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Fighting the arch-enemy
with
mathematics and climate models
SETA - 6 March 2009
Henk van den Brink
KNMI
Fighting the arch-enemy with mathematics and climate models – p.1
The Netherlands with dikes..
Fighting the arch-enemy with mathematics and climate models – p.2
The Netherlands without dikes..
Fighting the arch-enemy with mathematics and climate models – p.3
Why this research?
1. Dutch law states that sea dikes have towithstand the sealevel that is reached once in104 years
2. What is the effect of increased greenhousegases on the extreme sea levels?
Fighting the arch-enemy with mathematics and climate models – p.4
Sea level depends on:
astronomical tides (deterministic)
sea level rise (slow process)
storm surge (stochastic):windsea level pressure
Fighting the arch-enemy with mathematics and climate models – p.5
Fighting the arch-enemy withmathematics:
Fighting the arch-enemy with mathematics and climate models – p.6
Xp vsk (γ not fixed):
Fighting the arch-enemy with mathematics and climate models – p.7
Xp vsk (γ = 0):
Fighting the arch-enemy with mathematics and climate models – p.8
As a Gumbel plot:
1.5
2
2.5
3
3.5
4
4.5
5
5.5
-2 0 2 4 6 8
2 5 10 25 50 100 103 104
wat
er le
vel [
m]
Gumbel variate
Hoek van Holland
return period
observations 1888-2005GEV to observations
Gumbel to observations
Fighting the arch-enemy with mathematics and climate models – p.9
water level in Hoek van Holland:
1.5
2
2.5
3
3.5
4
4.5
5
5.5
-2 0 2 4 6 8
2 5 10 25 50 100 103 104
wat
er le
vel [
m]
Gumbel variate
Hoek van Holland
return period
observations 1888-2005GEV to observations
Gumbel to observations
large statistical uncertainty
Fighting the arch-enemy with mathematics and climate models – p.10
water level in Hoek van Holland:
1.5
2
2.5
3
3.5
4
4.5
5
5.5
-2 0 2 4 6 8
2 5 10 25 50 100 103 104
wat
er le
vel [
m]
Gumbel variate
Hoek van Holland
return period
observations 1888-2005GEV to observations
Gumbel to observations
large statistical uncertainty
is extrapolation allowed...?
Fighting the arch-enemy with mathematics and climate models – p.10
water level in Hoek van Holland:
1.5
2
2.5
3
3.5
4
4.5
5
5.5
-2 0 2 4 6 8
2 5 10 25 50 100 103 104
wat
er le
vel [
m]
Gumbel variate
Hoek van Holland
return period
observations 1888-2005GEV to observations
Gumbel to observations
large statistical uncertainty
is extrapolation allowed...?
→ need more data optimally » 10000 year...
Fighting the arch-enemy with mathematics and climate models – p.10
Fighting the arch-enemy withclimate models:
global models
Fighting the arch-enemy with mathematics and climate models – p.11
Fighting the arch-enemy withclimate models:
global models
does not contain measurements
Fighting the arch-enemy with mathematics and climate models – p.11
Fighting the arch-enemy withclimate models:
global models
does not contain measurements
results depend on CO2 concentrations
Fighting the arch-enemy with mathematics and climate models – p.11
Fighting the arch-enemy withclimate models:
generate meteorological data with climatemodels:
Fighting the arch-enemy with mathematics and climate models – p.12
Fighting the arch-enemy withclimate models:
generate meteorological data with climatemodels:
ECMWF seasonal forecasts(1600 yrs) ⇒
Fighting the arch-enemy with mathematics and climate models – p.12
Fighting the arch-enemy withclimate models:
generate meteorological data with climatemodels:
ECMWF seasonal forecasts(1600 yrs) ⇒ESSENCE (ECHAM5 MPI-OM)20×(1950-2000)=1000 yrs17×(1950-2100)=2550 yrs
=3550 yrs
Fighting the arch-enemy with mathematics and climate models – p.12
Fighting the arch-enemy withclimate models:
generate meteorological data with climatemodels:
ECMWF seasonal forecasts(1600 yrs) ⇒ESSENCE (ECHAM5 MPI-OM)20×(1950-2000)=1000 yrs17×(1950-2100)=2550 yrs
=3550 yrs
feed wave/surge-model with wind and pressurefrom climate model
Fighting the arch-enemy with mathematics and climate models – p.12
Advantages ofmodelswrt observations:
strongly improved extreme-value-statistics:(almost) no extrapolation neededassumptions of extrapolation can be checked
dynamical-physical properties can beinvestigated
influence of greenhouse effect can bedetermined
Fighting the arch-enemy with mathematics and climate models – p.13
Possibilities:
extreme windextreme surge
extreme wave heights
extreme precipitation
extreme temperature
river discharges.....simultaneous occurrences of extremes
Fighting the arch-enemy with mathematics and climate models – p.14
Example 1: Surge in Hoek van Holland
0
1
2
3
4
5
6
7
-2 0 2 4 6 8
2 5 10 25 100 103 104
surg
e [m
]
Gumbel variate
return period [years]
observationsECMWF
→uncertainty 4 times smaller!
