Zscheischler IMSC
Frequency of extreme hot and dry conditions during hottest months increases in the future IMSC Canmore Jakob Zscheischler and Sonia I. Seneviratne Institute for Atmospheric and Climate Science ETH Zurich, Switzerland
June 10, 2016
Return periods
Let N be the length of the time series (e.g. number of years), m the rank of the event. The univariate return period of the event is T =
N +1 m
[Chow , 1964].
Example: N = 99 years, m = 2 ⇒ T = 50 years.
Jakob Zscheischler (ETH Zurich)
1
Return periods
Let N be the length of the time series (e.g. number of years), m the rank of the event. The univariate return period of the event is T =
N +1 m
[Chow , 1964].
Example: N = 99 years, m = 2 ⇒ T = 50 years. Return periods are a proxy for the magnitude of an event.
Jakob Zscheischler (ETH Zurich)
1
Bivariate distributions Correlation = -0.85
[Schoelzel and Friedrichs 2008, NPG]
Jakob Zscheischler (ETH Zurich)
2
Bivariate distributions Correlation = -0.85
[Schoelzel and Friedrichs 2008, NPG]
Bivariate distribution can be written as F (x, y ) = C (FX (x), FY (y ))
[Sklar 1959]
with FX (x) = Pr (X ≤ x), FY (y ) = Pr (Y ≤ y ) the marginal distributions and C a bivariate Copula. Jakob Zscheischler (ETH Zurich)
2
Example for Archimedean copulas (Marginals are standard normal)
Clayton
allows lower tail dependence
Frank
no tail dependence
Gumbel
allows upper tail dependence
[Schoelzel and Friedrichs 2008, NPG]
Jakob Zscheischler (ETH Zurich)
3
Kendall distribution function and bivariate return periods The Kendall distribution function is defined as KC (t) = P(C (u, v ) ≤ t) measures the probability that a random event will appear in the region of I 2 defined by the inequality C (u, v ) ≤ t. Easily tractable for Archimedean Copulas.
Jakob Zscheischler (ETH Zurich)
4
Kendall distribution function and bivariate return periods The Kendall distribution function is defined as KC (t) = P(C (u, v ) ≤ t) measures the probability that a random event will appear in the region of I 2 defined by the inequality C (u, v ) ≤ t. Easily tractable for Archimedean Copulas. Kendall’s return period is then defined as [Salvadori et al. 2011] KRP =
1 1 − KC (t)
Jakob Zscheischler (ETH Zurich)
4
Kendall distribution function and bivariate return periods The Kendall distribution function is defined as KC (t) = P(C (u, v ) ≤ t) measures the probability that a random event will appear in the region of I 2 defined by the inequality C (u, v ) ≤ t. Easily tractable for Archimedean Copulas. Kendall’s return period is then defined as [Salvadori et al. 2011] KRP =
1 1 − KC (t)
Compute bivariate return periods of hot and dry seasons by fitting a copula to -P and T and computing Kendall’s return period. Return periods can be interpreted as the magnitude of an event.
Jakob Zscheischler (ETH Zurich)
4
Example: California
[AghaKouchak et al. 2014, GRL]
Return periods of concurrent dry and hot seasons Data: Means over November - April from 1896 to 2014.
Jakob Zscheischler (ETH Zurich)
5
Bivariate return periods depend on dependence structure
4
The same event has a lower return period if variables are more strongly correlated (here: correlation = 0.8).
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2
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−4
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−4
−2
0
Jakob Zscheischler (ETH Zurich)
2
4
6
Bivariate return periods depend on dependence structure
4
The same event has a lower return period if variables are more strongly correlated (here: correlation = 0.8).
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−2
0
2
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●
● ●
−4
●
−4
−2
0
Jakob Zscheischler (ETH Zurich)
2
4
6
Bivariate return periods depend on dependence structure
4
The same event has a lower return period if variables are more strongly correlated (here: correlation = 0.8).
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−2
0
2
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−4
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−4
−2
0
Jakob Zscheischler (ETH Zurich)
2
4
6
Bivariate return periods depend on dependence structure
4
The same event has a lower return period if variables are more strongly correlated (here: correlation = 0.8).
