1. Applied Hydrology Regional Frequency Analysis - Example Prof. Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

2. Estimating the return period of region-wide catastrophic rainfalls Ke-Sheng Cheng, Tsong-Hsiun Lien, Guan-Ming Su National Taiwan University 06/25/2013 2013 AOGS Conference 2

3. Introduction Occurrences of extraordinary rainfalls can complicate the work of hydrological frequency analysis. – Examples in Taiwan (Typhoon Morakot, 2009) Jia-Sien – 1040mm/24hr, 1601mm/48hr, 1856mm/72hr Weiliaoshan – 1415mm/24hr, 2216mm/48hr, 2564mm/72hr 06/25/2013 2013 AOGS Conference 3

4. Frequency analysis of 24-hr annual maximum rainfalls (AMR) at Jia-Sien station using 50 years of historical data – 1040mm/24 hours (by Morakot) excluding Morakot – 901 years return period – Return period inclusive of Morakot – 171 years 06/25/2013 2013 AOGS Conference 4 The same amount (1040mm/24 hours) was found to be associated with a return period of more than 2000 years by another study which used 25 years of annual maximum rainfalls.

5. Extraordinary rainfalls are extreme outliers. Whether outliers should be included/excluded in frequency is arguable. Random characteristics of extraordinary rainfalls – Occurrences of extraordinary rainfalls are very rare. – Within a not-too-long period, the probability of having repeated occurrences of extraordinary rainfalls at one station is very low. However, extraordinary rainfalls can occur at different locations. Regional frequency analysis (RFA) is adopted to deal with presence of extraordinary rainfalls in frequency analysis. 06/25/2013 2013 AOGS Conference 5

6. Previous studies have suggested that RFA performs better than the site-specific frequency analysis. However, how much confidence do we have? The main objectives of this study – To estimate the return period of catastrophic rainfalls using regional frequency analysis – To demonstrate the superior performance of RFA using stochastic simulation. 06/25/2013 2013 AOGS Conference 6

7. General procedures of regional frequency analysis 1.Data screening – Correctness check – Data should be stationary over time. 2.Identifying homogeneous regions – A set of characteristic variables are used for delineation of homogeneous regions. – Homogeneous regions are often determined by cluster analysis. 06/25/2013 2013 AOGS Conference 7

8. 3.Choice of an appropriate regional frequency distribution (GOF test) – GOF test using rescaled samples from different sites within the same homogeneous region. – The chosen distribution not only should fit the data well but also yield quantile estimates that are robust to physically plausible deviations of the true frequency distribution from the chosen frequency distribution. 06/25/2013 2013 AOGS Conference 8

9. 4.Parameter estimation of the regional frequency distribution – Estimating parameters of the site-specific frequency distribution. – Estimating parameters of the regional frequency distribution using record-length weighted average. 06/25/2013 2013 AOGS Conference 9

10. Study area and rainfall stations 06/25/2013 10 2013 AOGS Conference 28 rainfall stations in southern Taiwan. (1951 – 2010) Not all stations have the same record length. Annual maximum rainfalls (AMR) of various durations (1, 2, 6, 12, 18, 24, 48, 72 hours)

11. Homogeneous regions identification using Cluster analysis – Characteristic variables: Mean, standard deviation and coeff. of skewness of annual maximum rainfalls. – Cluster analysis was conducted for AMR of various durations. – Two homogeneous regions with 21 satations (region I) and 7 stations (region II), respectively, were identified. 06/25/2013 2013 AOGS Conference 11

12. (Mean, std dev, skewness) space of the gamma density 06/25/2013 2013 AOGS Conference 12 A 3-parameter distribution

13. Regional frequency analysis Delineating homogeneous regions 06/25/2013 2013 AOGS Conference 13

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16. Hot spots for occurrences of extreme rainfalls 06/25/2013 2013 AOGS Conference 16 1992 – 2010 Number of extreme typhoon events

17. Choice of an appropriate regional frequency distribution (GOF test) Site-specific rescaled annual max rainfalls – Rescaled with respect to site-specific mean and standard deviation Rescaled AMR is equivalent to the frequency factor, K. Rescaled AMR can be considered as an index variable with zero expectation and unity standard deviation. – Other studies also used as the index variable. 06/25/2013 17 2013 AOGS Conference

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19. Region I – Extreme value type I (EV1) distribution Region II – Log Pearson type III (LPT3) distribution – Considering the results of GOF tests for AMR of various durations – AIC, BIC and HQIC values were calculated for best-fit model selection. 06/25/2013 2013 AOGS Conference 19

20. Regional frequency analysis parameter estimation Method of L-moments for site-specific parameter estimation Regional parameter estimation Establishing regional growth curves for individual homogeneous regions – Region 1: Extreme Value type I – Region 2: Log Pearson type III (Model selection was based on the criterion of loss of information using AIC, BIC and HQIC.) 06/25/2013 2013 AOGS Conference 20

21. RFA results index variable 06/25/2013 2013 AOGS Conference 21

22. RFA results index variable 06/25/2013 2013 AOGS Conference 22

23. 24-hr, 100-year rainfall at the Jia-Sien station – 1018 mm (using (X-  )/  as the index variable) The 24-hr rainfall of Morakot (1040 mm) is associated with a return period of 115 years. – 1648 mm (using (X/  ) as the index variable) Site-specific frequency analysis 06/25/2013 2013 AOGS Conference 23 Which index variable performs better? Does RFA really perform better than the site- specific freq analysis? Or, just by chance?

24. Stochastic simulation Simulating n years (same as the record length of the historical data) of annual maximum rainfalls at each individual station, using site- specific distribution parameters. Such simulated data set is called a block of simulated samples. Generating 1000 blocks of simulated samples. Conducting site-specific frequency analysis and RFA for each block of simulated samples. Calculating 24-hr rainfalls of 5, 20, 50, 100, 200 years return period for each block of simulated samples. 06/25/2013 2013 AOGS Conference 24

25. RMSE comparison 06/25/2013 2013 AOGS Conference 25 RFA Site-specific using (X/  ) as the index variable using (X-  )/  as the index variable Probability for RFA (using (X-  )/  as the index variable) being superior = 0.77.

26. Further study Modeling dependence of extraordinary rainfall occurrences at different stations. 06/25/2013 2013 AOGS Conference 26

27. Conclusions Regional frequency analysis using (X-  )/  as the index variable is recommended to deal with extraordinary rainfalls (extreme outliers). It has been demonstrated through stochastic simulation that there is a high probability (0.77 in our study) that RFA performs better than site-specific frequency analysis. 06/25/2013 2013 AOGS Conference 27

28. Thanks for listening. Your comments and suggestions are most welcome. 06/25/2013 2013 AOGS Conference 28