Extreme Hurricane Winds in the United States

Thomas H. Jagger & James B. ElsnerDepartment of Geography

Florida State University

http://garnet.fsu.edu/~jelsner/www

University of Florida’s Winter Workshop on Environmental Statistics

January 12, 2007Gainesville, FL

• Maximum Likelihood Inference– Extreme hurricane winds– Calibrate geological records

• Bayesian Inference– US hurricanes and global temperature– US hurricanes and other covariates– Insured losses

Research QuestionsResearch Questions

1. What are the return levels of hurricane winds in the U.S. over 5, 10, 50, and 100 years?

2. Are return levels different for different regions? 3. What is the maximum possible hurricane wind

speed level? 4. How do these levels change under different climate

conditions? Although fewer hurricanes affect the United States when El Niño conditions are present, are they stronger?

5. Can we apply a similar analysis to insured hurricane losses?

6. What are the primary predictors of extreme insured losses?

7. How does the analysis of extreme insured losses differ from those of extreme winds?

We answer these questions by modeling the maximum wind speeds near the coast a using peaks-over-threshold model.

1. We estimate parameters using ML on data only over the reliable period from 1899-2004.

2. We then demonstrate the use of a Bayesian approach that allows us to incorporate an additional set of Atlantic hurricane data extending back to 1851.

3. We show that the same Bayesian extreme value model applied can be used to estimate extreme insured losses.

– We use log(loss).– Alternate truncated normal model used for yearly

losses.

• Coastal hurricanes are a serious social and economic concern for the United States. – Strong winds, heavy rainfall, and storm surge kill people and destroy

property. – Hurricane destruction rivals that from earthquakes.– Historically, 80% of all U.S. hurricane damage is caused by 20% of the most

intense hurricanes.

• The rarity of severe hurricanes implies that empirical estimates of return periods likely will be unreliable.

• Extreme value theory provides models for rare wind events and a justification for extrapolating to levels that are much greater than have already been observed. – Stuart Coles “An Introduction to the Statistical Modeling of Extreme Values”

• Definitive answers to questions about whether hurricanes will be more intense or more frequent in a future of global warming require long records.– The longest records available are near the coast.

The maximum possible hurricane wind speed is estimated to be 208 kt (183 kt) using the Bayesian (ML) model.

On average we can expect 132 (157) kt hurricane winds near the U.S. coast once every 10 (100) years.

Along the Florida coastline we can expect 108 kt (137) kt winds once every 5 (50) years on average.

Along the East coast we can expect 103 kt (120) kt hurricane winds once every 10 (300) years.

Return periods change with climate factors.

The return period for Hurricane Katrina is 21 years.

Global temperature may have some effect.

The expected worse case indicates a 94% probability of at least some loss during a year.

The expected worse case amount is $23.7 billion.

The expected best case indicates a 53% probability of at least some loss.

The expected best case amount is $1.1 billion.

The maximum 50-year single event loss under the worse case is $630 billion.

The maximum 50-year single event loss under the best case is $10 billion.

PoissonRegression

1993

RegressionTree

1996

DiscriminantModel

1997

TimeSeriesModel

1998

SingleChangePointModel

2000

WeibullModel

2001

SpaceTime

Model

2002

ExtremeValueModel

2006

ClusterModel

2003

TimeSeries

Regression

2008

MultipleChangePointModel

2009

SpaceTime

Regression

2010

Hurricane Type(Paths, Origin)

Hurricane Rate(Counts)

Hurricane Strength(Intensity)

Regional HurricaneActivity

Large Scale Predictors(AMO, NAO, ENSO)

Regional Scale Predictors(SST, SLP, etc)

● Darling (1991): empirical model to estimate local probabilities of hurricane wind speeds exceeding specified thresholds.

● Rupp and Lander (1996): method of moments on annual peak winds

over Guam to determine the parameters of an extreme value model leading to estimates of recurrence intervals for extreme typhoon winds.

● Heckert et al. (1998): peaks-over-threshold method and a reverse Weibull distribution to obtain mean recurrence intervals for extreme wind speeds at consecutive mileposts along the U.S. coastline.

● Chu and Wang (1998): various parametric distributions to model return periods for tropical cyclone wind speeds in the vicinity of Hawaii.