Fighting the arch-enemy with mathematics and climate models – p.15
Example 1: Surge in Hoek van Holland
1 febr 1953 ’26 dec 1987’
Fighting the arch-enemy with mathematics and climate models – p.16
Example 2: Maeslant closure barrier
Closure if level in Rotterdam ≥ 3 m NAP
Fighting the arch-enemy with mathematics and climate models – p.17
Example 2: Maeslant closure barrier
Closure if level in Rotterdam ≥ 3 m NAP
level influenced by:
Fighting the arch-enemy with mathematics and climate models – p.17
Example 2: Maeslant closure barrier
Closure if level in Rotterdam ≥ 3 m NAP
level influenced by:high tide at sea
Fighting the arch-enemy with mathematics and climate models – p.17
Example 2: Maeslant closure barrier
Closure if level in Rotterdam ≥ 3 m NAP
level influenced by:high tide at sealarge Rhine discharges
Fighting the arch-enemy with mathematics and climate models – p.17
Example 2: Maeslant closure barrier
0.2
0.5
1
2
5
10
0 0.2 0.4 0.6 0.8 1
retu
rn p
erio
d of
clo
sure
eve
nts
[yea
r]
sea level rise [m]
Fighting the arch-enemy with mathematics and climate models – p.18
Example 3: Petten’s seadike:
Fighting the arch-enemy with mathematics and climate models – p.19
Example 3: Petten’s seadike:
Dike fails if: dike load L + 0.3H > 7.6 [m]
Fighting the arch-enemy with mathematics and climate models – p.20
Example 4: CO2 effect on surge:
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
-2 0 2 4 6 8
2 5 10 25 50 100 103 104
skew
sur
ge [m
]
Gumbel variate
Vlissingen and Cuxhaven
return period
Vlissingen
Cuxhaven1950-20002050-2100
ESSENCE + WAQUA
Fighting the arch-enemy with mathematics and climate models – p.21
Is the extrapolation always valid?
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
-2 0 2 4 6 8 10 12
10610510410310050251052
sea
leve
l at K
ey W
est,
Flo
rida
[m]
Gumbel variate
return period [years]
hurricane Wilma, October 2005
observations 1971-2008
Fighting the arch-enemy with mathematics and climate models – p.22
Is the extrapolation always valid? (2)
5
10
15
20
-1 0 1 2 3 4 5 6 74
5
6
7
8
9
2 5 10 25 50 100 103
win
d sp
eed
(m/s
)
Bea
ufor
t sca
le
Gumbel scale
return period [years]
’Martin’, december 1999 in France (ERA40)
Fighting the arch-enemy with mathematics and climate models – p.23
Probability of ’outlier’:
-2 0 2 4 6 8 10
10 100 103 104
y
Gumbel variate x=-ln(-ln(F(y)))
return period T
∆Xn
x=ln
(n)
y=yn
g(x)
x=-ln
(-ln
(F(y
n)))
yrF(y)
Fighting the arch-enemy with mathematics and climate models – p.24
Application:
Determine ∆X̂n for every record/grid point
Require independence between outliers
Compare distribution of independent values of∆X̂n with theory
Fighting the arch-enemy with mathematics and climate models – p.25
Locations of outliers:
300˚ 310˚ 320˚ 330˚ 340˚ 350˚ 0˚ 10˚
40˚
50˚
60˚
70˚
Fighting the arch-enemy with mathematics and climate models – p.26
Distribution of outliers:
-2
-1
0
1
2
3
4
5
6
-2 -1 0 1 2 3 4 5 6
25010050201052∆∧ X
n
Gumbel variate
number of independent records m
Gumbel to uk
Gumbel to uGEV to uk
theory
Conclusion: Fit Gumbel to uk!
