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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:

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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?

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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

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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)

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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) D.B.Stephenson@reading.ac.uk 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)

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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

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Dispersion of monthly counts

Units: %

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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) D.B.Stephenson@reading.ac.uk 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

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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?

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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

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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

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Scatterplot of mean intensity versus counts

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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

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London daily summer temperatures

n=30*120=3600 daysBlack line = sample meanRed line = 99th percentile

G G’

O

O’

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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)

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Bias correctionO’=B(G’)

Change factorO’=C(O)

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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!

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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!

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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 attentiond.b.stephenson@exeter.ac.uk

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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.

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Extra slides ...

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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.