## 1. Introduction

This note estimates return periods of Atlantic basin hurricanes striking the U.S. coastline from Texas to Maine. Return period information helps set building design standards and insurance rates and also helps establish climatological norms. Here, our purpose is to estimate how frequently the United States experiences certain hurricane wind speeds and central pressures; we also estimate return periods of some memorable storms including Camille (1969), Andrew (1992), and Katrina (2005).

Under a uniform storm arrival pattern (mathematically, this is called a time-homogeneous Poisson process), an *L*-year hurricane strike is observed on the average of once every *L* years. That is, a storm of equal or greater magnitude will strike on the average of once every *L* years. This interpretation is slightly off in our case because the arrival rates of hurricanes vary within a season. Exact interpretations of return periods are given in section 2.

The data in this study contain Atlantic basin hurricanes striking the continental United States during the period 1900–2006. A strike is said to occur when the hurricane’s center of circulation crosses a continental landmass. In this study, the Florida Keys are viewed as part of the continental United States. The data were collected and cross-checked from various sources, including Neumann et al. (1999), Blake et al. (2005), Web pages supported by the National Oceanic and Atmospheric Administration (NOAA) and the National Hurricane Center (NHC), and the Hurricane Research Division’s hurricane database (HURDAT) Reanalysis Project (http://www.aoml.noaa.gov/hrd/hurdat/ushurrlist18512005-gt.txt).

We select 1900 as the study starting date to ensure storm count accuracy. While some work aims to improve accuracy for U.S. hurricane data (Landsea et al. 2004) and adjust counts for undetected storms (Solow 1989a), the 1900–2006 data are generally considered reliable. To elaborate, towns on the coast were likely dense enough to avoid missing (undercounting) landfalling hurricanes after 1900 (Murnane et al. 2000). The wind speeds for storms from 1900 to 1914 and from 1980 to present are considered accurate and are based on work in the HURDAT Reanalysis Project. For storms striking from 1915 to 1979, no official wind speed estimates are currently available. For storms during this period, wind speeds were estimated from the wind–pressure relationships described in Landsea et al. (2004) if there is an available central pressure; in other cases, the midpoint of the wind speed range for each storm’s Saffir–Simpson (SS) category was used. These midpoints (to the nearest 5 kt) are 75, 90, 105, and 125 kt for categories 1, 2, 3, and 4, respectively.

This study examines striking (landfalling) hurricanes only; hurricanes that do not strike the continental United States or tropical storms and depressions are not considered. For the purpose of this study, “striking hurricanes” include storms that produce hurricane-force winds at the coast, but the center of the eye does not cross the coast (Hurricane Ophelia in 2005 is one example). As a convention, hurricanes making two distinct strikes are counted as separate events. (Hurricane Andrew in 1992, e.g., struck southern Florida, reintensified in the Gulf of Mexico, and then struck Louisiana.) There are 214 hurricane strikes in our data. These storms and their landfalling central pressures and wind speeds are listed in Table 1. Storms with wind speeds derived from a wind–pressure relationship are marked with an asterisk; storms whose wind speeds are category midpoints are marked with double asterisks. Wind speeds were rounded to the nearest 5 kt.

There are 7 storms in our dataset with landfalling wind speeds below 64 kt that are included because their landfalling central pressures resulted in hurricane conditions. These storms are included in the central pressure distribution fits but not in the wind speeds distribution fits. Likewise, there are 25 storms for which the central pressures are not available.

## 2. Methodology

Our mathematical methods use Poisson processes to describe the arrival times of the hurricanes and extreme value techniques to model the wind speeds and central pressures of the hurricanes at their time of strike. These specifications are then used to estimate return periods. We now discuss these modeling aspects briefly.

Poisson processes and their variants have been widely used to describe hurricane counts in various regions of the tropics (Mooley 1981; Thompson and Guttorp 1986; Solow 1989a, b; Parisi and Lund 2000). While we refer the reader to these works for the basics on Poisson processes in hurricane modeling and to Ross (1996) for more mathematical aspects, Poisson models describe hurricanes well because the two “Poisson axioms” are approximately satisfied: 1) two distinct hurricanes are very unlikely to strike simultaneously, and 2) the hurricane strike counts in disjoint time intervals are approximately independent.

On axiom 2) above, the existence of patterns in the annual Atlantic hurricane counts is hotly debated (see Bove et al. 1998; Goldenberg et al. 2001; Elsner and Jagger 2004, 2006). Figure 1 displays the annual landfalling hurricane counts over the period of record. The simplest model for these annual counts is an independent Poisson random sequence (Poisson white noise). However, any pattern in the Fig. 1 landfall counts would imply that the Poisson white noise assumption is suboptimal. Sample correlations in the year-to-year hurricane counts support the white noise assumption. Elsner and Bossak (2001) conclude that historical Atlantic hurricane counts (not just U.S. landfalling storms) are essentially stationary, nor is there any significant shift (changepoint) in hurricane rates (Elsner et al. 2004). Neither of these studies includes data from the very active 2004 and 2005 seasons.

