Applied Hydrology

Regional Frequency Analysis - Example

Prof. Ke-Sheng ChengDepartment of Bioenvironmental Systems Engineering

National Taiwan University

Estimating the return period of region-wide catastrophic rainfalls

Ke-Sheng Cheng, Tsong-Hsiun Lien, Guan-Ming SuNational Taiwan University

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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

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• 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

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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.

• 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.

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• 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.

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General procedures ofregional 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.

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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.

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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.

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Study area and rainfall stations

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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)

• 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.

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(Mean, std dev, skewness) space of the gamma density

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A 3-parameter distribution

Regional frequency analysis• Delineating homogeneous regions

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Hot spots for occurrences of extreme rainfalls

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1992 – 2010Number of extreme typhoon events

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.

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/)( X

)/( X

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• 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.

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Regional frequency analysisparameter 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.)

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RFA results index variable

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/)( X

RFA resultsindex variable

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/X

• 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

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Which index variable performs better?

Does RFA really perform better than the site-specific freq analysis? Or, just by chance?

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

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• RMSE comparison

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RFA Site-specific 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.

Further study• Modeling dependence of extraordinary rainfall

occurrences at different stations.

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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.

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Thanks for listening.Your comments and suggestions

are most welcome.

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