , and , are taken to be those values that maximize the likelihood function L. The basic model fitted was Equation (1) with constants µ, σ and ξ. We also fitted Equation (2) with constants µ and σ. Model Gum0 is a submodel of Model GEV0. Thus, a standard way of determining the best fit model is the likelihood ratio test (Wilks, 1995; other criteria such as AIC or BIC reduce to the likelihood ratio test when models are nested). If Li is the maximum likelihood for model i and Lj is the maximum likelihood for model j, then under the simpler model, the test statistic λ = − 2log(Lj/Li) is assumed to be distributed as a chi-square variable with v degrees of freedom, where v is the difference in the number of parameters between the models i and j. In hypothesis testing problems, this would be asymptotically true as the amount of data tends to be infinite. Thus, at the 5% significance level, model j is preferred if . In practice, because annual maxima lack complete independence, this is likely to be conservatively interpreted (Nadarajah, 2005).
To investigate the existence of a trend in extreme rainfall with respect to time, we applied the following variations of Models GEV0 and Gum0:
a four-parameter model, where µ varies linearly with respect to time and ‘constant’ means that the parameter is not time dependent and is subject to the estimation;
a three-parameter model, where µ varies linearly with respect to time;
a five-parameter model, where µ varies quadratically with respect to time; and
a four-parameter model, where µ varies quadratically with respect to time. Thus, we employ a non-stationary extreme value model to reflect the context of climate change. In all of the four models, t0 denotes the year the records started.
A similar technique has been used by many researchers, including Katz et al. (2002), Nadarajah and Choi (2007), Feng et al. (2007), El-Adlouni et al. (2007) and Sugahara et al. (2009). See Khaliq et al. (2006) for a review of methods for non-stationary hydro-meteorological extremes.
Once the best models for the data are determined, the next step is to derive the return levels for rainfall. The T year return level, xT, is the level exceeded on average only once in every T years (Coles, 2001). If Model GEV0 is assumed, then by inverting F(xT) = 1 − 1/T, we obtain the following expression for the GEV distribution:
If Model Gum0 is assumed, then the corresponding expression is
By substituting , and into those xT, we have the maximum likelihood estimates of the return levels.
4. Results of the data analysis
All of the models were fitted to the AMP1 and AMP2 data from each of the 28 stations. The method of maximum likelihood estimation and the likelihood ratio test (Wilks, 1995) were used throughout this study.
The best fitted model, the estimates and the standard errors (SE) are presented in Tables II and III (only the stations for which the time-varying or GEV model performed the best are listed with the estimates). The extreme daily rainfall data for six stations (Incheon, Seoul, Pohang, Seosan, Andong and Jecheon) and the extreme 2-day rainfall data for five stations (Pohang, Gwangju, Seogwipo, Andong and Jecheon) exhibited evidence of non-stationary increasing trends. The estimate b̂ can be interpreted as the change in extreme rainfall from one year to another. The standard error of the estimated parameters was computed from the Fisher information matrix (Coles, 2001). The stationary Gumbel distribution provided a good fit to the AMP1 data for 18 stations and to the AMP2 data for 15 stations. The stationary GEV distribution provided a good fit to the AMP1 data for four stations and to the AMP2 data for eight stations.
Table II. Best fitting models, parameter estimates and standard errors computed from the AMP1 time series of the stations for which the best fitting model was not the stationary Gumble distribution
− 0.145 (0.10)
Table III. Best fitting models, parameter estimates, and standard errors computed from the AMP2 time series of the stations for which the best fitting model was not the stationary Gumble distribution
− 0.244 (0.11)
Table IV summarizes the return levels corresponding to 10, 20, 50, 100 and 200 years for the locations where a stationary model was chosen. The 95% confidence intervals for the return levels are also given. These intervals were computed by using the profile likelihood method for the GEV distribution and the delta method for the Gumble distribution (Coles, 2001). Three stations (Gangneung, Jangheung and Haenam) produced very high return levels. The contour maps of the estimated design values (unit: mm) corresponding to the 100-year return level are given in Figure 2 for AMP1 and in Figure 3 for AMP2. The eastern and south-western areas showed high return values. The sites with the highest return values were located in the south-western areas. This was due to the record-breaking rainfall from the influence of Typhoon Agnes between August 30 and September 4, 1981. During this period, the stations around Mokpo (Jangheung, Haenam and Goheung) had almost half the annual total precipitation in a single day (between 394 and 548 mm).
