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Transcript of 1 Grand Challenges in Modelling [Storm-Related] Extremes David Stephenson Exeter Climate Systems...
1
Grand Challenges in Modelling [Storm-Related] Extremes
David Stephenson Exeter Climate Systems
NCAR Research Colloquium, Statistical assessment of Extreme Weather Phenomenon under Climate Change, 14 June 2011 Acknowledgements:
2
Some grand challenges for climate science
How to quantify the collective risk of extremes? How to calibrate climate model extremes so that they resemble observed
extremes? How to understand and characterize complex spatio-temporal extremal
processes?
3
Extratropical storms Dec 1989 - Feb 1990 Tracks of maxima in 850mb vorticity
Example:Wind speed of 15 m/s and radius of 500 km vorticity of 6x10-5 /s.
Wind
speed u
r
u
rπ
π ru
AREA
NCIRCULATIOVORTICITY
.2
2
22
Why use vorticity?
• More prominent small-scale features allow earlier detection
• Much less sensitive to the background state
• Directly linked with low-level winds (through circulation) and precipitation (via vertical motion)
4
Data: storm tracks in 1950-2003 reanalysis• 355,460 eastward cyclone tracks identified using TRACK software• Extended 6-month winters (1 October - 31 March)• 6 hourly NCAR/NCEP reanalyses from 1950-2003
Mean transit counts (per month)Stormtracks of Dec 1989-Feb1990
(c) [email protected] 2004
Do extratropical storms cluster?
Transits +/-100 of Nova Scotia (45°N, 60°W)
Transits +/-100 of Berlin (52°N,12.5°E)
Yes! Especially over western EuropeProcesses that can lead to clusters: Random sampling Rate-varying process (hazard rate varies in time) Clustered process (one event spawns the next)
6
Blue curve = Poisson GLM trendGrey shading = 95% Conf Int.
Red curve = Lowess fit(local polynomial regression)lowess() or loess() in R
Counts of storms passing by London
Variance of counts is substantially greater than the mean (131%) Long-term trend has negligible effect on the overdispersion (3%)
"Not everything that can be counted counts, and not everything that counts can be counted" -- Albert Einstein
7
Dispersion of monthly counts
Units: %
12
n
sn
Mailier, P.J., Stephenson, D.B., Ferro, C.A.T. and Hodges, K.I. (2006): Serial clustering of extratropical cyclones, Monthly Weather Review, 134, pp 2224-2240
Substantial clustering over western Europe. Why??
(c) [email protected] 2004
Does varying-rate explain the clustering?
Poisson regression
= number of storms, e.g. monthly counts of windstorms.
= time-varying, flow dependent rate.
= large-scale teleconnection indices (covariates).
GLM maximum likelihood estimation of ß0, ßk
μn
( )kx t
01
)
logK
k kk=
n | x ~ Poisson(μ
μ = β + β x
9
Dependence of storm counts on patterns
NAOPEU SCA
PNA EAP EA/WR
Several patterns are required to capture regional storminess changes!
See Seierstad et al. (2007)
ˆk
10
Units: %
under
under
Do varying-rates explain the clustering?
12
n
sn
Clustering in counts is well accounted for by variations in rate related to time variations in teleconnections
Dispersion in counts Residual dispersion after regression on teleconnections
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Aggregate loss caused by extremesRenato Vitolo, Chris Ferro, Theo Economou, Alasdair Hunter, Ian Cook
1 2
2 2 2
is a random variable
is a random variable
Properties when X independent
of each other and N:
( )
N
N N X
N X N X
Y X X X NX
X
N
E(Y) E (NE(X)) μ
Var Y
Aggregate loss (and many extreme indices) are RANDOM SUMs - the sum of a random number of random values ...
Katz, R.W. 2002: Stochastic Modeling of Hurricane DamageJ. of Applied Meteorology, Vol 41, 754-762
1 2 3 ... N
Intensity/loss
time
12
Variations in London counts and mean intensity
N
Positive association between counts and mean magnitude thatis mainly due to interannual variations rather than trends.
x
13
Scatterplot of mean intensity versus counts
1
Dashed lines show isopleths
of aggregate intensity:
NY X X NX
364.0))/(,(
371.0)/,(
NYNcor
NYNcorx
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Correlation between mean intensity and counts
Robust positive correlation over most of northern Europe!
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The importance of correlation for aggregate lossSynthetic example: 10000 years simulated from Poisson and LogNormal distributions
Correlation 0.02 Mean Y 30.4 200yr Y 169.2
Correlation 0.19 Mean Y 28.6 200yr Y 257.7
200-year loss of 257 much greaterthan 169 obtained forno correlation!!
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Reinsurance actuarial assumptions1. The losses are identically distributed:
The distribution of X1,X2,...,XN does not
change in time:
Pr(Xi>u)=1-F(u) for i=1,2,...,N.
2. The losses are independent of one another:
The X1,X2,...,XN are independent of one another:
e.g. Pr(X1>u & X2>v)=Pr(X1>u)Pr(X2>v);
3. The losses are independent of the counts:
The X1,X2,...,XN are independent of the counts N.
CountsN
X1 X2 XN
dynamic background state
...
The assumptions are not valid for weather extremesconditional upon the dynamic state of the system!
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Calibration of model extremes Chun Kit Ho, Mat Collins, Simon Brown, Chris Ferro
DATA
Summertime daily mean air temperatures (15 May -15 Sep)
O=Observations 1970-1999 from E-OBS gridded dataset (Haylock et al., 2008)
G=HadRM3 standard run for 1970-1999 (25 km horizontal resolution) forced by HadCM3; SRES A1B scenario.
