1. Assessing High-Impact Weather Variations and Changes Utilizing Extreme Value Theory NCAR Earth System Laboratory National Center for Atmospheric Research Boulder, Co, USA NCAR is Sponsored by NSF and this work is partially supported by the Willis Research Network and the Research Program to Secure Energy for America Greg Holland 1

2. Presentation Summary The Issues Applying Extreme Value Theory – Reverse Weibull and Generalized Pareto Examples for Hurricanes Current and Future Climate – Example Application – Other Studies Suggest that Extreme Weather provides a Bellwether of Climate Transition and Change Holland ASP Extremes Colloquium 06112

3. The Issues Holland ASP Extremes Colloquium 06113 Frequency Value Highly skewed Short and noisy record May be truncated at high values: Real physical limitation Resolution/capacity/rarity May be truncated at low values: Observing practice High noise level Societal Importance α rarity Typical weather-system PDF

4. Early Applications of EVT Mearns, L. O., Katz, R. W., and Schneider, S. H., 1984: Extreme High-Temperature Events: Changes in their Probabilities with Changes in Mean Temperature. J. Clim. Appl. Meteorol. 23, 1601-1613. Katz, R. and B. Brown, 1992: Extreme Events in a Changing Climate: Variability is more Important than Averages. Clim. Change.,21, 289-302. Katz, R. and B. Brown, 1994: Sensitivity of Extreme Events to Climate Change: The Case of Autocorrelated Time Series. Environmetrics, 5, 451-462. Holland ASP Extremes Colloquium 06114

5. Our Current Applications of EVT Helping assess data reliability for meteorological extremes; Downscaling to build up an assessment of the tail of the severe weather distribution utilizing the truncated information that comes out of regional climate models; Enabling recognition and understanding of potential “regime” shifts in climate series; Assessing the “uniqueness” of specific extreme events, or combinations thereof, in the context of current and changing climate. Holland ASP Extremes Colloquium 06115

6. Choice of Distribution Holland ASP Extremes Colloquium 06116 Frequency Value Fisher-Tippett (-Gnedenko) Theorem (FTT) for Extreme Value Theory states that the normalized maximum of any set of random variables can be explained by one of three distributions: Weibull (Weibull 1939) Gumbel (Gumbel 1958) or Fréchet, (Fréchet 1927) (also known as Type 1, 2 or 3: Fisher and Tippett 1928, Gnedenko 1948). For working with just the tail of the distribution, the Generalized Pareto might be more appropriate.

7. Example Application: Hurricanes Holland ASP Extremes Colloquium 06117 Intensity Annual Frequency

8. Weibull Analysis We utilize the Reverse Weibull distribution for which the CDF and PDF are: Where: parameters a and b determine the scale and the shape, respectively b=1 is the exponential distribution b=2 is the Rayleigh distribution, and b=3.5 is an approximation of the normal distribution. 8Holland ASP Extremes Colloquium 0611

9. Sensitivity to Scale and Shape Variations 9Holland ASP Extremes Colloquium 0611

10. Sensitivity to Scale and Shape Variations 10Holland ASP Extremes Colloquium 0611 Hazard Rate Function for the Weibull distribution: (Hillier and Lieberman 1986) For b>1 the sensitivity increases with increasing x to a limit of 1-e -x. (Katz and Brown 1994) Frequency Value

11. Weibull Analysis Holland ASP Extremes Colloquium 061111 Probability of exceeding a threshold event: Exceedance probability decreases as the event becomes rarer (c/a increases), and/or the population less variable (b increases).

12. Weibull Analysis Holland ASP Extremes Colloquium 061112 (where: Γ is the Gamma function) Γ(n) α (n-1)! Given µ and σ we can make an estimate of a and b and thus the whole distribution.

