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  • View in gallery

    Hurricane Wilma at 0015 UTC 21 Oct 2005. (a) IR image. The current best-track intensity is 67 m s−1 and 924 hPa. [The square shows the area that is magnified in (c) and (d).] (b) The corresponding map of DAV values (deg2). The minimum value in the map of variances (1189 deg2) corresponds to the location 19.17°N, 85.63°W. The center of the storm reported by the NHC was located at 19.1°N and 85.8°W. (c),(d) As in (a),(b), respectively, but magnified and with the CIRA extended best-track R34 by quadrant overlaid.

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    Hovmöller diagram of the azimuthally averaged DAV signal (shading) for Hurricane Gustav with the observed wind radii and regression overlaid for (a) 34-, (b) 50-, and (c) 64-kt winds. The red dashed line is the symmetric observed wind radii (km), the thick black dashed line displays the regression line (km), and the thin black dashed line displays the intensity of the TC (kt).

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    MAEs for the R34, R50, and R64 wind radii using the symmetric model for the case employing all TCs, and the five bins tested during the cross-validation analysis.

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    The bias (km) for each radius for different maximum wind speed bins (kt) over all cases for the symmetric model. Numbers of samples for R34 and R64 in increasing intensity bin order are 1120, 839, 674, 668, 1126, and 281, respectively. Numbers of samples for R50 in increasing intensity bin order are 963, 738, 630, 668, 1126, and 281, respectively.

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    The MAE (km) for (a) northwest, (b) northeast, (c) southwest, and (d) southeast quadrants of the R34, R50, and R64 wind radii for all TCs and for the five bins tested during the cross-validation analysis. The sample sizes and mean intensity are the same as those reported in Table 2 for the symmetric model. Note that the sample size for each bin of the cross validation is one-fifth of the total.

  • View in gallery

    The bias (km) for each radius at different maximum wind speeds over all cases for each quadrant of the asymmetric model. Numbers of samples in each quadrant and for each radius are as in Fig. 4.

  • View in gallery

    Prediction of the R34 by quadrant for Hurricane Ike using all samples except Ike to train the regression. The azimuthally averaged DAV signal for each quadrant is plotted in color with values corresponding to the color bar on the right. The red dashed line is the observed 34-kt wind radii for that quadrant (km), the thick black dashed line displays the predicted R34 value (km) for that quadrant, and the thin black dashed line is the TC intensity (kt).

  • View in gallery

    As in Fig. 7, but for R50.

  • View in gallery

    As in Fig. 7, but for R64.

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    Comparison of the EBT winds (thick contours; R34, black; R50, blue; R64, green) with the asymmetric regression model (shading) for Hurricane Ike over a 7-day period for the 34-, 50-, and 64-kt wind radii: (a) 1800 UTC 5 Sep, (b) 1200 UTC 7 Sep, (c) 1200 UTC 10 Sep, and (d) 1800 UTC 12 Sep 2008.

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    As in Fig. 10, but for Hurricane Rita over a 6-day period for the 34-, 50-, and 64-kt wind radii: (a) 0006 UTC 19 Sep, (b) 0000 UTC 21 Sep, (c) 1200 UTC 22 Sep, and (d) 0000 UTC 24 Sep 2005.

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The Use of the Deviation Angle Variance Technique on Geostationary Satellite Imagery to Estimate Tropical Cyclone Size Parameters

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  • 1 Department of Chemistry, Physics, and Earth Science, Miami Dade College, Miami, Florida
  • | 2 School of Physical, Environmental, and Mathematical Sciences, University of New South Wales, Canberra, Australian Capital Territory, Australia
  • | 3 School of Engineering and Information Technology, University of New South Wales, Canberra, Australian Capital Territory, Australia
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Abstract

This study extends past research based on the deviation angle variance (DAV) technique that utilizes digital brightness temperatures from longwave infrared satellite images to objectively measure the symmetry of a tropical cyclone (TC). In previous work, the single-pixel DAV values were used as an objective estimator of storm intensity while maps of the DAV values indicated areas where tropical cyclogenesis was occurring. In this study the spatial information in the DAV maps is utilized along with information from the Cooperative Institute for Research in the Atmosphere’s extended best-track archive and the Statistical Hurricane Intensity Prediction Scheme model to create multiple linear regression models of wind radii parameters for TCs in the North Atlantic basin. These models are used to estimate both symmetric, and by quadrant, 34-, 50-, and 64-kt wind radii (where 1 kt = 0.51 m s−1 1) on a half-hourly time scale. The symmetric model assumes azimuthal symmetry and has mean absolute errors of 38.5, 23.2, and 13.5 km (20.8, 12.5, and 7.3 n mi) for the 34-, 50-, and 64-kt wind radii, respectively, which are lower than results for most other techniques except for those based on AMSU. The asymmetric model independently estimates radii in each quadrant and produces mean absolute errors for the wind radii that are generally highest in the northwest quadrant and lowest in the southwest quadrant similar to other techniques. However, as a percentage of the average wind radii from aircraft reconnaissance, all quadrants have similar errors.