Fighting the arch-enemy with mathematics and climate models – p.27
whole Northern Hemishpere (ERA40):
180˚ 240˚ 300˚ 0˚ 60˚ 120˚ 180˚
10˚
30˚
50˚
70˚
90˚
−2
−1
0 1
2 3
4 5
6
−2 −1 0 1 2 3 4 5 6
1001052
−2
0 2
4 6
8
−2 0 2 4 6 8
1031001052
−2
−1
0 1
2 3
4 5
6 7
−2 −1 0 1 2 3 4 5 6 7
1031001052
−2
0 2
4 6
8 1
0
−2 0 2 4 6 8 10
1041031001052
−2
−1
0 1
2 3
4 5
−2 −1 0 1 2 3 4 5
1001052
−2
−1
0 1
2 3
4 5
6
−2 −1 0 1 2 3 4 5 6
1001052
−2
−1
0 1
2 3
4 5
6
−2 −1 0 1 2 3 4 5 6
1001052
−2
0 2
4 6
8 1
0 1
2 1
4
−2 0 2 4 6 8 10 12 14
1061051041031001052
−2
−1
0 1
2 3
4 5
−2 −1 0 1 2 3 4 5
1001052
−2
−1
0 1
2 3
4 5
6
−2 −1 0 1 2 3 4 5 6
1001052
−2
−1
0 1
2 3
4 5
6
−2 −1 0 1 2 3 4 5 6
1001052
−2
0 2
4 6
8 1
0 1
2 1
4
−2 0 2 4 6 8 10 12 14
1061051041031001052
−2
−1
0 1
2 3
4 5
6 7
−2 −1 0 1 2 3 4 5 6 7
1031001052
−2
−1
0 1
2 3
4 5
6
−2 −1 0 1 2 3 4 5 6
1001052
−2
−1
0 1
2 3
4 5
6
−2 −1 0 1 2 3 4 5 6
1001052
−2
0 2
4 6
8
−2 0 2 4 6 8
1031001052
−1
0 1
2 3
4 5
6 7
−1 0 1 2 3 4 5 6 7
1031001052
−2
−1
0 1
2 3
4 5
6 7
−2 −1 0 1 2 3 4 5 6 7
1031001052
−2
0 2
4 6
8
−2 0 2 4 6 8
1031001052
−2
0 2
4 6
8
−2 0 2 4 6 8
1031001052
−2
−1
0 1
2 3
4 5
6 7
−2 −1 0 1 2 3 4 5 6 7
1031001052
−2
0 2
4 6
8 1
0
−2 0 2 4 6 8 10
1041031001052
−2
−1
0 1
2 3
4 5
6
−2 −1 0 1 2 3 4 5 6
1001052
−2
0 2
4 6
8
−2 0 2 4 6 8
1031001052
Fighting the arch-enemy with mathematics and climate models – p.28
1887
-yea
rE
SS
EN
CE
data
set:
180˚ 240˚ 300˚ 0˚ 60˚ 120˚ 180˚
−90˚
−70˚
−50˚
−30˚
−10˚
10˚
30˚
50˚
70˚
90˚
−2
−1
0 1
2 3
4
−2 −1 0 1 2 3 4
1052
−2
−1
0 1
2 3
4 5
6
−2 −1 0 1 2 3 4 5 6
1001052
−3
−2
−1
0 1
2 3
4 5
6
−3 −2 −1 0 1 2 3 4 5 6
1001052
0 5
10
15
20
0 5 10 15 20
1071061051041031001052
0 5
10
15
20
25
30
35
0 5 10 15 20 25 30 35
1071061051041031001052
−2
0 2
4 6
8 1
0
−2 0 2 4 6 8 10
1041031001052
−3
−2
−1
0 1
2 3
4 5
6
−3 −2 −1 0 1 2 3 4 5 6
1001052
−2
0 2
4 6
8
−2 0 2 4 6 8
1031001052
−3
−2
−1
0 1
2 3
4 5
−3 −2 −1 0 1 2 3 4 5
1001052
−2
−1
0 1
2 3
4 5
6
−2 −1 0 1 2 3 4 5 6
1001052
−3
−2
−1
0 1
2 3
4 5
6
−3 −2 −1 0 1 2 3 4 5 6
1001052
−2
0 2
4 6
8
−2 0 2 4 6 8
1031001052
−2
0 2
4 6
8
−2 0 2 4 6 8
1031001052
0 5
10
15
20
0 5 10 15 20
1071061051041031001052
−2
0 2
4 6
8 1
0
−2 0 2 4 6 8 10
1041031001052
−2
0 2
4 6
8 1
0
−2 0 2 4 6 8 10
1041031001052
−2
−1
0 1
2 3
4 5
6
−2 −1 0 1 2 3 4 5 6
