Assessing High-Impact Weather
Variations and Changes Utilizing
Extreme Value Theory
Greg Holland
NCAR Earth System Laboratory
National Center for Atmospheric Research
Boulder, Co, USA
NCAR is Sponsored by NSF and this work is partially supported by the
Willis Research Network and the Research Program to Secure Energy for America
1
Presentation Summary
• The Issues
• Applying Extreme Value Theory
– Reverse Weibull and Generalized Pareto Examples
for Hurricanes
• Current and Future Climate
– Example Application
– Other Studies
• Suggest that Extreme Weather provides a
Bellwether of Climate Transition and Change
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The Issues
Frequency
Highly skewed
Short and noisy record
May be truncated at high values:
• Real physical limitation
• Resolution/capacity/rarity
May be truncated at low values:
• Observing practice
• High noise level
Societal Importance α rarity
Value
Typical weather-system PDF
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Early Applications of EVT
• Mearns, L. O., Katz, R. W., and Schneider, S. H., 1984: Extreme
High-Temperature Events: Changes in their Probabilities with
Changes in Mean Temperature. J. Clim. Appl. Meteorol. 23,
1601-1613.
• Katz, R. and B. Brown, 1992: Extreme Events in a Changing
Climate: Variability is more Important than Averages. Clim.
Change.,21, 289-302.
• Katz, R. and B. Brown, 1994: Sensitivity of Extreme Events to
Climate Change: The Case of Autocorrelated Time Series.
Environmetrics, 5, 451-462.
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Our Current Applications of EVT
• Helping assess data reliability for meteorological
extremes;
• Downscaling to build up an assessment of the tail of
the severe weather distribution utilizing the
truncated information that comes out of regional
climate models;
• Enabling recognition and understanding of potential
“regime” shifts in climate series;
• Assessing the “uniqueness” of specific extreme
events, or combinations thereof, in the context of
current and changing climate.
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Choice of Distribution
Frequency
For working with just the tail of the distribution,
the Generalized Pareto might be more
appropriate.
Value
Fisher-Tippett (-Gnedenko) Theorem (FTT) for Extreme Value Theory
states that the normalized maximum of any set of random variables can
be explained by one of three distributions:
• Weibull (Weibull 1939)
• Gumbel (Gumbel 1958) or
• Fréchet, (Fréchet 1927)
(also known as Type 1, 2 or 3: Fisher and Tippett 1928, Gnedenko 1948).
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Example Application: Hurricanes
1995-2008 a=149 b=46 lifmin=0
1
1
b
1935-2007
a=35
b=1.9
0.8
Normalized Frequency
Normalized Frequency
0.9
0.7
0.6
0.5
Intensity
0.4
0.3
0.2
0.1
0
20
40
60
80
Intensity (m/s)
100
120
0.8
0.6
Annual
Frequency
0.4
0.2
0
5
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15
20
Annual Number
25
30
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Weibull Analysis
We utilize the Reverse Weibull distribution
for which the CDF and PDF are:
CDF  f ( x)  1  e
 x
 
a
bx
PDF  f '( x)   
aa
b
b 1
e
 x
 
a
b
Where:
• parameters a and b determine the scale and the shape, respectively
• b=1 is the exponential distribution
• b=2 is the Rayleigh distribution, and
• b=3.5 is an approximation of the normal distribution.
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Sensitivity to Scale and Shape
Variations
0.03
a=35, b=1.9
0.025
Probability
0.02
0.015
0.01
a=40, b=1.9
0.005
0
0
a=35, b=2.3
20
40
60
Scale
80
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120
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Sensitivity to Scale and Shape
Variations
Hazard Rate Function for the Weibull distribution:
b 1
f (x)
b x 
H(x) :
  
1  F(x) a a 
(Hillier and Lieberman 1986)
Frequency

For b>1 the sensitivity
increases with increasing
x to a limit of 1-e-x .
(Katz and Brown 1994)
Value
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Weibull Analysis
Probability of exceeding a threshold event:
P ( E{x  c})  1  f (c)  e
c
 
