## Abstract

A simple technique is developed that enables the radius of maximum wind of a tropical cyclone to be estimated from satellite cloud data. It is based on the characteristic cloud and wind structure of the eyewall of a tropical cyclone, after the method developed by Jorgensen more than two decades ago. The radius of maximum wind is shown to be partly dependent on the radius of the eye and partly on the distance from the center to the top of the most developed cumulonimbus nearest to the cyclone center. The technique proposed here involves the analysis of high-resolution IR and microwave satellite imagery to determine these two parameters. To test the technique, the derived radius of maximum wind was compared with high-resolution wind analyses compiled by the U.S. National Hurricane Center and the Atlantic Oceanographic and Meteorological Laboratory. The mean difference between the calculated radius of maximum wind and that determined from observations is 2.8 km. Of the 45 cases considered, the difference in 50% of the cases was ≤2 km, for 33% it was between 3 and 4 km, and for 17% it was ≥5 km, with only two large differences of 8.7 and 10 km.

## 1. Introduction

To determine *r _{m}*, the radius of maximum wind for a tropical cyclone, one needs to analyze the strong surface winds in its inner core and in its eyewall. There are a number of methods that have been developed to measure or estimate surface winds from satellite sensors but they are not reliable in the inner core of a tropical cyclone. The Special Sensor Microwave Imager (SSM/I) can only be used in cloudless sky and cannot therefore be used within the central dense overcast (CDO) or in the eyewall of a tropical cyclone (Goodberlet et al. 1989). Satellite-based microwave scatterometers can only produce a reasonably good estimate of light to moderate surface winds in areas of no or slight precipitation (Weissman et al. 2002; Yueh et al. 2003). They are therefore not reliable for estimating the strong winds associated with tropical cyclones. Surface winds can also be estimated from satellite cloud track winds deduced from geostationary satellites (Dunion and Velden 2002; Velden et al. 2005), but these also are only useful outside the CDO.

Another sensor on board a polar-orbiting satellite that can produce high-resolution surface wind fields over the ocean is the Wind Field Synthetic Aperture Radar (WiSAR). It measures the small-scale ocean surface roughness from which can be determined the direction and the speed of the surface wind (Lehner et al. 2006). WiSAR waves can penetrate cloud layers and rain and can operate day and night in all weather conditions and are therefore well suited to determining the radius of maximum wind in a tropical cyclone. The only problem with WiSAR is that the data are only available when the tropical cyclone is along the satellite track and are therefore available only twice a day.

Surface winds inside the CDO are also obtainable from reconnaissance planes that fly at an altitude of about 3 km. The measured 3-km winds are reduced to the surface by using an empirically derived relationship (Franklin et al. 2003). Using these flight data, the U.S. National Hurricane Center can determine the maximum surface wind and the radius of maximum wind. But these flight wind data are now only available in the North Atlantic when a hurricane is within the flight range of the reconnaissance aircraft. In the central North Atlantic and in other tropical cyclone basins there is no method to directly measure the maximum winds or the radius of maximum wind.

Recently, Kossin et al. (2007) used reconnaissance flight data to investigate the relationship that exists between the eye size (*r _{e}*) and

*r*. They assumed

_{m}*r*to be the mean radius of the −45°C isotherm in the cloud-top brightness temperature analysis. Their mean absolute error in determining

_{e}*r*was 4.7 km.

_{m}Hsu and Babin (2005) have suggested that the radius of maximum winds in a tropical cyclone is the distance between the coldest cloud-top temperature surrounding the eye and the warmest temperature in the eye. They evaluated their hypothesis on only one cyclone, however. Here, we describe below a simple and easy-to-use technique for estimating *r _{m}* from the color-enhanced imagery of high-resolution IR satellite cloud data, from microwave Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI; Simpson et al. 1988; Lonfat et al. 2004) and TRMM Precipitation Radar (PR) data (Iguchi et al. 2000).