1001052
−3
−2
−1
0 1
2 3
4 5
−3 −2 −1 0 1 2 3 4 5
1001052
−3
−2
−1
0 1
2 3
4 5
−3 −2 −1 0 1 2 3 4 5
1001052
−3
−2
−1
0 1
2 3
4 5
6
−3 −2 −1 0 1 2 3 4 5 6
1001052
−2
0 2
4 6
−2 0 2 4 6
1031001052
−2
0 2
4 6
8 1
0 1
2 1
4
−2 0 2 4 6 8 10 12 14
1061051041031001052
0 5
10
15
0 5 10 15
1061051041031001052
−2
0 2
4 6
8
−2 0 2 4 6 8
1031001052
−3
−2
−1
0 1
2 3
4 5
6
−3 −2 −1 0 1 2 3 4 5 6
1001052
−3
−2
−1
0 1
2 3
4 5
−3 −2 −1 0 1 2 3 4 5
1001052
−2
−1
0 1
2 3
4 5
6
−2 −1 0 1 2 3 4 5 6
1001052
−3
−2
−1
0 1
2 3
4
−3 −2 −1 0 1 2 3 4
1052
−3
−2
−1
0 1
2 3
4 5
−3 −2 −1 0 1 2 3 4 5
1001052
−3
−2
−1
0 1
2 3
4 5
6
−3 −2 −1 0 1 2 3 4 5 6
1001052
−2
0 2
4 6
8 1
0 1
2 1
4
−2 0 2 4 6 8 10 12 14
1061051041031001052
−2
0 2
4 6
8 1
0 1
2
−2 0 2 4 6 8 10 12
1051041031001052
0 5
10
15
20
0 5 10 15 20
1071061051041031001052
−3
−2
−1
0 1
2 3
4 5
6
−3 −2 −1 0 1 2 3 4 5 6
1001052
−2
0 2
4 6
−2 0 2 4 6
1031001052
−2
−1
0 1
2 3
4 5
−2 −1 0 1 2 3 4 5
1001052
−3
−2
−1
0 1
2 3
4
−3 −2 −1 0 1 2 3 4
1052
−2
−1
0 1
2 3
4 5
−2 −1 0 1 2 3 4 5
1001052
−2
0 2
4 6
−2 0 2 4 6
1031001052
−2
0 2
4 6
8
−2 0 2 4 6 8
1031001052
−2
0 2
4 6
8 1
0 1
2
−2 0 2 4 6 8 10 12
1051041031001052
−2
0 2
4 6
8
−2 0 2 4 6 8
1031001052
−2
0 2
4 6
8
−2 0 2 4 6 8
1031001052
−3
−2
−1
0 1
2 3
4 5
−3 −2 −1 0 1 2 3 4 5
1001052
−2
0 2
4 6
8
−2 0 2 4 6 8
1031001052
−2
−1
0 1
2 3
4
−2 −1 0 1 2 3 4
1052
−3
−2
−1
0 1
2 3
4 5
6
−3 −2 −1 0 1 2 3 4 5 6
1001052
−3
−2
−1
0 1
2 3
4 5
6
−3 −2 −1 0 1 2 3 4 5 6
1001052
−2
0 2
4 6
8
−2 0 2 4 6 8
1031001052
−2
0 2
4 6
8 1
0 1
2
−2 0 2 4 6 8 10 12
1051041031001052
−2
0 2
4 6
8
−2 0 2 4 6 8
1031001052
−3
−2
−1
0 1
2 3
4 5
6
−3 −2 −1 0 1 2 3 4 5 6
1001052
−2
−1
0 1
2 3
4 5
−2 −1 0 1 2 3 4 5
1001052
−3
−2
−1
0 1
2 3
4 5
6
−3 −2 −1 0 1 2 3 4 5 6
1001052
Fig
hting
the
arc
h-e
nem
ywith
math
em
atics
and
clim
ate
models
–p.2
9
Extreme precipitation:
Wilson&Toumi (2005): R = κ(qρw)zm
R precipitation
κ efficiency/fraction
q specific humidity
w vertical velocity
ρ density
zm level
independent variables q, w, κ:
Pr(R > r) = exp[−(r
R0
)2/3
]
Fighting the arch-enemy with mathematics and climate models – p.30
Extreme precipitation (2):
R Weibull-distributed with k = 2/3
R2/3 exponential-distributed
fast convergence to Gumbel-distribution for R2/3
fit Gumbel distribution to R2/3!
