1 McGill University Department of Civil Engineering and Applied Mechanics Montreal, Quebec, Canada.

1

McGill UniversityMcGill UniversityDepartment of Civil Engineering Department of Civil Engineering

andandApplied MechanicsApplied Mechanics

Montreal, Quebec, CanadaMontreal, Quebec, Canada

2

STATISTICAL MODELING AND STATISTICAL MODELING AND ANALYSIS OF ANALYSIS OF

EXTREME PRECIPITATION PROCESSESEXTREME PRECIPITATION PROCESSES

Van-Thanh-Van NguyenVan-Thanh-Van Nguyenandand

Tan-Danh NguyenTan-Danh NguyenDepartment of Civil Engineering and Applied Mechanics Department of Civil Engineering and Applied Mechanics

McGill University McGill University Montreal, Quebec, CanadaMontreal, Quebec, Canada

andandOURANOS, Climate Change ConsortiumOURANOS, Climate Change Consortium

Montreal, Quebec, CanadaMontreal, Quebec, Canada

316-17 October 2003, Victoria, BC

OUTLINEOUTLINE

INTRODUCTIONINTRODUCTION OBJECTIVESOBJECTIVES METHODOLOGYMETHODOLOGY APPLICATIONSAPPLICATIONS CONCLUSIONSCONCLUSIONS


INTRODUCTIONINTRODUCTION Extreme Extreme stormsstorms and floods and floods account for more losses than any account for more losses than any

other natural disaster [other natural disaster [both both in terms of loss of lives and in terms of loss of lives and economic costs: Saguenay (Quebec) flood damages = economic costs: Saguenay (Quebec) flood damages = CAD CAD $800 million dollars; $800 million dollars; US average annual flood damages = US average annual flood damages = US$2.1 billion dollarsUS$2.1 billion dollars].].

Information on extreme rainfalls and floods is Information on extreme rainfalls and floods is essentialessential for for planning, design, and management of water-resource systems.planning, design, and management of water-resource systems.

Design Rainfall Design Rainfall = the maximum= the maximum amount amount of precipitation falling at of precipitation falling at a given point (or over a given area) for a specified a given point (or over a given area) for a specified durationduration and and return period return period Frequency analysis of extreme rainfall events. Frequency analysis of extreme rainfall events.

Climate variability and change Climate variability and change will have important impacts on will have important impacts on the hydrologic cyclethe hydrologic cycle, and in particular , and in particular extreme storm and flood extreme storm and flood eventsevents How to quantify these impacts? How to quantify these impacts?


DOWNSCALING METHODSDOWNSCALING METHODS

GCM

RCM or LAM(Dynamic

Downscaling)

StatisticalModels

(StatisticalDownscaling)

StochasticWeather

Generators

RegressionModels

Weather Typing orClassification

ImpactModels

(HydrologicModels)

low resolution high resolution

1 kmday, hour, minute

~ 300 kmmonth, season, year


……

The choice of an estimation method The choice of an estimation method depends on the availability of historical depends on the availability of historical data:data: Gaged SitesGaged Sites Sufficient long historical Sufficient long historical

records (> 20 years?) records (> 20 years?) At-site MethodsAt-site Methods.. Partially-Gaged SitesPartially-Gaged Sites Limited data Limited data

records records Regionalization MethodsRegionalization Methods.. Ungaged SitesUngaged Sites Data are not available Data are not available

Regionalization MethodsRegionalization Methods. .


Issues Related to Estimation Issues Related to Estimation of Extreme Rainfall Events:of Extreme Rainfall Events: At-site methodsAt-site methods

Current practice:Current practice: Annual maximum series (AMS) using 2- Annual maximum series (AMS) using 2-parameter Gumbel/Ordinary moments method, or using 3-parameter Gumbel/Ordinary moments method, or using 3-parameter GEV/ L-moments method.parameter GEV/ L-moments method.

Regionalization methodsRegionalization methods Current practice:Current practice: GEV/Index-flood method. GEV/Index-flood method.

Similarity (or homogeneity) of sites?Similarity (or homogeneity) of sites? How to define groups of homogeneous sites? What are How to define groups of homogeneous sites? What are

the classification criteria?the classification criteria? No general agreement on the choice of a suitable distribution No general agreement on the choice of a suitable distribution

model for extreme rainfalls.model for extreme rainfalls. What are the impacts of climate variability and change on What are the impacts of climate variability and change on

annual maximum series?annual maximum series?


…… The “scale” problemThe “scale” problem

The properties of a variable The properties of a variable depend ondepend on the the scale of measurement or observationscale of measurement or observation..

Are there Are there scale-invariancescale-invariance properties? And properties? And how to determine these scaling properties?how to determine these scaling properties?

Existing methods are limited to Existing methods are limited to the specific the specific time scaletime scale associated with the data used. associated with the data used.

Existing methods Existing methods cannotcannot take into account take into account the properties of the physical process over the properties of the physical process over different scales.different scales.