## 2. A technique for determining *r*_{m} from satellite cloud data

_{m}

Jorgensen (1984a, b) has studied the eyewall structure of tropical cyclones, using a comprehensive set of satellite cloud pictures, photographs of eyewall clouds, and radar and reconnaissance flight data along 146 flight legs across the eye and eyewall of five hurricanes. His schematic representation of a cross section of the eyewall of North Atlantic Hurricane Allen on 5 August 1980 is shown in Fig. 1a. More recent studies (e.g., Marks and Houze 1987; Marks et al. 1992; Corbosiero et al. 2005) have found rather similar structures in their analyses, although unlike in Fig. 1a downdrafts are also typically observed in regions close to the main updraft.

In Fig. 1a a few letters have been added to help in the following discussion. We use “A” as the inner edge of the eyewall at the surface and “T” as the top of the most developed cumulonimbus nearest to the cyclone center. Let the coordinates of “A” and “T” be denoted, respectively, by (*r _{e}*, 0) and (

*r*,

_{t}*h*) (see also Fig. 1b). The line AT is inclined to the horizontal by an angle

_{t}*ϕ*that may be as large as 45°–75° for intense storms with small eye radius (

*r*) and

_{e}*r*, and as small as 25° for weaker storms or for storms with large

_{m}*r*and

_{e}*r*(Jorgensen 1984a). Let the straight line AT be inclined to the horizontal by an angle

_{m}*ϕ*. The maximum updraft

*w*within this cumulonimbus is along BC, which represents the variation of

_{m}*w*with height in the low and middle levels, B being the maximum

_{m}*w*at the base of the cloud at radius

_{m}*r*. Here, DE represents the variation with height of the radius of maximum tangential wind (

_{b}*V*)

_{θ}*and is inclined at an angle to the horizontal equal to or slightly different from*

_{m}*ϕ*, denoted by

*ψ*. It is assumed that the radius of maximum wind varies linearly with height: although fine-resolution model simulations indicate some departure from linearity of this feature (e.g., Zhang and Kieu, 2006); since we are only taking the two end points in our calculation, this will not affect our final results.

In Fig. 1a, the vertical line TEF passes through E, the maximum tangential wind (*V _{θ}*)

*at level E. Let the coordinates of E and F be (*

_{m}*r*,

_{t}*h*) and (

_{V}*r*, 0), respectively. Because D is the maximum wind at the surface, the coordinates of D are (

_{t}*r*, 0). Note also that

_{m}*r*, the radius of maximum precipitation rate in the lowest 2–3 km of the troposphere, measured from radar observations, is on average greater by 3 km than

_{r}*r*but less than

_{m}*r*(Jorgensen 1984b). According to Jorgensen (1984a, b),

_{t}*r*,

_{e}*r*, and

_{m}*r*have all preferred locations relative to each other and always occur in the same order, so that

_{r}Now, if we assume that angles TAF and EDF in Fig. 1a are nearly the same, then

so that

However, if *ψ* ≠ *ϕ*, then an error will be introduced in estimating *r _{m}* from Eq. (3). The maximum error introduced from this inequality can be estimated from Fig. 1b. In the case that angle EDF is

*ψ*′, where

*ψ*′ = (

*ϕ*±

*α*), then the radius of maximum wind would be

*r*′

_{m}, and (

*r*′

_{m}−

*r*

_{m}) would be equal to

*h*[1/tan(

_{υ}*ϕ*+

*α*) – 1/tan

*ϕ*]. There are not many evaluations of the parameters

*h*,

_{υ}*ϕ*, and

*ψ*. However, if we use an estimate of

*h*from diagrams published in Jorgensen (1984 a,b), then

_{υ}*h*may vary between 4 and 7 km. For a maximum value of

_{υ}*α*of 10°, the range of (

*r*′

_{m}−

*r*

_{m}) is between 1.0 and 1.4 km when

*ϕ*= 75° and between 1.1 and 1.8 km when

*ϕ*= 60°. For a maximum value of

*α*of 20°, the maximum value of (

*r*′

_{m}−

*r*

_{m}) would vary between 2.2 and 3.0 km when

*ϕ*= 75°, between 2.0 and 4.3 km when

*ϕ*= 60°, and between 3.7 and 8.0 km when

*ϕ*= 45°. The estimated ranges of (

*r*′

_{m}−

*r*

_{m}) for different values of

*ϕ*and

*α*are given in Table 1.