● Jagger et al. (2001): maximum likelihood (ML) estimation to determine a linear regression for the scale and shape parameters of the Weibull distribution for hurricane wind speeds in coastal counties.

● Pang et al. (2001): Bayesian approach to estimating parameters from a Weibull distribution using wind speed data.

We enhance previous research efforts by

1. Interpolating 6-hourly positions and intensities hourly. • For each tropical storm in each region, find maximum wind

speed using interpolated values.• Interpolation remove some boundary bias.

2. Examining the effect of climate variables on the distribution of extreme winds. ● The model employed by Jagger et al. (2001) captures the

variation of hurricane frequency as a function of climate variables using the Weibull distribution, which is appropriate for wind speeds above some threshold but not necessarily for the most extreme winds.

● Here we attempt to put extreme hurricane winds in the context of climate variability and climate-change.

1. Demonstrating the feasibility of a Bayesian approach for adding older, less reliable, data into the analysis.

Generalized Pareto Distribution

Limit family for extremes

P(x

> v

| x

> u

)

60 100 140 180

0.0

0.2

0.4

0.6

0.8

1.0shape = 0shape = +0.5shape = -0.5

v [kt]

GPD threshold estimation

• Which observations do we keep?

• Graphical Tools:– Mean residual life plot

Smallest value.

– , 0 versus threshold plot (not shown).

• Thresholds: – 83 kt for Florida and Gulf.– 64 kt for East.– 96 kt for Entire Coast.

• Best Track HURDAT center fixes• Time period:1851 through 2004.

– 10 kt rounding error prior to 1900

– 5 kt 1900 and later

– 1900-2004 data used in initial analysis, 1851-1899 used in Bayesian analysis.

• Interpolated to hourly fixes using natural splines.– Initial smoothing: Use polynomial

local fit: Savitsky-Golay

– Try smoothing splines with uniform Wmax ± 2.5 kt distribution on true Wmax.

Data Sets:Regions

Anatomy of a Tropical Storm

Atlantic Sea Surface Temperature (SST)

ºC

Hurricane Season Global Warming

1860 1880 1900 1920 1940 1960 1980 2000

Year

-0.4

-0.2

0.0

0.2

0.4

0.6

Degrees C

Global Temperature Anomoly July through October

Results:Results: Models for ExtremesModels for Extremes

Curves are based on an extreme value model and asymptote to finite levels as a consequence of the shape parameter having a negative value. Parameter estimates are made using the ML approach. The thin lines are the 95% confidence limits. The return level is the expected maximum hurricane intensity over p-years. Points are empirical estimates and fall close to the curves.

Ret

urn

leve

l plo

ts b

y re

gio

n

Return level plots for the entire U.S. coast (Region 4) by climate factors.

Curves are based on an extreme value model using a ML estimation procedure. Data is partitioned separately by predictor. Red (blue) lines and points indicate above (below) normal climate conditions for each predictor.

15

More hurricanes with

La Ni ña

More extreme hurricanes with

El Ni ñoENSO NAO

SST

More hurricanes with

-NAO

More strong hurricanes with

high SST

More strong hurricanes with global warming

15

More hurricanes with

La Ni ña

More extreme hurricanes with

El Ni ñoENSO NAO

SST

More hurricanes with

-NAO

More s trong hurricanes with

high SST

More s trong hurricanes with global warming

15

Theoretical predictions of the influence of global warming on hurricane activity suggest:– Increased maximum intensity.– Increased level of storminess is uncertain.

Impact of Global Warming?

Return Level (kt)

1 10 100 1,0002 3 4 5 678 2 3 4 5 67 2 3 4 5 6

Return Period (yr)

60

80

100

120

140

160

180

200

Warm years

Cold years

For a given return period (> 5 yr), warm

years result in higher return levels.

Entire coast

137 kt

Magnitude of the difference in return level is consistent with climate

models

Warm Years

ColdYears

Saffir-Simpson Category

14 yr

No change in frequency of

weaker hurricanes

11

Hurricane Katrina

Bayesian Inference

Bayesian Inference

Bayesian Inference of Extremes

Bayesian Inference of Extremes:

Hurricane Intensity Component

Hurricane Frequency Component

Covariate Component

Observed maximum

wind speed.

True maximum

wind speed.

