Regional frequency analysis for consecutive hour rainfall using L-moments approach in Jeju Island, Korea
- Kanak Kanti Kar^{1}Email authorView ORCID ID profile,
- Sung-Kee Yang^{1},
- Jun-Ho Lee^{1} and
- Fahad Khan Khadim^{2}
Received: 23 March 2017
Accepted: 5 June 2017
Published: 28 June 2017
Abstract
Background
Extreme rainfall events are enormously frequent and abrupt in tropical areas like the Jeju Island of South Korea, impacting the hydrological functions as well as the social and economic situation. Rainfall magnitude and frequency distribution related information are essential for water system design, water resources management and hydro-meteorological emergencies. This study therefore has investigated the use of L-moments approach for hourly regional rainfall frequency estimation so as to ensure better accuracy and efficiency of the estimation process from the usually limited data sets.
Results
The Hancheon catchment was considered as the primary study domain and several best fitted statistical tools were used to analyze consecutive hour rainfall data from five hydro-meteorological stations (Jeju, Ara, Eorimok, Witsaeorum and Jindallaebat) adjacent to the area. The cluster analysis and discordancy measure categorized the Hancheon catchment in three regions (1, 2 and 3). Based on the L-moments heterogeneity and goodness-of-fit measure, Gumbel and generalized extreme value (GEV) distribution were identified as robust distributions for the study area. The RMSE ratios for the catchment were found as 0.014 to 0.237 for Gumbel and 0.115 to 0.301 for GEV distribution. The linear regression analysis of the different rainfall quantiles inferred r-square values from 0.842 to 0.974.
Conclusions
The L-moments and other statistical information derived from the study can be useful for important hydrological design considerations in connection with flood risk management, mitigation and safety; whereas the methodological framework of the study may be suitable for other small scaled catchment areas with high slope.
Keywords
Regional frequency analysis Consecutive hour rainfall L-moments Heterogeneity measure Jeju IslandBackground
Extreme hydro-meteorological occurrences, such as heavy rainfall, floods, storms and typhoons are regarded as being the most costly natural disaster risk and leading research hotspots, bearing wide scope of scientific applications relevant to the field of hydrology and water resources engineering (Bruce 1994; Obasi 1994). The coastal parts of East Asia are extensively and continuously hit by climatic disasters and leaving substantial effects on the hydrological functions (Jun 1989; Shabri et al. 2011; Chang et al. 2012; Cai et al. 2014). Driven by this, the study attempts to carry out a reliable estimation of extreme rainfall occurrences and corresponding regional frequency analysis (RFA) using the L-moments approach, so as to ensure efficient design and control of the hydrological systems of South Korea.
The Hancheon catchment of Jeju Island, South Korea has been considered as the study domain, which covers an area of 37.39 sq. km, exhibiting dynamic and distinct hydrological characteristics. Over the years, the Jeju Island experienced several typhoon events and especially in the last decade, a number of typhoons such as, Typhoon Nari, Khanun, Borlaben, Sanba and Nakri hit the Island, killing a total number of 11 people and causing property damages worth around 1.41 million USD ($).
The regional frequency analysis is a continuously developing insight used by local disaster management departments, while many researchers also find the approach as a contemporary method to define identical hydrological regions, viewing it as a modification over the typical probability moments (e.g. Bradley 1998; Parida et al. 1998; Fowler and Kilsby 2003; Kumar and Chatterjee 2005; Wallis et al. 2007; Noto and Loggia 2009; Saf 2009; Shahzadi et al. 2013; Devi and Choudhury 2013; Liu et al. 2015). Um et al. (2010) studied five distribution models to examine extreme rainfall events in Jeju Island using elevation and geographic coordinates as modeling inputs. The study discussed multiple non-linear form, linear regressions, an intensity-duration-frequency (IDF) relationship curve and obtained model accuracies in the range of 18–86%. There has been a number of identical studies up to now, exploring and updating different working methods for RFA of extreme rainfall occurrences, the most prominent ones are Cluster analysis (Easterling 1989; Venkatesh and Jose 2007), L-moments analysis (Hosking 1990), L-moments associated with cluster analysis (Schaefer 1990; Guttman 1993; Wallis et al. 2007; Satyanarayana and Srinivas 2008), spatial correlation analysis (Gadgil and Yadumani 1993), homogeneity test (Wiltshire 1986) and regional frequency analysis techniques (Eslamian and Feizi 2007; Ngongondo et al. 2011; Hossein and Arash 2014; Zhang and Hall, 2004). Due to the data shortage and hydro-meteorological complexity, the L-moment approach for RFA was carried out to ensure efficient estimation of extreme rainfall occurrences, taking account of the spatial variability of the study area.