Recent research allows the annual Poisson mean parameter, denoted by *λ* and also called an arrival rate, to depend on covariate factors such as time, the North Atlantic Oscillation (NAO), the Southern Oscillation index (SOI), and the Atlantic multidecadal oscillation (AMO) (see Van den Dool et al. 2006, and the references therein for how the NAO and AMO influence climate in North America). In addition to these covariates we also modeled the Bivariate El Niño–Southern Oscillation Time series (BEST). It is important to note that the BEST is a univariate time series and that “bivariate” refers to the fact that it is calculated from two data series (the SOI and Niño-3.4). (Data for the NAO, the SOI, the AMO, and the BEST were taken from links at http://www.cdc.noaa.gov/ClimateIndices/List/ and http://www.cru.uea.ac.uk/cru/data/pci.htm. The units of all covariates are in standard deviations.)

*t*of the study is modeled as a Poisson random variable with mean

*λ*, where

_{t}*α*is a linear trend slope and the

*β*s are regression coefficients. Poisson regression methods are used to statistically fit and assess such models (Davison 2003 gives an overview). Elsner (2003), Elsner and Bossak (2004), McDonnell and Holbrook (2004), and Elsner and Jagger (2004, 2006) employ such techniques and find that the NAO is the only significant predictor among the NAO, the AMO, and the SOI.

_{i}In our Poisson regression fittings with the storm strike data through 2006, we also find that the estimates of *α*, *β*_{3}, and *β*_{4} are statistically insignificant (judged as zero) at the 95% confidence level. This was gauged by an all-subsets regression technique; that is, every possible combination of factors was examined. Only the NAO and the BEST were statistically significant in our model. That some of the covariates are insignificant is perhaps not unexpected. In particular, hurricanes, our population of interest, are composed of the strongest of the tropical cyclones. Moreover, it is known that correlating extremes to covariates is more difficult than correlating means to covariates (McCormick and Qi 2000 make the notions precise). Hence, an analysis containing all tropical storms may stand a better chance in fingerprinting the AMO as a legitimate covariate influencing storm counts. It is also not surprising that the SOI is insignificant in the presence of the BEST; indeed, the SOI is one of the components of the BEST.

*t*is modeled as

*t*were taken as the May–June averages; these values are plotted in Fig. 2 in anomalies of standard deviations. Both time series are modeled as zero mean Gaussian white noise with variances

*σ*

^{2}

_{ZNAO}= 0.881 for {NAO

*}, and*

_{t}*σ*

^{2}

_{ZBEST}= 0.567 for {BEST

*}. These models were selected by the model selection criteria and normality assessments in Brockwell and Davis (1991). Additionally, {NAO*

_{t}*} and {BEST*

_{t}*} show no clear correlation.*

_{t}No significant trend in the hurricane counts is seen in the data through 2006. This result is consistent with Landsea (2005). The trend estimator and one standard error is *α̂* = −0.0026 ± 0.0023, which has a *p* value of 0.28 (in a test of *α* = 0 against *α* ≠ 0). An estimate of the long-run annual average of landfalling hurricanes is *λ̂* = *n*^{−1}_{yr}Σ^{nyr}_{t=1}*λ̂ _{t}*, which is about 2 storms per year. Here,

*n*

_{yr}= 107 is the number of years of observations.

*f*(·). This density is estimated from kernel density techniques:

_{D}*d*is the day of year on which the

_{i}*i*th storm struck (the year is not relevant), and

*K*is a Gaussian kernel function defined by

*K*(

*x*) = exp{ −

*x*

^{2}/2}/

*π*

*h*= 33.80 days was selected here [Parisi and Lund (2000) provide more details on kernel smoothing methods and the Atlantic hurricane arrival seasonality].

*W*is the landfalling wind speed for the

_{i}*i*th storm (see Embrechts et al. 1997; Coles 2001; Wilks 2006 for more on peaks over threshold methods and Pareto distributions). The parameters of this distribution are

*σ*> 0 and

*ξ*, and

*u*is a fixed threshold that we take as 64 kt for the wind speeds. The notation uses

*x*

_{+}= max(

*x*, 0). Landfalling central pressures are modeled similarly, except that the Pareto distribution is fitted to {1002 −

*P*} (

_{i}*u*= −1002 mb) to reverse the natural ordering in pressures. The estimated wind speed parameters and standard errors are

*σ̂*= 32.751 ± 2.712 and

*ξ̂*= −0.3122 ± 0.039; those for the central pressures are

*σ̂*= 48.109 ± 3.963 and

*ξ̂*= −0.4242 ± 0.0377. The method of maximum likelihood was used to estimate these parameters. The model fits the data reasonably well; specifically the thresholds

*u*= 64 kt and

*u*= −1002 mb were gauged as adequate via the mean excess plots of Davison and Smith (1990) and Parisi and Lund (2000). Implications of the negative

*ξ*s in the model fits are that wind speeds can be no larger than 169 kt and central pressures can be no lower than 888 mb at the storm strike time. These bounds do not apply to storms over open waters. The 169-kt bound is slightly less than the 185-kt bound used in Murnane et al. (2000). Time-varying Pareto parameters were considered but were not ultimately needed. In fact, simple linear regression fits of the striking wind speeds and central pressures on the year of arrival and the relative NAO and BEST levels did not reveal any significant relationships at the 95% significance level. Table 2 summarizes all parameters in our hurricane model.

