104 years. 2. What is the effect of increased greenhouse gases on the extreme
sea levels? Fighting the arch-enemy with mathematics and climate models – p.4
...
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 to withstand the sealevel that is reached once in 104 years 2. What is the effect of increased greenhouse gases on the extreme sea levels?
Fighting the arch-enemy with mathematics and climate models – p.4
Fighting the arch-enemy with mathematics and climate models – p.5
Fighting the arch-enemy with mathematics:
Fighting the arch-enemy with mathematics and climate models – p.6
Xp vs k (γ not fixed):
Fighting the arch-enemy with mathematics and climate models – p.7
Xp vs k (γ = 0):
Fighting the arch-enemy with mathematics and climate models – p.8
As a Gumbel plot: Hoek van Holland
2
103
104
observations 1888-2005 GEV to observations Gumbel to observations
5.5 5 water level [m]
5
return period 10 25 50 100
4.5 4 3.5 3 2.5 2 1.5 -2
0
2
4 Gumbel variate
6
8
Fighting the arch-enemy with mathematics and climate models – p.9
water level in Hoek van Holland: Hoek van Holland
2
103
104
observations 1888-2005 GEV to observations Gumbel to observations
5.5 5 water level [m]
5
return period 10 25 50 100
4.5 4 3.5 3 2.5 2 1.5 -2
0
2
4
6
8
Gumbel variate
large statistical uncertainty
Fighting the arch-enemy with mathematics and climate models – p.10
water level in Hoek van Holland: Hoek van Holland
2
103
104
observations 1888-2005 GEV to observations Gumbel to observations
5.5 5 water level [m]
5
return period 10 25 50 100
4.5 4 3.5 3 2.5 2 1.5 -2
0
2
4
6
8
Gumbel variate
large statistical uncertainty is extrapolation allowed...?
Fighting the arch-enemy with mathematics and climate models – p.10
water level in Hoek van Holland: Hoek van Holland
2
103
104
observations 1888-2005 GEV to observations Gumbel to observations
5.5 5 water level [m]
5
return period 10 25 50 100
4.5 4 3.5 3 2.5 2 1.5 -2
0
2
4
6
8
Gumbel variate
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 with climate models:
global models
Fighting the arch-enemy with mathematics and climate models – p.11
Fighting the arch-enemy with climate models:
global models does not contain measurements
Fighting the arch-enemy with mathematics and climate models – p.11
Fighting the arch-enemy with climate 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 with climate models: generate meteorological data with climate models:
Fighting the arch-enemy with mathematics and climate models – p.12
Fighting the arch-enemy with climate models: generate meteorological data with climate models: ECMWF seasonal forecasts (1600 yrs) ⇒
Fighting the arch-enemy with mathematics and climate models – p.12
Fighting the arch-enemy with climate models: generate meteorological data with climate models: ECMWF seasonal forecasts (1600 yrs) ⇒ ESSENCE (ECHAM5 MPI-OM) 20×(1950-2000)=1000 yrs 17×(1950-2100)=2550 yrs =3550 yrs
Fighting the arch-enemy with mathematics and climate models – p.12