Bayesian Bayesian Extreme Value ModelsExtreme Value Models

Bayesian model for coastal hurricane wind speeds

Intercept: j=1 X[,1]=1 for all

WinBUGS Code

How Gibbs Sampling Works

Gibbs sampling algorithm in two

parameter dimensions starting from an initial point

and completing three iterations.

(0)

(1)

(2)

(3)

The contours in the plot represent the joint distribution of  and the labels (0),

(1) etc., denote the simulated values. One iteration of the algorithm is

complete after both parameters are revised. Each parameter is revised

along the direction of the coordinate axes---problematic if the two

parameters are correlated (contours compressed) as movement along the

axes tend to produce small changes in parameter values.

POT WinBUGS code Part IModel{

for(j in 1:M) {

#For each year calculate log sigma and xi: lsigma2[j] <- inprod(ls.x[],X[j,]) xi2[j] <- inprod(xi.x[],X[j,]) #Threshold Crossing Rate for each year: H[j] ~ dpois(lambda[j]) log(lambda[j]) <- inprod(tc[],X[j,])

} #Extreme value for each exceedance, note censoring:

for( i in 1:N){

yy[i]~dGPD(u,sigma[i],xi[i])I(ys[i],yl[i])

offset[i] <- Yr[i] - Yr0 lsigma[i] <- lsigma2[offset[i]] sigma[i] <- exp(lsigma[i]) xi[i] <- xi2[offset[i]] yl[i] <- y[i] + e[i] ys[i] <- y[i] - e[i]

}

POT WinBUGS code Part II#Initializations and Missing Data model #In our case Np=2, intercept and global temperature

for(k in 1:Np) { #Parameter initializations for model parameters #tau.xx[], data used for model selection:

tc[k] ~ dnorm(0,tau.tc[k]) ls.x[k] ~ dnorm(0,tau.ls[k])

xi.x[k] ~ dnorm(0,tau.xi[k])

#Missing data model, normal no trend, for(j in 1:M) { X[j,k] ~ dnorm(xmu[k],xtau[k]) }

#Missing data model initializations: xmu[k] ~ dnorm(0,.001) xtau[k] ~ dgamma(.01,.01)

} } #Close Model (Demo: GPD1)

Posterior Densities

Model Output Comparison Data

Probability Parameter is less than zero

tc[1] tc[2] ls.x[1] ls.x[2] xi.x[1] xi.x[2] 1.000000 0.277300 0.000000 0.128425 0.961925 0.150350

GT Scale GT Shape

Maximum Likelihood values (MLE) from Coles ismev gpd.fit():Bayesian samples: burn in 5000, sample 40000:

MLE se mean sd MC_error 2.5% median 97.5% tc[1] -0.381 0.113 -0.395 0.108 0.00078 -0.613 -0.394 -0.188 tc[2] 0.287 0.451 0.255 0.430 0.00291 -0.600 0.263 1.086ls.x[1] 3.167 0.138 3.134 0.143 0.00462 2.853 3.136 3.407ls.x[2] 0.699 0.601 0.724 0.624 0.01945 -0.472 0.726 1.965xi.x[1] -0.264 0.088 -0.205 0.103 0.00337 -0.372 -0.215 0.028xi.x[2] 0.543 0.425 0.532 0.538 0.01691 -0.572 0.541 1.572

Regions Parameters Covariates+ Intercept

-9 5 -9 0 -8 5 -8 0 -7 5 -7 0

lo n g itu d e

25

30

35

40

45

latitude

H ourly In terpo lated Positions in R egions of S tudy

R e g io n 1 : G u lf C o a stR e g io n 2 : Flo rid aR e g io n 3 : Ea st C o a stR e g io n 4 : N e w En g la n d

Region 4: Northeast Coast

Region 3: Southeast Coast

Region 2: FloridaRegion 1: Gulf Coast

Hourly interpolated

hurricanepositions

(1851-2004)

Bayesian Model for Coastal Hurricane Winds 2

Hurricane Intensity Component

Hurricane Frequency Component

Covariate Component

Observed maximum

wind speed.

True maximum

wind speed.

Results: Raw Climatology

P(>0) = 0.22 P(>0) < 0.01

P(>0) < 0.01 P(>0) < 0.01

Assuming the model is correct, the data support a super-intense hurricane threat only in the Gulf of Mexico.