The parameter *r _{t}* in Eq. (1) is the distance between the cyclone center and the coldest cloud-top temperature nearest to the cyclone center. It can be evaluated in high-resolution IR satellite cloud data, or in TRMM 85-GHz (85H) imagery. An example of a TRMM 85H imagery is shown in Fig. 2. In this picture the TRMM imagery is overlaid over the corresponding Geostationary Operational Environmental Satellite (GOES) visible (VIS) imagery taken about 1 h earlier. In the temperature analysis, blue is very warm and dark red is very cold. The semicircular dark blue arc at the center of the CDO represents very warm temperature or part of the eye, while the nearby green patch represents the top of the middle-level stratiform cloud. In this case, the center of the outer edge of the semicircular dark blue arc is the cyclone center and the radius of the outer edge of the dark blue semicircular area is

*r*. The nearest dark red cold band to the northeast of the cyclone center, as indicated by the arrow, indicates a band of highest cloud tops. Parameter

_{e}*r*is the shortest distance from the cyclone center to the inner edge of the dark red cold band. Although the determination of

_{t}*r*, and therefore of

_{t}*r*, is based on an arc of either the lowest brightness temperature or intense precipitation rate, it is assumed that the distance of the maximum wind from the cyclone center is the same around the cyclone; that is, the deduced

_{m}*r*is axisymmetric.

_{m}Thus, if *r _{e}*,

*r*, and a mean value of

_{t}*h*/

_{υ}*h*are known, then the radius of maximum wind can be estimated from Eq. (3). Hence, this method requires a well-defined eye in the satellite imagery. We assume here that

_{t}*h*/

_{υ}*h*and thereby the radius of maximum wind are axisymmetric. The use of a mean value of

_{t}*h*/

_{υ}*h*instead of one that varies from storm to storm introduces an error, but we show below that a good estimate of

_{t}*r*is obtained in spite of this approximation.

_{m}## 3. Data

Table 2 gives observed values of parameters relevant to the calculation of the RMW, observed RMW, and calculated RMW. Three series of data have been used to determine the mean value of (*h _{υ}*/

*h*) in Table 2. The first series of data gives

_{t}*r*, the radius of the eye, and RMW, the radius of maximum wind as observed by reconnaissance flights in the North Atlantic, for three storms: Anita, Allen, and Frederic. These are read off from Figs. 4a–d in Jorgensen (1984a), which give plots of tangential velocity and radar reflectivity versus radius. As shown in Fig. 1a,

_{e}*r*was assumed to be at the highest top nearest to the cyclone center of the 10-dB

_{t}*Z*radar reflectivity isoyet.

The second series of data was obtained for two recent North Atlantic hurricanes, Katrina and Wilma, in August 2005. For these storms, *r _{e}* was obtained from the 6-hourly National Hurricane Center hurricane warning advisories (information online at http://www.nhc.noaa.gov). Sometimes these were 2–3 h away from the time of the satellite pictures when

*r*was determined, but such an independent assessment of

_{t}*r*was preferred. Based on aircraft reconnaissance, these estimates are typically accurate to about 5% (R. J. Deatherage 2008, personal communication). RMW was obtained from the same Web site as for

_{e}*r*, from the archived analyzed surface wind field of the Atlantic Oceanographic and Meteorological Laboratory (AOML) (Powell et al. 1998). Errors in RMW from aircraft reconnaissance can be as small as 1 km (J. L. Franklin 2008, personal communication), although the location of RMW can vary rapidly with the changing structure of the storm. It is worth noting that for these surface wind analyses the winds were collected during a period of 2–3 h before and after the valid time of the wind analysis.