Corresponding author address: E. Ritchie, School of Physical, Environmental, and Mathematical Sciences, University of New South Wales, P.O. Box 7916, Canberra BC 2610, Australia. E-mail: e.ritchie@adfa.edu.au

Abstract

This study extends past research based on the deviation angle variance (DAV) technique that utilizes digital brightness temperatures from longwave infrared satellite images to objectively measure the symmetry of a tropical cyclone (TC). In previous work, the single-pixel DAV values were used as an objective estimator of storm intensity while maps of the DAV values indicated areas where tropical cyclogenesis was occurring. In this study the spatial information in the DAV maps is utilized along with information from the Cooperative Institute for Research in the Atmosphere’s extended best-track archive and the Statistical Hurricane Intensity Prediction Scheme model to create multiple linear regression models of wind radii parameters for TCs in the North Atlantic basin. These models are used to estimate both symmetric, and by quadrant, 34-, 50-, and 64-kt wind radii (where 1 kt = 0.51 m s−1 1) on a half-hourly time scale. The symmetric model assumes azimuthal symmetry and has mean absolute errors of 38.5, 23.2, and 13.5 km (20.8, 12.5, and 7.3 n mi) for the 34-, 50-, and 64-kt wind radii, respectively, which are lower than results for most other techniques except for those based on AMSU. The asymmetric model independently estimates radii in each quadrant and produces mean absolute errors for the wind radii that are generally highest in the northwest quadrant and lowest in the southwest quadrant similar to other techniques. However, as a percentage of the average wind radii from aircraft reconnaissance, all quadrants have similar errors.

Corresponding author address: E. Ritchie, School of Physical, Environmental, and Mathematical Sciences, University of New South Wales, P.O. Box 7916, Canberra BC 2610, Australia. E-mail: e.ritchie@adfa.edu.au

1. Introduction

Larger storms may have increased storm surge, increased oceanic upwelling, a wider area in which flooding and damaging winds can occur (Irish et al. 2008; Price 1981), and a larger area of rainfall (Kidder et al. 2005; Matyas 2009). Thus, a robust and accurate depiction of the extent of damaging winds is important both to directly prepare for storm impacts and also to properly initialize numerical weather prediction (NWP) forecasts of future tropical cyclone (TC) motion, intensity, wind structure, and secondary models for wave fields and storm surge (Fiorino and Elsberry 1989; Fovell et al. 2009; Sampson et al. 2013). The size of a TC also has important implications for TC bogusing techniques. The insertion of a synthetic bogus vortex into a model is used extensively (Leslie and Holland 1995; Pu and Braun 2001; Kwon et al. 2002; Kwon and Cheong 2010), and knowledge of TC size in the data-sparse oceans would be useful in customizing the bogused vortex for individual forecasts (e.g., Davidson et al. 2014).

Operational centers, including the National Hurricane Center (NHC), the Joint Typhoon Warning Center (JTWC), and the Japan Meteorological Agency (JMA), among others, routinely provide forecasts of the wind field structure of active TCs within their areas of responsibility. Forecasts of the wind radii, representing the maximum radial extent of the 34-, 50-, and 64-kt winds (where 1 kt = 0.51 m s−1; hereafter referred to as R34, R50, and R64, respectively) in four quadrants circling the TC (northeast, southeast, southwest, and northwest quadrants), are routinely constructed.

At present, the estimates of wind radii analyses and forecasts are derived primarily from NWP predictions or statistical models (e.g., Knaff et al. 2007) and are augmented by any available observations. However, there are few objective, in situ, or remotely sensed data available to directly characterize the size and structure of the wind field over the oceans. Those data that are available (e.g., from buoys or satellite-based scatterometers) are nonuniform in time and space, presenting significant challenges to the forecaster who must produce wind radii forecasts at regular intervals. Furthermore, NWP forecasts of wind radii, although regular, and timed to meet forecaster warning requirements, are only slowly becoming skillful compared with climatology (Sampson and Knaff 2015). Climatology and persistence (CLIPER) models provide basic guidance and serve as a control for the verification of other techniques (Knaff and Landsea 1997; Aberson 1998; Knaff et al. 2003, 2007).

There have been recent advances in real-time data analysis capabilities that have made the wind radii forecast less subjective. In areas where aircraft reconnaissance is not routinely available, surface stations, buoys, and remotely sensed surface and near-surface winds [e.g., Quick Scatterometer (QuikSCAT), WindSat, Oceansat-2 (OSCAT), Advanced Scatterometer (ASCAT), and the Special Sensor Microwave Imager (SSM/I)] are routinely used (Halpern 1993; Rienecker et al. 1996; Mears et al. 2001; Ebuchi et al. 2002). The satellite microwave scatterometers perform best at lower wind speeds where precipitation is sparse (Zeng and Brown 1998; Weissman et al. 2002), and these observations are nonuniform in time and space. Another method of deducing the wind field is through the use of geostationary satellite cloud-track winds (Velden et al. 2005). This method can be used to calculate low-level winds when the upper-level cirrus shield is not obscuring the low-level clouds. These winds may then be reduced to surface values using estimations from Dunion et al. (2002) and Dunion and Velden (2002). Wind field analyses are also archived in all basins based on measurement from a combination of some of the previously mentioned satellite techniques (Kossin et al. 2007).

Currently, there are a few real-time, objective techniques for estimating wind radii. Mueller et al. (2006) used infrared satellite data, the radius of maximum wind, and the maximum sustained wind speeds to derive a multiple linear regression model to obtain both symmetric and asymmetric wind fields. Mean absolute errors (MAEs) for R34, R50, and R64 were not provided. Kossin et al. (2007) used a technique similar to that of Mueller et al. (2006) and obtained MAEs of 50.3, 41.4, and 29.7 km for R34, R50, and R64 respectively.

Knaff et al. (2011) used multiple satellite platforms to estimate the surface wind fields. Visual inspection of their Fig. 9 displays symmetric estimates of surface wind radii with MAEs of approximately 65, 33, and 24 km for R34, R50, and R64, respectively. Asymmetric values in each of the four quadrants are also displayed in their Fig. 9. Lower MAEs are found in the southwest quadrants and higher values in the other three quadrants when compared with symmetric values. This technique was updated in Knaff et al. (2015) to include a higher-resolution infrared (IR) image principle component analysis, which results in a better depiction of the inner-core wind field structure. However, R34, R50, and R64 error statistics were not reported in this study for comparison.