Region 1: Gulf Coast: Shape

Region 2: Florida: Shape

Region 3: Southeast: Shape Region 4: Northeast: Shape

Probability that hurricane intensity is

unbounded

Fre

qu

ency

Dis

trib

uti

on

Freq

uen

cy Distrib

utio

n

Important Results: Conditional Climatology

log()

log()

log()

log()

log()

log ()

Fre

qu

ency

Dis

trib

uti

on

Freq

uen

cy Distrib

utio

n

Region 1: AMO: Scale Region 3: AMO: Threshold

Region 1: NAO: Threshold Region 2: NAO: Threshold

Region 1: SOI: Threshold Region 3: SOI: Threshold

Stronger Hurricanes

More Hurricanes

More Hurricanes

More Hurricanes

More Hurricanes

More Hurricanes

Simulated Hurricane Seasons

• A Monte Carlo procedure is employed on the posterior samples to generate a large number (50K) simulated hurricane seasons based on sampled covariate and parameter values.

Results from the Gulf Coast show that the simulated data match the empirical data through Category 4 wind speeds but for winds in excess of 135 kt (68 m/s) the simulated data indicate a higher frequency (by a factor of 2 to 3).

Category (SS) > I II III IV V V+ V++

Threshold (kt) 80 83 96 114 135 150 170

Empirical data* 0.463 0.418 0.313 0.142 0.037 0.007 0.007

Simulation* 0.469 0.423 0.279 0.143 0.054 0.028 0.015

*mean annual exceedence rate

Region 1: Gulf Coast

• Modeled 100-year return levels do not appear to match empirical evidence.

– Might be a problem with the regression structure of the shape parameter. Using simpler models where we replace the covariates with discrete factors (above/below normal) we produce a better match.

• Convergence is not guaranteed when modeling the log() and scale as a linear combination of covariates.

– Problem in region 4 (NE) where there are fewer hurricanes.

• Threshold is not estimated in the current model.

– Behrens, Lopes and Gamerman(2004) "Bayesian analysis of extreme events with threshold estimation“ Statistical Modelling, 4, 227-244.

• The nature of the shape parameter is such that its value is a function of the support of the underlying distribution.

– Attempts to model this using a Reversible Jump Markov Chain MC approach reduces this problem but introduces significant autocorrelation into the sampler.

– Model assigns positive probability for discrete values of xi, uses RJMCMC

Outstanding Issues

• To be useful to risk models, the relationship between climate and hurricane activity needs to be forecast in advance of the season.

Fortunately, three important climate variables related to hurricanes can be used in a prediction model; but each variable enters the prediction model in a unique way.

NAO: Natural precursor signal to hurricane activity.

SST: Slowly varying (persistent).

ENSO: Can be predicted with some skill by dynamical models.

16

Predicting Insured LossesPredicting Insured Losses

21

22

Peaks Over Threshold

Small LossEvents(36.5%)

Large LossEvents(63.5%)

99.4%Losses

0.6%Losses

Reference line indicates 80/17 splitWe split losses (red line: $100 Million)Allows us to examine significant events

Splitting Insured Losses

24

Large Loss Potential Evenly Distributed

SST

SS

T

25

Preseason Predictors Insured Loss Model

Model Distributions:log(loss): Truncated Normal. dnorm(,1/2)Rate: Poisson dpois()

May June averaged values of predictors

+ NAO, -SST

- NAO, +SST

26

Yearly Insured Losses Depends on Climate

SST

27

Extreme Loss Model and Results

Predictors set at maximum values of covariates with least

favorable climate:

+NAO, +SOI

-NAO, -SOI

Model Distributions:log(loss): GPD distribution. dGPD(u,,)

u=9 (log(1 Billion))Rate: Poisson dpois()

● The yearly model and extreme loss model use the same Peaks over Threshold approach.

● Yearly model used SST and NAO predictors with truncated Normal distributions:– Smallest DIC for truncated Normal, with SST and NAO predictors– Better estimates of single year insured losses.

● Extreme Loss Model uses SST, NAO and SOI with GPD distribution:– SST and NAO used in regression for logarithm of threshold

crossing rate– Log(loss) assumed to have GPD distribution for loss > $1 Billion.