Fighting the arch-enemy with climate models: generate meteorological data with climate models: ECMWF seasonal forecasts (1600 yrs) ⇒ ESSENCE (ECHAM5 MPI-OM) 20×(1950-2000)=1000 yrs 17×(1950-2100)=2550 yrs =3550 yrs feed wave/surge-model with wind and pressure from climate model
Fighting the arch-enemy with mathematics and climate models – p.12
Advantages of models wrt observations: strongly improved extreme-value-statistics: (almost) no extrapolation needed assumptions of extrapolation can be checked dynamical-physical properties can be investigated influence of greenhouse effect can be determined
Fighting the arch-enemy with mathematics and climate models – p.13
Possibilities: extreme wind extreme 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 return period [years] 2 5 10 25 100 103
7
104
observations ECMWF
6
surge [m]
5 4 3 2 1 0 -2
0
2
4
6
8
Gumbel variate
→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 sea large Rhine discharges
Fighting the arch-enemy with mathematics and climate models – p.17
Example 2: Maeslant closure barrier return period of closure events [year]
10
5
2
1
0.5
0.2 0
0.2
0.4 0.6 sea level rise [m]
0.8
1
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: Vlissingen and Cuxhaven
2 5
5
return period 10 25 50 100
103
1950-2000 2050-2100
4.5
104
Cuxhaven
skew surge [m]
4 3.5 3
Vlissingen
2.5 2 1.5 1 0.5 -2
0
2
4 Gumbel variate
6
8
ESSENCE + WAQUA Fighting the arch-enemy with mathematics and climate models – p.21
Is the extrapolation always valid? 2
sea level at Key West, Florida [m]
3
return period [years] 5 10 25 50100 103 104
105
106
hurricane Wilma, October 2005
2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2
observations 1971-2008
1 -2
0
2
4
6
8
10
12
Gumbel variate Fighting the arch-enemy with mathematics and climate models – p.22
Is the extrapolation always valid? (2) 2
return period [years] 5 10 25 50 100
103 9
8
15 7
Beaufort scale
wind speed (m/s)
20
6
10
5 4
5 -1
0
1
2 3 4 Gumbel scale
5
6
7
’Martin’, december 1999 in France (ERA40)
Fighting the arch-enemy with mathematics and climate models – p.23
Probability of ’outlier’: return period T 10 100 yr F(y)
103
104
g(x)
y=yn
x=ln(n)
x=-ln(-ln(F(yn)))
y
∆Xn
-2
0
2
4
6
8
10
Gumbel variate x=-ln(-ln(F(y))) Fighting the arch-enemy with mathematics and climate models – p.24
Application: ˆ n for every record/grid point Determine ∆X Require independence between outliers Compare distribution of independent values of ˆ n with theory ∆X
Fighting the arch-enemy with mathematics and climate models – p.25
Locations of outliers: 70˚
60˚
50˚
40˚ 300˚
310˚
320˚
330˚
340˚
350˚
0˚
10˚
Fighting the arch-enemy with mathematics and climate models – p.26
Distribution of outliers: number of independent records m 2 5 10 20 50 100 250 6
Gumbel to uk Gumbel to u GEV to uk theory
5 4
∧ ∆Xn
3 2 1 0 -1 -2 -2
-1
0
1 2 3 Gumbel variate
4
5
6
Conclusion: Fit Gumbel to uk ! Fighting the arch-enemy with mathematics and climate models – p.27
whole Northern Hemishpere (ERA40): 10
2
100
5 10
103
100
2
5 10
103
100
2
5 10
103
100
6 5
4 3 2
2
−2
−1
0
100
1
2
10
2
3
4
5 10
−2
5
−1
