Estimating the return period of multisite rainfall extremes – An example of the Taipei City Prof. Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering, National Taiwan University

Outline Introduction Improving accuracy of hydrological frequency analysis Presence of outliers (extraordinary rainfall extremes) Spatial correlation (non-Gaussian random field simulation) Frequency analysis of multi-site rainfall extremes Stochastic simulation of multivariate event-maximum rainfalls. 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Introduction Frequent occurrences of disasters induced by heavy rainfalls in Taiwan Landslides Debris flows Flooding and urban inundation Increasing occurrences of rainfall extremes (some of them are record breaking) in recent years Almost all of the long-duration rainfall extremes (longer than 12 hours) were produced by typhoons. 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Examples of annual max events in Taiwan Design durations

Proportion of rainfall stations observing annual max rainfalls.

Problems in rainfall frequency analysis In Taiwan there exists a large number of rain gauges which are maintained by CWB and WRA. Most of them have near or less than 30 years record length . Design rainfalls play a key role in studies related to climate change and disaster mitigation. Problems in rainfall frequency analysis Short record length (less than 30 years) (small sample size) Record breaking rainfall extremes (presence of extreme outliers) Typhoon Morakot 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Typhoon Morakot (2009)  World record 0805-0810 2467 2361 1825 1624  World record 0805-0810 cumulative rainfall  Morakot 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

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

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 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. 06/25/2013 2013 AOGS Conference

Catastrophic storm rainfalls (or extraordinary rainfalls) are considered as extreme outliers. Whether or not such rainfalls should be included in site-specific frequency analysis is often disputed. 24-hr annual max. rainfalls (Morakot) of 2009 甲仙 1077 mm 泰武 1747 mm 大湖山 1329 mm 阿禮 1237 mm 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Concurrent occurrences of extraordinary rainfalls at different rain gauges Several stations had 24-hr rainfalls exceeding 100-yr return period. Site-specific events of 100-yr return period. What is the return period of the event of multi-site 100-yr return period? (100)^4 = 100,000,000 years (4 sites), assuming extreme rainfalls at different sites are independent. Redefining extreme events multi-site extreme events w.r.t. specified durations and rainfall thresholds Spatial covariation of rainfall extremes 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Spatial covariation of rainfall extremes By using site-specific annual maximum rainfall series for frequency analysis implies a significant loss of valuable information. Delineating homogeneous regions in the study area 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Study area and rainfall stations 15 rainfall stations in northern Taiwan. Annual maximum and typhoon event-maximum rainfalls of various durations (1, 2, 6, 12, 18, 24, 48, 72 hours) 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Event-maximum rainfalls of various design durations Event-max 1, 2, 6, 12, 18, 24, 48, 72-hr typhoon rainfalls at individual sites. Approximately 120 events 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Delineation of homogeneous regions (K-mean cluster analysis) 24 classification features (8 design durations x 3 parameters – mean, std dev, skewness) Normalization of individual features Two homogeneous regions Subregion 1 (mountainous area, higher rainfalls) Subregion 2 (plain area, lower rainfalls) 先講資料分群的目的及重要性： 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

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

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Rescaled variables for regional frequency analysis – Frequency factors 標準化的目的：在於移除絕對位置的影響。 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Goodness-of-fit test and parameters estimation L-moment ratio diagrams (LMRD) for goodness-of-fit test Pearson Type 3 distribution Parameters estimation Method of L-moments Record-length-weighted regional parameters 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

L-moment-ratio diagram GOF test Sample-size dependent acceptance region 甲仙（2）， t = 24hr PT3 Gaussian EV-1 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

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Covariance structure of the random field Covariance matrices are semi-positive definite. Experimental covariance matrices often do not satisfy the semi-positive definite condition. Modeling the covariance structure of frequency factors (event-max rainfalls) by variogram modeling. 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Relationship between semivariogram and covariance function Influence range Sill γ(h) C(h) 表示可以由半變異數得共變異數 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Semi-variograms of EMRs of different durations 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

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Stochastic Simulation of Multi-site Event-Max Rainfalls Pearson type III (Non-Gaussian) random field simulation Covariance Transformation Approach Covariance matrix of multivariate PT3 distribution Covariance matrix of multivariate standard Gaussian distribution Multivariate Gaussian simulation (Sequential Gaussian simulation) Transforming simulated multivariate Gaussian realizations to PT3 realizations 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

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Transforming Gaussian realizations to gamma realizations 12/1/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

12/1/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

An approximation of the chi-squared distribution by the standard normal distribution, known as the Wilson-Hilferty approximation, is given as follows (Patel and Read, 1996) where w represents the standard normal deviate and y is the corresponding chi-squared variate. 12/1/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

The above equation is an one-to-one mapping function between x and w. Transformation of the standard normal deviate w to gamma variate x can thus be derived as The above equation is an one-to-one mapping function between x and w. 12/1/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

10,000 samples were generated. Each simulation run yields one sample of t-hr multi-site event maximum rainfalls. Simulated samples preserve the spatial covariation of multi-site EMRs as well as the marginal distributions. 10,000 samples were generated. Multi-site t-hr EMRs of 10,000 typhoon events. The number of typhoons vary from one year to another. Annual count of typhoons is a random variable. 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Determination of t-hr rainfall of T-yr return period Average number of typhoons per year, m = 3.69 Return period, T=100 years Exceedance probability of the event-max rainfalls, pE=1/(100*3.69) Expected value of random sums 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Validation of the simulation results Comparing site-specific ECDF of historical and simulated data 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