Another objective technique applies retrieval methods to the Advanced Microwave Sounding Unit (AMSU) (Demuth et al. 2004; Demuth et al. 2006) data, which can be used when the coverage of the polar-orbiting satellite swath is adequate over the TC. The AMSU has a resolution of approximately 48 km at nadir and can be used to retrieve a coarse warm-core structure. After using a variety of transformations of the original AMSU-derived parameters, the surface wind radii and other wind fields may be retrieved (Kidder et al. 2000; Knaff et al. 2000; Spencer and Braswell 2001; Brueske and Velden 2003; Knaff et al. 2004; Demuth et al. 2004; Bessho et al. 2006; Demuth et al. 2006). Demuth et al. (2006) used a multiple regression technique where the dependent data (i.e., maximum sustained winds, minimum pressure, wind radii) are from the Cooperative Institute for Research in the Atmosphere (CIRA) extended best-track and the independent data rely on AMSU-derived parameters of the retrieved pressure, winds, temperature, moisture, and satellite resolution. From this information they were able to derive the azimuthally averaged wind radii for R34, R50, and R64. Their regression model resulted in azimuthally averaged wind radii MAEs of 31.3, 24.6, and 12.6 km, respectively, compared to 6 yr of reconnaissance-based data. While these are perhaps the best results of all the methods that currently exist, the two limitations of this method are the low resolution of the AMSU measurements, which probably limits the accuracy at R50 and R64, and the low sampling of tropical cyclones by AMSU.

Finally, Knaff et al. (2016) describe a technique that fits a rankine vortex to operational estimates of the TC location, intensity, motion, and overall size as estimated by the radius of 5-kt winds. The R34, R50, and R64 results are then extracted from the fitted wind field. They report larger errors (MAE = 54, 32, and 20 km for R34, R50, and R64, respectively) than for the satellite-based techniques for an independent test using the 2012–14 hurricane seasons. However, the advantage of this system is that an estimate of the wind parameters can be provided to forecasters very early in the forecast cycle.

Clearly, there are several techniques currently available for estimating wind radii, each with their own strengths and weaknesses. Techniques that use measurements from polar-orbiting satellites such as the AMSU approaches lack consistency since the TC is not sampled regularly. Techniques that utilize geostationary satellites have the advantage of regular sampling; however, the physical relationships that tie the structure of the radiances at the top of the cloud field to the surface TC wind field are not as clear. Some techniques may be more accurate at certain wind radii than other techniques, and some will have biases that offset others. Ultimately, it may be that an ensemble of different techniques will provide the most accurate estimates of storm size in a manner similar to the satellite consensus (SATCON) technique for objective TC intensity estimation (Velden and Herndon 2014). Here, we present an independent technique that utilizes IR imagery to estimate the surface wind radii and construct a two-dimensional surface wind field.

The current study extends research based on the deviation angle variance (DAV) technique (Piñeros et al. 2008), which utilizes digital brightness temperatures from IR satellite images to objectively measure the symmetry of a TC solely based on a comparison of the gradient vectors of brightness temperatures from an actual TC with the gradient vectors of an ideal, symmetric vortex. The DAV technique has already been utilized to estimate TC intensity, objectively track cloud clusters, and as a means of identifying cyclogenesis in the Atlantic, eastern North Pacific, and western North Pacific basins (Ritchie et al. 2012, 2014; Wood et al. 2015) Since the gradient of the brightness temperatures is a measure of the cloud structure, and therefore indicative of the surface forcing for cloud formation, there is potentially information contained within the spatial structure of the DAV maps currently created for genesis and cloud cluster tracking that directly infers the wind field structure at the surface. For example, Fig. 1 shows an example for Hurricane Wilma (2005) with the extended best-track R34 by quadrant overlaid. There is an indication of a correlation between the radial extent of the lower values of DAV (<2250 deg2) and the R34; that is, the farther extension of the lower DAV values corresponds to a larger R34. In this study, the spatial–temporal structure of the DAV signal for TCs is utilized (Piñeros et al. 2010) along with wind radii from the CIRA extended best-track (EBT) dataset (Demuth et al. 2006) and information from the Statistical Hurricane Intensity Prediction Scheme (SHIPS) model (DeMaria and Kaplan 1994, 1999; DeMaria et al. 2005) to develop a multiple linear regression technique that objectively calculates the wind radii for a given TC on a half-hourly basis. Models are developed that separately predict the symmetric and quadrant components of the wind radii and provide objective wind field estimates in real time that may be used by forecasters as well as for NWP model initialization. The estimated wind radii are then tested against NHC best-track wind radii and error statistics are calculated for all estimates.

Fig. 1.
Fig. 1.

Hurricane Wilma at 0015 UTC 21 Oct 2005. (a) IR image. The current best-track intensity is 67 m s−1 and 924 hPa. [The square shows the area that is magnified in (c) and (d).] (b) The corresponding map of DAV values (deg2). The minimum value in the map of variances (1189 deg2) corresponds to the location 19.17°N, 85.63°W. The center of the storm reported by the NHC was located at 19.1°N and 85.8°W. (c),(d) As in (a),(b), respectively, but magnified and with the CIRA extended best-track R34 by quadrant overlaid.

Citation: Weather and Forecasting 31, 5; 10.1175/WAF-D-16-0056.1

The structure of the paper is as follows. In section 2 the data and methodology used to estimate wind radii are presented. Results using the regression to extract both the symmetric and asymmetric components of the wind field are presented and discussed in section 3. A summary and conclusions are provided in section 4.