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−2
0
2
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−4
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−4
−2
0
Jakob Zscheischler (ETH Zurich)
2
4
6
Bivariate return periods depend on dependence structure
4
The same event has a lower return period if variables are more strongly correlated (here: correlation = 0.8).
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● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ●● ●●●● ● ●● ●●● ●● ●● ● ●●●●● ● ● ●●● ● ●● ●● ●● ● ● ● ●●● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ●●● ● ●●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ●●●● ●●●● ● ● ● ● ●●● ●● ● ●● ●● ● ● ●●● ● ● ●● ● ● ● ●● ●●● ●●●●● ● ● ● ●● ● ● ●● ● ●● ● ●●● ●● ●● ● ●● ● ●● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ●● ●●● ● ●● ●●● ● ●● ● ● ●●● ● ●● ● ● ●● ● ●●● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ●● ●● ● ●● ● ●●●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ●●●● ● ● ● ● ●● ● ● ●● ● ●●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ●● ●● ● ●● ●● ● ●● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ●●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ●●●● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ●● ● ● ●●●● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ●●● ●●●●● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ●● ● ●●● ●● ● ● ●● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ●● ● ●● ● ●● ● ●● ●● ● ●● ●● ●● ● ● ● ●● ●● ● ● ●● ●● ● ● ● ●● ● ● ● ● ●● ●●●● ●● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ●● ●● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ●●● ● ● ● ●●●● ●● ●● ●● ●● ●● ● ● ● ● ● ●●●● ●● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●●● ●● ● ●●● ●● ●● ●● ● ●● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ●● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ●●● ●● ● ● ● ●● ● ●● ● ● ● ● ●●● ● ●● ● ●● ● ●●●●● ● ●● ● ● ●● ● ● ●● ●● ● ●● ● ●● ● ● ● ● ● ● ● ●● ●●● ●● ●● ●● ●●● ●● ● ● ●● ● ● ●● ● ●●●● ●● ●● ● ●● ● ● ● ● ●● ●● ● ● ●● ● ●● ●●● ● ●● ● ●● ● ● ●● ●● ●● ● ● ●●● ●● ●●● ● ●● ● ●● ● ● ●● ● ●●●●●●● ● ● ● ●● ●● ● ● ●● ●● ● ●●● ● ●● ●● ● ● ●●● ● ●●●●● ●●● ● ●● ● ● ●●●●● ● ●●●●● ● ●● ●● ●●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ●● ●●● ● ● ●●●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ●● ●● ● ●●●● ●● ● ●● ● ● ●● ● ●● ● ● ● ●●●● ● ● ● ● ● ● ● ●
−2
0
2
●
●
−4
●
−4
●
● ●
●
shuffled data original data 50 yr return period 19 yr return period −2
●
●
●
●
●
● ●
● ●
0
Jakob Zscheischler (ETH Zurich)
2
4
6
Bivariate return periods depend on dependence structure 0.9
relative difference (in %)
0.71
0.81
80
correlation 0.43 0.52 0.62
60
0.24
0.33
40
0.05
0.14
20
% 10
20
50 100 250 return periods [years]
500
1000
Ignoring the dependence between variables can lead to a strong overestimation of bivariate return periods Jakob Zscheischler (ETH Zurich)
7
Temperature and precipitation from CRU at Berlin (JJA)
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● ●
20
●
18
●
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●
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●
● ●●
●
● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ●● ●● ● ●● ●● ● ● ● ●●● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●●●● ●●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ●● ● ● ●
