## 1. Introduction

The distribution of earth-relative winds around a hurricane is not symmetric; the winds on the right side of the storm (relative to storm motion) are typically stronger than the winds on the left side. George and Gray (1976) found that the wind asymmetry is greatest at the radius of maximum winds, and decreases outward. Most of the asymmetry between the winds on the right and left sides of hurricanes is due to storm motion. Note that throughout this paper the term *radius of hurricane- force winds* will refer to the *maximum* radius of hurricane-force winds.

There is a dearth of meteorological research concerning the systematic documentation of the maximum radii of hurricane-force winds in North Atlantic hurricanes. Schwerdt et al. (1979) examined the interrelationships of parameters that influence the strength and regional variation of hurricane wind fields. The parameters used in their study were central pressure, peripheral pressure, radius of maximum winds, forward speed of the storm, track direction, wind inflow angle, latitude, and longitude. The authors concluded that wind fields cannot be estimated through statistical relations between pairs of parameters; the relationships are curvilinear, and more than two parameters are involved in the development of hurricane wind fields.

In an important early study, Merrill (1984) examined that size of tropical cyclones in both the Pacific and Atlantic basins and introduced the measure of the radius of the outer closed isobar (ROCI). The ROCI was found to be only weakly related to the storm intensity. In another related study, Croxford and Barnes (2002) continued to examine the relationship of inner core strength (the mean tangential winds between 65 and 140 km outward form the storm center) to intensity and to intensification. This study might be thought of as helping to define the range of the inner core strength as well as its relationship to other commonly observed hurricane variables.

Weatherford and Gray (1988a,b) examined typhoons and deduced average winds at large radial distances from storm centers, and the relationship of those winds with other storm parameters. Due to data paucity, they assumed symmetry in storms. Theses authors also found only small correlations between the radial extent of storms and central pressure.

Researchers have attempted to approximate the radial profile of surface winds in hurricanes using forms such as the modified Rankine vortex and the Holland B profile (Holland, 1980). Because of the difficulties involved in obtaining direct observations, however, these profiles have been created using boundary layer knowledge obtained through diagnostic and empirical calculation and extrapolation (Holland 1995; Vickery and Twisdale 1995).

Details of the wind structure in hurricanes is incomplete. This study attempts to complete that picture through a comprehensive examination of hurricane- force wind radii using flight-level data gathered in 33 North Atlantic hurricanes which occurred between 1977 and 1999. The advantage of using flight-level data adjusted to the surface to determine hurricane-force wind radii is that data resolution is much greater than that of the surface observational network.

## 2. Data sources

Only two flight-level datasets for the North Atlantic basin are available. One is an archive of National Oceanic and Atmospheric Administration (NOAA) flight- level observations from hurricanes of the 1950s and 1960s. Shea and Gray's (1973) research using the NOAA flight-level data did not include any analysis of hurricane-force wind radii.

The other dataset and the primary data source for the present study is an archive of Atlantic basin flight-level hurricane and tropical storm data for the years 1977– 99 maintained by the Hurricane Research Division (HRD). This archive contains postseason processed data obtained through NOAA and U.S. Air Force research flights. The primary data for each storm represented in the flight-level data archive are held in files, each of which contains a profile of the storm for one inward or outward radial pass. Most of the data in the HRD flight- level data archive are from flights at the 850- and 700- hPa levels. Reconnaissance missions are not conducted over land due to safety considerations.

The track direction is a key parameter in this study. The dynamic storm center is located by a procedure described in Willoughby and Chelmow (1982). Wind and pressure data are input into an algorithm used to locate the axis of rotation at flight level. At typical airspeeds (about 125 m s^{−1}), the spatial interval between observations is 125 m. At every 125 m and at the closest point of approach to the center (which can be determined within 0.5–1.0 km), lines normal to the wind are constructed. These lines converge in a region 1–2 km wide rather than a point because hurricanes are somewhat asymmetric, and observational data contain errors. The coordinates of the eye are then chosen by least squares methods. Once a series of successive centers has been located, a track is constructed by fitting cubic splines to the coordinates of the storm center using time as a parameter. The resulting track is a smooth function with continuous first and second derivatives. Cubic spline interpolation has the effect of removing small-scale fluctuations in the position of the eye of the hurricane.

