Report

Assessing High-Impact Weather Variations and Changes Utilizing Extreme Value Theory Greg Holland 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 1 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 0611 2 The Issues Frequency 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 Value Typical weather-system PDF Holland ASP Extremes Colloquium 0611 3 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 0611 4 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 0611 5 Choice of Distribution Frequency For working with just the tail of the distribution, the Generalized Pareto might be more appropriate. 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). Holland ASP Extremes Colloquium 0611 6 Example Application: Hurricanes 1995-2008 a=149 b=46 lifmin=0 1 1 b 1935-2007 a=35 b=1.9 0.8 Normalized Frequency Normalized Frequency 0.9 0.7 0.6 0.5 Intensity 0.4 0.3 0.2 0.1 0 20 40 60 80 Intensity (m/s) 100 120 0.8 0.6 Annual Frequency 0.4 0.2 0 5 Holland ASP Extremes Colloquium 0611 10 15 20 Annual Number 25 30 7 Weibull Analysis We utilize the Reverse Weibull distribution for which the CDF and PDF are: CDF f ( x) 1 e x a bx PDF f '( x) aa b b 1 e x a b 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. Holland ASP Extremes Colloquium 0611 8 Sensitivity to Scale and Shape Variations 0.03 a=35, b=1.9 0.025 Probability 0.02 0.015 0.01 a=40, b=1.9 0.005 0 0 a=35, b=2.3 20 40 60 Scale 80 Holland ASP Extremes Colloquium 0611 100 120 9 Sensitivity to Scale and Shape Variations Hazard Rate Function for the Weibull distribution: b 1 f (x) b x H(x) : 1 F(x) a a (Hillier and Lieberman 1986) Frequency For b>1 the sensitivity increases with increasing x to a limit of 1-e-x . (Katz and Brown 1994) Value Holland ASP Extremes Colloquium 0611 10 Weibull Analysis Probability of exceeding a threshold event: P ( E{x c}) 1 f (c) e c a b Exceedance probability decreases as the event becomes rarer (c/a increases), and/or the population less variable (b increases). Holland ASP Extremes Colloquium 0611 11 Weibull Analysis 1 Mean : a 1 b (where: Γ is the Gamma function) 2 2 Variance : a 1 b 2 2 Γ(n) α (n-1)! • Given µ and σ we can make an estimate of a and b and thus the whole distribution. Holland ASP Extremes Colloquium 0611 12 Application to Atlantic Hurricane Intensity Weibull Probability Plot 1 0.9 0.99 0.96 0.90 1935-2007 a=35 b=1.9 0.8 0.7 Log-Log Weibull Probability Plot 0.75 0.6 Probability Normalized Frequency 0.999 Intensity PDF 0.5 0.4 0.50 0.25 0.3 1935-2007 a=35 b=1.9 0.2 0.10 0.1 0 20 0.05 40 60 80 Intensity (m/s) 100 PDF (Normalized, HURDAT Smoothed) 120 32 42 48 Hurricane Intensity (m/s) 58 69 CDF with Weibull fit: a=35, b=1.9 Holland ASP Extremes Colloquium 0611 13 1 1980-1994 a=32, b=1.9 Mean=28.4 SD=15.5 PE69=1.4% PE58=4.5% PE32=36.8% 0.9 Normalized Frequency 0.8 0.7 0.8 0.6 0.5 0.4 0.3 0.7 0.6 0.5 0.4 0.3 0.2 0.2 0.1 0.1 0 20 40 60 80 Intensity (m/s) 100 1995-2007 a=35, b=1.6 Mean=31, SD=20 PE69=5.1% PE48=19.1% PE32=42.1% 0.9 Normalized Frequency 1 Application to Current Climate: 1980-1994 vs 1994-2007 120 0 20 40 60 80 Intensity (m/s) 100 120 • 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) Holland ASP Extremes Colloquium 0611 14 Application to Current Climate 1994-2008 a=343 b=19 1995-2008 a=149 b=46 lifmin=0 1 Observed Weibull Estimated Weibull Original 0.6 0.4 0.2 0 30 40 50 60 70 Intensity (m/s) 80 90 100 Normalized Frequency Normalized Frequency Intensity PDF 0.8 1 a b Annual Frequency PDF 0.8 0.6 0.4 0.2 0 5 10 15 20 Annual Number 25 30 Weibull original is for 1980-1994 data Observed is 1995-2008 data (Data from IBTrACS) Holland ASP Extremes Colloquium 0611 15 Application to Truncated Model PDFs The Issue: Regional Climate Models cannot run over long time scales at resolutions sufficient to resolve weather extremes Holland ASP Extremes Colloquium 0611 16 Background: Regional Climate Modeling 12 km 36 km 150 km Time Slices Holland ASP Extremes Colloquium 0611 17 NRCM Hurricane Intensity Change Model projected increase in TC intensity, but can only resolve up to Cat 2. Wind speed distribution shifting to right How do we assess the extremes? Cannot use EVT extrapolation or quantile mapping Holland ASP Extremes Colloquium 0611 18 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 0611 19 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 0611 20 Example: NRCM Hurricane Changes Holland ASP Extremes Colloquium 0611 21 Application to NRCM Hurricane Predictions PE69=Cat5 PE58=Cat4,5 PE48=Major Hurricanes PE32=Hurricanes Holland ASP Extremes Colloquium 0611 22 Alternative Generalized Pareto Distribution (where a,b,c are the scale, shape and location parameters) 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 (Suzuki 2011) Holland ASP Extremes Colloquium 0611 23 Comparison with Alternative Approaches Holland ASP Extremes Colloquium 0611 24 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. Holland ASP Extremes Colloquium 0611 25 Yamada et al 2010 Cloud-resolving simulation over limited time period, with and without Greenhouse Warming contribution. Holland ASP Extremes Colloquium 0611 26 Hallegatte et al (2007) Annual probability of landfall increase: Cat1=17% , Cat2=33%, Cat3=44%, Cat 4=58%, and Cat5=215%. Holland ASP Extremes Colloquium 0611 27 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 Holland ASP Extremes Colloquium 0611 28 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 0611 29 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 Thank You Holland ASP Extremes Colloquium 0611 30