● ●
16
Temperature [°C]
● ● ● ● ● ● ● ●
●
● ● ●
●
● ● ● ●●● ● ● ● ● ● ● ● ●● ● ● ● ●● ●
● ●● ●
● ● ●
●
●
●●
● ●
140
shuffled data original data 20 yr return period 13.5 yr return period 120
100
80
60
40
20
Precipitation [mm]
Jakob Zscheischler (ETH Zurich)
8
Global analysis
I
Data: seasonal T and P from CMIP5 (1901-2100), 2.5◦
I
fit (Archimedian) copulas to T and -P over 3 hottest months
I
compute bivariate return periods of concurrent dry and hot seasons
I
compare 20th and 21st century (RCP8.5)
I
compare with CRU
Jakob Zscheischler (ETH Zurich)
9
Correlations between T and P
60
80
CMIP5 Ensemble Mean T-P correlations during hottest months (20th century)
20
40
0.5
−20
0
0.0
−60
−40
−0.5
−150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
10
Correlations between T and P
80
CMIP5 Ensemble Mean T-P correlations during hottest months (21st century)
60
0.6
40
0.4
20
0.2
0
0.0
−20
−0.2
−40
−0.4
−60
−0.6
−150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
11
Correlations between T and P
60
80
CMIP5 Ensemble Mean T-P correlations during hottest months (detrended, 21st century)
20
40
0.5
−20
0
0.0
−60
−40
−0.5
−150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
12
Observation: correlation between T and P intensify
80
CMIP5 Ensemble Mean difference in T-P correlations during hottest months (detrended, 21st-20th century)
60
0.15
40
0.10
20
0.05
0
0.00
−20
−0.15
−60
−0.10
−40
−0.05
−150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
13
Change in correlations leads to higher frequency of very dry and hot seasons What is the return period of a 100yr event derived from uncorrelated T and P data in the 20th and 21st century? (SREX regions)
Jakob Zscheischler (ETH Zurich)
14
Change in correlations leads to higher frequency of very dry and hot seasons
0.0
What is the return period of a 100yr event derived from uncorrelated T and P data in the 20th and 21st century? (SREX regions) ●
−0.2
● ●
● ●
−0.4
T−P correlation
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●
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●
●
●
● ●●
● ●
●●
−0.6
● ●●
●
50
●●
● ●
●
● ●●
●
45
● ●
40
● ●
●
35
●
●●
● ●
●
●●
●
● ●
●
●
●
●
●
●
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●
● ●●
●
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● ●
● ●
●
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30
100 year event
55
20th century 21th century
●●
ALA
CGI
WNA CNA
ENA CAM AMZ
NEB WSA SSA
NEU
CEU MED SAH WAF
EAF
Jakob Zscheischler (ETH Zurich)
SAF
NAS WAS CAS
TIP
EAS
SAS
SEA
NAU
SAU
14
Change in correlations leads to higher frequency of very dry and hot seasons
0.0
What is the return period of a 100yr event derived from uncorrelated T and P data in the 20th and 21st century? (SREX regions) ●
−0.2
● ●
● ●
−0.4
T−P correlation
●
●●
●
● ●
●●
● ●
●●
●
●
● ● ●
● ●
●● ●
●
●●
●●
●
●
●
●
● ●
●
●
●
● ●●
● ●
●●
−0.6
● ●●
●
50
●●
● ●
●
● ●●
●
45
● ●
40
● ●
●
35
●
●●
● ●
●
●●
●
● ●
●
●
●
●
●
●
● ●
●●
●
● ●●
●
● ●●
● ●
● ●
●
●●
30
100 year event
55
20th century 21th century
●●
ALA
CGI
WNA CNA
ENA CAM AMZ
NEB WSA SSA
NEU
CEU MED SAH WAF
EAF
Jakob Zscheischler (ETH Zurich)
SAF
NAS WAS CAS
TIP
EAS
SAS
SEA
NAU
SAU
14
Frequency of very dry and hot seasons increases
80
What is the return periods of a 100yr event of the 20th century in the 21st century?