_{e}As discussed above, *r _{t}* was obtained from analyses of enhanced IR satellite cloud imagery or TRMM 85H imagery provided online (http://www.nrlmry.navy.mil/tc_pages/tc_home.html). There were 16 sets of data for these two hurricanes.

The third series of data were obtained from archived satellite data of North Atlantic tropical cyclones in 2004. RMW was obtained as in the second series of data, while *r _{e}* was estimated from imagery of colored analyses of enhanced IR satellite cloud data, from TMI imagery of either brightness temperature or derived rain rate, or from TRMM PR images. An example of the enhanced IR imagery is shown in Fig. 3a, of TMI-derived rain rate in Fig. 3b, while Fig. 4 shows a typical TRMM PR image. In the first picture, the dark brown circular spot in the middle of the CDO is the warmest spot or the eye of the cyclone, and a good estimate of

*r*can be made. Note, though, that the type of enhancement used in Fig. 3a is not suitable for determining

_{e}*r*because there is not enough temperature resolution in the red area surrounding the cyclone center to determine the location of the nearest band of coldest cloud tops from the cyclone center.

_{t}In Fig. 3b, the TMI imagery shows the distribution of the precipitation rate around the cyclone: The blue region is rain free and the dark red areas are areas of greatest precipitation rate. The semicircular red band in the middle of the CDO is the eyewall. In this case also the eyewall is not a complete circular band of clouds. The center of the circle, of which the inner edge of this semicircular red band forms part, is the center of the hurricane, and half of the diameter of this circle is *r _{e}*. In general, the eye is blue but in this case the yellow eye indicates it is raining within the eye with a precipitation rate of about 7 mm h

^{−1}. The eye radius as measured from this imagery was found to be 9 km. With

*r*, measured from a TRMM image, of 65 km,

_{t}*r*was estimated to be 34 km. The nearest red or heavy rainband to the north east of the cyclone center was 52 km from the cyclone center; that is

_{m}*r*, was 52 km. Thus,

_{r}*r*,

_{e}*r*,

_{m}*r*, and

_{r}*r*satisfy Eq. (1).

_{t}Note when the eye is circular, a good estimate of *r _{e}* can be made. However, when the eye is elliptical, for instance if the ellipticity is caused by wavenumber 2 vortex Rossby waves (Montgomery and Kallenbach 1997), half of the longest axis has been taken as

*r*. The accuracy of determining

_{e}*r*and

_{e}*r*on the computer screen is estimated to be ±1.5 km.

_{t}## 4. Results

Columns 1–8 in Table 2 give, respectively, the name of the hurricane; the time and date of the AOML wind analysis or the time of the TMI rainfall intensity imagery; *r _{e}* (as determined by the U.S. National Hurricane Center, but when this was not available

*r*was estimated from any available enhanced satellite cloud imagery); RMW (obtained from AOML-analyzed surface wind fields);

_{e}*r*(obtained from either high-resolution digitized IR data or TRMM 85H imagery); (

_{t}*h*/

_{υ}*h*) calculated from Eq. (2);

_{t}*r*[calculated by using a mean value of 0.6 for (

_{m}*h*/

_{υ}*h*) in Eq. (3)]; and |(RMW −

_{t}*r*)|. In Table 2 the times with an asterisk indicate that

_{m}*r*for that row was obtained from TRMM 85H or TRMM PR imagery. In the following analysis of the absolute error, cases when the difference between the time of the RMW and that of the TMI is greater than 4 h are not considered. Despite the difficulties in estimating some of the parameters and two rather large values of |(RMW −

_{e}*r*)|, (10 km during Wilma at 0945 UTC 24 Oct 2005 was dissipating rapidly and 8.7 km when Allen apparently had double eyewalls), the mean value of |(RMW −

_{m}*r*)| is 2.8 km. Of the 58 cases considered, 50% of |(RMW −

_{m}*r*)| are equal to or less than 2 km, 33% are between 3 and 4 km, and 17% are equal to or greater than 5 km. This compares with a mean absolute error for the clear-eye method of Kossin et al. (2007) of 4.7 km, although our data sample is considerably smaller than theirs.