EMR AMR 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

EMR AMR 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Estimating return period of multisite Nari rainfall extremes Four rain gauges within the Taipei City recorded 24-hr rainfalls exceeding 600mm during Typhoon Nari in 2001. Define a multi-site extreme event over 600 mm 24-hr rainfalls at all four sites Among the 10,000 simulated events, 19 events satisfied the above requirement. Average number of typhoons per year, m=3.69 Multi-site extreme event return period T = 143 years. 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Further look at the simulation results Preserving the spatial pattern of rainfall extremes 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Summary We developed a stochastic approach for simulation of multi-site event-max rainfalls to cope with the problems of outliers and short record length in hydrological frequency analysis. By increasing the sample size and considering the spatial covariation of EMRs, the return periods of site-specific and multi-site rainfall extremes can be better estimated. 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

References Hsieh, H.I., Su, M.D., Wu, Y.C., Cheng, K.S., 2016. Water shortage risk assessment using spatiotemporal flow simulation. Geoscience Letters, 3:2, DOI 10.1186/s40562-016-0034-7. Hsieh, H.I., Su, M.D., Cheng, K.S., 2014. Multisite Spatiotemporal Streamflow Simulation - With an Application to Irrigation Water Shortage Risk Assessment. Terrestrial, Atmospheric, Oceanic Sciences, 25(2): 255-266. Wu, Y.C., Liou, J.J., Cheng, K.S., 2012. Establishing acceptance regions for L-moments based goodness-of-fit tests for the Pearson type III distribution. Stochastic Environmental Research and Risk Assessment, 26: 873-885, DOI 10.1007/s00477-011-0519-z. Wu, Y.C., Hou, J.C., Liou, J.J., Su, Y.F., Cheng, K.S., 2012. Assessing the impact of climate change on basin-average annual typhoon rainfalls with consideration of multisite correlation. Paddy and Water Environment, 10(2): 103-112, DOI 10.1007/s10333-011-0271-5. 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Liou, J. J. Su, Y. F. , Chiang, J. L. , Cheng, K. S. , 2011 Liou, J.J. Su, Y.F., Chiang, J.L., Cheng, K.S., 2011. Gamma random field simulation by a covariance matrix transformation method. Stochastic Environmental Research and Risk Assessment, 25(2): 235 – 251, DOI: 10.1007/s00477-010-0434-8. Cheng, K.S., Hou, J.C., Liou, J.J., Wu, Y.C., Chiang, J.L., 2011. Stochastic Simulation of Bivariate Gamma Distribution – A Frequency-Factor Based Approach. Stochastic Environmental Research and Risk Assessment, 25(2): 107 – 122, DOI 10.1007/s00477-010-0427-7. Liou, J.J., Wu, Y.C., Cheng, K.S., 2008. Establishing acceptance regions for L-moments-based goodness-of-fit test by stochastic simulation. Journal of Hydrology, Vol. 355, No.1-4, 49-62. (doi:10.1016/j.jhydrol.2008.02.023). Cheng, K.S., Chiang, J.L., and Hsu, C.W., 2007. Simulation of probability distributions commonly used in hydrologic frequency analysis. Hydrological Processes, 21: 51 – 60. 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Thank you for listening. Your comments and suggestions are most welcome! 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Rationale of BVG simulation using frequency factor From the view point of random number generation, the frequency factor can be considered as a random variable K, and KT is a value of K with exceedence probability 1/T. Frequency factor of the Pearson type III distribution can be approximated by Standard normal deviate [A] 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

General equation for hydrological frequency analysis 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

The gamma distribution is a special case of the Pearson type III distribution with a zero location parameter. Therefore, it seems plausible to generate random samples of a bivariate gamma distribution based on two jointly distributed frequency factors. [A] 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Gamma density 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Assume two gamma random variables X and Y are jointly distributed. The two random variables are respectively associated with their frequency factors KX and KY . Equation (A) indicates that the frequency factor KX of a random variable X with gamma density is approximated by a function of the standard normal deviate and the coefficient of skewness of the gamma density. 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

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Thus, random number generation of the second frequency factor KY must take into consideration the correlation between KX and KY which stems from the correlation between U and V. 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Conditional normal density Given a random number of U, say u, the conditional density of V is expressed by the following conditional normal density with mean and variance . 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

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Flowchart of BVG simulation (1/2) 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Flowchart of BVG simulation (2/2) 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

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[B] 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

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Frequency factors KX and KY can be respectively approximated by where U and V both are random variables with standard normal density and are correlated with correlation coefficient . 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Correlation coefficient of KX and KY can be derived as follows: 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

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Since KX and KY are distributed with zero means, it follows that 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

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It can also be shown that Thus, 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

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We have also proved that Eq. (B) is indeed a single-value function. 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Proof of Eq. (B) as a single-value function 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Therefore, 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

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The above equation indicates increases with increasing , and thus Eq The above equation indicates increases with increasing , and thus Eq. (B) is a single-value function. 12/02/2016 Dept of Bioenvironmental Systems Engineering, NTU

Conceptual description of a gamma random field simulation approach