Fighting the arch-enemy with mathematics and climate models – p.31
Extreme precipitation (3)
310˚ 315˚ 320˚ 325˚ 330˚ 335˚ 340˚ 345˚ 350˚ 355˚ 0˚ 5˚ 10˚ 15˚ 20˚ 25˚ 30˚ 35˚ 40˚ 45˚ 50˚
20˚
25˚
30˚
35˚
40˚
45˚
50˚
55˚
60˚
65˚
70˚
GEV shape parameter = 0 (Gumbel)1−day sums, annual maxima of R 2/3
−2.5 −1.5 −0.5 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5
−2 −1 0 1 2 3 4 5 6 7 8 9
10
−2 −1 0 1 2 3 4 5 6 7 8 9 10
Fighting the arch-enemy with mathematics and climate models – p.32
Extreme precipitation (4)
-2
0
2
4
6
8
10
-2 0 2 4 6 8 10
DX
n
Gumbel variate
GEV to R (k=free)GEV to R (k=0.10)Gumbel to R2/3
theory
Fighting the arch-enemy with mathematics and climate models – p.33
Extreme precipitation (5)
0
20
40
60
80
100
120
140
160
-2 0 2 4 6 8 10 12
10510410310050251052
prec
ipita
tion
[mm
/day
]
Gumbel variate
return period [years]
annual maximaGumbel to R2/3
GEV to R (k=0.10)GEV to R
MANSTON, England (1961-2005 – 19730920)Fighting the arch-enemy with mathematics and climate models – p.34
Back to sea levels:
use 17 runs of ESSENCE data (1950-2100)
feed surge model (WAQUA) with wind andpressure from ESSENCE
time series for 19 coastal stationsapply extreme value statistics to 50-year timeseries
17 × 19 × 3 = 969 recordsrequire 3-day interval between extremeevents
Fighting the arch-enemy with mathematics and climate models – p.35
Back to sea levels (2):
1˚ 2˚ 3˚ 4˚ 5˚ 6˚ 7˚ 8˚ 9˚
51˚
52˚
53˚
54˚
−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
−2
−1
0
1
2
3
4
5
∆X −2 −1 0 1 2 3 4 5
Gumbel variate
Fighting the arch-enemy with mathematics and climate models – p.36
For observations:
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
201052∆∧ X
n
Gumbel variate
number of independent records m
Gumbel to subseriestheory
Fighting the arch-enemy with mathematics and climate models – p.37
Example for Scheveningen:
1
2
3
4
5
6
7
-2 0 2 4 6 8
2 5 10 25 50 100 103 104
sea
leve
l at S
chev
enin
gen
[m]
Gumbel variate
return period [years]
observations 1896-2005Gumbel to observations
GEV to observations
Fighting the arch-enemy with mathematics and climate models – p.38
Conclusion:
climate models are helpful tool for analysis of(never observed) extremes
Gumbel distribution optimal model for (all?)meteorological variables
not in tropicssimple power transformation needed
Fighting the arch-enemy with mathematics and climate models – p.39
Questions....?
Fighting the arch-enemy with mathematics and climate models – p.40