The track determined using the algorithm described earlier is an absolute track, a track which lies within a coordinate frame fixed to the earth. To determine the relative track (one in a coordinate frame which moves with the hurricane), the absolute track is differentiated to obtain the center's velocity, and the winds are then transformed into storm-relative coordinates.

The Atlantic Hurricane Database (HURDAT; Jarvinen et al. 1984), maintained by the Tropical Prediction Center provides storm motion data and hurricane classification by central pressure. This database includes 6- hourly latitudinal and longitudinal coordinates of the circulation center, maximum 1-min averaged sustained surface wind speed to the nearest 5 kt, and central pressure (when available) in millibars for Atlantic storms from 1851 to the present. The HURDAT dataset does not contain information regarding size of tropical storms or hurricanes.

## 3. Methodology

The Saffir–Simpson scale is referenced in this research for classification of hurricanes. This scale is widely used to categorize hurricanes by the maximum 1-min averaged near-surface (10 m) with speed, or the minimum central surface pressure. The Saffir–Simpson scale shown in Table 1 is based on Simpson (1974). Historically, minimum sealevel pressure was the basis of classification; however, the maximum sustained wind has become the more accepted parameter, and is now the basis of the official classification. Although the trend of stronger winds with decreasing central pressure is well established, this relationship is not strong enough to avoid inconsistencies. Generally a hurricane is classified by its strongest winds or by its intensity upon landfall. Here, we consider the hurricane intensity at the time of aircraft penetration. Because of the uncertainty that any series of penetrations will actually record the maximum wind, and the uncertainty introduced by calculating the maximum sustained wind from the flight- level data, we have used the instantaneous minimum pressure deduced from the flight-level data for the purpose of hurricane classification. In the data stratification, all storms with deduced surface-level minimum pressures greater than or equal to 965 mb are called *minimal hurricanes* and all hurricanes with minimum deduced minimum central pressures les than 965 mb are considered *major hurricanes.*

### a. Composite of radial profiles

A composite of radial profiles from 33 Atlantic hurricanes for which HRD flight-level data are available was generated. The use of the composite technique minimizes the effects of variation between individual storms. HURDAT was referenced to determine the interval over which these storms were of hurricane intensity; only those flight profiles obtained during these periods are used in this study. For all 33 hurricanes examined, aircraft missions were flown at the 850 or 700 hPa levels, or both. However, for storms into which research missions were flown at multiple pressure levels, only data from the level with the greatest number of observations are retained in the composite. More flight data are available for some storms than others because the number and/or duration of research missions varies depending on the lifespan of the storm and consideration of research priorities. However, the sample of hurricane profiles at varying intensities is sufficiently large that this bias is not likely to appreciably affect the final results of the analysis.

### b. Quadrants

To provide a common framework for analysis, each radial flight profile is categorized by quadrant with respect to direction of storm motion. The orientation of the quadrants is then rotated 45°, and the same radial flight profiles are again categorized by quadrant. The use of overlapping quadrants provides a means of checking the reliability of the analysis. A graphical depiction of the overlapping quadrants relative to storm track is shown in Fig. 1. The front quadrant is centered in the direction of storm motion while the right-front quadrant, for example, is centered 45° to the right of storm motion. Note that there may be more observations for some quadrants than for others, especially near landfall; because reconnaisance missions are not conducted over land due to safety considerations, data may be obtained only while the storm center is over water.

The angle between flight direction and mean storm direction during a radial flight pass determines the quadrant by which data are stratified. Both mean storm direction and flight direction are taken with respect to 0° facing toward the north, increasing in the clockwise direction. Flight direction is taken as if the radial flight pass is always outward.

Mean storm direction during a radial flight pass is found using the appropriate position file, which containts the *x*- and *y*-components of the storm velocity at 10-min intervals. The mean storm heading is then calculated based on the averaged *x*- and *y*-components of the storm velocities at the beginning and end of this portion of the position file.

### c. Earth-relative winds

*A*(based on Schwerdt et al. 1979):

*A*

*T*

^{0.63}

*T*

^{0.07}

_{0}

*α*

*σ*

*α,*which is taken relative to the mean heading of the radial flight pass, is calculated as follows:

It should be noted that as *V* approaches zero, the arctangent term becomes undefined; it is assumed in this case that the second term on the right-hand side of the preceding equation is set to 90°. Also, *T*_{0} = 1 when the data are in knots. The *α* − *σ* represents the angle between the surface wind direction relative to the heading of the aircraft over the radial flight leg, and the mean storm heading over the duration of a radial flight pass. This angle varies azimuthally and with distance from the storm center. The convention used in this study is that *α* = 0° for surface tangential and radial winds blowing toward the north, and *α* increases in the clockwise direction. The same convention is used for mean storm heading.