13. Application to Atlantic Hurricane Intensity PDF (Normalized, HURDAT Smoothed) CDF with Weibull fit: a=35, b=1.9 13Holland ASP Extremes Colloquium 0611 Intensity PDF Log-Log Weibull Probability Plot

14. Application to Current Climate: 1980-1994 vs 1994-2007 Mean intensity and SD change of 2-3 and 4-5 m/s 3-5 times increase in probability of Cat 5 hurricanes! Equivalent to moving from 1 Cat 5 every 3-5 years to 1 every year. (Caveat on ability to fit the observed distribution) 14Holland ASP Extremes Colloquium 0611

15. Application to Current Climate Holland ASP Extremes Colloquium 061115 Weibull original is for 1980-1994 data Observed is 1995-2008 data (Data from IBTrACS) Intensity PDF Annual Frequency PDF

16. Application to Truncated Model PDFs Holland ASP Extremes Colloquium 061116 The Issue: Regional Climate Models cannot run over long time scales at resolutions sufficient to resolve weather extremes

17. Background: Regional Climate Modeling Holland ASP Extremes Colloquium 061117 150 km 12 km 36 km Time Slices

18. NRCM Hurricane Intensity Change Wind speed distribution shifting to right Model projected increase in TC intensity, but can only resolve up to Cat 2. How do we assess the extremes? 18Holland ASP Extremes Colloquium 0611 Cannot use EVT extrapolation or quantile mapping

19. Application to Model Data We make the following assumptions: – the modeled changes are indicative of the changes that would have been simulated in the full distribution, were it resolved – there is no process that will cause a change in the unresolved tail of the distribution without any signal in the resolved component. Note: the truncated model will underestimate changes in mean and SD – Thus this approach provides a conservative assessment. Holland ASP Extremes Colloquium 061119

20. Method Fit a Reflected Weibull to the observed distribution for a specified historical period and calculate the exceedance probability for the extremes of interest Calculate the mean and standard deviation changes from current to future climate model predictions; Apply these changes to assess the changes to the Reflected Weibull shape and strength parameters; Apply these new parameters to fill out an estimated distribution and to calculate changes in exceedance probability. Holland ASP Extremes Colloquium 061120

21. Example: NRCM Hurricane Changes Holland ASP Extremes Colloquium 061121

22. Application to NRCM Hurricane Predictions PE69=Cat5 PE58=Cat4,5 PE48=Major Hurricanes PE32=Hurricanes 22Holland ASP Extremes Colloquium 0611

23. Alternative Generalized Pareto Distribution Holland ASP Extremes Colloquium 061123 Results: Exceedance probability decreases for weak storms Exceedance probability increases for intense storms Cat 5 by 29% for 2045-2055 compared to 32% for Weibull (where a,b,c are the scale, shape and location parameters) (Suzuki 2011)

24. Comparison with Alternative Approaches Holland ASP Extremes Colloquium 061124

25. Bender et al (2009) Used GFDL Hurricane Model applied to future storms: – Mean intensity increase ~2.5 m/s – 78% increase in Cat4-5 numbers Applying EVT to 1995-2007 conditions (same period as Bender et al): – Mean intensity increase of 2.5 m/s gives ~35% increase in Cat4-5; – Adding increase in SD of 5 m/s gives ~60% increase in Cat4-5. 25Holland ASP Extremes Colloquium 0611

26. Yamada et al 2010 26Holland ASP Extremes Colloquium 0611 Cloud-resolving simulation over limited time period, with and without Greenhouse Warming contribution.

27. Hallegatte et al (2007) Annual probability of landfall increase: Cat1=17%, Cat2=33%, Cat3=44%, Cat 4=58%, and Cat5=215%. 27Holland ASP Extremes Colloquium 0611

28. Hallegatte et al ctd The nonlinearity of intense hurricane changes is a clear illustration of the multiplicative effect of extreme events: – “a limited change in the characteristics of the mean (+13% in average maximum wind speed at landfall) can have a dramatic impact on the distribution tail (+215% increase in category-5 landfall probability)”. Moreover, because socio-economic impacts of hurricanes are highly nonlinear with respect to hurricane intensity, such an evolution may translate into an even larger change for hurricane damages 28Holland ASP Extremes Colloquium 0611

29. Severe Weather as a Bellwether of Climate Transition and Change This analysis indicates that: – Weather extremes will respond to climate variability and change at much higher amplitude than will the general weather-system population – Thus, weather extremes provide a bellwether of climate transition and change Holland ASP Extremes Colloquium 061129

30. Summary Application of EVT to weather extremes can provide a valuable tool for: – diagnosing extremes – assessing potential signals for climate variability and change – downscaling assessment for any weather extreme EVT predicts that changes in extremes will be >>> than those in the mean or variance: – extreme hurricanes increase 2-3-fold for a 5-10% increase in the mean and SD – supported by both observations and modeling – Implies climate variability and change will first appear in extremes, thus Weather extremes may provide a bellwether for climate variability and change Holland ASP Extremes Colloquium 061130 Thank You