_{m}The calculated *r _{m}* values from Eq. (3) have been plotted against the corresponding NHC-analyzed RMWs in Fig. 5. It is likely that some of the larger scattering of the data points is due to the fact that the times of the wind and of the cloud data sometimes differ by up to 5 h. Two of the larger errors (Allen at 1500 UTC 5 Aug 1980 and Wilma at 0945 UTC 24 Oct 2005) may have been caused by the presence of double eyewalls, where our method would be expected to perform poorly. Otherwise, the agreement is excellent.

It is worth noting that Hsu and Babin’s (2005) suggestion, namely that the RMW is equal to *r _{t}*, the distance between the coldest cloud-top temperature surrounding the eye and the warmest temperature in the eye, is not borne by the data presented in Table 2. There are only 19 out of 58 cases that have

*r*and RMW differing by 5 km or less. These occur when

_{t}*r*and RMW are small. For medium and large

_{e}*r*and RMW, the difference between

_{e}*r*and RMW is even greater, varying between 15 and 35 km.

_{t}## 5. Discussion and conclusions

A simple technique for diagnosing the radius of maximum wind from satellite imagery has been proposed and tested against reconnaissance observations, showing good agreement. The skill of this method is comparable to or slightly better than that of Kossin et al. (2007), although here we analyze a smaller sample of storms. In addition, our technique uses microwave imagery, where the eye is often more clearly defined than it is in IR images. One issue that should be examined in future work is the effect that vertical wind shear would have on the accuracy of this technique, as shear would displace *r _{t}*. Lajoie (2007) argues that this displacement would also have a compensating effect on

*r*and

_{e}*r*, thus suggesting that this method might still give a reasonable estimate of RMW, but this has yet to be tested quantitatively. Another issue is the influence of asymmetries on the calculation of

_{m}*r*, as our method assumes that derived quantities are axisymmetric.

_{m}There are a number of potential applications of this method. RMW or *r _{m}* is an important parameter for diagnosing and forecasting the maximum wind of a tropical cyclone. For example, studies estimating the climatological impact of tropical cyclones on wave fields and storm surge usually have to assume a fixed radius of maximum winds in ocean regions where there are no routine observations of this quantity (e.g., McInnes et al. 2003). Using the method outlined in this paper, a diagnosed record of tropical cyclone structure parameters can thereby be constructed for regions of the tropics where such information has been difficult to obtain to date.

There are other, more theoretical, applications. Lajoie (2007) has developed a simple mathematical model that can determine the mean gradient-level wind averaged around the cyclone at a radius of 1° latitude. It uses *r _{m}* as a necessary parameter to determine the sustained mean maximum surface wind at

*r*, and the radial distribution of the sustained mean surface wind along different orientations with respect to the direction of motion of the cyclone. The model has been used for three tropical cyclones operating in the Australian region to successfully diagnose their maximum winds when they pass over a meteorological station, the radial distribution of the winds ahead and at the rear of the cyclone, as well as the time variation of the radial profile of the mean surface wind. Details are contained in forthcoming publications.

_{m}## Acknowledgments

The authors thank the Australian Bureau of Meteorology, particular Noel Davidson and Jeff Callaghan, for supplying some of the data used in this project. We also thank the University of Melbourne, which supported part of this work. Ian Simmonds made some comments on an earlier version of this work. We thank NOAA, AOML, the U.S. Naval Research Laboratory, NASA, and JAXA for use of their analyzed data. Comments by three anonymous reviewers improved the manuscript.

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## Footnotes

*Corresponding author address:* Dr. Kevin Walsh, School of Earth Sciences, University of Melbourne, Parkville, VIC 3010, Australia. Email: kevin.walsh@unimelb.edu.au