Transcript Recent Advances in the Modelling of Extremes and FLoods

McGill University
Department of Civil Engineering
and
Applied Mechanics
Montreal, Quebec, Canada
1
STATISTICAL MODELING AND
ANALYSIS OF
EXTREME PRECIPITATION PROCESSES
Van-Thanh-Van Nguyen
and
Tan-Danh Nguyen
Department of Civil Engineering and Applied Mechanics
McGill University
Montreal, Quebec, Canada
and
OURANOS, Climate Change Consortium
Montreal, Quebec, Canada
2
OUTLINE





INTRODUCTION
OBJECTIVES
METHODOLOGY
APPLICATIONS
CONCLUSIONS
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INTRODUCTION




Extreme storms and floods account for more losses than any
other natural disaster [both in terms of loss of lives and
economic costs: Saguenay (Quebec) flood damages = CAD
$800 million dollars; US average annual flood damages =
US$2.1 billion dollars].
Information on extreme rainfalls and floods is essential for
planning, design, and management of water-resource systems.
Design Rainfall = the maximum amount of precipitation falling at
a given point (or over a given area) for a specified duration and
return period  Frequency analysis of extreme rainfall events.
Climate variability and change will have important impacts on
the hydrologic cycle, and in particular extreme storm and flood
events  How to quantify these impacts?
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DOWNSCALING METHODS
RCM or LAM
(Dynamic
Downscaling)
Stochastic
Weather
Generators
GCM
Statistical
Models
(Statistical
Downscaling)
Weather Typing or
Classification
Impact
Models
(Hydrologic
Models)
Regression
Models
low resolution
~ 300 km
month, season, year
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high resolution
1 km
day, hour, minute
5
…

The choice of an estimation method
depends on the availability of historical
data:
Sites  Sufficient long historical
records (> 20 years?)  At-site Methods.
 Partially-Gaged Sites  Limited data
records  Regionalization Methods.
 Ungaged Sites  Data are not available 
Regionalization Methods.
 Gaged
16-17 October 2003, Victoria, BC
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Issues Related to Estimation
of Extreme Rainfall Events:
At-site methods
 Current practice: Annual maximum series (AMS) using 2parameter Gumbel/Ordinary moments method, or using 3parameter GEV/ L-moments method.
 Regionalization methods
 Current practice: GEV/Index-flood method.
Similarity (or homogeneity) of sites?
How to define groups of homogeneous sites? What are
the classification criteria?
 No general agreement on the choice of a suitable distribution
model for extreme rainfalls.
 What are the impacts of climate variability and change on
annual maximum series?

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…

The “scale” problem
 The
properties of a variable depend on the
scale of measurement or observation.
 Are there scale-invariance properties? And
how to determine these scaling properties?
 Existing methods are limited to the specific
time scale associated with the data used.
 Existing methods cannot take into account
the properties of the physical process over
different scales.
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OBJECTIVES



To propose new modelling methods that can
take into account the scaling properties of the
extreme rainfall process.
To demonstrate the importance of scaling
consideration in the estimation of extreme
rainfalls.
To propose new regional estimation methods
of extreme rainfalls for ungaged sites.
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METHODOLOGY

Scaling Methods (for partially-gaged and
ungaged sites)
 The
scaling concept:
f (t )  C ( ). f ( t )
C ( )  

k  E{ f (t )}   (k ) t
k
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k
10
Generalized Extreme-Value (GEV)
Distribution.

The cumulative distribution function:
   ( x   ) 1/  
F ( x )  exp  1 
 
  
 
 The quantile:


X ( F )    1  [  ln F ] 

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
The first three moments of GEV distribution:
 1  A  B.1
 2  A2  2 A. B. 1  B 2 . 2
 3  A3  3 A2 . B. 1  3 A. B 2 . 2  B 3 . 3
 A    / 
B   / 
 1   (  1 )
  2   ( 2  1 )
  3   ( 3  1 )
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The Scaling GEV Distribution
 (t )   (t )

 (t )   . (t )

 (t )   . (t )

X T (t )   . X T (t )
where
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

1 (t )

1 (t )
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Estimation of Extreme Rainfalls
for Partially-Gaged Sites
 Rainfall
data are not always available
for the time and space scales of
interest.
 Short time interval rainfall extremes are
important for small watersheds, but not
always available.
 Daily rainfall data are widely available.
Daily scale  shorter time scales ?
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Methods of estimation of short-duration
extreme rainfalls from long-duration
extreme rainfalls

Method 1.
Basic equation.

where
X T ( t )   . X T ( t )
  1( t )
 
 1( t )
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…
Method 2
Basic equation:

 (t )
1 /  ( t )
X T (t )   (t ) 
[1  (  ln p )
]
 (t )
 Parameters are estimated by the method of
moments.
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...
Data used:



Rainfall duration:
from 5 minutes to 1 day.
Raingage network: 14 stations in Quebec.
Record lengths:
from 15 yrs. to 48 yrs.
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Observation of scaling regime :
k (t)
k (t )   (k )t
 (k )
STATION DORVAL
1000000
Non-central moments
100000
3rd order moment.
10000
2nd order moment.
1000
100
1st order moment.
10
1
Durations
5 min
1 hour
k=1
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k=2
1 day
t
k=3
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Scaling characteristics  (k)  k  
(k)
1.2
Scaling exponent
1.0
0.8
0.6
0.4
0.2
1
2
Orders of moment
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3
k
19
Results on scaling regimes:




Non-central moments are scaling.
Two scaling regimes:
 5-min. to 1-hour interval.
 1-hour to 1-day interval.
The slope of the straight line is the
estimate of the scaling exponent
b(k).
Relationship between  (k) and k,
for k = 1 to 3, are linear.
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Based on these results, two
estimations were made:


5-min. extreme rainfalls from 1-hr rainfalls.
1-hr. extreme rainfalls from 1-day rainfalls.
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5-min extreme rainfalls from 1-hr extreme
rainfalls.
STATION DORVAL
5-Min AM Rainfalls
25
20
15
10
5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Probability
Obser.
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Method 1
Method 2
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1-hr extreme rainfalls from 1-day
extreme rainfalls.
STATION DORVAL
50
45
AM hourly rainfall (mm)
40
35
30
25
20
15
10
5
0
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Probability
Obser.
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Method 1
Method 2
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k (t )   (k )t
 (k )
McGILL
1000000
M3 = 328179 T -2.289
10000
M2 = 4245 T -1.5506
1000
M1 = 60.639 T -0.7897
100
10
1
1
M1
M2
M3
10
100
Duration (hrs)
1
 (k)  k  
2
3
4
0.0
Order of moments
Scaling exponents
Raw moments (in./100)
100000
-0.5
y = -0.7684 x
-1.0
-1.5
-2.0
-2.5
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Results on the estimation methods:


Extreme rainfalls estimated in two cases by
two methods were in good agreement with
observations.
Method 2 provided more accurate estimates
than method 1, especially at the two
extremes.
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Regional estimation of daily
extreme rainfalls for ungaged sites



Homogeneous sites are defined based on the
similarity of rainfall occurrences (e.g., strong
correlation of the number of rainy hours).
Regional relations between statistical moments of
daily extreme rainfalls and the mean number of rainy
hours are developed for the homogeneous group.
Statistical moments of daily extreme rainfalls at an
ungaged site are estimated using these regional
relations  Distribution of daily extreme rainfalls is
estimated for the ungaged site.
M 1  0.619 N R
M1  2.087 e0.025 M2
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M 3  200.38 e0.013 M2
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BREBEUF
DORVAL
14
14
Observed
Observed
12
At site
Regional method
1-day extreme rainfall (x 0.01 in./hr)
1-day extreme rainfall (x 0.01 in./hr)
12
10
8
6
4
2
At site
Regional method
10
8
6
4
2
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1.0
0.0
0.1
0.2
0.3
0.4
Probability
0.6
0.7
0.8
0.9
1.0
McGILL
HUBERT
16
14
Observed
14
Observed
12
At site
At site
1-day extreme rainfall (x 0.01in./hr)
1-day extreme rainfall (x 0.01 in./hr)
0.5
Probability
Regional method
10
8
6
4
2
Regional method
12
10
8
6
4
2
0
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Probability
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0.8
0.9
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Probability
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Results on the regional estimation
method


Regional estimates are comparable with
corresponding at-site estimates.
Good agreement between the estimates
(both at-site and regional) with the
observations indicates the feasibility of
the proposed regional estimation
method.
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CONCLUSIONS



Consideration of scaling properties of hydrologic
processes could lead to the development of more
accurate and more reliable estimation methods.
Consideration of scaling properties of hydrologic
processes could provide better understanding of
the physical phenomenon studied.
The GEV distribution is suitable for regional
estimation of extreme rainfalls and floods.
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CONCLUSIONS (Continued)


It is feasible to assess the homogeneity of
extreme rainfall conditions at different
locations based on the similarity of rainfall
occurrences.
Problems related to the estimation of
extreme rainfalls are still far from being
completely solved.  integrated physicalstatistical approaches?
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...
Thank You!
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Slides required for
presentations
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I (mm/hr)
True image
time (hr)
I (mm/hr)
time (hr)
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UNKNOWN TRUE
IMAGE
A
A1
A2
Α1  Α2  Α
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
Common probability distributions:
 Two-parameter distribution:
Gumbel distribution
Normal
Log-normal (2 parameters)
 Three-parameter distributions:
Beta-K distribution
Beta-P distribution
Generalized Extreme Value distribution
Pearson Type 3 distribution
Log-Normal (3 parameters)
Log-Pearson Type 3 distribution
Generalized Gamma distribution
Generalized Normal distribution
Generalized Pareto distribution
 Four-parameter distribution
Two-component extreme value distribution
 Five-parameter distribution:
Wakeby distribution
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