Return periods can be estimated from the above model via simulation. For preciseness, the return period of a hurricane with a landfalling wind speed of *w* kt is defined as the expected time (a statistical average) that one must wait, starting from 1 January of a given year, until a hurricane with a wind speed of *w* kt or greater makes landfall.

A single simulation run must generate a fair draw of a “level *w*” return period. To do this, one first generates {NAO* _{t}*} and {BEST

*} over a suitably long time horizon (the length of this time horizon is not overly relevant for this discussion). From the two covariate series, one then generates a time series of*

_{t}*λ*s via (2.1). We then generate the number of storms

_{t}*N*in each year

_{t}*t*as a Poisson random variable with parameter

*λ*. Given that

_{t}*N*=

_{t}*k*for a fixed year

*t*, the day of arrival of the

*k*storms within the calendar year is generated as the order statistics of

*k*independent draws from the arrival time density in (2.2). For each storm, wind speeds and central pressures are then generated from the distributions fitted in (2.3). We do not vary these distributions for the day of storm arrival for the reasons discussed in Parisi and Lund (2000).

The above procedure will generate a random sequence of hurricane landfalling times and storm strength characteristics that realistically match those seen in the observed data. The waiting time for the simulation run is merely the first time that a hurricane with wind speeds of *w* or greater is encountered. By empirically averaging waiting time draws over many independent simulations—the number of which is taken as one hundred thousand to minimize sampling error—we arrive at an estimate of the wind speed *w* return period.

## 3. Results

Table 3 lists estimated return periods for storms of various wind speed magnitudes. For example, one waits an average of 0.9 yr for a Saffir–Simpson (SS) category 1 or stronger storm, which has wind speeds of 64 kt or higher, to make landfall (as measured from 1 January). The nonencounter probability listed is the estimated probability that no SS 1 storm or greater makes landfall in a calendar year. For example, the chance that no hurricane (SS 1 or higher) makes landfall in a given year is about 17%. Major storms (SS 3 and higher) have a return period of about 2.0 yr, with a probability of about 0.45 occurring annually (one or more landfalls in a given year). Table 4 displays estimated return periods for central pressures of the storms. Their interpretations are similar to the wind speed return periods.

The covariates NAO and BEST are not overly important in the return period debate. For example, a Saffir–Simpson category 5 hurricane strike return period is 23.1 yr when these are taken into account, and 22.5 yr when they are ignored.

The return periods in Tables 3 and 4 apply to the continental United States as a whole. We have also partitioned the storms into three regions of strike location: the Gulf of Mexico (excluding Florida), the East Coast, and Florida. Return periods for these subregions are presented in Table 5 and have the same interpretations. Category 5 hurricanes striking the Atlantic coast north of Florida were deemed impossible by the fitted model.

Table 6 exhibits return periods of some memorable Atlantic basin storms by both striking wind speed and central pressure. That Katrina is roughly a 4-yr storm (based on wind speed, 13 yr based on central pressure) may seem surprisingly low, but perhaps not so when only the storm’s meteorological characteristics are considered. As the central pressure is arguably a better measure of overall storm strength, central pressure return periods are probably better measures of overall severity.

The 1935 Labor Day Florida Keys storm was the most severe in our dataset. With a 265-yr wind speed return period and a 102-yr central pressure return period, it presses the fitted model boundaries. We believe this is due in part to the extreme southern latitude of this landfalling storm. Another storm of this intensity would likely again require a very southern landfalling latitude, with the Florida Keys or the Brownsville, Texas, region being the most likely hosts.

## 4. Summary

Return periods of continental U.S. hurricane strikes were estimated from Poisson processes and extreme value techniques. Incorporating the NAO, the BEST, the SOI, and the AMO does not seem to greatly impact return period estimates, with only the NAO and BEST influencing the results at all. The hypothesis that hurricane strike frequencies are increasing in time is also statistically rejected.

## Acknowledgments

This work was supported in part by National Science Foundation Grant DMS 0304407. The authors also thank three reviewers—their comments substantially improved this note.

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U.S. hurricanes used in the study with their wind speeds (WS) and central pressures (CP).

Model parameter estimates with std errors.

U.S. hurricane wind speed return periods and nonencounter probabilities.

U.S. hurricane central pressure return periods and nonencounter probabilities. Classification by central pressures was discontinued in the 1990s.

Regional return periods in years.

Return periods of some notable U.S. landfalling hurricanes.