The primary intent of this study is to carry out an efficient RFA of extreme rainfall by applying the L-moment approach initially developed by Hosking and Wallis (1997). The specific aims of this research study are as follows: a) to carry out a RFA method of 6-h, 12-h, 24-h maximum consecutive rainfall series using L-moment approach, b) to evaluate the accuracy of design rainfall and reliability on goodness-of-fit test by consecutive hour rainfall (likely to be more accurate than daily and monthly rainfall), and c) to provide an appropriate estimation with 90% confidence intervals for the uncertainty analysis. This information perceived from the study may provide useful probability distribution upshots for extreme rainfall events.
Study area and data description
List and type of the five rainfall stations’ utilized for analysis
Rainfall station | GPS point | Region | Elevation (m) | Period of data collection | Record length (year) | One-day max. Rainfall (mm) |
---|---|---|---|---|---|---|
Jeju | 33°31′ N 126°31′ E | Northern | 20 | 1964–2013 | 50 | 615.6 |
Ara | 33°27′ N 126°33′ E | North-Eastern | 379 | 2001–2013 | 13 | 838.5 |
Eorimok | 33°22′ N 126°32′ E | Mountain | 972 | 1995–2013 | 19 | 909.5 |
Witsaeorum | 33°23′ N 126°29′ E | Mountain | 1673 | 2003–2013 | 11 | 1396.5 |
Jindallaebat | 33°21′ N 126°30′ E | Mountain | 1490 | 2003–2013 | 12 | 1183.5 |
Methods
L-moments: Theoretical background
Where, τ_{2} is the measure of covariance (scale), τ_{3} is the measure of skewness (shape) with values ranging from 0 to 1, and τ_{4} is the measure of kurtosis (peakedness). Notable, these ratio estimator equations and their graphical diagrams are particularly good to identify the distributional properties of highly skewed data. Thus, following the above equations rainfall of 6-h, 12-h and 24-h L-moments ratio for each region has shown in this study.
Data screening by discordancy measure
Where, u_{i} = vector of L-CV, L-Skewness and L-Kurtosis; S is covariance matrix of u_{i} and \( \overset{-}{\mathrm{u}} \) is the mean vector of u_{i}.
Regional heterogeneity test
For H statistics criterion, Hosking and Wallis (1993) suggested that the region is reasonably homogeneous if H < 1, possibly homogeneous region if 1 ≤ H < 2 and absolutely heterogeneous region if H ≥ 2.
Goodness-of-fit measure
Here, \( {\mathrm{t}}_4^{\mathrm{R}} \) is an average L-Kurtosis value of the data from a given region, \( {\uptau}_4^{\mathrm{Dist}} \) is a theoretical L-Kurtosis value for a fitted distribution and σ_{4} is the standard deviation value that obtained from simulated data. For an approximate 90% confidence level, the acceptable goodness-of-fit is found at |Z^{Dist}| ≤ 1.64.
Estimation of regional rainfall quantiles
The frequency distribution procedure of maximum consecutive hour rainfall data in a homogeneous region consist of similar quantile distribution (Dalrymple 1960). In the simulations, quantile estimated for various robust probability distributions were calculated. If the quantile estimates consisted of regional growth curve Q^{m}(F), i site’s non-exceedance probability F and site scaling factor, then the T-year quantile of the normalized regional distribution is computed by: Q_{i}(F) = l_{1}q(F); where q is common dimensionless function. For simulation of a homogeneous region, the regions had the same number of stations, data record length, heterogeneity and L-moments ratio as the observed data. During simulation, quantiles error, root mean square error (RMSE) and 90% error bounds were estimated from that assessment can be provided the accuracy level.
All kinds of statistical analysis and graphical representation for this study were done in R statistical program of 3.2.0 and MS excel 2007 version. The L-moment approach (lmomRFA 3.0–1 version) was also used in R package, developed by Hosking (2009).