Streamlines of hurricane winds are not circles around the hurricane center; they spiral inward. The inflow angle is the angle between the wind direction and a tangent to a circle concentric with the hurricane center. The asymmetry factor used in this study accounts for the inflow angle and is more accurate, especially for large inflow angles, than a formulation which assumes that hurricane winds blow in a circular fashion around the circulation center.

The −20° term, which has been added, accounts for a mean backing of the winds between the flight level and the surface. Because the asymmetry factor includes a specification of wind direction, one must account for vertical variation in wind direction. A mean backing of 20° in wind direction between flight level and the surface accounts for most of this vertical variation and represents a good first approximation (F. Marks, 2000, personal communication).

### d. Reducing flight-level winds to the surface

A major problem faced in hurricane forecasting is how to adjust flight-level observations to make them representative of sustained surface winds.

Based on boundary layer research flights and recent airborne Doppler radar measurements, the vertical profile of the horizontal wind component in the boundary layer of a hurricane is essentially logarithmic up to the base of the level of maximum winds (usually between 500 and 1800 m). Above this level, hurricane winds typically decrease with height due to weakening pressure gradient. Thus, under certain conditions, a wind observation at 3000 m may be less than a surface wind measurement.

Powell et al. (1996) maintain that the radius of maximum wind increases with height. Thus, there is a variation of *R*_{max} with height, and a failure to account for this increase with height will cause the radius of maximum wind adjusted to the surface from flight-level data to be too far from the center of the storm. The research of Jorgensen (1984) and Marks et al. (1992) supports a slope of the eyewall with height. Gray and Shea's (1976) research also supports a slope in the radius of maximum wind, but only with weaker storms. However, it is assumed in this study, which utilizes flight data obtained only at the 700- and 850-hPa levels, the the *R*_{max} is the same at flight level and at the surface, regardless of tilt.

There are additional assumptions and limitations involved in using flight-level wind measurements to inter surface winds. Adjustment of flight-level winds to the surface requires an assessment of the maximum wind speed level. Research aircraft may not fly at the level of strongest winds; however, every attempt is made to fly through the strongest winds at flight level. According to Powell et al. (1991), one must assume that the level of maximum winds coincides with the altitude of the aircraft.

Large differences may exist between the winds at flight level and the surface, especially when data are gathered above the top of the boundary layer. The location of the strongest winds aloft and at the surface may be different. Surface mesoscale features may be weak or absent at flight level. Downdrafts may affect flow at the surface without influencing flight-level winds at all. Furthermore, surface wind speeds may be greater than wind speeds aloft in areas of downdrafts; in rainbands and areas of convection, the level of maximum winds may be much lower than flight level (Powell 1987).

Some of the solutions to the problem of adjusting flight-level winds to the surface are reviewed in Powell (1980). The basic techniques reviewed include the application of boundary layer models to flight-level measurements, and rough estimation methods. Powell founds errors of 9%–10% in adjusted surface winds for all methods, with the largest errors at the inner core of the eyewall. This suggests that the use of a uniform flight reduction factor to adjust flight-level winds to the surface is not more prone to error than other methods.

Uniform surface reduction factors (one for each level at which flight data are gathered) are used in this study to adjust flight-level winds to the surface. These surface reduction factors are based on the work of Franklin et al. (2000, 2003). Franklin et al. (2003) used GPS-based dropwindsondes (with 5-m vertical resolution and 0.5– 2.5 m s^{−1} accuracy) to study the mean wind structure in three areas of the lowest 3000 m of the hurricane: in the eyewall, in convection outside the eyewall, and completely outside convection. Surface reduction factors for a given pressure level are based on the mean GPS dropwindsonde profiles within each of these regions of the hurricane. The wind speeds at all levels within each profile are divided by the wind speed at the normalization level, and all normalized profiles are then averaged together.