a
b
Exceedance probability decreases as the event becomes rarer (c/a
increases), and/or the population less variable (b increases).
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Weibull Analysis
 1
Mean :   a  1  
 b
(where: Γ is the Gamma
function)
 2
2
Variance :   a   1    
 b
2
2
Γ(n) α (n-1)!
• Given µ and σ we can make an estimate of
a and b and thus the whole distribution.
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Application to Atlantic Hurricane
Intensity
Weibull Probability Plot
1
0.9
0.99
0.96
0.90
1935-2007
a=35
b=1.9
0.8
0.7
Log-Log Weibull
Probability Plot
0.75
0.6
Probability
Normalized Frequency
0.999
Intensity PDF
0.5
0.4
0.50
0.25
0.3
1935-2007
a=35
b=1.9
0.2
0.10
0.1
0
20
0.05
40
60
80
Intensity (m/s)
100
PDF (Normalized, HURDAT Smoothed)
120
32
42 48
Hurricane Intensity (m/s)
58
69
CDF with Weibull fit: a=35, b=1.9
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1
1980-1994
a=32, b=1.9
Mean=28.4 SD=15.5
PE69=1.4%
PE58=4.5%
PE32=36.8%
0.9
Normalized Frequency
0.8
0.7
0.8
0.6
0.5
0.4
0.3
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
20
40
60
80
Intensity (m/s)
100
1995-2007
a=35, b=1.6
Mean=31, SD=20
PE69=5.1%
PE48=19.1%
PE32=42.1%
0.9
Normalized Frequency
1
Application to Current Climate:
1980-1994 vs 1994-2007
120
0
20
40
60
80
Intensity (m/s)
100
120
• Mean intensity and SD change of 2-3
and 4-5 m/s
• 3-5 times increase in probability of Cat 5
hurricanes!
• Equivalent to moving from 1 Cat 5 every
3-5 years to 1 every year.
(Caveat on ability to fit the observed distribution)
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Application to Current Climate
1994-2008 a=343 b=19
1995-2008 a=149 b=46 lifmin=0
1
Observed
Weibull Estimated
Weibull Original
0.6
0.4
0.2
0
30
40
50
60
70
Intensity (m/s)
80
90
100
Normalized Frequency
Normalized Frequency
Intensity PDF
0.8
1
a
b
Annual
Frequency
PDF
0.8
0.6
0.4
0.2
0
5
10
15
20
Annual Number
25
30
Weibull original is for 1980-1994 data
Observed is 1995-2008 data
(Data from IBTrACS)
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Application to Truncated Model PDFs
The Issue: Regional Climate
Models cannot run over long time
scales at resolutions sufficient to
resolve weather extremes
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Background: Regional Climate
Modeling
12 km
36 km
150 km
Time Slices
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NRCM Hurricane Intensity Change
Model projected
increase in TC
intensity, but can only
resolve up to Cat 2.
Wind speed
distribution shifting
to right
How do we assess the
extremes?
Cannot use EVT extrapolation or quantile mapping
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Application to Model Data
• We make the following assumptions:
– the modeled changes are indicative of the changes
that would have been simulated in the full
distribution, were it resolved
– there is no process that will cause a change in the
unresolved tail of the distribution without any
signal in the resolved component.
• Note: the truncated model will underestimate
changes in mean and SD
– Thus this approach provides a conservative
assessment.
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Method
• Fit a Reflected Weibull to the observed distribution
for a specified historical period and calculate the
exceedance probability for the extremes of interest
• Calculate the mean and standard deviation changes
from current to future climate model predictions;
• Apply these changes to assess the changes to the
Reflected Weibull shape and strength parameters;
• Apply these new parameters to fill out an estimated
distribution and to calculate changes in exceedance
probability.
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Example: NRCM Hurricane Changes
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Application to NRCM
Hurricane Predictions
PE69=Cat5
PE58=Cat4,5
PE48=Major Hurricanes
PE32=Hurricanes
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Alternative Generalized Pareto
Distribution
(where a,b,c are the scale, shape and
location parameters)
Results:
• Exceedance probability decreases for
weak storms
• Exceedance probability increases for
intense storms
• Cat 5 by 29% for 2045-2055
compared to 32% for Weibull
(Suzuki 2011)
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Comparison with Alternative
Approaches
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Bender et al (2009)
• Used GFDL Hurricane Model applied to future
storms:
– Mean intensity increase ~2.5 m/s
– 78% increase in Cat4-5 numbers
• Applying EVT to 1995-2007 conditions (same
period as Bender et al):
– Mean intensity increase of 2.5 m/s gives ~35%
increase in Cat4-5;
– Adding increase in SD of 5 m/s gives ~60%
increase in Cat4-5.
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Yamada et al 2010
Cloud-resolving simulation over limited time period, with
and without Greenhouse Warming contribution.
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Hallegatte et al (2007)
Annual probability of landfall increase: Cat1=17% ,
Cat2=33%, Cat3=44%, Cat 4=58%, and Cat5=215%.
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Hallegatte et al ctd
• The nonlinearity of intense hurricane changes is a
clear illustration of the multiplicative effect of
extreme events:
– “a limited change in the characteristics of the mean (+13% in average
maximum wind speed at landfall) can have a dramatic impact on the
distribution tail (+215% increase in category-5 landfall probability)”.
• Moreover, because socio-economic impacts of
hurricanes are highly nonlinear with respect to
hurricane intensity, such an evolution may translate
into an even larger change for hurricane damages
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Severe Weather as a Bellwether of
Climate Transition and Change
• This analysis indicates that:
– Weather extremes will respond to climate
variability and change at much higher amplitude
than will the general weather-system population
– Thus, weather extremes provide a bellwether of
climate transition and change
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Summary
• Application of EVT to weather extremes can provide a
valuable tool for:
– diagnosing extremes
– assessing potential signals for climate variability and change
– downscaling assessment for any weather extreme
• EVT predicts that changes in extremes will be >>>
than those in the mean or variance:
– extreme hurricanes increase 2-3-fold for a 5-10% increase in the mean
and SD
– supported by both observations and modeling
– Implies climate variability and change will first appear in extremes, thus
• Weather extremes may provide a bellwether for
climate variability and change
Thank You
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