Results and discussions
Rainfall availability
Stationary and independence test
Summary of trend analysis of maximum hourly rainfall series using Mann-Kendall test
No | Station | Trend value | p-value |
---|---|---|---|
1 | Jeju | 1.67 | 0.01 |
2 | Ara | 1.28 | 0.02 |
3 | Eorimok | 3.40 | 0.02 |
4 | Jindallaebat | −0.47 | 0.04 |
5 | Witsaeorum | −0.16 | 0.03 |
Identification of homogeneous region by cluster based analysis
Region 1 (Jeju and Ara station) is situated in the urban portion of northern part of Jeju Island with an average elevation of 253 m, recording an average annual rainfall of around 1835 mm. Region 2 (Eorimok station) is located in the middle portion of Hancheon catchment, which is a semi urban area with an average elevation of 950 m, recording an average rainfall of 2436 mm. Region 3 (Witsaeorum and Jindallaebat station) is situated near Hallasan Mountain with an average elevation of 1570 m and recording an average rainfall of around 2361 mm. The rainfall characteristics of region 3 are fully influenced by tropical and mountainous winds.
Estimation of L-moments, homogeneity test and best fitted distribution
Discordance, heterogeneity measure and best fitted distribution for three regions
Region | Discordance (D _{ i }) | Heterogeneity measure | Best fitted distribution | Z ^{ Dist } value | ||
---|---|---|---|---|---|---|
H1 | H2 | H3 | ||||
1 | Jeju (1.61), Ara (0.95) | 0.64 | −0.53 | −1.97 | Gumbel | 0.54 |
2 | Eorimok (1.53) | −0.36 | 0.72 | −1.78 | Gumbel | 1.25 |
3 | Jinadallaebat (1.37) Witsaeorum (1.28) | −0.13 | −1.40 | −2.11 | GEV | 1.03 |
Estimation of regional growth curves
Simulation results of estimated regional quantiles, RMSE and corresponding 90% error bounds values
Region | Distribution | Return period (year) | F | q(F) | RMSE | Error bound (mm) | |
---|---|---|---|---|---|---|---|
Lower | Upper | ||||||
1 | Gumbel | 5 | 0.8 | 1.129 | 0.014 | 1.046 | 1.170 |
10 | 0.9 | 1.347 | 0.032 | 1.263 | 1.481 | ||
20 | 0.95 | 1.526 | 0.057 | 1.514 | 1.596 | ||
50 | 0.98 | 1.721 | 0.072 | 1.608 | 1.739 | ||
70 | 0.985 | 1.876 | 0.084 | 1.758 | 1.924 | ||
80 | 0.987 | 2.039 | 0.094 | 1.982 | 2.145 | ||
100 | 0.999 | 2.275 | 0.105 | 2.239 | 2.303 | ||
2 | Gumbel | 5 | 0.8 | 1.134 | 0.077 | 1.027 | 1.243 |
10 | 0.9 | 1.632 | 0.148 | 1.522 | 1.801 | ||
20 | 0.95 | 1.859 | 0.165 | 1.839 | 1.954 | ||
50 | 0.98 | 2.102 | 0.174 | 2.01 | 2.227 | ||
70 | 0.985 | 2.846 | 0.203 | 2.621 | 2.981 | ||
80 | 0.987 | 3.64 | 0.218 | 3.105 | 3.764 | ||
100 | 0.999 | 4.023 | 0.237 | 3.978 | 4.135 | ||
3 | GEV | 5 | 0.8 | 1.079 | 0.115 | 0.960 | 1.092 |
10 | 0.9 | 1.923 | 0.197 | 1.799 | 2.163 | ||
20 | 0.95 | 2.754 | 0.243 | 2.548 | 2.936 | ||
50 | 0.98 | 3.628 | 0.266 | 3.332 | 3.847 | ||
70 | 0.985 | 4.507 | 0.279 | 4.395 | 4.715 | ||
80 | 0.987 | 5.706 | 0.285 | 5.389 | 5.902 | ||
100 | 0.999 | 7.656 | 0.301 | 7.459 | 7.829 |
Regional quantile analysis
Results of the consecutive hour (6-h, 12-h, 24-h) regional rainfall quantile for five station
Station | Consecutive hour | Non-exceedance probability (return period, year) | ||||||
---|---|---|---|---|---|---|---|---|
5 | 10 | 20 | 50 | 70 | 80 | 100 | ||
0.800 | 0.900 | 0.950 | 0.980 | 0.985 | 0.987 | 0.999 | ||
Jeju | 6 | 165.12 | 193.32 | 224.38 | 251.88 | 272.88 | 279.92 | 283.32 |
12 | 208.02 | 233.62 | 259.72 | 285.02 | 298.14 | 308.19 | 312.22 | |
24 | 246.17 | 268.19 | 289.24 | 308.31 | 326.35 | 341.05 | 344.65 | |
Ara | 6 | 252.05 | 282.02 | 303.39 | 323.27 | 328.85 | 330.85 | 333.97 |
12 | 400.77 | 442.71 | 471.45 | 497.04 | 503.96 | 506.42 | 510.20 | |
24 | 548.88 | 621.50 | 672.30 | 708.57 | 731.33 | 735.89 | 742.94 | |
Eorimok | 6 | 163.23 | 181.40 | 199.52 | 211.81 | 224.15 | 230.93 | 235.11 |
12 | 185.15 | 208.19 | 226.42 | 241.89 | 256.45 | 265.21 | 270.64 | |
24 | 227.03 | 253.18 | 272.91 | 288.67 | 301.06 | 307.18 | 311.03 | |
Witsaeorum | 6 | 243.76 | 267.47 | 287.90 | 306.16 | 321.45 | 332.16 | 337.09 |
12 | 307.39 | 337.54 | 363.72 | 383.59 | 400.87 | 410.60 | 414.27 | |