2. Data and methodology

a. Preprocessing of brightness temperatures and tropical cyclone selection

The data used for this study are digital brightness temperatures from longwave (10.7 μm) IR satellite images. Approximately 4700 half-hourly images from the Geostationary Operational Environment Satellite-12 (GOES-12) imager are processed from 21 TCs for the years 2004–10 in the North Atlantic basin. Tropical cyclones are chosen based on the existence of aircraft reconnaissance data within 3 h of an NHC best-track time. For the purposes of the regression, each case required at least 48 h of semicontinuous aircraft reconnaissance, so they are confined to the Atlantic basin west of 55°W where aircraft reconnaissance is routinely performed. Occasionally, aircraft will fly storms in the eastern or central North Pacific, but only at the discretion of the NHC and Central Pacific Hurricane Center (CPHC). Aircraft-reconnaissance data were collected from low-altitude flights. The National Oceanic and Atmospheric Administration (NOAA) WP-3D and the U.S. Air Force WC-130 typically fly radial legs in and out of the core in a figure four pattern. Most of the flights were conducted at an altitude of 3 km, and the wind was measured by inertial and GPS navigation systems. Flight-level winds were then reduced to surface values using empirical relationships (Franklin et al. 2003) and used in the postanalysis of the NHC best-track wind radii (since 2004). Table 1 provides a list of the TCs, the dates for which reconnaissance was available for each TC, and the maximum intensity that each storm reached for the period of study. Tropical cyclones from the years 2004–10 in the North Atlantic are used because of the abundance of TCs with reconnaissance during this period, especially in the years 2004–05 when there were a large number of TCs within the vicinity of the continental United States. This also coincides with the period when NHC began including wind radii information in the best-track archive.

Table 1.

All TCs used in the study along with the dates sampled by aircraft reconnaissance. The third and fourth columns list maximum wind (kt) and minimum sea level pressure (hPa), respectively, achieved by each TC.

Table 1.

Piñeros et al. (2008) developed an automated system based on the variance in the orientation angle of the brightness temperature gradient calculated from geostationary infrared imagery—the deviation angle variance or DAV—to infer TC intensity. For the intensity estimation, the DAV is calculated over a training period using a nine-point average at the center of the vortex. These DAV values are used to build a parametric curve that relates DAV values to wind intensity for current or predicted IR images. Testing in the North Atlantic basin has resulted in a root-mean-square error of 13.9 kt compared to the best track (Ritchie et al. 2012) with comparable performances also demonstrated in the eastern and western North Pacific basins (Ritchie et al. 2014).

The single-point DAV calculation for intensity estimation was extended in Piñeros et al. (2010) to detect developing disturbances by using each pixel in the satellite image in turn as the reference center, calculate the DAV, and then map that value back to the reference pixel, creating a “map of variances” that corresponds directly to the original infrared image (e.g., Fig. 1). Lower values of DAV indicate higher levels of axisymmetry, and these areas are most likely to be found near the center of an already existing disturbance. In this study, we calculate the map of variances for each half-hourly IR image in the dataset using a radius of 500 km for the calculation, which was determined to provide the best relationship with the wind radii. For more information on the DAV and its use in TC estimation, see Piñeros et al. (2008, 2010, 2011).

Once the map of the variances is obtained, the DAV signal is azimuthally averaged in 10-km increments to a radial distance of 600 km using interpolated NHC best-track centers at half-hourly resolution to match the resolution of the DAV maps. Interpolation of the best-track centers may not provide an exact estimate of the TC location at off-synoptic times. However, TC locations matching the time of the imagery were needed and the slight error introduced by this methodology is likely to only slightly affect the results because of the azimuthal averaging. The DAV signal has high temporal variation (e.g., Piñeros et al. 2008) whereas the best-track wind radii vary smoothly. Therefore, various filters are tested to determine which filtered DAV signal has the best correlation with the wind radii. A 24-h past-running mean is found to have the best correlation with the best-track wind radii and is applied to the signal to smooth the fields. For the first 24 h, the satellite images are averaged until 24 h has occurred, and then the running mean moves with each image. This may be the source of some of the initial error in the signal as the DAV signal fluctuates more in the first few hours, until the filter is properly established. Using this method, smoothed spatiotemporal maps are produced for each of the 21 TCs.

Finally, information from the CIRA extended best-track file, which provides the same quadrant wind radii information as the NHC best-track archive (post 2004), and the SHIPS model is used to conduct the regression analysis. For the purposes of this study, the symmetric wind radii are calculated by averaging the quadrant values for each radius including zeros. Finally, in order to match the 6-hourly SHIPS and best-track information with the temporal resolution of the satellite images, these data are linearly interpolated to half-hour intervals.

The TC wind field is known to be disrupted as the TC moves over land (Powell 1982, 1987; Powell et al. 1991; Powell and Houston 1996, 1998; Wong and Chan 2007). Proximity to land (including islands) is tested in the regression analysis with minimal impact. The root-mean-square error (RMSE) improves by <1 (0.5) km when TCs within 50 (100) km of land are ignored. Because of its small impact, the land mask is not used in this study.

b. Multiple linear regression and symmetric analysis

A multiple linear regression model is used to develop a relationship among the axisymmetric DAV signal, variables from the EBT data, and environmental parameters from the SHIPS model. The prediction equation for the multiple linear regression model is
e1
where b0 is the intercept or the regression coefficient, x is a predictor variable, and each of the k predictor variables has its own coefficient bk, analogous to the slope of a line (Wilks 2006, 179–252). Potential predictors aside from the DAV value are obtained from the SHIPS model and best-track data. These included parameters that past research has found may be related to TC size, such as higher latitudes corresponding to larger storms; sea surface temperatures (SSTs); the maximum wind speed (Vmax); relative humidity in the lower, middle, and upper troposphere; and storm age, calculated as the time since the TC became a tropical storm (Shea and Gray 1973; Merrill 1984; Weatherford and Gray 1988; Cocks and Gray 2002; Kimball and Mulekar 2004; Kimball 2006; Hill and Lackmann 2009; Wang 2009).