40
60
140
20
120
0
100
−20
80
−40
60
−60
years −150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
15
Caution: Models overestimate correlation between T and P in hottest months
60
80
CMIP5 Ensemble Mean (20th century)
20
40
0.5
−20
0
0.0
−60
−40
−0.5
−150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
16
Caution: Models overestimate correlation between T and P in hottest months
60
80
CRU (20th century)
20
40
0.5
−20
0
0.0
−60
−40
−0.5
−150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
17
Caution: Models overestimate correlation between T and P in hottest months
80
CRU-CMIP5 (20th century)
60
0.6
40
0.4
20
0.2
0
0.0
−20
−0.6
−60
−0.4
−40
−0.2
−150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
18
Conclusions I
bivariate return periods depend on the correlation of the underlying variables, if variables are more strongly correlated, events where both variables are extreme are more likely
Jakob Zscheischler (ETH Zurich)
19
Conclusions I
bivariate return periods depend on the correlation of the underlying variables, if variables are more strongly correlated, events where both variables are extreme are more likely
I
correlation between T and P during the hottest 3 months will intensify (according to CMIP5)
Jakob Zscheischler (ETH Zurich)
19
Conclusions I
bivariate return periods depend on the correlation of the underlying variables, if variables are more strongly correlated, events where both variables are extreme are more likely
I
correlation between T and P during the hottest 3 months will intensify (according to CMIP5)
I
extreme hot and dry 3-month periods will occur more often in the future
Jakob Zscheischler (ETH Zurich)
19
Conclusions I
bivariate return periods depend on the correlation of the underlying variables, if variables are more strongly correlated, events where both variables are extreme are more likely
I
correlation between T and P during the hottest 3 months will intensify (according to CMIP5)
I
extreme hot and dry 3-month periods will occur more often in the future
Remarks: I
models overestimate correlation between T and P
I
CMIP5 data was detrended
Jakob Zscheischler (ETH Zurich)
19
Conclusions I
bivariate return periods depend on the correlation of the underlying variables, if variables are more strongly correlated, events where both variables are extreme are more likely
I
correlation between T and P during the hottest 3 months will intensify (according to CMIP5)
I
extreme hot and dry 3-month periods will occur more often in the future
Remarks: I
models overestimate correlation between T and P
I
CMIP5 data was detrended
Thank you! Jakob Zscheischler (ETH Zurich)
19
Return periods (hot and dry) in hottest/driest months dry
0.5
original T detrended P detrended T & P detrended
0.0
0.0
Bivariate
0.5
1.0
1.0
1.5
1.5
2.0
hot
+T
mean(log(return time)) 0.5 1.0 1.5 2.0
original T detrended
0.0
0.0
Univariate
mean(log(return time)) 0.5 1.0 1.5 2.0
2.5
+T
1900
0.2
0.4
0.6
0.8
1.0
1.2
−P
original P detrended 1920
1940
0.0
Univariate
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
−P
1960
1980
2000
Jakob Zscheischler (ETH Zurich)
1900
1920
1940
1960
1980
2000
20
Return periods (cold and wet) in coldest/wettest months wet
1.0 0.5
original T detrended P detrended T & P detrended
0.0
0.0
Bivariate
0.5
1.0
1.5
1.5
2.0
cold
mean(log(return time)) 0.5 1.0 1.5
original T detrended
0.0
0.0
Univariate
−T mean(log(return time)) 0.5 1.0 1.5 2.0
−T
+P
0.6
0.8
0.4
0.6 1900
0.2
0.4
original P detrended 1920
1940
0.0
0.2 0.0
Univariate
0.8
1.0
1.0
1.2
1.2
+P
1960
1980
2000
Jakob Zscheischler (ETH Zurich)
1900
1920
1940
1960
1980
2000
21
Return periods (cold and wet) in coldest/wettest months wet
1.0 0.5
original T detrended P detrended T & P detrended
0.0
0.0
Bivariate
0.5
1.0
1.5
1.5
2.0
cold
mean(log(return time)) 0.5 1.0 1.5
original T detrended
0.0
0.0
Univariate
−T mean(log(return time)) 0.5 1.0 1.5 2.0
−T
+P
0.6
0.8
0.4
0.6 1900
0.2
0.4
original P detrended 1920
0.0
0.2 0.0
Univariate
0.8
1.0
1.0
1.2
1.2
+P
1940
1960
1980
2000
1900
1920
1940
1960
1980
2000
Data from Global Energy Balance Archive; Wild (2012, BAMS) Jakob Zscheischler (ETH Zurich)
21
Bivariate versus univariate return periods
At the global scale, bivariate return periods are strongly correlated with univariate return periods of temperature.
What about other levels of aggregation?
Jakob Zscheischler (ETH Zurich)
22
Bivariate versus univariate return periods
correlation univariate vs bivariate return periods 0.2 0.4 0.6 0.8
1.0
Average correlation between bivariate and univariate return periods for different steps of aggregation.
0.0
temperature precipitation quadrants global interquartile range 0
20
40 60 spatial resolution [degree]
80
Jakob Zscheischler (ETH Zurich)
23
T varies globally, P varies locally
[Ahlstr¨ om et al 2015]
Jakob Zscheischler (ETH Zurich)
24
June 10, 2016
Return periods
Let N be the length of the time series (e.g. number of years), m the rank of the event. The univariate return period of the event is T =
N +1 m
[Chow , 1964].
