Estimating Design Rainfalls Using
Dynamical Downscaling Data
Ke-Sheng Cheng
Department of Bioenvironmental Systems Engineering
Mater Program in Statistics
National Taiwan University

Outline• Introduction• GEV modeling of the extreme values• Mixture distribution of order statistics• Demonstration by stochastic simulation• Demonstration using observed rainfall datain Taiwan• Conclusions

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International Workshop on Climate Downscaling Studies 20172Introduction• Design rainfalls– Design rainfalls represent rainfall depths ofspecific storm durations (e.g., 1, 2, 6, 12, 24, 48,
72 hours) which, on average, occur once in every
T (5, 20, 100, 200, 1000) years.
– Design rainfalls are essential for hydrologicalmodeling and water resources planning andengineering design.
– Estimation of design rainfalls under climatechange requires downscaling GCM rainfall
outputs to hourly (or even sub-hourly) scale.
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International Workshop on Climate Downscaling Studies 20173• Previous work – Stochastic Storm Rainfall
Simulation Model (SSRSM)– Annual count of storm occurrences (Poissondistribution)– Bivariate distribution of storm duration andevent-total rainfall depth (Non-Gaussianbivariate distribution)– Temporal variation of hourly rainfalls of
individual storms (Bivariate truncated-gamma
distribution + Markov process)– Meiyu (MCS), summer convective storms,typhoons, and winter frontal rainfalls are treatedseparately.International Workshop on Climate Downscaling Studies 201710/03/2017
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Annual counts of storm events estimated by ANN
FrontalMaiyuTyphoon

ConvectiveNorthCenterSouth

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Storm characteristics (average duration of typhoon)

Gauge observations

MRI (1979 - 2003)

Source:NCDR, Taiwan

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MRI (2015 – 2039)

MRI (2075 - 2099)

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Storm characteristics (average event-total rainfalls of typhoon)

Gauge observations

MRI (1979 - 2003)

Source:
NCDR, Taiwan

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MRI (2015 – 2039)

MRI (2075 - 2099)

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Stochastic storm rainfall process
Storm characteristics•Duration•Event-total depth•Inter-arrival(or inter-event) time
•Time variation of rain-rates

Inter-arrival time

Inter-arrival time

RainrateTotaldepth

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InternationalWorkshop on Climate
Downscaling Studies Duration
Duration
Duration
Duration2017

Time(hr)
8Season-specific storm characteristics

Rainfalls (mm)FrontalJan- April
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Storm type

PeriodFrontalNov - April

Mei-YuMay - June

Convective

July - October

TyphoonJuly - October

Mei-YuConvective,
TyphoonFrontal

May - June

July - October

Nov - Dec

International Workshop on Climate Downscaling Studies
20179Each simulation run yields an annual sequenceof hourly rainfalls. 500 runs were generated foreach rainfall station.
Time of storm occurrences(Duration, total depth) bivariate simulation
first-order Truncated Gamma-Markov simulation
Rainrate

Hourly rainfall
sequence
TotaldepthTime(hr)Duration

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Duration

Duration

DurationInternational Workshop on Climate Downscaling Studies 201710

Examples of hourly rainfall sequence(Kaoshiung)

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Hyetograph Simulation results(Typhoons)

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Time-to-peak and peak rainfall
percentage (Typhoons)• Empirical cumulative distribution functions

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Impact on design storm depths

TainanKaoshiung(Projection period: 2020-2039)

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Introduction• Most rainfall frequency analyses wereconducted using annual maximum series.
However, for stations with short record
length, i.e. small sample size, (for example,less than 30 years) and with presence ofoutliers, results of rainfall frequency analysis
will be less reliable.
• Annual maximum rainfalls of longer
durations ( > 12 hours) in Taiwan were
mostly produced by typhoon or meiyu events.
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• We propose using event-maximum rainfalls
for rainfall frequency analysis.
• By using the event-maximum rainfalls (ofvarious design durations), the sample size canbe increased.
• The annual count of events (in our case,typhoons) can vary from one year to anotherand can be considered as a Poisson randomvariable.
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International Workshop on Climate Downscaling Studies 201716• The average annual count of typhoons inTaiwan is roughly 3.4 (depending onlocations). Mean value of the Poisson
distribution.
Can be readily provided by• The basic idea is tobase our rainfall
high resolution dynamic
frequency analysisdownscalingon the probabilitydata.
distribution of t-hr event-maximum rainfalls
and the Poisson distribution whichcharacterizes the occurrence of typhoons. [Amixture distribution]
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• For stations whose annual maximum rainfalls
were produced by different storm types(typhoons and meiyu events), a storm-type
mixture of mixture distributions needs to beconsidered.