The preliminary step in the method compares each TC with the azimuthally averaged DAV signal for each radius along with the symmetric component of the EBT wind radii using the EBT centers (Fig. 2). After all the data are processed, the DAV signals with the lowest RMSE corresponding to each of the three wind radii are used in the regression model. The next step consists of a screening regression known as forward selection (Wilks 2006, 179–252), where each of the predictors is numerically tested for the strength of their linear relationship to the predictand. After the predictor with the best linear relationship to the predictand is selected, the process is repeated with the remaining variables. This continues until the predictors display diminished improvements within the regression equation. Once additional predictors improve the regression equation by less than 1%, they are ignored. For the symmetric wind field model, TC age, SHIPS SST, and intensity (R64) were the most valuable predictands after DAV.

Fig. 2.
Fig. 2.

Hovmöller diagram of the azimuthally averaged DAV signal (shading) for Hurricane Gustav with the observed wind radii and regression overlaid for (a) 34-, (b) 50-, and (c) 64-kt winds. The red dashed line is the symmetric observed wind radii (km), the thick black dashed line displays the regression line (km), and the thin black dashed line displays the intensity of the TC (kt).

Citation: Weather and Forecasting 31, 5; 10.1175/WAF-D-16-0056.1

c. Asymmetric components of the wind radii

The asymmetric model is similar in principle to the symmetric model for wind radii. In this method the DAV is azimuthally averaged in each of the four quadrants (northeast, southeast, southwest, and northwest) at each radius at 10-km increments. The multiple linear regression procedure, as described in the previous section, is then carried out on the individual quadrant DAVs. The same regression parameters (TC age, SSTs, and intensity) are found to produce the best results.

d. Statistical analysis using cross validation

The large sample size (4708 half-hourly satellite images) does allow for the statistical testing of the data. A cross-validation method is used to investigate the forecast precision of the wind radii. In the cross-validation exercise, 80% of the data are used to create the multiple linear regression parameters, and the linear regression model is then tested on the remaining 20% of the data. This method is applied 5 times by leaving out successive 20% increments of data and using the other 80% to create the multiple linear regression models. Two tests were performed. First, the data were left in temporal order and successive 20% pieces were withheld from the training and used for testing. In this case considerable variability was found in the results because although there are a large number of samples in the dataset, they represent relatively few TCs (21), and it is possible for an individual TC to dominate the statistics in a cross-validation exercise. In the second test, the data are first randomized to ensure that no single TC dominates the statistics. The method is used for both the symmetric and asymmetric calculations of the different wind radii. Both RMSEs and MAEs for the symmetric and four quadrants of the asymmetric model are calculated, although to rationalize the data provided generally only the MAE is presented.

3. Results

a. The symmetric regression model

The TCs listed in Table 1 represent a wide range of intensities ranging from tropical storm (TS) through category 5 on the Saffir–Simpson scale. The mean intensity of the sample set used in the study after linear interpolation is 91 kt. Table 2 displays the wind radii, the RMSEs, the MAEs, and the best linear predictors. For the R50 wind radii, the sample size is slightly smaller because the 50-kt values were missing for TCs Paula and Richard in our dataset even though they were greater than 64 kt in intensity and did have 64-kt wind radii recorded.

Table 2.

Results from the symmetric multiple linear regression models. Regression results for all TCs are in boldface. Regression results for hurricane intensity and above are in italics. Numbers of samples in each category and the mean intensity of the sample are provided in parentheses.

Table 2.

Typically, for the symmetric regression, adding more than two predictors yielded negligible returns, and thus only the top two after DAV were used. The DAV value that correlates best with the R34 accounts for 86.4% of the signal in the multiple linear regressions. The only two predictors that improve the DAV–R34 relationship are TC age and SST, which is evaluated at the center location of the TC in the SHIPS model. TC age shows the largest improvement, explaining 12% of the variance and a positive linear relationship to the wind radii, confirming past research that, as a storm matures, the wind field tends to increase in size (Cocks and Gray 2002). The SSTs display negative correlation with the R34 wind radii and explain only 1.6% of the variance. It is often observed that as a stronger TC moves over cooler SSTs, the TC weakens and the wind field spreads out, especially during extratropical transition (Brand and Guard 1979; Evans and Hart 2008), although direct sensitivity to latitude was negligible. The MAE for the R34 wind radii training and testing over the entire dataset is 38.5 km, representing an average error of 18.1% when compared to the extended best-track values. The R2 value is 0.72 and the p values have confidence levels above 99.9%, indicating a strong relationship between the predictors and the predictand. For the subset of instances where the TC is greater than or equal to hurricane intensity, the DAV also accounts for the majority of the signal, with SST and TC age accounting for rapidly diminishing yields. The hurricane subset produces a slightly lower MAE of 35.7 km.

The R50 predictors display similar results to the R34 predictors, with the DAV accounting for 80% of the variance. TC age improves the model a large amount when compared with its contribution to the R34 model, explaining 18.2% of the variance, with SSTs again contributing a small amount (1.9%). In contrast with the R34 hurricane intensity and above subset, all the predictors for the R50 hurricane subset separately improve the relationship. However, TC age and SSTs improve the model by the greatest amount. The model is improved by less than 1% by the other predictors. The MAE for the R50 wind radii training and testing over the entire dataset is 23.2 km, representing an average error of 20% when compare to the extended best-track values with an R2 value of 0.79 and p values with confidence levels above 99.9%. The subset of instances where the TC is at hurricane intensity and above produces a slightly reduced MAE of 21.8 km.