The operational surface wind reductions made by the Tropical Prediction Center are used in this study. J. Franklin (2000, personal communication) provided the flight-reduction factors shown in Table 2. Although separate flight-reduction factors for the eyewall and outer vortex were provided, Franklin suggested using an average of the two for the purposes of this study. Thus, for each pressure level, an average of the factors for the eyewall and outer vortex has been used to reduce flight- level winds to the surface. Note that as a result, the reduction factor used in this study might be too low in the eyewall and too high in the outer vortex. Furthermore, in regions of convective downdrafts, wind speeds at the surface may be higher than those aloft. However, at low levels, the reduction factors are the same within and outside the eyewall in any case.

### e. Minimum surface pressure

Pressure data are not provided as part of the HRD flight-level data archive. Typically, when pressure data from dropsonde reports or surface observations are not available, the HURDAT 6-hourly minimum sea level pressure values are extrapolated from aircraft flight data through the use of pressure–height curves. We employ the same procedure here.

*p*

_{s}

*H*

_{700}

*p*

_{s}is the surface pressure in hectopascals, and

*H*

_{700}is geopotential height of the 700-hPa surface in tens of feet.

Pressure–height information at all levels is sensitive to errors in aircraft altimetry, but according to Jordan (1958) there is no evidence to suggest that over a large sample of flight data these errors should not be random. The preceding 700-hPa regression was tested using 65 dropsonde records from the 1955 and 1956 hurricane seasons. Surface pressures determined using the preceding regression were compared to the corresponding surface pressures measured by dropsonde. For latitudes south of 30°N, 60% of predicted surface pressures were within 1 hPa of observed pressures, and 90% were within 3 hPa.

*p*

_{s}

*H*

_{850}

*H*

_{850}is the height of the 850-hPa surface in tens of feet.

### f. Maximum wind speeds, radii of maximum winds, and radii of hurricane-force winds

For each radial flight pass, wind data are first converted to earth-relative coordinates; then the maximum earth-relative wind speed, *V*_{max}, and the radius of maximum wind, *R*_{max}, are retrieved directly from the generated radial profile. Notice that *V*_{max} and *R*_{max} are not used in the typical sense, where there is only one value per storm. Short flights are problematic; the highest wind speed along any given azimuth may occur further in or out from the center of the storm in a region not sampled by the aircraft. By assuming that *V*_{max} is equivalent to the maximum wind speed in the radial flight profile regardless of the length of the flight, one unnecessarily introduces biased values of *V*_{max} and *R*_{max} into the composite. Therefore, if *V*_{max} occurs near the beginning or end of a short flight leg, the profile is discarded from the composite.

An upper bound for *R*_{max} for a typical hurricane is taken to be 50 km (Croxford and Barnes 2002; Samsury and Zipser 1995). Radial flight profiles where *R*_{max} exceeds 50 km are considered outliers and removed from the composite. Profiles with a radius of maximum wind greater than 50 km are more characteristic of a hurricane which is undergoing eyewall replacement. Thirty-two percent of the samples were removed because the wind maximum was beyond 50 km. As a result, 3 out of 38 hurricanes for which flight-level data were available were not included in this study. The final composite of radial flight passes in this study should be representative of typical hurricane structure.

A moving average of the earth-relative wind speed is found using a 10-km window. To compute the average, data must be available for at least half of the points within the window. The radius of hurricane-force winds, *R*_{hf}, is found by starting at *R*_{max} and moving outward from the center of the storm and locating the radius where the moving average of earth-relative wind speed drops below the hurricane-force threshold (see Fig. 2). Using a moving average to locate the largest radius of hurricane-force winds is advantageous for two reasons. First, it is easier to see when the winds fall below the hurricane-force threshold when some of the variability has been removed. The averages and standard deviations of data used in the moving-average window indicate that there is little difference in the radii of hurricane- force winds between the smoothed and unsmoothed wind profiles. Second, small-scale and transient features are removed. If, for example, there is a convective cell which causes winds in excess of hurricane force over a small distance, the effects of the cell will be removed from the analysis.

Problems still arise when trying to determine the radius of hurricane-force winds. In an effort to maintain consistency throughout the analysis, a framework has been developed in order to deal with these problems. First, there may be multiple radii at which the wind speed profile drops below hurricane force. In the case of multiple *R*_{hf}, the outermost value is taken (see Fig. 3).