Example: N = 99 years, m = 2 ⇒ T = 50 years.
Jakob Zscheischler (ETH Zurich)
1
Return periods
Let N be the length of the time series (e.g. number of years), m the rank of the event. The univariate return period of the event is T =
N +1 m
[Chow , 1964].
Example: N = 99 years, m = 2 ⇒ T = 50 years. Return periods are a proxy for the magnitude of an event.
Jakob Zscheischler (ETH Zurich)
1
Bivariate distributions Correlation = -0.85
[Schoelzel and Friedrichs 2008, NPG]
Jakob Zscheischler (ETH Zurich)
2
Bivariate distributions Correlation = -0.85
[Schoelzel and Friedrichs 2008, NPG]
Bivariate distribution can be written as F (x, y ) = C (FX (x), FY (y ))
[Sklar 1959]
with FX (x) = Pr (X ≤ x), FY (y ) = Pr (Y ≤ y ) the marginal distributions and C a bivariate Copula. Jakob Zscheischler (ETH Zurich)
2
Example for Archimedean copulas (Marginals are standard normal)
Clayton
allows lower tail dependence
Frank
no tail dependence
Gumbel
allows upper tail dependence
[Schoelzel and Friedrichs 2008, NPG]
Jakob Zscheischler (ETH Zurich)
3
Kendall distribution function and bivariate return periods The Kendall distribution function is defined as KC (t) = P(C (u, v ) ≤ t) measures the probability that a random event will appear in the region of I 2 defined by the inequality C (u, v ) ≤ t. Easily tractable for Archimedean Copulas.
Jakob Zscheischler (ETH Zurich)
4
Kendall distribution function and bivariate return periods The Kendall distribution function is defined as KC (t) = P(C (u, v ) ≤ t) measures the probability that a random event will appear in the region of I 2 defined by the inequality C (u, v ) ≤ t. Easily tractable for Archimedean Copulas. Kendall’s return period is then defined as [Salvadori et al. 2011] KRP =
1 1 − KC (t)
Jakob Zscheischler (ETH Zurich)
4
Kendall distribution function and bivariate return periods The Kendall distribution function is defined as KC (t) = P(C (u, v ) ≤ t) measures the probability that a random event will appear in the region of I 2 defined by the inequality C (u, v ) ≤ t. Easily tractable for Archimedean Copulas. Kendall’s return period is then defined as [Salvadori et al. 2011] KRP =
1 1 − KC (t)
Compute bivariate return periods of hot and dry seasons by fitting a copula to -P and T and computing Kendall’s return period. Return periods can be interpreted as the magnitude of an event.
Jakob Zscheischler (ETH Zurich)
4
Example: California
[AghaKouchak et al. 2014, GRL]
Return periods of concurrent dry and hot seasons Data: Means over November - April from 1896 to 2014.
Jakob Zscheischler (ETH Zurich)
5
Bivariate return periods depend on dependence structure
4
The same event has a lower return period if variables are more strongly correlated (here: correlation = 0.8).
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● ●
−4
●
−4
−2
0
Jakob Zscheischler (ETH Zurich)
2
4
6
Bivariate return periods depend on dependence structure
4
The same event has a lower return period if variables are more strongly correlated (here: correlation = 0.8).
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●
●
−2
0
2
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●
● ●
−4
●
−4
−2
0
Jakob Zscheischler (ETH Zurich)
2
4
6
Bivariate return periods depend on dependence structure
4
The same event has a lower return period if variables are more strongly correlated (here: correlation = 0.8).
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−2
0
2
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−4
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−4
−2
0
Jakob Zscheischler (ETH Zurich)
2
4
6
Bivariate return periods depend on dependence structure
4
The same event has a lower return period if variables are more strongly correlated (here: correlation = 0.8).
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−2
0
2
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−4
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−4
−2
0
Jakob Zscheischler (ETH Zurich)
2
4
6
Bivariate return periods depend on dependence structure
4
The same event has a lower return period if variables are more strongly correlated (here: correlation = 0.8).