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GEV modeling of the extreme values

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Mixture distribution of order statistics
for annual maximum rainfalls• Order statistic– Given a random sample of size n (),the max order statistic
satisfies– Annual maximum rainfalls can be considered as
the maximum order statistic of individual years.
– However, the annual count of events has aPoisson distribution. Thus, annual maximum
rainfalls of different years have different
probability distributions.
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– Mixture distribution modeling. Annual count ofevents (k) has a Poisson distribution.

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International Workshop on Climate Downscaling Studies 201722• Three approaches (for calculation of T-yr
rainfall, XT)– Derive the probability distribution of annual
maximum rainfalls from the event-maximum
rainfalls and then find the 1-(1/T) quantitle of theannual maximum distribution. [Approach 1 Mixture distribution approach]– Calculate XT directly from the probabilitydistribution of event-maximum rainfalls[Approach 2 - Adjusted exceedance probabilityapproach]

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– Simulate many years (N = 10,000) of typhoon
occurrences and event-maximum rainfalls.
Extract N annual maximum rainfalls from thesimulation results and find XT . [Approach 3 Simulation approach]

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Demonstration by stochastic
simulation• Annual count of events has a Poisson
distribution (mean = 3.6)
• 1-hour event-maximum rainfall has a gamma
distribution with mean = 96 mm and stdev =76 mm. (scale = 60, shape = 1.6)
• We simulated 100,000 years of typhoon
occurrences and event-maximum rainfalls
using R.

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1-hr annual maximum

161
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1-hr event maximum96
Can be considered as the theoretical
distribution of annual maximam rainfalls
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• Rainfalls of various return periods (in years)(seq(20,200,10)) [Approaches 2 and 3]

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• Comparison between Approaches 1 and 3

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• The above simulation results demonstratethat annual maximum rainfalls are
asymptotically and theoretically a mixture
distribution of order statistics.
• Practically, dynamic downscaling data are
available for 25 or 30-year period.

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Demonstration using observed rainfall
data in Taiwan (limited record length)• 44 years of event-maximum rainfalls oftyphoons (頭汴坑) in Taiwan.
Return period in years

AMS-based

Typhoons-dominated 24-hour
Annual maximum rainfalls

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Mixture distribution – Poisson
distribution of annual count of typhoons
Return period in years

Typhoons-dominated 24-hour
Annual maximum rainfalls

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Design rainfall depth (mm)Typhoons-dominated24-hour Annual maximum rainfalls

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Return period (years)

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Demonstration using observed rainfall
data in Taiwan (limited record length)• 42 years of event-maximum rainfalls oftyphoons (Jia-Yi) in Taiwan.
Return period in years

AMS-based

Typhoons and Meiyu mixture of
24-hour annual maximum rainfalls
P(typhoon)=0.64, P(meiyu)=0.36 (tr=24)

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• A mixture of mixture distributions oftyphoons and meiyu event-maximumrainfalls.

Typhoons and Meiyu mixture of24-hour Annual maximum
rainfalls

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Design rainfall depth (mm)Typhoons and Meiyu mixture of
24-hour Annual maximum rainfalls
P(typhoon)=0.64, P(meiyu)=0.36 (tr=24)

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Return period (years)

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Conclusions• We demonstrate that the annual maximum
rainfalls in Taiwan (which are produced bytyphoons or meiyu events) can be characterized
by a mixture distribution.
• For stations with short record length, using
event-maximum rainfalls for rainfall frequency
analysis can provide good estimates of design
rainfalls.
• By adopting the proposed approach, design
rainfalls of the projection period can be derived
from high resolution dynamic downscaling data.
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Thanks for your attention.

