1. 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

2. 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

3. 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

4. Examples of annual max events in Taiwan
Design durations

6. Proportion of rainfall stations observing annual max rainfalls.

7. 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

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

9. 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

10. 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

11. 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 mm
泰武 mm
大湖山 1329 mm
阿禮 mm
12/02/2016
Dept of Bioenvironmental Systems Engineering, NTU

12. 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
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Dept of Bioenvironmental Systems Engineering, NTU

13. 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
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Dept of Bioenvironmental Systems Engineering, NTU

14. 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

15. 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
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Dept of Bioenvironmental Systems Engineering, NTU

16. 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：
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Dept of Bioenvironmental Systems Engineering, NTU

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

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Dept of Bioenvironmental Systems Engineering, NTU

19. 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。
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Dept of Bioenvironmental Systems Engineering, NTU

20. 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

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

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23. 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.
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Dept of Bioenvironmental Systems Engineering, NTU

24. 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
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Dept of Bioenvironmental Systems Engineering, NTU

25. Semi-variograms of EMRs of different durations
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33. 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
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Dept of Bioenvironmental Systems Engineering, NTU

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Dept of Bioenvironmental Systems Engineering, NTU

35. 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.

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

37. 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.

38. 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.

39. 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

40. 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

41. Validation of the simulation results Comparing site-specific ECDF of historical and simulated data
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Dept of Bioenvironmental Systems Engineering, NTU

42. EMR
AMR
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Dept of Bioenvironmental Systems Engineering, NTU

43. EMR
AMR
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Dept of Bioenvironmental Systems Engineering, NTU

44. 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

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

46. 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

47. References
Hsieh, H.I., Su, M.D., Wu, Y.C., Cheng, K.S., Water shortage risk assessment using spatiotemporal flow simulation. Geoscience Letters, 3:2, DOI /s
Hsieh, H.I., Su, M.D., Cheng, K.S., Multisite Spatiotemporal Streamflow Simulation - With an Application to Irrigation Water Shortage Risk Assessment. Terrestrial, Atmospheric, Oceanic Sciences, 25(2):
Wu, Y.C., Liou, J.J., Cheng, K.S., Establishing acceptance regions for L-moments based goodness-of-fit tests for the Pearson type III distribution. Stochastic Environmental Research and Risk Assessment, 26: , DOI /s z.
Wu, Y.C., Hou, J.C., Liou, J.J., Su, Y.F., Cheng, K.S., Assessing the impact of climate change on basin-average annual typhoon rainfalls with consideration of multisite correlation. Paddy and Water Environment, 10(2): , DOI /s
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Dept of Bioenvironmental Systems Engineering, NTU

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

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

50. 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]
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Dept of Bioenvironmental Systems Engineering, NTU

51. General equation for hydrological frequency analysis
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Dept of Bioenvironmental Systems Engineering, NTU

52. 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]
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Dept of Bioenvironmental Systems Engineering, NTU

53. Gamma density
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Dept of Bioenvironmental Systems Engineering, NTU

54. 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.
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Dept of Bioenvironmental Systems Engineering, NTU

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56. 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.
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Dept of Bioenvironmental Systems Engineering, NTU

57. 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
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60. Flowchart of BVG simulation (1/2)
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61. Flowchart of BVG simulation (2/2)
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Dept of Bioenvironmental Systems Engineering, NTU

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63. [B]
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66. 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
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Dept of Bioenvironmental Systems Engineering, NTU

67. Correlation coefficient of KX and KY can be derived as follows:
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70. Since KX and KY are distributed with zero means, it follows that
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72. It can also be shown that
Thus,
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Dept of Bioenvironmental Systems Engineering, NTU

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

75. Proof of Eq. (B) as a single-value function
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Dept of Bioenvironmental Systems Engineering, NTU

76. Therefore,
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Dept of Bioenvironmental Systems Engineering, NTU

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81. 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.
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Dept of Bioenvironmental Systems Engineering, NTU

82. Conceptual description of a gamma random field simulation approach