The R64 predictors exhibit large contrasts to the R34 and R50 predictors with every predictor individually increasing the relationship between the DAV signal and the R64 wind radii. The top three predictors are DAV (60%), TC age (20%), and the maximum sustained winds (20%). Similar to the R50 hurricane subset, additional predictors improve the model by less than 1%. The MAE for the R64 wind radii training and testing over the entire dataset is 13.5 km, representing an average error of 23.7% when compare to the extended best-track values with an R2 value of 0.84 and p values with confidence levels above 99.9%. We note that the current model has no a priori knowledge of the maximum intensity of the tropical cyclone at the time of the wind radii prediction. Thus, there will be some error associated with the cases where the model predicts a positive R64 even though the TC intensity is less than 64 kt. Such results would be filtered out in an operational system but are included in the statistics shown here. The subset of instances where the TC is at hurricane intensity and above produces a slightly increased MAE of 14.8 km, probably because of the very small sample size at these intensities.

The symmetric components of the R34, R50, and R64, respectively, with time are shown in Fig. 2 for Hurricane Gustav (2008). Hurricane Gustav is a good example of a TC that varies in both intensity and the size of the wind radii through its sample period, and is indicative of the performance obtained with this method. In each of the panels, the DAV signal (shading) closely mimics the shape of the wind field (red dashed line). The added predictors increase this accuracy and the final regressed estimation of the wind field radii (black dashed line) shows good agreement with the EBT values.

Both the randomized and ordered cross-validation independent testing results are shown in Fig. 3 and Table 3. These results demonstrate how in general the dependent test MAE value lies within the range of the independent testing MAE values. There is more variability in the cross-validation errors when the samples are kept in temporal order; that is, some samples produce much better results and some produce worse results. The randomization removes this potential bias. R34 MAEs range from 32.0 to 48.5 km for the ordered sample and from 36.8 to 39.9 km for the randomized sample with an average that is close to the overall sample results (Table 3). The R50 MAEs range from 20.4 to 30.0 km for the ordered sample and from 22.4 to 23.6 km for the randomized sample with an average MAE that is again close to the overall sample results. The R64 MAEs range from 12.8 to 16.7 km for the ordered sample and from 12.9 to 14 km for the randomized sample with an average, which is again very close to the MAE for the dependent testing. The R2 values steadily increase from R34 to R64, and p values of each subset tested for all wind radii display confidence levels above 99.9% (not shown). It is not surprising that the randomization of the sample resulted in errors that were more constrained and very similar to the overall results. The ordered sample results are likely more representative of what might be expected for individual years as particular conditions may bias particular years or portions of years. In general, the results suggest that our model is consistent and produces similar results when forecasting for subsets of the data for each wind radius. Furthermore, the results from our technique compare favorably with those from the other techniques cited in the literature (Table 4).

Fig. 3.
Fig. 3.

MAEs for the R34, R50, and R64 wind radii using the symmetric model for the case employing all TCs, and the five bins tested during the cross-validation analysis.

Citation: Weather and Forecasting 31, 5; 10.1175/WAF-D-16-0056.1

Table 3.

MAE cross-validation results for all three symmetric wind radii. Values provided include the results for the ordered and randomized samples. Total numbers of samples for each radius are as in Table 2. Each cross-validation sample is one-fifth the total sample.

Table 3.
Table 4.

MAEs for those studies cited from the literature.

Table 4.

To test whether there is dependence in the sample to the intensity of the tropical cyclone independent of size, and size of the tropical cyclone independent of intensity, the dataset was divided first into the intensity categories of tropical storm and then category 1–5 hurricanes, and the MAE was calculated for each bin. Next, the dataset was divided by size into R34 less than 125 km, R34 between 125 and 215 km, between 215 and 305 km, between 305 and 395 km, and greater than 395 km, and again, the MAE was calculated for each bin. The results are shown in Table 5. When MAEs are binned by intensity categories, there does not appear to be any consistent pattern with increasing intensity and the individual MAEs are within ±6 km of the overall MAE. For the size bins, MAEs are relatively low for TCs with R34 < 300 km, which includes approximately 86% of the dataset. For the largest tropical cyclones it is clear that the technique systematically underpredicts the R34. This is partly because of the smaller number of samples at this very large size. We are working on strategies to improve the technique at the large end of the scale.

Table 5.

Mean absolute errors for the predicted symmetric R34 by TC size and TC intensity. The number of samples in each bin is indicated in parentheses.

Table 5.

Finally, the bias for each radius for different hurricane categories is shown in Fig. 4. In general, biases are higher and positive (large bias) for tropical storms and category 1 hurricanes. Note the positive bias for the tropical storm bin for the R64. This is mostly due to the lack of filtering in the model of wind radii values for TCs of lower intensity than that threshold. Such results would be filtered out in an operational system but are included in the statistics calculated here for completeness. Biases are negative (small wind radii bias) for category 2 and higher hurricanes. The rather large and positive bias for R64 in the category 5 bin is almost certainly because of the small number of cases at that intensity category. However, in general, the biases are relatively small (<20 km).

Fig. 4.
Fig. 4.

The bias (km) for each radius for different maximum wind speed bins (kt) over all cases for the symmetric model. Numbers of samples for R34 and R64 in increasing intensity bin order are 1120, 839, 674, 668, 1126, and 281, respectively. Numbers of samples for R50 in increasing intensity bin order are 963, 738, 630, 668, 1126, and 281, respectively.