● Mean Residual Life Plot used to estimate threshold.– NAO used in log() regression– SOI used in regression

● More Information: “Forecasting U.S. Insured Hurricane Loses”, available at our website, in “Forecasting Insured Losses” in press.

Comments on POT BUGS Models:

● Model mixes well.● Model must be initialized carefully.

– Once model compiles it runs smoothly.– Posterior mean and MLE sometimes very different.– MLE may be approximated by posterior sample values at

minimum deviance…, but not well in this model.● Return level sampling not recommended. ● Data and R code using BRugs available.● POT model using U.S. Hurricane loss data

– Demo and R/OpenBUGS code available. ● Examples of truncated normal distributions in POT type model

as well as truncated normal/GPD.

Evidence of Prehistoric HurricanesEvidence of Prehistoric Hurricanes

peat layer

sand layer1954 H

peat layer

peat layer

peat layer

sand layer1938 H

sand layer1635 H

Courtesy: Jeff Donnelly

A sediment core from a back barrier marsh in New England.– These data are currently not used in risk models.– Need calibration.

Increasing Radius beginning at 45 km

• Combining extreme value theory with a Bayesian specification provides a practical way to assess return periods of extreme hurricane winds.– Results are expressed in terms of posterior distributions of the

parameters of interest.– Less reliable, but still useful information is incorporated into the model

in a natural way.– A large number of simulated hurricane seasons can be generated by

repeated sampling from the posterior distributions. – Analysis of hurricane winds suggests the possibility that the highest

winds estimated for early storms may be under reported.– Results suggests the possibility that the highest winds estimated for

early storms may be under reported. • The approach can be used to better understand the

projected impact of global warming on extreme hurricane winds.

• The POT approach can be extended to insured losses and the logarithm of extreme losses can be modeled using a GPD distribution.

Summary

32

Taken as a random event along the Gulf coast, the return period of Hurricane Katrina is 21 years.

Katrina might be a sign of things to come as the observed and modelled effect of global warming appears to start at Katrina’s intensity.

Preseason climate signals provide information about the nature of the upcoming hurricane season.

The hurricane risk index quantifies this information geographically.

Insured losses (both expected and maximum) show a statistical link to preseason climate signals.

More InformationMore Information

Google hurricane climate

http://garnet.fsu.edu/~jelsner/www

jelsner@fsu.edu; tjagger@fsu.edu

35

Future Work I• Use more data and new covariates

– Incorporate historical climate data with geological evidence and historical data sets. (last slide)

– Use proxy data for covariates– Use Quasi-Biennial Oscillation (QBO) Zonal Wind

Index .• Only from 1953, very predictable, especially west

phase• 50 mb, East phase, more vertical wind shear.• El-Nino may interact with West descent phase.

– New wind index: • Saunders, M. A. and A. S. Lea, 2005: Seasonal

prediction of US landfalling hurricane activity from 1 August, Nature 434, 1005–1008.

Future Work II• Use alternative prior formulation:

– Choose three return periods, rp[1:3]= [2,10, 100] years

– Assign reasonable multivariate prior to the return levels for rp[1:3] with rl[1] < rl[2] < rl[3] (using rl[1], rl[2]-rl[1],rl[3]-rl[2])

– Use rl[i]=u+ ¢((z[i]¢u)-1 ) with z[i]=1/log(rp[i]/(rp[i]-1)) ¼ rp[i]-1/2

– Solve for u, , as a function of rl[1:3], rp[1:3]

– Coles, S. G. and J. A. Tawn, (1996): A Bayesian analysis of extreme rainfall data. Appl. Statist., 45, 463-478

Future Work III• Additional analyses using extreme value theory and

Bayesian methods:– Use Bayesian model averaging and median models

selection– Use RJMCMC to examine spatial and temporal parameter

changes, e.g. smoothing splines (WinBUGS 1.4 RJMCMC)– Apply to pressure, rainfall and sea surge.– Apply to hurricane rapid intensification (RI) process.

• Bivariate GPD, maximum intensity, intensification per storm• Western Pacific RI density, conditional probability of intensification.• How does global temperature affect RI.

– Use spatial model for distribution of GPD parameters.

• Spatial and Temporal Models:– Clustering models on a lattice. (Jagger Dissertation)– Point process models (Cox Processes). (R/Splus)– Apply to hurricane activity and intensity full models.