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Projections in coarse
spatial and time scales.
(200 – 300 km; monthly)

24 GCMs

RCM

Projections in finer spatial scale.
(5km; monthly)

From GCM outputs to design
storm depths – a problem of scalemismatch (both temporal andspatial)
Design rainfall depths
For example, 24-hr, 100-year rainfall depth
Characteristics of extreme storm events

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• Characteristics of storm events• Number of storm events• Duration of a storm event• Total rainfall depth• Time variation of rainfall intensities• These characteristics are random in natureand can be described by certain probability
distributions.
• Although the realized values of these storm
characteristics of individual storm events
represent weather observations, theirprobability distributions are climate (longterm and ensemble) properties.
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• A GCM – stochastic model integratedapproach– Climatological projection by GCMs
• Changes in the means of storm characteristics• For examples,– Average number of typhoons per year
– Average duration of typhoons– Average event-total rainfall of typhoons– Hydrological projection by a stochastic storm
rainfall simulation model• Generating realizations of storm rainfall process using
storm characteristics which are representative of theprojection period.
• Preserving statistical properties of the all storm
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Projections in coarse
spatial and time scales.
(200 – 300 km; monthly)

24 GCMs

RCM

Projections in finer spatial scale.
(5km; monthly)

Weather Generator(Richardson type)

Conceptual flowchart

Projections in finer time scale.
(5km; daily)

ANN

Projections in point(spatial) and hourly(time) scales.

Stochasticstormrainfall
simulation

Characteristics of storm events
1 Number of storm events
2 Onset of storm occurrences
3 Duration of a storm event
4 Total rainfall depth
5 Time variation of rainfall intensity

Design rainfall depths
For example, 24-hr, 100-year rainfall depth
Characteristics of extreme storm events

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Climate change scenarios andGCM outputs• Emission scenario: A1B• Baseline period: 1980 – 1999
• Projection period– Near future: 2020 – 2039
– End of century: 2080 – 2099

• GCM model: 24 GCMs statisticaldownscaling• Hydrological scenario: changes in storm
characteristics
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Changes in monthly rainfalls (Statistical downscaling,Ensemble average with standard deviation adjustment)
Taipei area

Near future (2020 – 2039)
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Stochastic Storm Rainfall Simulation
Model (SSRSM)• Simulating occurrences of storms and their rainfallrates• Preserving seasonal variation and temporalautocorrelation of rainfall process.
• Duration and event-total depth• Inter-event times• Percentage of total rainfalls in individual intervals(Storm hyetographs)

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• Simulating occurrences of storm events ofvarious storm types– Number of events per year
• Poisson distribution for typhoon and Mei-Yu– Inter-event time
• Gamma or log-normal distributions

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• Simulating joint distribution of duration andevent-total depth– Bivariate gamma distribution (e.g. typhoons)– Log-normal-Gamma bivariate– Non-Gaussian bivariate distribution wastransformed to a corresponding bivariate
standard normal distribution with desired
correlation matrix.

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Bivariate gamma (X,Y)

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• Simulating percentages of total rainfalls
in individual intervals (Simulation ofstorm hyetographs)– Based on the simple scaling property• Durations of all events of the same storm types are
divided into a fixed number of intervals (e.g. 24 intervals).
• For a specific interval, rainfall percentages of different
events are identically and independently distributed (IID).
• Rainfall percentages of adjacent intervals are correlated.

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Modeling the storm hyetographProbability density of x(15)10/03/2017

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Taking all the above
properties into account,
we propose to modelthe dimensionlesshyetograph by atruncated gamma
Markov process.

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Truncated gamma
density (parametersestimation,including thetruncation level)

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– Rainfall percentages should sum to 100%
• Truncated gamma distributions• Conditional simulation is necessary• 1st order Markov process– Conditional simulation of first order truncatedgamma Markov process

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CWB Raingauge Network
More than 400 automatedrainfall stations establishedsince early 1990.
Until now, most stations haverecord length less than orclose to 20 years.
If rainfall frequency analysis
requires at least 40 years ofannual maximum rainfalls,
then we have to wait foranother 20 years.
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