Table IV. Return levels and 95% confidence intervals computed from the time series of AMP1 for various return periods (unit: mm)
The goodness of fit of these models was examined by quantile and density plots. In a quantile plot, the observed quantile is plotted against the quantile predicted by the fitted model. A density plot compares the fitted density of the model with a non-parametric version computed directly from the annual maximum daily data. The various diagnostic plots that assess the accuracy of the GEV model fitted to the Daegu AMP1 data are shown in Figure 4. These plots were drawn by using Coles' R program (Coles, 2001). The corresponding density estimate was consistent with the histogram of the data. Consequently, all four diagnostic plots supported the fitted GEV model.
The graphical displays of the fits of the AMP1 data for Pohang (1949–2007), Incheon (1904–2007) and Andong (1973–2007) are shown in Figure 5. The increasing trend for Andong was clear and remarkable. Each figure shows the original annual maximum daily rainfall data superimposed with an estimate of the median of the extreme rainfall data; the interval estimates of 50-year, 100-year, 200-year return levels are also shown. The median of the extreme rainfall (or the 2-year return level) data was estimated by
The interval estimates from the lower T-year return level to the upper T-year return level are given by
For example, the upper 20-year return level represents the 0.995 quantile. The lower 20-year return level corresponds to the 0.005 quantile. Thus, approximately 99% of data are expected to be between these two quantities. Assuming that this trend would continue into the future, these curves can be projected beyond the range of the data to provide predictions. Note that if p = exp(−1); thus, can be interpreted as the exp(−1)≈ 37% percentile of extreme rainfall (Nadarajah, 2005).
Tables V–X provide the projected return levels of extreme daily rainfall for 2020, 2050 and 2100. For obvious reasons, we considered only those locations (six stations) that exhibited non-stationary behaviour with respect to extreme rainfall (AMP1). As shown in the tables, we estimated the median of the extreme rainfall data and the three intervals (T = 20, 40 and 200) given by Equation (12). The general pattern indicates an increasing trend in extreme rainfall, which is consistent with the results from a regional climate model derived by the A1B emission forcing of IPCC AR4. For example, Boo et al. (2006) demonstrated that by the end of 21st century, extremely heavy precipitation above 50 mm/day would increase by approximately 57 and 168% in the southern and northern areas of Korea, respectively. However, the typical resolution of the present regional climate models is approximately 20–50 km, which is still insufficient for adequately investigating future climate projections for the local stations covered in this study.
Table V. Return levels of extreme rainfall (AMP1) for Seoul
T = 20
T = 40
T = 200
Table VI. Return levels of extreme rainfall (AMP1) for Incheon
T = 20
T = 40
T = 200
Table VII. Return levels of extreme rainfall (AMP1) for Pohang
T = 20
T = 40
T = 200
Table VIII. Return levels of extreme rainfall (AMP1) for Seosan
T = 20
T = 40
T = 200
Table IX. Return levels of extreme rainfall (AMP1) for Andong
T = 20
T = 40
T = 200
Table X. Return levels of extreme rainfall (AMP1) for Jecheon
T = 20
T = 40
T = 200
The spatial pattern of the estimated return values shown in Figure 2 may be partially associated with the high mountainous topography in the eastern area of the Korean Peninsula, reflecting the effect of local topography on precipitation in association with global warming (Giorgi et al., 1994). However, the dependency between temperature and precipitation in Korea may also be attributable to the small return values in the mid-interior area of the Korean Peninsula, which is another high mountainous region. Boo et al. (2004) demonstrated that changes in the precipitation, temperature, and drought indices for the Korean Peninsula had significantly positive correlation coefficients and that areas exhibiting lower levels of temperature increases correspond well to those areas with small return values in Figure 2.
The increasing trends of extreme precipitation shown in this study and its relationship to low-frequency phenomena such as El Nino and Southern Oscillation (ENSO) and Typhoons need to be further examined to confirm the consistency of this study's results with those of previous research. Despite diverse features in terms of the intensity and amplitude of ENSO projections, Yamaguchi and Noda (2006) found that most atmosphere/ocean-coupled climate models (CGCMs) tend to produce El Nino-like changes in the sea surface temperature (SST), reflecting global warming effects. Meanwhile, it has been generally acknowledged that a wet East Asian summer monsoon (EASM) is preceded by a warm ENSO phase in the previous winter and followed by a cold phase in the following fall, that is, East Asian summer precipitation coincides with the ENSO decaying season (Chang et al., 2000). With respect to warm SSTs and their relation to EASMs, most CGCMs in IPCC AR4 have shown increases in precipitation with strong EASMs under global warming scenarios (Lee et al., 2007), which is consistent with the present findings. Increases in extreme precipitation over Korea may also be associated with changes in Typhoon activity, which is expected intensify with reduced frequency (NIMR, 2009). A physical interpretation of the relationship between extreme climate events and global warming is beyond the scope of this study, but it is found in Hegerl et al. (2007).