Citation: Weather and Forecasting 31, 5; 10.1175/WAF-D-16-0056.1

b. The asymmetric regression model

The asymmetric regression model predicts the wind radii in each of the four quadrants separately (Fig. 5). As mentioned above, the analysis utilizes the azimuthally averaged DAV within each TC quadrant and develops separate regression equations for each quadrant. For each of the wind radii, the northeast and northwest quadrants have the highest MAEs, which is not surprising since these quadrants have the largest wind radii on average. When taken as a percentage of the average wind radius in each quadrant, they typically have lower relative errors than the other quadrants. The southwest quadrant consistently has the lowest MAEs, likely an artifact of the general motion of TCs to the west and north and the smaller average wind radii in that quadrant in the North Atlantic basin.

Fig. 5.
Fig. 5.

The MAE (km) for (a) northwest, (b) northeast, (c) southwest, and (d) southeast quadrants of the R34, R50, and R64 wind radii for all TCs and for the five bins tested during the cross-validation analysis. The sample sizes and mean intensity are the same as those reported in Table 2 for the symmetric model. Note that the sample size for each bin of the cross validation is one-fifth of the total.

Citation: Weather and Forecasting 31, 5; 10.1175/WAF-D-16-0056.1

The R34 wind radius has MAEs of 51.3, 46.7, 36.9, and 55.7 km for the northeast, southeast, southwest, and northwest quadrants, respectively. The R50 wind radius has MAEs of 32.3, 30.9, 22.8, and 33.5 km while the R64 wind radius has MAEs of 19.1, 16.5, 12.8, and 16.9 km for the same quadrants, respectively (Fig. 5). The R2 values are above 0.53 for all wind radii and quadrants. Confidence levels are above the 99.9% level in all cases as well.

The MAEs of the asymmetric wind radii for the hurricane intensity and the above subset are similar to those of the entire set. They are slightly lower for the R34 and R50 wind radii and slightly larger for the R64 wind radius (not shown). As with the entire set, the southwest quadrant consistently has the lowest MAEs and the northeast and northwest quadrants have the highest. The R2 values range from 0.40 to 0.67 for all wind radii and quadrants, and confidence levels are above the 99.9% level for all quadrants.

Finally, the bias for each quadrant and each wind radii for different intensity categories is shown in Fig. 6. In general, biases are higher and positive (large bias) for lower-intensity TCs. Biases go negative (small wind radii bias) for category 2 and higher and then are generally small and positive for the category 5 hurricanes, most likely because of the small number of cases at that intensity. However, in general, the biases are small (<10 km) for each R64 quadrant.

Fig. 6.
Fig. 6.

The bias (km) for each radius at different maximum wind speeds over all cases for each quadrant of the asymmetric model. Numbers of samples in each quadrant and for each radius are as in Fig. 4.

Citation: Weather and Forecasting 31, 5; 10.1175/WAF-D-16-0056.1

Similar to the symmetric model, cross-validation testing is done for each wind radii in each quadrant by withholding 20% of the samples, training with the remaining 80%, and then testing on the withheld 20% of the samples. The cross-validation testing produces MAEs, which are close to, and envelope, the MAE for all samples for R34 and R50. In the case of R64, the cross-validation MAEs are consistently higher by 3–7 km on average than the MAEs for the whole sample (Fig. 5). This is possibly because of the higher number of zeros in this dataset compared with the results for R34 and R50, which could be problematic when subsetting the sample by one-fifth. The R2 values for the cross validation of the asymmetric regression all have confidence levels above 99.9%.

As an added test to investigate the ability of the asymmetric regression model to estimate wind radii for TCs, the data from TC Ike are withheld, and the regression parameters are derived from the remaining TCs. TC Ike remained above hurricane intensity for the entire observation period; therefore, the regression model of hurricane intensity and the above subset is used. As displayed in Figs. 7, 8, and 9 for R34, R50, and R64, respectively, the regression is able to model the general shape and distance of each wind radius in each quadrant, though the observed wind radii can display some large swings while the regression model tends to have a smoother shape to it (e.g., Fig. 7, southeast quadrant). The wind fields from the Ike test were created so the asymmetric regression model may be compared to the best-track archive wind radii. Figure 10 displays four different times throughout the 6–7-day period representing the beginning, middle, and end of Ike’s evolution. The comparison demonstrates that the regression model is able to estimate the general size, shape, and size growth of the vortex throughout the time period, albeit with some errors. In a similar test the data from Hurricane Rita are withheld and the regression parameters are derived from the other TCs in the dataset (Fig. 11). The test again shows that the asymmetric regression captures the general size, and period of growth of the TC, although during the second half of the time period R34 from the best-track data is noticeably larger than that computed by the regression.

Fig. 7.
Fig. 7.

Prediction of the R34 by quadrant for Hurricane Ike using all samples except Ike to train the regression. The azimuthally averaged DAV signal for each quadrant is plotted in color with values corresponding to the color bar on the right. The red dashed line is the observed 34-kt wind radii for that quadrant (km), the thick black dashed line displays the predicted R34 value (km) for that quadrant, and the thin black dashed line is the TC intensity (kt).

Citation: Weather and Forecasting 31, 5; 10.1175/WAF-D-16-0056.1

Fig. 8.
Fig. 8.

As in Fig. 7, but for R50.

Citation: Weather and Forecasting 31, 5; 10.1175/WAF-D-16-0056.1

Fig. 9.
Fig. 9.

As in Fig. 7, but for R64.