0
100
1
2
2
3
4
5 10
5
0
1
−2 −1
0 0
1
2
2
100
3
4
5
5 10
6
7
−1
0
1
100
2
2
3
4
5
5 10
6
−2 −1
7 3
100
10
0 2
1
2
5 10
3
4
5
6
3
100
7 4
10
10
6
5
4 0
0
3
4
5 10
5
6
−1
0
1 2
2
3
4
5 10
5
6
−2
−1
0
100
1 2
2
3
4
5 10
5
−2
−2 −1
−1 −2
100
−2 −1
6
0 2
100
1
2
5 10
3
4
5
6
7
−2
0
2
3
100
4
2
10
6
8
5 10
10
100
4
4
4
3
4
3
2
2
2
1
2
1
6
7
104
−1
0
1
2 5 10
100
0
4
2
3
103
4 104
5
6
105
106
−1
0
1
2 5 10
100
0
4
2
3
103
4 104
5
6
105
106
−1 −2
−1
0 2
1 5 10
2
3
4
5
6
103
100
−2
−2
−1 −2
−2
0 2
2 5 10
4
6
−1
0 2
1 5 10
2
3
4
5
6
103
100
6
10
4
2
2
2
6 4
0
0 0
2
4
6
8
10
−2
2
6
8
10
12
14
−2
2
6
8
10
12
−2
14
−2
−2
−2
0
−2
−2
0
0 −2 −2
0
2
2
2
4
4
6
4
4
8
8
6
10
−2
8 103
100
6
8
−2
−1 −2 −2
8
5 103
6
4
8
3 100
8
2
12
10
1 5 10
12
0 2
14
−2 −1
30˚
14
−2
−2 −1
−1
0
0
0
0
0
0
1
1
1
2
2
3
3
3
4
4
5
6
5
6
2
2
6
10
1
5
0
8
100
−2
−1 −1
3
6
5 10
−2
8
5
6
7
2
4
6
2
5
0
6
−2
50˚
−2
−2
−2
−1
0
0
0
0
1
2
1
1
1
2
2
2
2
2
3
3
3
4
3
4
4
4
4
6
8
5
−1 −2 −1
10
6 3
7
5
6
5 10
4
6
3
8
2
2
5
1
6
0
5
−1
6
−2
70˚
−2 −1
−2
−2
−2
−1
−1
−1
0
0
0
0
1
1
1
2
1
1
2
3
3
2
2
3
4
4
3
3
4
5
4
4
5
6
6
5
7
2
100
6
10
5
5
7
2
100
7
5 10
5
2
5
90˚
0
2
4
6
8
−2
0
2
4
6
8
−2
0
2
4
6
8
10˚ 180˚
240˚
300˚
0˚
60˚
120˚
180˚
Fighting the arch-enemy with mathematics and climate models – p.28
5 10
100
2
10
2
5
10
2
4
3
4
5 3
5
10
2 1 0
0
0
1
2
3
4
−2
−1
0
1
2
2
5 10
3
4
5
6
−3
−2
−1
0
1
2
5 10
2
3
4
5
−1
−1 −3
−2
−1
0
1
2
3
4
−2
−2 −3
−3
−2 −3
−2 −1
−2
−1
0 −1
−1 −2 −2
−1
0
0
1
0
1
1
2
1
1
2
3
2
2
2
3
4
5
4
2
3
100
6
5 10
3
2
4
10
5
5
4
2
−3
−2
−1
0
1
2
3
4
−2
−1
0
1
2
3
4
70˚ 2
5 10
100
2
100
2
5 10
100
5
5
10
3
3
1
1
1 1
1 4
5
6
−3 −2 −1
0
100
1
2
3
5 10
4
5
6
1
5 10
2
3
4
5
6
−3 −2 −1
0 −2 −3
103
100
−2
−1
0
1
2
5 10
2
3
4
5
−2
−1
0
100
1 2
5
6
0 2
2
3
5 10
4
5 103
100
−3 −2 −1
0
1 2
2
3
4
5 10
5
6
100
1
1 5 10
4
6
100
103
−2
0
2
2 5 10
100
0
4
4
103
6
104
106
12 10
−3 −2 −1
−2 −3 −2 −1
105
0
1
2
3
2 510
100 103
0
4
4
5
104
105
6
−2
0
106
2
2
4
5 10
6
−3 −2 −1
103
100
1
2
5 10
3
4
5
6
103
100
4 2
2
6
20
−2
0
2
4
6
8
−2
2
6
8
10
12
14
2 510 100 103 104 105 106
2
6
2 5 10
8
10
103
100
12
104
0 −2
−2 −2
14 105
10
20
35
15
2 510 100 103 104 105 106 107
−2
0
2
2 5 10
4
6 103
100
8 104
−2
105
0
2
4
2 5 10
12
15
6 103
100
8 104
105
0
2
4
6
8
8 2 0
0 25
30
103
35
0
104
5 2
10
15
20
103
100
0
104
5 2
10
5 10
15
−2
103
100
0
2
4
6
8
10
−2
12
2 510 100 103 104 105 106 107
−2
0
2
4
2
6
8
5 10
10
12
103
100
−2
0
2
4
2
6
8
5 10
10
12
103
100
5 10
6
8
−2
0
2
10
2
4
6
8 103
100
4 2 0
0
10 104
−2
0
2 2
4
6
5 10
0
5
10 2
15
20
5 10
−2
0
100
2
2
4
5 10
6
8 103
100
−2
0
2 2
5
4
6
5 10
8
100
4
4
1
2
2
2 1
2
6
−2
0
2 2
4
6
8
5 10
10
−3 −2 −1
0
1
100
2
2
3
4
5
5 10
6
0
1
2
5 10
2
3
4
5
6 103
100
−2
0
2
4
2
6
8
5 10
−3 −2 −1
0
100
1
2
2 5
3
4
5
10
6
100
0
2
4
2
5 10
6
8
−1
0
1
2
3
4
5
6
0 −1
−1 −3 −3
−2
−1
0
1
2
3
4
5
−2
0
2
4
6
−2
−2
−2
−2 −2
−3
−2
−1
0
1
2
3
−70˚ 5 10
100
2
5 10
100
2
5
10
100
2
5 10
4
5
103
100
−1
0
1
2
3
4
5 10
5
100
5 2
4
0
−3
−2
−1
0
1
2
3
4
5
−1
−1 −3
−2
−1
0
1
2
3
4
5
−2
−1
0
1
2
3
4
5
6
−2
−2
−2
−3
−3
−2
−2
0
−3 −2 −1
0
−1
−1
