1. 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. 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. 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 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. 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

5. Do extratropical storms cluster? Transits +/-10 0 of Nova Scotia (45°N, 60°W)‏ Transits +/-10 0 of Berlin (52°N,12.5°E)‏  Yes! Especially over western Europe Processes that can lead to clusters: Random sampling Rate-varying process (hazard rate varies in time) Clustered process (one event spawns the next)

6. 6 Blue curve = Poisson GLM trend Grey 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. 7 Dispersion of monthly counts Units: % 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??

8. 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 ß 0, ß k

9. 9 Dependence of storm counts on patterns NAO PEU SCA PNA EAP EA/WR  Several patterns are required to capture regional storminess changes! See Seierstad et al. (2007)

10. 10 Units: % under Do varying-rates explain the clustering?  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

11. 11 Aggregate loss caused by extremes Renato Vitolo, Chris Ferro, Theo Economou, Alasdair Hunter, Ian Cook 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 Damage J. of Applied Meteorology, Vol 41, 754-762 1 2 3...N Intensity/loss time

12. 12 Variations in London counts and mean intensity Positive association between counts and mean magnitude that is mainly due to interannual variations rather than trends.

13. 13 Scatterplot of mean intensity versus counts

14. 14 Correlation between mean intensity and counts  Robust positive correlation over most of northern Europe!

15. 15 The importance of correlation for aggregate loss Synthetic example: 10000 years simulated from Poisson and LogNormal distributions Correlation0.02  Mean Y30.4  200yr Y169.2 Correlation0.19  Mean Y28.6  200yr Y257.7 200-year loss of 257 much greater than 169 obtained for no correlation!!

16. 16 Reinsurance actuarial assumptions 1. The losses are identically distributed: The distribution of X 1,X 2,...,X N does not change in time: Pr(X i >u)=1-F(u) for i=1,2,...,N. 2. The losses are independent of one another: The X 1,X 2,...,X N are independent of one another: e.g. Pr(X 1 >u & X 2 >v)=Pr(X 1 >u)Pr(X 2 >v); 3. The losses are independent of the counts: The X 1,X 2,...,X N are independent of the counts N. Counts N X1X1 X2X2 XNXN dynamic background state... The assumptions are not valid for weather extremes conditional upon the dynamic state of the system!

17. 17 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 extreme daily temperatures might change in future since they have big impacts on society. e.g. Heat-related mortality in London

18. 18 London daily summer temperatures n=30*120=3600 days Black line = sample mean Red line = 99th percentile GG’ O O’

19. 19 Probability density functions Black line = pdf of obs data 1970-1999 Blue line = pdf of climate data 1970-1999 Red line = pdf of climate data 2070-2099 O,GO,G,G’ O O’

20. 20 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. F o’ = F G’ 2. Bias correction Assume O’=B(G’) where B(.)=F o -1 (F G (.)) 3. Change factor Assume O’=C(O) where C(.)=F G’ -1 (F G (.)) 4. Other e.g. EVT fits to tail and then adjust EVT parameters Calibration strategies G O O’ G’ O’=B(G’) O’=C(O)

21. 21 Bias correction O’=B(G’) Change factor O’=C(O)

22. 22 Example: mean warming for London (no shape change)  Two approaches give different future mean temperatures!

23. 23 Effect of calibration on extremes Change in 10-summer level 2040-69 relative to 1970-99 No calibration T g’ - T o Bias correction Location, scale & shape Change factor Location + scale  Substantial differences between different predictions!

24. 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 G  O 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 attention d.b.stephenson@exeter.ac.uk

25. 25 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.

26. 26 Extra slides...

27. 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

28. 28 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 statistical models 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?

29. 29 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 = y d(x, y) = d(y, x) d(x, z) ≤ d(x, y) + d(y, z)

30. 30 Exeter Storm Risk Group David Stephenson, Renato Vitolo, Chris Ferro, Mark Holland Alef 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.