Second, it is possible that data are missing from the wind profile within an interval which includes hurricane-force wind speeds. In this case, a linear interpolation of the moving-average curve is performed between the end points of the interval over which the data are missing. It is then possible to find the radius at which the interpolated moving-average curve drops below the hurricane-force threshold.

Finally, wind speeds may be greater than hurricane force throughout a radial profile. In this case, a linear regression of the earth-relative wind speeds is performed over the last 20 km of the profile. This regression is then used to extrapolate the wind speed profile out far enough from the center of the storm that the wind speed curve falls below the hurricane-force threshold (see Fig. 4). An inverse relationship between wind speed and *R* from *V*_{max} outward, where *R* is the radial distance in kilometers from the storm center, might more accurately represent the shape of the wind speed profile. However, extrapolation over short distances using a linear regression works just as well, and eliminates the problem of deciding from which peak to start the extrapolation in the case of multiple local wind speed maxima. Radial profiles are discarded from the analysis if the extrapolation is carried out over a distance larger than 50 km, if data are not available for at least 20 km between *R*_{max} and the end of the flight, or if there is an upward trend in the earth-relative wind speed curve at the end of the flight.

The final dataset, with outliers and problematic radial profiles removed as described earlier, contains 1070 radial flight passes from 33 North Atlantic hurricanes which occurred between 1977 and 1999. The names of the hurricanes represented in the composite of observations, the times over which hurricane intensity was sustained (according to HURDAT), and the maximum earth-relative wind speed and minimum central pressure at the surface derived from the flight-level data are shown in Table 3.

Radial profiles retained in the dataset are considered to be in one of two groups based on pressure: minimal hurricanes—representing catagories 1 and 2 on the Saffir–Simpson scale, and major hurricanes—representing categories 3, 4, and 5. The number of category 1 hurricanes represented in the analysis is smaller (by 2) than the number for which flight-level data are available because profiles in which the moving average of the observed wind speed does not exceed 33 m s^{−1} are not included, even though the unfiltered wind speed may exceed hurricane force and the central pressure may indicate a hurricane. The number of profiles representing category 5 hurricanes is low because category 5 hurricanes are rare events.

## 4. Analysis and results

### a. Stratification by pressure

Each radial profile in the composite of observations is categorized by the value of the minimum central pressure. Estimates of minimum central pressure based on the extrapolation of height data to the surface are fairly reliable; they are accurate to within about 5 hPa, as mentioned previously. While of greater interest, knowledge of maximum surface-level wind speed is likely to be known less accurately than central pressure.

It can be seen in Table 3 that the Saffir–Simpson categories of hurricane intensity based on the minimum central pressure and maximum earth-relative wind speed for any given hurricane do not always agree. Hurricanes are now categorized by the maximum sustained surface wind speed at any location around the storm, the minimum central pressure over the lifetime of the storm was used historically. In this study, the minimum central pressure extrapolated from the height data was determined for each radial profile during the period over which the storm maintained hurricane intensity; classification of hurricanes was then based on the lowest extrapolated value of minimum central pressure over all radial flight profiles from any given hurricane.

### b. Statistical analysis

A statistical analysis was conducted using the 730 radial profiles out of 1070 in the composite which contained hurricane-force winds; this analysis was conducted separately for each set of overlapping quadrants.

The average, standard deviation, and median of the following storm parameters were calculated for each quadrant: minimum central pressure, maximum earth- relative wind speed, radius of maximum winds, and radius of hurricane-force winds. The statistics for each quadrant are presented in Tables 4 and 5. At the bottom of each table are the number of profiles in each quadrant which contain hurricane-force winds (HF), and the number of profiles which do not contain hurricane-force winds (no HF).

Trends for these two groupings emerge. As expected, pressure decreases and maximum wind speed increases as hurricanes intensify, and the pressure gradient tightens as the minimum central pressure decreases. The number of radial profiles representing major hurricanes is much larger than those representing minor hurricanes, and the range of central pressures and maximum wind speeds is also larger for major hurricanes. Thus, the standard deviation of minimum central pressure and maximum wind speed is larger for major hurricanes.

The average radius of maximum winds is smaller for more intense hurricanes than weaker ones. However, the standard deviation of *R*_{max} is not much different between minimal and major hurricanes. The average radius of hurricane-force winds is much larger for more intense storms, and the standard deviation of *R*_{hf} also increases substantially for stronger hurricanes. This reflects a large degree of variability among major hurricanes.

