Hannover, 28 November 2006 Fehlergrenzen von Extremwerten des Wetters Errors bounds in extreme weather analyses Manfred Mudelsee University of Leipzig, Germany Climate Risk Analysis, Halle (S), Germany
Hannover, 28 November 2006 Fehlergrenzen von Extremwerten des Wetters Errors bounds in extreme weather analyses Manfred Mudelsee University of Leipzig,
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Hannover, 28 November 2006
Fehlergrenzen von
Extremwerten des Wetters
Errors bounds in
extreme weather analyses
Manfred Mudelsee
University of Leipzig, Germany
Climate Risk Analysis, Halle (S), Germany
What‘s it all about? Changing risk.
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
2% 5%
Present Future
What‘s it all about? Changing risk.
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
2% 5%
Present Future
Past Present
Message 1
Climate science: no certainty,
no proofs.
Rather:
hypothesis tests,
parameter estimates.
Message 2
Parameter estimates
(e.g., of flood risk)
without realistic error bars
are useless.
Basics
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
2% 5%
Theoretical example:
o daily runoff values
o one year, n = 365
What is the maximum value in a year?
Basics
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
2% 5%
Theoretical example:
5% > 3500 m3 s–1
return period =
20 years
Objective
Return period estimation using data
risk estimation
temporal changes
expected damages
Structure of talk
1 Return period estimation
2 Statistical uncertainties
3 Systematic uncertainties
Example: Elbe
1 Return period estimation
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
2% 5%
f(x)
x
1 Return period estimation
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
2% 5%
f(x)
x
Johnson et al. (1995) Continuous Univariate Distributions, Vol. 2, Wiley.
1 Return period estimation
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
2% 5%
f(x)
x
1 Return period estimation
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
2% 5%
f(x)
x
maximize LGEV maximize likelihood
that GEV model produced data
1 Return period estimation: Example
Elbe, Dresden, 1852–2002, summer,
annual maxima (n = 151)
1 8 5 0 1 9 0 0 1 9 5 0 2 0 0 0
Y e a r
0
1 0 0 0
2 0 0 0
3 0 0 0
4 0 0 0
5 0 0 0R u n o f f ( m 3
s – 1 )
0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0
R u n o f f ( m 3 s – 1 )
P D F
3 0 0 0 3 5 0 0 4 0 0 0 4 5 0 0 5 0 0 0
1 %
HQ100 = 3921 m3 s–1
2 Statistical uncertainties
n finite
GEV parameter estimation errors > 0
return period estimation error > 0
How large is error?
1. Theoretical derivation
2. Simulation
Johnson et al. (1995)
2 Statistical uncertainties: Simulation
Jackknife simulation:
Step 1: Remove randomly one point
Step 2: Fit GEV, estimate return period
Step 3: Go to Step 1 until 400
simulated return periods exist
Step 4: Take STD over simulations
2 Statistical uncertainties: Example
Elbe, Dresden, 1852–2002, summer,
annual maxima (n = 151)
Jackknife simulations of HQ100:
3886 3962 3895 3903 3960
3902 3920 3911 3957 3961
3953 3959 3886 3961 3959
3936 3892 3838 3957 3871
HQ100 = 3921 m3 s–1
Mean = 3923 m3 s–1 STD = 38 m3
s–1
3 Systematic uncertainties
3.1 Model suitability
fitted GEV
P D F
0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0
R u n o f f ( m 3 s – 1 )
3 0 0 0 3 5 0 0 4 0 0 0 4 5 0 0 5 0 0 0
1 %
empirical (kernel density)
3 Systematic uncertainties
3.2 Data errors: WQ relation
100 200 300 400W a te r sta g e (cm )
0
40
80
120
160
200
Ru
no
ff (m
3 s
–1)
1918–1986
1987–2003
Mudelsee et al. (2006) Hydrol. Sci. J. 51:818–833.Werra
3 Systematic uncertainties
3.2 Data errors: Simulation
Step 1: Qsim(i) = Q(i) + δQWQ(i)
Step 2: Combine Qsim(i) with
jackknife
3 Systematic uncertainties
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
1 000 2 000 3 000 4 000
Runoff (m 3 s– 1)
2% 5%
Present Future
3.3 Instationarity
3 Systematic uncertainties
Mudelsee et al. (2003) Nature 425:166–169.
3.3 Instationarity1000 1200 1400 1600 1800 2000
0.00.10.20.30.4
Occ
urre
nce
rate
(yr
-1)
123
Mag
nitu
de
0 .0
0 .1
Occ
urre
nce
rate
(yr
-1)
123
Mag
nitu
de
1000 1200 1400 1600 1800 2000Year
1200 1400 1600 1800 2000
0.0
0.1
0.2
0.3
Occ
urre
nce
rate
(yr
-1)
123
Mag
nitu
de
0 .0
0 .1
0.2
Occ
urre
nce
rat
e (
yr-1
)
123
Mag
nitu
de
1200 1400 1600 1800 2000Year
a E lbe, w in ter
b
c E lbe, sum m er
d h
g O der, sum m er
f
e O der, w inter
3 Systematic uncertainties
3.3 Instationarity = the real
challenge!