0
2
1
0
0
1
1
2
1
1
2
3
2
2
3
3
6
4
4
4
5
4 3
4 3
−2
2
8
5
6
2 5
5
100
6
−2
−3
−2
−2
−1
0
0
−1
0
0
0
1
2
1
1
2
1
2
2
2
2
4
3
4
3
3
3
4
6
4
5
−3 −2 −1
−2 −3 −2 −1
100
5
5 103
4
4
5
3 100
4
2
6
1
5 10
5
0
8
2
6
−3 −2 −1
−50˚
−3 −2 −1
−3 −2 −1
−2
−3 −2 −1
0
0
0
0
0
0
2
1
1
4
2
4
3
3
3
6
4
6
8 4 3
−2
−2
8
100
5
5 10
5
6
0
−2
10
100
6
4
2
5
2
8
0
6
−2
−30˚
6
−2
−2
0
0
0
5
2
2
2
2
4
10
4
4
4
6
6
15
8
6
6
8
20
10
5 10
8
20 100
6
15
8
10
2 5 10
8
5
10
0
−10˚
−2
0
−2
0
5
0
5
2
4
6 5
4
10
20 15 10
6
10
15
25
8
30
10
10
10
5
3 4 5 6 7 2510100 1010 101010
12
0
10˚
12
−2
−2
−2
0
0
0
0
2
0
2
5
4
4
2
6
10
0 2
4
4
8
15
6
20
−3 −2 −1
8
8
2
2 8
0
6
−2
8
6
6
5
14
4
12
3
10
2
8
1
2 510 100 103 104 105 106 107
14
0
0
0 −2
0 −2
−3 −2 −1 −3 −2 −1
30˚
0
0
0
1
2
2
2
2
2
2
4
3
3
3
4
4
4
4
4
6
5
−1
−1 −2 −3 −3 −2 −1
103
100
8
6
2
6
3
5
2 5 10
6
1 2
6
0
0
0
−3 −2 −1 −1
−3 −2 −1
0
0
1 0 −1 −2 −2
50˚
1887-year ESSENCE dataset:
2
2
2
2
2
2
3
3
3
3
4
4
4
5
5
6
100
4
4
4
5
5
6
100
5
100
6
5 10
6
2
−2
−1
0
1
2
3
4
5
−2
0
2
4
6
8
−3 −2 −1
0
1
2
3
4
5
6
−90˚ 180˚
240˚
300˚
0˚
60˚
120˚
180˚
Fighting the arch-enemy with mathematics and climate models – p.29
90˚
Extreme precipitation: Wilson&Toumi (2005): R = κ(qρw)zm
R κ q w ρ zm
precipitation efficiency/fraction specific humidity vertical velocity density
level independent variables q, w, κ:
r 2/3 Pr(R > r) = exp[−( ) ] R0 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
Fighting the arch-enemy with mathematics and climate models – p.32
Extreme precipitation (4) 10
GEV to R (k=free) GEV to R (k=0.10) Gumbel to R2/3 theory
8
DXn
6 4 2 0 -2 -2
0
2
4
6
8
10
Gumbel variate Fighting the arch-enemy with mathematics and climate models – p.33
Extreme precipitation (5) 2
return period [years] 5 10 25 50100 103
104
105
160
precipitation [mm/day]
140 120 100 80 60 40
annual maxima Gumbel to R2/3 GEV to R (k=0.10) GEV to R
20 0 -2
0
2
4 6 Gumbel variate
8
10
12
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 and pressure from ESSENCE time series for 19 coastal stations apply extreme value statistics to 50-year time series 17 × 19 × 3 = 969 records require 3-day interval between extreme events
Fighting the arch-enemy with mathematics and climate models – p.35
Back to sea levels (2): 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
−2 −1
0
4.5
5.0
5.5
53˚ 5 4
∆X
3
52˚
2 1 0 −1 −2 1
2
3
4
5
Gumbel variate
51˚ 1˚
2˚
3˚
4˚
5˚
6˚
7˚
8˚
9˚
Fighting the arch-enemy with mathematics and climate models – p.36
For observations: number of independent records m 2 5 10
20
3 2.5 2 1.5 ∧ ∆Xn
1 0.5 0 -0.5 -1 -1.5 -1.5 -1 -0.5
Gumbel to subseries theory 0
0.5
1
1.5
2
2.5
3
Gumbel variate
Fighting the arch-enemy with mathematics and climate models – p.37
Example for Scheveningen: 2
sea level at Scheveningen [m]
7
return period [years] 5 10 25 50 100
103
104
observations 1896-2005 Gumbel to observations GEV to observations
6 5 4 3 2 1 -2
0
2
4
6
8
Gumbel variate 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 tropics simple 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