Distributions of hurricane-force wind radii for minimal and major hurricanes, stratified by quadrant, are shown in Figs. 5 and 6. Estimates of *R*_{hf} are especially prone to errors introduced through uncertainties in inferring surface wind speeds from flight-level data. As mentioned earlier, aircraft flight patterns may be such that the maximum winds are not sampled, and the flight- reduction factor used to convert flight-level winds to the surface may be incorrect. For these reasons, hurricane- force wind radii are grouped into bins 5 km wide to reduce the effect of wind speed errors on the overall analysis.

Radii of hurricane-force winds are not normally distributed. The histograms shown in Figs. 5 and 6 are skewed toward smaller values of *R*_{hf}. This effect is more pronounced for major hurricanes. In addition, the distributions of *R*_{hf} are much broader for stronger hurricanes, reflecting the larger standard deviations in Tables 4 and 5. For major hurricanes, the range of possible *R*_{hf} values over all quadrants is about 155 km, from about 15 to 170 km. The possible values of hurricane-force wind radii for minimal hurricanes range from about 15 to 90 km, about half the range of hurricane force wind radii for major hurricanes. The horizontal distributions of hurricane-force wind radii would be narrower if only the inner radius of hurricane-force winds were taken (neglecting an outer maxima in winds when it occurred) and if there were no dispersion in the relationship between center pressure and wind speed.

### c. Relationships among storm parameters

Scatterplots of important storm parameters for minimal and major hurricanes are shown in Figs. 7–14. Only plots corresponding to one set of quadrants are shown; scatterplots from the overlapping set of quadrants are similar. It can be seen in these figures that all obvious outliers have been removed by rejecting from the composite those profiles in which *R*_{max} is greater than 50 km. Thus, the averages and standard deviations discussed in the previous section are not likely to have been significantly affected by anomolous data.

Although the focus of this paper is on hurricane-force wind radii, a general comment on the relationship between maximum wind speed and central pressure is useful. As illustrated in the lower-left panel of Fig. 11, for example, there is a clear indication that for the most intense hurricanes, the highest winds speeds are associated with the smallest radii of maximum winds. This is consistent with cyclostrophic balance. For a given central pressure, as the radius of maximum winds increases, the pressure gradient decreases and the maximum wind speed decreases. However, as the central pressure increases, the tendency is for the maximum winds to decrease, again, principally because the pressure gradient also decreases.

For minimal hurricanes, it appears that there may be a linear relationship between *R*_{hf} and *R*_{max}; hurricane-force wind radii seem to increase with *R*_{max} (see upper-left plots in Figs. 7–10). A positive relationship between *R*_{hf} and *V*_{max} is also supported in most quadrants. Correlation coefficients also suggest a negative relationship between *R*_{hf} and the minimum central pressure.

For major hurricanes, a much weaker but positively correlated relationship exists between *R*_{hf} and *R*_{max}, and a negative correlation exists between *R*_{hf} and minimum central pressure (see upper-left plots in Figs. 11–14 and Tables 6 and 7). Storms with a larger radius of maximum winds in any radial profile also have hurricane-force winds which extend farther out from the center of the storm. For major hurricanes, the range of possible *R*_{hf} values is broader than for minimal hurricanes, although there is more clustering of hurricane-force wind radii around smaller values of *R*_{max}. The radius of hurricane- force winds tends to be larger for storms with lower minimum central pressures (see upper-right plots in Figs. 7–14), which is supported by the statistical analysis. There also seems to be a relationship between minimum central pressure and *R*_{max} (see lower-left plots in Figs. 7–14); storms with lower central pressures have smaller radii of maximum winds. This conclusion is also supported by the statistical analysis.

Although higher wind speed is correlated with lower central pressure, there is significant dispersion of points in the scatterplot of *R*_{max} as a function of *V*_{max} (see lower- right plots in Figs. 7–14). However, the variability among the radial profiles for a given pressure category indicates only a weak relationship between higher wind speeds and smaller radii of maximum winds. At least some of the dispersion is because of a pronounced peak in the wind field and a contraction of the radius of maximum winds which occurs as a hurricane intensifies. For a decaying storm, there is a corresponding decrease in wind speed as the radius of maximum winds slowly increases.