OBJECTIVESOBJECTIVES To propose new modelling methods that can To propose new modelling methods that can

take into account take into account the scaling propertiesthe scaling properties of the of the extreme rainfall process.extreme rainfall process.

To demonstrate To demonstrate the importance of scaling the importance of scaling considerationconsideration in the estimation of extreme in the estimation of extreme rainfalls.rainfalls.

To propose new To propose new regional estimation methodsregional estimation methods of extreme rainfalls for ungaged sites.of extreme rainfalls for ungaged sites.


METHODOLOGYMETHODOLOGY Scaling MethodsScaling Methods (for partially-gaged and (for partially-gaged and

ungaged sites)ungaged sites) The scaling concept: The scaling concept:

kkk tktfE

C

tfCtf

)()}({

)(

)().()(


Generalized Extreme-Value (GEV) Generalized Extreme-Value (GEV) Distribution.Distribution.

The cumulative distribution function:The cumulative distribution function:

The quantile: The quantile:

/1

)(1exp)(

xxF

]ln[1)( FFX


The The first three momentsfirst three moments of GEV distribution: of GEV distribution:

1 1

22

12

2

33 2

12

23

3

1

2

3

2

3 3

1

2 1

3 1

A B

A A B B

A A B A B B

A

B

.

. . .

. . . . .

/

/

( )

( )

( )


The Scaling GEV DistributionThe Scaling GEV Distribution


Estimation of Extreme Rainfalls Estimation of Extreme Rainfalls for Partially-Gaged Sitesfor Partially-Gaged Sites

Rainfall data are Rainfall data are not always availablenot always available for the time and space scales of for the time and space scales of interest.interest.

Short time interval rainfall extremes areShort time interval rainfall extremes are importantimportant for small watersheds, but for small watersheds, but not not always availablealways available..

DailyDaily rainfall data are rainfall data are widely availablewidely available.. Daily scale Daily scale shorter time scales ? shorter time scales ?


Methods of estimation of short-duration extreme rainfalls from long-duration extreme rainfalls MethodMethod 1. 1.Basic equation.Basic equation.

wherewhere

X t X t

tt

T T( ) . ( )

( )( )

1

1


…… MethodMethod 2 2Basic equation:Basic equation:

Parameters are estimated by the method of moments.Parameters are estimated by the method of moments.

])ln(1[)(

)()()( )(/1 t

T pt

tttX


......

Data used:Data used: Rainfall duration:Rainfall duration: from 5 minutes to 1 day. from 5 minutes to 1 day. Raingage network:Raingage network: 14 stations in Quebec. 14 stations in Quebec. Record lengths:Record lengths: from 15 yrs. to 48 yrs. from 15 yrs. to 48 yrs.


Observation of scaling regime : Observation of scaling regime :

STATION DORVAL

1

10

100

1000

10000

100000

1000000

Durations

Non

-cen

tral

mom

ents

k = 1 k = 2 k = 35 min 1 hour 1 day

3rd order moment.

2nd order moment.

1st order moment.

k (t)

t

)()()( kk tkt


Scaling characteristicsScaling characteristics

Orders of moment

Sca

ling

exp

onen

t

0.2

0.4

0.6

0.8

1.0

1.2

1 2 3

k

( k )

kk)(


ResultsResults on scaling regimes:on scaling regimes:

Non-central moments are Non-central moments are scalingscaling. . TwoTwo scaling regimes: scaling regimes:

5-min. to 1-hour interval.5-min. to 1-hour interval. 1-hour to 1-day interval.1-hour to 1-day interval.

The slope of the straight line is the The slope of the straight line is the estimate of the estimate of the scaling exponentscaling exponent b(k)b(k)..

Relationship between Relationship between (k)(k) and and kk,, for for kk = = 1 to 31 to 3, , are are linearlinear..


Based on these results, two Based on these results, two estimations were made:estimations were made:

5-min.5-min. extreme rainfalls from extreme rainfalls from 1-hr1-hr rainfalls rainfalls.. 1-hr.1-hr. extreme rainfalls from extreme rainfalls from 1-day1-day rainfalls. rainfalls.


5-min5-min extreme rainfalls from 1-hr extreme extreme rainfalls from 1-hr extreme rainfalls. rainfalls.

STATION DORVAL

Probability

5-M

in A

M R

ain

falls

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Obser. Method 1 Method 2


1-hr extreme rainfalls from 1-day 1-hr extreme rainfalls from 1-day extreme rainfalls.extreme rainfalls.

STATION DORVAL

0

5

10

15

20

25

30

35

40

45

50

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Probability

AM

hou

rly

rain

fall

(mm

)

Obser. Method 1 Method 2


McGILL

M1 = 60.639 T -0.7897

M2 = 4245 T -1.5506

M3 = 328179 T -2.289

1

10

100

1000

10000

100000

1000000

1 10 100

Duration (hrs)

Raw

mom

ents

(in

./100

)

M1 M2 M3

y = -0.7684 x

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

1 2 3 4

Order of moments

Sca

ling

expo

nent

s

)()()( kk tkt

kk)(


Results on the estimation methods:Results on the estimation methods:

Extreme rainfalls estimated in two cases by two Extreme rainfalls estimated in two cases by two methods were in methods were in good agreementgood agreement with with observations.observations.