G’=Future 30-year time slices from HadRM3 standard run forced by HadCM3; SRES A1B scenario
• 2010-2039• 2040-2069• 2070-2099
We would like to know how extremedaily temperatures might change in futuresince they have big impacts on society.
e.g. Heat-related mortality in London
18
London daily summer temperatures
n=30*120=3600 daysBlack line = sample meanRed line = 99th percentile
G G’
O
O’
19
Probability density functions
Black line = pdf of obs data 1970-1999Blue line = pdf of climate data 1970-1999Red line = pdf of climate data 2070-2099
O,G O,G,G’
OO’
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How to infer distribution of O’ from distributions of O, G and G’?
1. No calibration Assume O’ and G’ have identical distributions (i.e. perfect model!)
i.e. Fo’ = FG’
2. Bias correction Assume O’=B(G’) where B(.)=Fo
-1 (FG(.))
3. Change factorAssume O’=C(O) where C(.)=FG’
-1 (FG(.))
4. Othere.g. EVT fits to tail and then adjust EVT parameters
Calibration strategies
G
O O’
G’
O’=B(G’)
O’=C(O)
22
Example: mean warming for London (no shape change)
ngmean warmi 6.3 i.e.
6.19)61.1565.19(35.3
01.395.15
:gives dataour for Which
)()'(
)'(
))'(()'('
)(
)(
Correction Bias
'
1
C
C
OE
G
GFFGBO
xFxF
xFxF
GGG
OO
GG
OO
GO
G
GG
O
OO
ngmean warmi 14 i.e.
1.20)61.1595.15(35.3
03.465.19
:gives dataour for Which
)()'(
)'(
))'(()'('
)(
)(
factor Change
''
'
''
1'
'
''
C.
C
OE
O
OFFOCO
xFxF
xFxF
GOG
GG
GG
GG
GG
G
GG
G
GG
Two approaches give different future mean temperatures!
23
Effect of calibration on extremesChange in 10-summer level 2040-69 relative to 1970-99
No calibrationTg’ - To
Bias correctionLocation, scale & shape
Change factorLocation + scale
Substantial differences between different predictions!
24
Summary Storm-related extremes cluster in time due to dynamic modulation of the
rate by large-scale circulation patterns. Correlation exists between counts and mean intensity of extremes – this
has large implications for aggregate losses Climate model extremes are not statistically distributed like observed
extremes and so calibration is required. Different “rational” calibration strategies lead to substantially different
predictions of future mean and extremes! Change factor transformation G G’ is more linear than bias correction
transformation GO Bias correction of extremes can be validated for present day climate whereas
change factor can only be used for future changes. Much more statistical research is required in this area in order to infer
reliable estimates of future weather and climate extremes.
We need to develop WELL-SPECIFIED PHYSICALLY INFORMED STATISTICAL MODELS of the EXTREMAL PROCESSES
Thank you for your [email protected]
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References
Mailier, P.J., Stephenson, D.B., Ferro, C.A.T. and Hodges, K.I. (2006): Serial clustering of extratropical cyclones, Monthly Weather Review, 134, pp 2224-2240. 21 citations and growing!
Seierstad, I.A., Stephenson, D.B., and Kvamsto, N.G. (2007): How useful are teleconnection patterns for explaining variability in extratropical storminess? Tellus A, 59 (2), pp 170–181
Kvamsto, N-G., Song, Y., Seierstad, I., Sorteberg, A. and D.B. Stephenson, 2008: Clustering of cyclones in the ARPEGE general circulation model, Tellus A, Vol. 60, No. 3.), pp. 547-556.
Vitolo, R., Stephenson, D.B., Cook, I.M. and Mitchell-Wallace, K. (2009): Serial clustering of intense European storms, Meteorologische Zeitschrift, Vol. 18, No. 4, 411-424.
27
Statistical methods used in climate science Extreme indices – sample statistics Basic extreme value modelling
GEV modelling of block maxima GPD modelling of excesses above high
threshold Point process model of exceedances
More complex EVT models Inclusion of explanatory factors
(e.g. trend, ENSO, etc.) Spatial pooling Max stable processes Bayesian hierarchical models + many more
Other stochastic process models
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Extreme indices are useful and easy but … They don’t always measure extreme
values in the tail of the distribution! They often confound changes in rate
and magnitude They strongly depend on threshold and
so make model comparison difficult They say nothing about extreme
behaviour for rarer extreme events at higher thresholds
They generally don’t involve probability so fail to quantify uncertainty (no inferential model)
More informative approach: model the extremal process using statisticalmodels whose parameters are then sufficient to provide complete summaries of all other possible statistics (and can simulate!)
See: Katz, R.W. (2010) “Statistics of Extremes in Climate Change”, Climatic Change, 100, 71-76
Frich indices …
Oh really?
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Furthermore … indices are not METRICS!One should avoid the word “metric” unless the statistic has distance properties! Index, sample/descriptive statistic, or measure is a more sensible name!
Oxford English Dictionary: Metric - A binary function of a topological space which gives, for any two points of the space, a value equal to the distance between them, or a value treated as analogous to distance for analysis.
Properties of a metric:d(x, y) ≥ 0 d(x, y) = 0 if and only if x = yd(x, y) = d(y, x) d(x, z) ≤ d(x, y) + d(y, z)
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Exeter Storm Risk GroupDavid Stephenson, Renato Vitolo, Chris Ferro, Mark HollandAlef Sterk, Theo Economou, Alasdair Hunter, Phil Sansom
Key areas of interest and expertise:
Mathematical and statistical modelling of extratropical and tropical cyclones and the quantification of risk using concepts from
extreme value theory, dynamical systems theory, stochastic processes, etc.