The storm parameters which are most closely related to the radius of hurricane-force winds, ranked in descending order by the magnitudes of the correlation coefficients (see Tables 6 and 7), are radius of maximum winds, maximum wind speed, and minimum central pressure, respectively. The correlation coefficient indicates that there may be a weak linear relationship between *R*_{hf} and *V*_{max}; in fact, the graphs reveal a weak positive relationship between these two variables.

Hurricane-force winds may extend further out in storms with larger eye diameters. The radius of hurricane-force winds may be related through *V*_{max} to the height and/or shape of the wind speed profile. There is a clear positive correlation between storm *V*_{max} and *R*_{hf}.

To assess the significance and confidence of the correlations found in Tables 6 and 7, a Student's *t* test was performed using the appropriate degrees of freedom. From these results the corresponding *p* values were found, and the results summarized. A two-sided distribution was assumed, which makes the results more conservative. If the correlation coefficient had a significance value of 95% or greater, it was indicated by the letter H (for highly significant); a significance between 80% and 95% was indicated by the letter M (for moderately significant), and a significance less than 80% was indicated by the letter L (for low significance). In general, there is the expected relationship between high correlation and high confidence, with the number of samples increasing the levels of significance and confidence.

### d. Toward a predictive model

*y*

_{i}

*β*

_{1}

*x*

_{i1}

*β*

_{2}

*x*

_{i2}

*β*

_{m}

*x*

_{im}

*x*

_{ij}is the observed value of the

*j*th of

*m*other variables associated with the

*i*th

*y*-value and

*β*

_{;t7j}is a parameter which specifies how the independent variable

*y*is related to the

*j*th dependent variable,

*x*

_{j}(Freund and Minton 1979). Mathematically, the

*β*

_{;t7j}are standardized regression coefficients which result when the regression is based on variables which have been transformed to unit variance. Regression coefficients are functions of units of measurement of the independent and dependent variables, whereas standardized regression coefficients are independent of the units of measurement. These parameters are estimated using the method of least squares (Aiken and West 1991). Each

*β*coefficient indicates how many standard deviation changes in

*y*are associated with one standard deviation change in

*x*with all other

*x*

_{j}held constant. Multiplication of an ordinary regression coefficient by

*s*

_{xi}/

*s*

_{y}, where

*s*denotes standard deviation, yields the corresponding standardized regression coefficient (Freund and Minton 1979).

The regression analysis is conducted using both two and three independent variables for minimal and major hurricanes for each set of overlapping quadrants. The results of the analysis can be seen in Tables 8 and 9. In the top-half of each panel, the regression with the two independent variables *R*_{max} and *V*_{max} is displayed. The regression with the two independent variables mentioned earlier, plus minimal central pressure is displayed in the bottom-half of each panel. Here, *β* × *r*_{xy} denotes the standardized regression coefficient multiplied by the correlation coefficient between the variables *x* and *y.* The value *R*^{2}, the amount of variance explained by the combined effects of the independent variables, is equal to the sum of *β* × *r*_{xy} for each independent variable; *R*^{2} is not equal to the sum of the squared correlation coefficients because there may also be an interaction between independent variables.

By taking an average value of *R*^{2} over all quadrants, it can be seen that *V*_{max} and *R*_{max} account for about 75% of the variance in minimal hurricanes, and for about 50% of the variance in major hurricanes. Enough variance in the hurricane-force wind radii is explained using the storm parameters *V*_{max} and *R*_{max} to warrant the development of a model to predict *R*_{hf} as a function of these parameters. There may be more unexplained variance with major hurricanes, at least in part because when there are two radii where the winds go from hurricane force to below 33 m s^{−1}, *R*_{hf} is taken at an outer radius; this occurs more often for major hurricane wind profiles. A regression analysis which includes a third independent variable, minimum central pressure, accounts for only another 2% of the variability in hurricane-force wind radii; adding pressure as an independent variable is not particularly useful in explaining additional variance. There is a strong relationship between maximum wind speed and minimum central pressure indicated by the correlation coefficient between these two variables. Thus, it is unlikely that the effects of *V*_{max} are held constant when the relationship between pressure and *R*_{hf} is explored.

### e. Cumulative probability distributions

Wind data from the 730 radial flights in which hurricane-force winds were found are used to construct horizontal distributions of hurricane-force winds at the surface. One reason that the composite of 730 radial flight profiles used in this study are analyzed once for each of two sets of overlapping quadrants is to increase the number of data points available in constructing cumulative probability distributions.