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● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ●● ●●●● ● ●● ●●● ●● ●● ● ●●●●● ● ● ●●● ● ●● ●● ●● ● ● ● ●●● ● ●● ● ● ●● ● ● ●● ● ● ●● ● ● ●●● ● ●●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ●●●● ●●●● ● ● ● ● ●●● ●● ● ●● ●● ● ● ●●● ● ● ●● ● ● ● ●● ●●● ●●●●● ● ● ● ●● ● ● ●● ● ●● ● ●●● ●● ●● ● ●● ● ●● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ●● ●●● ● ●● ●●● ● ●● ● ● ●●● ● ●● ● ● ●● ● ●●● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ●● ●● ● ●● ● ●●●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ●●●● ● ● ● ● ●● ● ● ●● ● ●●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ●● ●● ● ●● ●● ● ●● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ●●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ●●●● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ●● ● ● ●●●● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ●●● ●●●●● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ●● ● ●●● ●● ● ● ●● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ●● ●● ● ● ● ● ●● ● ●● ● ●● ● ●● ●● ● ●● ●● ●● ● ● ● ●● ●● ● ● ●● ●● ● ● ● ●● ● ● ● ● ●● ●●●● ●● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ●● ●● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ●●● ● ● ● ●●●● ●● ●● ●● ●● ●● ● ● ● ● ● ●●●● ●● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●●● ●● ● ●●● ●● ●● ●● ● ●● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ●● ● ●● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ●●● ●● ● ● ● ●● ● ●● ● ● ● ● ●●● ● ●● ● ●● ● ●●●●● ● ●● ● ● ●● ● ● ●● ●● ● ●● ● ●● ● ● ● ● ● ● ● ●● ●●● ●● ●● ●● ●●● ●● ● ● ●● ● ● ●● ● ●●●● ●● ●● ● ●● ● ● ● ● ●● ●● ● ● ●● ● ●● ●●● ● ●● ● ●● ● ● ●● ●● ●● ● ● ●●● ●● ●●● ● ●● ● ●● ● ● ●● ● ●●●●●●● ● ● ● ●● ●● ● ● ●● ●● ● ●●● ● ●● ●● ● ● ●●● ● ●●●●● ●●● ● ●● ● ● ●●●●● ● ●●●●● ● ●● ●● ●●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ●● ●●● ● ● ●●●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ●● ●● ● ●●●● ●● ● ●● ● ● ●● ● ●● ● ● ● ●●●● ● ● ● ● ● ● ● ●
−2
0
2
●
●
−4
●
−4
●
● ●
●
shuffled data original data 50 yr return period 19 yr return period −2
●
●
●
●
●
● ●
● ●
0
Jakob Zscheischler (ETH Zurich)
2
4
6
Bivariate return periods depend on dependence structure 0.9
relative difference (in %)
0.71
0.81
80
correlation 0.43 0.52 0.62
60
0.24
0.33
40
0.05
0.14
20
% 10
20
50 100 250 return periods [years]
500
1000
Ignoring the dependence between variables can lead to a strong overestimation of bivariate return periods Jakob Zscheischler (ETH Zurich)
7
Temperature and precipitation from CRU at Berlin (JJA)
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● ●
20
●
18
●
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●
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●
● ●●
●
● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ●● ●● ● ●● ●● ● ● ● ●●● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●●●● ●●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ●● ● ● ●
● ●
16
Temperature [°C]
● ● ● ● ● ● ● ●
●
● ● ●
●
● ● ● ●●● ● ● ● ● ● ● ● ●● ● ● ● ●● ●
● ●● ●
● ● ●
●
●
●●
● ●
140
shuffled data original data 20 yr return period 13.5 yr return period 120