Citation: Weather and Forecasting 31, 5; 10.1175/WAF-D-16-0056.1

Fig. 10.
Fig. 10.

Comparison of the EBT winds (thick contours; R34, black; R50, blue; R64, green) with the asymmetric regression model (shading) for Hurricane Ike over a 7-day period for the 34-, 50-, and 64-kt wind radii: (a) 1800 UTC 5 Sep, (b) 1200 UTC 7 Sep, (c) 1200 UTC 10 Sep, and (d) 1800 UTC 12 Sep 2008.

Citation: Weather and Forecasting 31, 5; 10.1175/WAF-D-16-0056.1

Fig. 11.
Fig. 11.

As in Fig. 10, but for Hurricane Rita over a 6-day period for the 34-, 50-, and 64-kt wind radii: (a) 0006 UTC 19 Sep, (b) 0000 UTC 21 Sep, (c) 1200 UTC 22 Sep, and (d) 0000 UTC 24 Sep 2005.

Citation: Weather and Forecasting 31, 5; 10.1175/WAF-D-16-0056.1

4. Discussion and conclusions

Multiple linear regression models for the symmetric, and quadrant, 34-, 50-, and 64-kt wind radii were developed and tested using 21 TCs from the years 2004–10, a period that had extensive in situ observations. Approximately 4700 half-hourly infrared images from GOES-12 were used to create DAV maps, which provide information on the symmetric organization of a tropical system. These maps were then azimuthally averaged about the TC center and used to develop two multiple linear regression models for TC wind radii. The symmetric model utilizes the full azimuthally averaged DAV along with information from the best-track archive and the SHIPS model, to estimate the symmetric 34-, 50-, and 64-kt wind radii. The asymmetric model utilizes the azimuthally averaged DAV by quadrant, along with information from the best-track archive and the SHIPS model to estimate the wind radii by quadrant.

Training and testing on the same set of data (dependent testing) produces MAEs of 38.5, 23.2, and 13.5 km (20.8, 12.5, and 7.3 n mi) for R34, R50, and R64, respectively, for the symmetric model. The errors using a cross-validation technique to produce independent testing datasets were consistent with these results. Confidence levels are above 99.9% for all cases presented here. The asymmetric (by quadrant) model produced MAEs for R34 in the northeast, southeast, southwest, and northwest quadrants of 51.3, 46.7, 36.9, and 55.7 km (27.7, 25.2, 19.9, and 30.1 n mi), respectively; for R50 of 32.3, 30.9, 22.8, and 33.5 km (17.5, 16.7, 12.3, and 18.1 n mi) respectively; and for R64 of 19.1, 16.5, 12.8, and 16.9 km (10.3, 8.9, 6.9, and 9.1 n mi), respectively. The asymmetric regression model using the subset of data for hurricane intensity and above only has similar MAEs with slightly lower errors for the R50 (not shown).

The use of the best-track archive as the standard for this study may place some limitations on how well the true wind radii can be determined. The dataset used in the study comprises only periods that were observed by aircraft reconnaissance within a 3-h period of a best-track recording and that were also semicontinuous (at least 48 h of contiguous samples). Another factor is that aircraft reconnaissance is often focused on the inner radii, especially for large TCs (Moyer et al. 2007), and thus may not always measure the R34. In addition, in recent years, direct measurements of the surface winds have been made using the Stepped Frequency Microwave Radiometer (SFMR; Brennan et al. 2009). However, prior to 2007, the surface winds were not directly measured by the WC-130 aircraft; rather, they were inferred from flight level. Consequently, while the dataset used in this study for both training and validation represents one of the very best wind radii datasets that is currently available, there is some inherent uncertainty that will introduce uncertainty into the technique. Furthermore, because the dataset is focused on cases west of 55°W in order to ensure semicontinuous reconnaissance, the data are skewed toward more intense tropical cyclones compared with climatology. Knaff and Sampson (2015) find that objective techniques tend to perform better on more intense tropical cyclones. While our binning by intensity category does not seem to show this, it may be that our errors will increase if we test on the entirety of the best-track dataset both because this will include more weaker systems and because the entire best-track wind radii dataset most likely has more uncertainty than the selection used here.

The technique presented here for measuring the wind radii is objective and, because it uses the Geostationary Operational Environment Satellite imagery as its primary predictor, wind radii estimates can be made every half hour although it is likely that smoothing the data to a 3- or 6-h interval will be of the most use operationally. In the future the regression model will be extended to those ocean basins where there is no aircraft reconnaissance so that the size and strength of TC wind fields can be estimated on a regular basis. Validation will be more problematic in these basins and extensive use of data sources such as scatterometer data will be necessary to validate the models in those cases. Furthermore, while we have developed specifically symmetric and quadrant models to best match with the best-track archive and operational needs, the DAV map includes finer-scale structure that can be used to develop a more realistic wind field. This is the focus of current work. Finally, we hope that a future extension of this work will include the development of synthetic vortices created with this regression model to generate a more realistic structure for the bogus vortex in NWP models, which may increase the accuracy of the track, intensity, rainfall, and surge forecasts.

Acknowledgments

Much of the research in this paper was accomplished while all three authors were at The University of Arizona. Extended best-track and SHIPS development data were provided by the Cooperative Institute for Research in the Atmosphere. The NHC Best-Track archive (HURDAT2) is made available by the National Hurricane Center. GOES infrared imagery was obtained from The University of Arizona’s Department of Atmospheric Science archive. The paper has been improved by the thoughtful comments of C. Velden, J. Knaff, and an anonymous reviewer. Funding for this work was partially provided by the U.S. Office of Naval Research under Grants N00014-10-1-0416 and N00014-13-1-0365.

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