24 | 418.49 | 452.76 | 478.54 | 498.62 | 518.11 | 531.48 | 539.54 | |
Jindallaebat | 6 | 183.46 | 209.58 | 229.37 | 244.70 | 257.49 | 268.84 | 276.60 |
12 | 318.26 | 352.45 | 374.80 | 391.29 | 407.08 | 420.66 | 427.51 | |
24 | 452.03 | 484.46 | 509.57 | 530.73 | 544.10 | 551.75 | 555.18 |
The station analysis showed that the Eorimok station’s probable rainfall (163.23 mm to 311.03 mm) is lower than the other stations. This is because of the Eorimok station’s location within the transitional steep slopes between forest and hilly regions, which yields less rainfall accumulation due to varying elevations and slopes. Above all, the L-moments technique depicts accurate predictions of all kinds of statistical analysis and as such, the method can be suggested for policies and decision makings pertaining to hydrological catchment design.
Conclusions
The maximum consecutive hour rainfall data was analyzed using L-moments approach, to study the spatial homogeneity, probability distributions and as well as regional frequency estimates. The entailed careful data screening from historical rainfall events, carried out using cluster based dendogram analysis. From ward’s classification, three reasonably homogeneous regions were suggested for Hancheon catchment (Jeju and Ara in region 1, Eorimok in region 2 and Jindallaebat and Witsaeorum in region 3). After heterogeneity measure test, no limited discordant values were seen for the data sets. The L-moments ratio values varied within 0.1 to 0.4, which were considered as the statistical thresholds for the regional frequency analysis. The study concluded that Gumbel and generalized extreme value (GEV) distribution are more successful and reliable models for Hancheon catchment, which is marked by the relatively lower RMSE values (at 90% probability level). The analysis showed better rainfall predictions for region 1 (error bound between 1.046 to 2.303 mm); whereas for the region 2 (error bound between 1.027 to 4.135 mm) and region 3 (error bound between 0.960 to 7.829 mm) significant errors were found. Considering the spatial variations of hydro-meteorological and topographic characteristics, the rainfall estimates for different regions can be considered as useful hydrological design attributes. In spite of the statistical and design related findings researched thoroughly in this study a number of limitations still persist, leaving potentials for future identical research. The datasets used to develop the statistical analysis were limited, for which the scope of the study was confined within a definite statistical approach (L-moments). With more availability of hydro-meteorological data, more statistical as well as locally established mathematical tools could be taken into consideration which could further emphasize on the perfection of the technique. Furthermore, the study was carried out using five rainfall stations only. With more rainfall stations, relatively finer homogeneous clusters may be developed, which would further improve the local accuracies of rainfall estimates. However, given the size and locations of data availability, the study inferred good results. Moreover, the methodological framework used in the study is not only applicable for the Jeju Island but can also be implied in other similar areas where the rainfall data records are limited and the land slope is steep.
Declarations
Acknowledgments
This research was supported by a grant from regional development research program (15RDRP-B076272-02) funded by the Ministry of Land, Infrastructure and Transport of Korean Government.
Authors’ contributions
KKK structured the full manuscript. He was also responsible for using R Programming and developed the statistical analysis. SKY supervised to prepare methodology from different techniques and prepared the discussion section. JHL carried out the rainfall data analysis. FKK was responsible to develop the study background and prepare draft on the study area section. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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