100
80
60
40
20
Precipitation [mm]
Jakob Zscheischler (ETH Zurich)
8
Global analysis
I
Data: seasonal T and P from CMIP5 (1901-2100), 2.5◦
I
fit (Archimedian) copulas to T and -P over 3 hottest months
I
compute bivariate return periods of concurrent dry and hot seasons
I
compare 20th and 21st century (RCP8.5)
I
compare with CRU
Jakob Zscheischler (ETH Zurich)
9
Correlations between T and P
60
80
CMIP5 Ensemble Mean T-P correlations during hottest months (20th century)
20
40
0.5
−20
0
0.0
−60
−40
−0.5
−150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
10
Correlations between T and P
80
CMIP5 Ensemble Mean T-P correlations during hottest months (21st century)
60
0.6
40
0.4
20
0.2
0
0.0
−20
−0.2
−40
−0.4
−60
−0.6
−150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
11
Correlations between T and P
60
80
CMIP5 Ensemble Mean T-P correlations during hottest months (detrended, 21st century)
20
40
0.5
−20
0
0.0
−60
−40
−0.5
−150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
12
Observation: correlation between T and P intensify
80
CMIP5 Ensemble Mean difference in T-P correlations during hottest months (detrended, 21st-20th century)
60
0.15
40
0.10
20
0.05
0
0.00
−20
−0.15
−60
−0.10
−40
−0.05
−150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
13
Change in correlations leads to higher frequency of very dry and hot seasons What is the return period of a 100yr event derived from uncorrelated T and P data in the 20th and 21st century? (SREX regions)
Jakob Zscheischler (ETH Zurich)
14
Change in correlations leads to higher frequency of very dry and hot seasons
0.0
What is the return period of a 100yr event derived from uncorrelated T and P data in the 20th and 21st century? (SREX regions) ●
−0.2
● ●
● ●
−0.4
T−P correlation
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●
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●
●
●
● ●●
● ●
●●
−0.6
● ●●
●
50
●●
● ●
●
● ●●
●
45
● ●
40
● ●
●
35
●
●●
● ●
●
●●
●
● ●
●
●
●
●
●
●
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●
● ●●
●
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● ●
● ●
●
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30
100 year event
55
20th century 21th century
●●
ALA
CGI
WNA CNA
ENA CAM AMZ
NEB WSA SSA
NEU
CEU MED SAH WAF
EAF
Jakob Zscheischler (ETH Zurich)
SAF
NAS WAS CAS
TIP
EAS
SAS
SEA
NAU
SAU
14
Change in correlations leads to higher frequency of very dry and hot seasons
0.0
What is the return period of a 100yr event derived from uncorrelated T and P data in the 20th and 21st century? (SREX regions) ●
−0.2
● ●
● ●
−0.4
T−P correlation
●
●●
●
● ●
●●
● ●
●●
●
●
● ● ●
● ●
●● ●
●
●●
●●
●
●
●
●
● ●
●
●
●
● ●●
● ●
●●
−0.6
● ●●
●
50
●●
● ●
●
● ●●
●
45
● ●
40
● ●
●
35
●
●●
● ●
●
●●
●
● ●
●
●
●
●
●
●
● ●
●●
●
● ●●
●
● ●●
● ●
● ●
●
●●
30
100 year event
55
20th century 21th century
●●
ALA
CGI
WNA CNA
ENA CAM AMZ
NEB WSA SSA
NEU
CEU MED SAH WAF
EAF
Jakob Zscheischler (ETH Zurich)
SAF
NAS WAS CAS
TIP
EAS
SAS
SEA
NAU
SAU
14
Frequency of very dry and hot seasons increases
80
What is the return periods of a 100yr event of the 20th century in the 21st century?