Time-dependent GEV parameters
Work in progress ...
Message 1
Climate science: no certainty,
no proofs.
Rather:
hypothesis tests,
parameter estimates.
Message 2
Parameter estimates
(e.g., of flood risk)
without realistic error bars
are useless.
Message 2
Parameter estimates
(e.g., of flood risk)
without realistic error bars
are useless.
Case 1 Q100 = 3921 m3 s–1 ± ???
Case 2 Q100 = 3921 m3 s–1 ± 38 m3
s–1
Case 3 Q100 = 3921 m3 s–1 ± 300 m3
s–1
THANKS!
Example 2: Extremes, Xout(T)
Elbe, winter, class 2–3
hCV = 35 yr
00 . 10 . 20 . 30 . 4O c c u r r e n c e
r a t e ( y r – 1 )
1 5 0 0 2 0 0 0
Bootstrap resample(with replacement,same size)
Elbe, winter, class 2–3
hCV = 35 yr
00 . 10 . 20 . 30 . 4O c c u r r e n c e
r a t e ( y r – 1 )
1 5 0 0 2 0 0 0
1 5 0 0 2 0 0 0
Example 2: Extremes, Xout(T)
Bootstrap resample(with replacement,same size)
Elbe, winter, class 2–3
hCV = 35 yr
00 . 10 . 20 . 30 . 4O c c u r r e n c e
r a t e ( y r – 1 )
1 5 0 0 2 0 0 0
1 5 0 0 2 0 0 0
Example 2: Extremes, Xout(T)
Bootstrap resample(with replacement,same size)
2ndBootstrap resample
00 . 10 . 20 . 30 . 4O c c u r r e n c e
r a t e ( y r – 1 )
1 5 0 0 2 0 0 0
1 5 0 0 2 0 0 0
1 5 0 0 2 0 0 0
Elbe, winter, class 2–3
hCV = 35 yr
Example 2: Extremes, Xout(T)
Bootstrap resample(with replacement,same size)
2ndBootstrap resample
2000Bootstrap resamples
00 . 10 . 20 . 30 . 4O c c u r r e n c e
r a t e ( y r – 1 )
1 5 0 0 2 0 0 0
1 5 0 0 2 0 0 0
1 5 0 0 2 0 0 0
Elbe, winter, class 2–3
hCV = 35 yr
Example 2: Extremes, Xout(T)
Elbe, winter, class 2–3
hCV = 35 yr
Bootstrap resample(with replacement,same size)
2ndBootstrap resample
2000Bootstrap resamples
00 . 10 . 20 . 30 . 4O c c u r r e n c e
r a t e ( y r – 1 )
1 5 0 0 2 0 0 0
1 5 0 0 2 0 0 0
1 5 0 0 2 0 0 0
Example 2: Extremes, Xout(T)
Elbe, winter, class 2–3
hCV = 35 yr
00 . 10 . 20 . 30 . 4O c c u r r e n c e
r a t e ( y r – 1 )
1 5 0 0 2 0 0 0
Example 2: Extremes, Xout(T)
90% bootstrap confidence band
Elbe, winter, class 2–3
hCV = 35 yr
00 . 10 . 20 . 30 . 4O c c u r r e n c e
r a t e ( y r – 1 )
1 5 0 0 2 0 0 0
Example 2: Extremes, Xout(T)
90% bootstrap confidence band
Cowling et al. (1996)J. Am. Statist. Assoc. 91:1516.
Mudelsee et al. (2004)J. Geophys. Res. 109:D23101.
1000 1200 1400 1600 1800 2000
0.00 .10 .20 .30 .4
Occ
urre
nce
rate
(yr
-1)
123
Mag
nitu
de
0 .0
0 .1
Occ
urre
nce
rate
(yr
-1)
123
Mag
nitu
de
1000 1200 1400 1600 1800 2000Year
1200 1400 1600 1800 2000
0.0
0 .1
0 .2
0 .3
Occ
urre
nce
rate
(yr
-1)
123
Mag
nitu
de
0 .0
0 .1
0 .2
Occ
urre
nce
rate
(yr
-1)
123
Mag
nitu
de
1200 1400 1600 1800 2000Year
a E lbe, w in ter
b
c E lbe, sum m er
d h
g O der, sum m er
f
e O der, w inter
Example 2: Extremes, Xout(T)
Mudelsee et al. (2003) Nature 425:166.
References
http://www.uni-leipzig.de/~meteo/MUDELSEE/
http://www.climate-risk-analysis.com
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