Method 2 provided Method 2 provided more accuratemore accurate estimates estimates than method 1, especially at the two extremes. than method 1, especially at the two extremes.


Regional estimation of daily Regional estimation of daily extreme rainfalls for ungaged sitesextreme rainfalls for ungaged sites Homogeneous sitesHomogeneous sites are defined based on the are defined based on the similarity of rainfall occurrences similarity of rainfall occurrences (e.g.,(e.g., strong strong

correlationcorrelation of the number of rainy hours). of the number of rainy hours). Regional relationsRegional relations between statistical moments of daily extreme rainfalls and the mean between statistical moments of daily extreme rainfalls and the mean

number of rainy hours are developed for the homogeneous group.number of rainy hours are developed for the homogeneous group. Statistical moments of daily extreme rainfalls at an Statistical moments of daily extreme rainfalls at an ungaged siteungaged site are estimated using these are estimated using these

regional relations regional relations Distribution of daily extreme rainfalls is estimated for the ungaged site.Distribution of daily extreme rainfalls is estimated for the ungaged site.

RNM 619.012025.0

1 087.2 MeM 2013.03 38.200 MeM



BREBEUF

0

2

4

6

8

10

12

14

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Probability

1-da

y ex

trem

e ra

infa

ll (x

0.0

1 in

./hr)

Observed

At site

Regional method

DORVAL

0

2

4

6

8

10

12

14

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Probability

1-da

y ex

trem

e ra

infa

ll (x

0.0

1 in

./hr)

Observed

At site

Regional method

HUBERT

0

2

4

6

8

10

12

14

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Probability

1-da

y ex

trem

e ra

infa

ll (x

0.0

1 in

./hr)

Observed

At site

Regional method

McGILL

0

2

4

6

8

10

12

14

16

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Probability

1-da

y ex

trem

e ra

infa

ll (x

0.0

1in.

/hr)

Observed

At site

Regional method


Results on the regional estimation Results on the regional estimation methodmethod

RegionalRegional estimates are estimates are comparablecomparable with with corresponding corresponding at-siteat-site estimates. estimates.

Good agreementGood agreement between the estimates between the estimates (both at-site and regional) with the (both at-site and regional) with the observations indicates the feasibility of observations indicates the feasibility of the proposed regional estimation the proposed regional estimation method.method.


CONCLUSIONSCONCLUSIONS

Consideration of scaling properties of hydrologic Consideration of scaling properties of hydrologic processes could lead to the development of processes could lead to the development of more more accurate and more reliableaccurate and more reliable estimation methods. estimation methods.

Consideration of scaling properties of hydrologic Consideration of scaling properties of hydrologic processes could provide processes could provide better understandingbetter understanding of of the physical phenomenon studied. the physical phenomenon studied.

The The GEV distributionGEV distribution is suitable for regional is suitable for regional estimation of extreme rainfalls and floods.estimation of extreme rainfalls and floods.


CONCLUSIONSCONCLUSIONS (Continued)(Continued)

It is It is feasiblefeasible to assess the to assess the homogeneityhomogeneity of of extreme rainfall conditions at different extreme rainfall conditions at different locations based on the locations based on the similarity of rainfall similarity of rainfall occurrencesoccurrences..

Problems related to the estimation of extreme Problems related to the estimation of extreme rainfalls are still rainfalls are still far from being completely far from being completely solved.solved. integrated physical-statistical integrated physical-statistical approaches?approaches?


......

Thank You!Thank You!


Slides required for Slides required for presentationspresentations


I (mm/hr)

time (hr)

I (mm/hr)

time (hr)

True image



UNKNOWN TRUEIMAGE

A

A1A2

ΑΑΑ 21


Common probability distributions:Common probability distributions: Two-parameter distributionTwo-parameter distribution::

Gumbel distributionGumbel distribution NormalNormal Log-normal (2 parameters)Log-normal (2 parameters)

Three-parameter distributionsThree-parameter distributions:: Beta-K distributionBeta-K distribution Beta-P distributionBeta-P distribution Generalized Extreme Value distributionGeneralized Extreme Value distribution Pearson Type 3 distributionPearson Type 3 distribution Log-Normal (3 parameters)Log-Normal (3 parameters) Log-Pearson Type 3 distributionLog-Pearson Type 3 distribution Generalized Gamma distributionGeneralized Gamma distribution Generalized Normal distributionGeneralized Normal distribution Generalized Pareto distributionGeneralized Pareto distribution

Four-parameter distributionFour-parameter distribution Two-component extreme value distributionTwo-component extreme value distribution

Five-parameter distribution:Five-parameter distribution: Wakeby distributionWakeby distribution

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1 McGill University Department of Civil Engineering and Applied Mechanics Montreal, Quebec, Canada.

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