Figures 15 and 16 depict the cumulative probability distributions of hurricane-force wind radii for minimal and major hurricanes. The total number of composites in the dependent dataset is 1070; within each quadrant, there is a certain probability that no hurricane-force winds are found (which can be calculated using the information in Tables 4 and 5 concerning the number of profiles with and without hurricane-force winds). Thus, Figs. 15 and 16 show the cumulative probabilities that hurricane-force winds occur within various radii assuming hurricane-force winds are present at all.

The labeled contours denote the probability (in terms of a percentage) that hurricane-force winds will be contained within that radius. Alternatively, 100% minus the contour value denotes the probability that hurricane- force winds will occur outside that radius. Notice that these distributions are open-tailed; there is no contour within which there is a 100% chance hurricane-force winds will occur.

Both Figs. 15 and 16 depict an asymmetry in the earth-relative wind field, with hurricane-force wind radii extending farther out in some quadrants than others. However, there are two differences between the cumulative probability distributions of hurricane-force wind radii for minimal and major hurricanes which are immediately striking. First, the distribution is broader for major hurricanes; contours of the same value will generally occur further outward from the center of the storm for major hurricanes. Second, the wind asymmetries have a different character. Wind fields of minimal hurricanes appear to be fairly symmetric; hurricane-force wind radii tend to be largest in the right-rear quadrant, and smallest in the left-rear quadrant. In major hurricanes, the asymmetry is rather marked, especially toward the outermost contours; hurricane-force wind radii are largest in the front quadrant, and smallest in the left quadrant.

When producing figures of this type, there are inherent assumptions involving the placement of data and contours. The data within each quadrant in the cumulative probability distributions are placed along a radial which bisects the quadrant. The fact that this radial was chosen is somewhat arbitrary; these data could be placed along a radial anywhere in the quadrant with no less uncertainty. The use of overlapping quadrants produces twice as many points and, thus, reduces some of the uncertainty in generating contours. Although there is some smoothing of the probability distributions that results from a 45° overlap between quadrants, there are no significant differences between quadrants, such as values of hurricane-force wind radii which oscillate around a mean from quadrant to quadrant. Thus, there is nothing in the figures which suggests an unreliable analysis.

## 5. Conclusions

Radial profiles of earth-relative wind speeds are generated using the Hurricane Research Division flight-level data archive of 1977–99 North Atlantic hurricanes. Two particularly useful aspects of this study emerge. First, the two-parameters radius of maximum wind and maximum wind speed explain enough variance in hurricane-force wind radii, especially for minimal hurricanes, to warrant the development of a model which would predict hurricane-force wind radii based on these two storm parameters. Second, hurricane-loss modelers can benefit from the cumulative probability distributions which are generated.

The feasibility of fitting the wind data to a function or model which would predict radii of hurricane-force winds can perhaps be improved. One could try to curve- fit the wind speed data, and then determine how various storm parameters might be related to the resulting curve. Perhaps curve-fitting only a smaller radial belt of the wind speed profile encompassing the radius of hurricane-force winds would yield more fruitful results.

## Acknowledgments

Hugh E. Willoughby, director of the Hurricane Research Division, generously provided the HRD flight-level data archive. Appreciation is also extended to James L. Franklin of the Tropical Prediction Center for sharing his calculations of flight-level reduction factors. Bret Whissel contributed invaluable computer programming expertise, for which we are deeply grateful. Very constructive reviews came from Dr. Chris Landsea and another anonymous reviewer. This work was partially supported by Contract 1338- 847-41 from Florida International University.

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The Saffir–Simpson scale

Surface reduction factors (to the nearest 5%) for the eyewall and outer vortex

Hurricane information. Dates and times refer to interval(s) over which hurricane strength sustained. Saffir–Simpson categories according to maximum sustained wind speed and minimum central pressure determined from flight-level data are in parentheses

Statistics for minimal hurricanes

Statistics for major hurricanes

Correlations and their significance among various storm parameters by quadrant (for each set of overlapping quadrants) for minimal hurricanes. Here, HF and no HF refer to the number of profiles which do and do not contain hurricane-force winds, respectively

The same as in Table 6 except for major hurricanes

Summary regression results for minimal and major hurricanes

Summary regression results for minimal and major hurricanes