40
60
140
20
120
0
100
−20
80
−40
60
−60
years −150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
15
Caution: Models overestimate correlation between T and P in hottest months
60
80
CMIP5 Ensemble Mean (20th century)
20
40
0.5
−20
0
0.0
−60
−40
−0.5
−150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
16
Caution: Models overestimate correlation between T and P in hottest months
60
80
CRU (20th century)
20
40
0.5
−20
0
0.0
−60
−40
−0.5
−150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
17
Caution: Models overestimate correlation between T and P in hottest months
80
CRU-CMIP5 (20th century)
60
0.6
40
0.4
20
0.2
0
0.0
−20
−0.6
−60
−0.4
−40
−0.2
−150
−100
−50
0
50
Jakob Zscheischler (ETH Zurich)
100
150
18
Conclusions I
bivariate return periods depend on the correlation of the underlying variables, if variables are more strongly correlated, events where both variables are extreme are more likely
Jakob Zscheischler (ETH Zurich)
19
Conclusions I
bivariate return periods depend on the correlation of the underlying variables, if variables are more strongly correlated, events where both variables are extreme are more likely
I
correlation between T and P during the hottest 3 months will intensify (according to CMIP5)
Jakob Zscheischler (ETH Zurich)
19
Conclusions I
bivariate return periods depend on the correlation of the underlying variables, if variables are more strongly correlated, events where both variables are extreme are more likely
I
correlation between T and P during the hottest 3 months will intensify (according to CMIP5)
I
extreme hot and dry 3-month periods will occur more often in the future
Jakob Zscheischler (ETH Zurich)
19
Conclusions I
bivariate return periods depend on the correlation of the underlying variables, if variables are more strongly correlated, events where both variables are extreme are more likely
I
correlation between T and P during the hottest 3 months will intensify (according to CMIP5)
I
extreme hot and dry 3-month periods will occur more often in the future
Remarks: I
models overestimate correlation between T and P
I
CMIP5 data was detrended
Jakob Zscheischler (ETH Zurich)
19
Conclusions I
bivariate return periods depend on the correlation of the underlying variables, if variables are more strongly correlated, events where both variables are extreme are more likely
I
correlation between T and P during the hottest 3 months will intensify (according to CMIP5)
I
extreme hot and dry 3-month periods will occur more often in the future
Remarks: I
models overestimate correlation between T and P
I
CMIP5 data was detrended
Thank you! Jakob Zscheischler (ETH Zurich)
19
Return periods (hot and dry) in hottest/driest months dry
0.5
original T detrended P detrended T & P detrended
0.0
0.0
Bivariate
0.5
1.0
1.0
1.5
1.5
2.0
hot
+T
mean(log(return time)) 0.5 1.0 1.5 2.0
original T detrended
0.0
0.0
Univariate
mean(log(return time)) 0.5 1.0 1.5 2.0
2.5
+T
1900
0.2
0.4
0.6
0.8
1.0
1.2
−P
original P detrended 1920
1940
0.0
Univariate
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
−P
1960
1980
2000
Jakob Zscheischler (ETH Zurich)
1900
1920
1940
1960
1980
2000
20
Return periods (cold and wet) in coldest/wettest months wet
1.0 0.5
original T detrended P detrended T & P detrended
0.0
0.0
Bivariate
0.5
1.0
1.5
1.5
2.0
cold
mean(log(return time)) 0.5 1.0 1.5
original T detrended
0.0
0.0
Univariate
−T mean(log(return time)) 0.5 1.0 1.5 2.0
−T
+P
0.6
0.8
0.4
0.6 1900
0.2
0.4
original P detrended 1920
1940
0.0
0.2 0.0
Univariate
0.8
1.0
1.0
1.2
1.2
+P
1960
1980
2000
Jakob Zscheischler (ETH Zurich)
1900
1920
1940
1960
1980
2000
21
Return periods (cold and wet) in coldest/wettest months wet
1.0 0.5
original T detrended P detrended T & P detrended
0.0
0.0
Bivariate
0.5
1.0
1.5
1.5
2.0
cold
mean(log(return time)) 0.5 1.0 1.5
original T detrended
0.0
0.0
Univariate
−T mean(log(return time)) 0.5 1.0 1.5 2.0
−T
+P
0.6
0.8
0.4
0.6 1900
0.2
0.4
original P detrended 1920
0.0
0.2 0.0
Univariate
0.8
1.0
1.0
1.2
1.2
+P
1940
1960
1980
2000
1900
1920
1940
1960
1980
2000
Data from Global Energy Balance Archive; Wild (2012, BAMS) Jakob Zscheischler (ETH Zurich)
21
Bivariate versus univariate return periods
At the global scale, bivariate return periods are strongly correlated with univariate return periods of temperature.
What about other levels of aggregation?
Jakob Zscheischler (ETH Zurich)
22
Bivariate versus univariate return periods
correlation univariate vs bivariate return periods 0.2 0.4 0.6 0.8
1.0
Average correlation between bivariate and univariate return periods for different steps of aggregation.
0.0
temperature precipitation quadrants global interquartile range 0
20
40 60 spatial resolution [degree]
80
Jakob Zscheischler (ETH Zurich)
23
T varies globally, P varies locally
[Ahlstr¨ om et al 2015]
Jakob Zscheischler (ETH Zurich)
24