Feng et al. (2007) analysed the annual maximum precipitation in China by using a generalized extreme value distribution modified to explore linear temporal trends. They found that more than 12% of the stations studied had significant linear trends. Decreasing trends were observed mainly in northern China, in the Yangtze River basin, and north-western China. The present study observed only increasing trends for Korea: six stations (21%) for AMP1 and five stations (18%) for AMP2 (out of the 28 locations studied).
Although the Gumbel distribution with a constant location parameter was selected as the best-fit model for many stations, extrapolating return levels for very high years was problematic. Because there were only two parameters in the Gumbel distribution, it might not have been flexible enough to estimate such high return levels (i.e., more than 100 years). Thus, it may be reasonable to consider distributions with three or more (e.g. 4 or 5) parameters, such as the Pearson type-III, log-logistic, 4-parameter kappa and Wakeby distributions (Hosking and Wallis, 1997), in addition to the GEV distribution.
Climatologically, the cause of the dependencies in the annual maxima is unclear. Decadal or multi-decadal circulation regimes that affect the occurrence of extreme rain events in the region can be investigated. In this regard, further studies are warranted to address this question. In addition, the identification of 5 or 6 trends in the 28 stations might have been a random occurrence, particularly if the series of the stations were correlated. The present findings can be verified by resampling analysis.
It is unclear whether the stations for which the time-varying model performed the best were biased by one or more specific meteorological events. Because the data underlying the 28 stations might have been correlated, a ‘field significance’ analysis (Wilks, 2006) would be useful. An application of a spatial extreme model (Buishand et al., 2008; Davidson, 2009), which takes into account the spatial dependency of extreme events, is left for future research.
This study's model allows only the location parameter to vary with time by setting the scale and shape parameters (σ and ξ) as constants. We can, however, modify the model so that the scale and shape parameters could be time-varying, which would lead to σ(t) and ξ(t). In addition, the techniques presented here can be adapted for the generalized Pareto distribution to model the excess rainfall over the threshold. Sugahara et al. (2009) used a non-stationary generalized Pareto distribution to detect changes in extreme daily rainfall in Sao Paulo, Brazil.
The r-largest GEV distribution is another technique in modelling r-largest order statistics (heavy precipitation). We know that, compared with the GEV distribution for a maxima data set, the interpretation of the parameters is unaltered but their precision improves because of the inclusion of extra information (Coles, 2001). Thus, the r-largest GEV distribution with time-varying location parameters may be more useful in modelling heavy rainfall in the course of climatic change. Actually, Zhang et al. (2003) used the r-largest GEV distribution to investigate trends in precipitation and extreme temperature. They found the method of covariates (like the above models) to be superior to parametric and non-parametric counterparts in investigating trends.
Heavy rainfall is common in Korea during the warm season (from late June through early September) because of synoptic disturbances, typhoons or convective systems within the air mass (Lee et al., 1998). We performed an extreme value analysis of heavy daily and 2-day rainfall data for 28 stations in Korea. For each of the 28 stations, we identified a model that best described the behaviour of extreme rainfall. Based on the chosen model, we provided the return levels and the 95% confidence intervals corresponding to 10, 20, 50, 100 and 200 years for the stations for which a stationary model was chosen.
We generated contour maps of the estimated design values (unit: mm) corresponding to the 100-year return level for AMP1 (annual maxima of daily precipitation), shown in Figure 2, and for AMP2 (annual maxima of 2-day precipitation), shown in Figure 3. The eastern and south-western areas of the maps showed high return values. In addition, there was evidence of non-stationary increasing trends in extreme daily rainfall for six stations (Incheon, Seoul, Pohang, Seosan, Andong and Jecheon) and extreme 2-day rainfall for five stations (Pohang, Gwangju, Seogwipo, Andong and Jecheon). This trend is consistent with the results from a regional climate model derived by the A1B emission forcing of IPCC AR4. For the six stations, we projected the return levels of extreme daily rainfall for the years 2020, 2050 and 2100. The stationary Gumbel distribution provided a good fit to the AMP1 data for 18 stations and to the AMP2 data for 15 stations. The stationary GEV distribution provided a good fit to the AMP1 data for four stations and to the AMP2 data for eight stations.
The authors are grateful to the three reviewers for their valuable and constructive comments. This work was funded by the Korea Meteorological Administration Research and Development Program under Grant RACS_20102015.