## Abstract

The ground-based velocity track display (GBVTD) was developed to deduce a three-dimensional primary circulation of landfalling tropical cyclones from single-Doppler radar data. However, the cross-beam component of the mean wind cannot be resolved and is consequently aliased into the retrieved axisymmetric tangential wind . Recently, the development of the hurricane volume velocity processing method (HVVP) enabled the independent estimation of ; however, HVVP is potentially limited by the unknown accuracy of empirical assumptions used to deduce the modified Rankine-combined vortex exponent . By combing the GBVTD with HVVP techniques, this study proposes a modified GBVTD method (MGBVTD) to objectively deduce from the GBVTD technique and provide a more accurate estimation of and via an iterative procedure to reach converged and cross-beam component of solutions. MGBVTD retains the strength of both algorithms but avoids their weaknesses. The results from idealized experiments demonstrate that the MGBVTD-retrieved cross-beam component of is within 2 m s^{−1} of reality. MGBVTD was applied to Hurricane Bret (1999) whose inner core was captured simultaneously by two Weather Surveillance Radar-1988 Doppler (WSR-88D) instruments. The MGBVTD-retrieved cross-beam component of from single-Doppler radar data is very close to that from dual-Doppler radar synthesis using extended GBVTD (EGBVTD); their difference is less than 2 m s^{−1}. The mean difference in the MGBVTD-retrieved from the two radars is ~2 m s^{−1}, which is significantly smaller than that resolved in GBVTD retrievals (~5 m s^{−1}).

## 1. Introduction

A landfalling tropical cyclone (TC) is one of the most devastating and deadly natural disasters along coastal regions of many countries. Accurately monitoring inner-core structure and its evolution before and after landfall is crucial for the protection of life and property. Doppler weather radar is the only platform that can capture the three-dimensional (3D) structure of landfalling TCs with high spatial (~1 km) and temporal (~6 min) resolutions. Donaldson (1970) found that a vortex produces a Doppler velocity dipole signature with opposite parity in a plan position indicator (PPI) mode. Based on the location and magnitude of this Doppler velocity dipole, Wood and Brown (1992) developed a pattern-recognition algorithm to estimate the three critical characteristics of vortex structure, including the center, the radius of maximum wind , and the maximum wind speed. However, this method cannot quantitatively provide the detailed 3D circulation of a TC.

Lee et al. (1994) proposed a robust single-Doppler wind retrieval technique, called the velocity track display (VTD), to deduce the primary circulations of TCs at different altitudes in real time from an airborne tail Doppler radar on board the National Oceanic Atmospheric Administration WP-3D aircraft. To study the landfalling TCs using coastal radars, Lee et al. (1999) reformulated the VTD equations for a ground-based Doppler radar, called the ground-based VTD (GBVTD). Recently, successful applications of GBVTD to several landfalling TCs (Lee et al. 2000; Lee and Bell 2007; Zhao et al. 2008) have demonstrated its ability in monitoring and warning. A series of GBVTD extensions—including ground-based extended VTD (GB-EVTD; Roux et al. 2004), extended GBVTD (EGBVTD; Liou et al. 2006), generalized VTD (GVTD; Jou et al. 2008), and gradient VTD (GrVTD; Wang et al. 2012)—expanded the GBVTD analysis into multiple flight legs for airborne Doppler radar, multiple ground-based Doppler radars and direct use of aliased radial velocity data, and so on. However, the cross-beam component of the mean wind is neglected in all of the aforementioned VTD family of technique and is consequently aliased into the retrieved axisymmetric tangential wind .

Several methods have been developed to estimate the independently, including the hurricane volume velocity processing method (HVVP; Harasti 2003), using the storm motion as a proxy of mean wind (Harasti et al. 2004), extended-HVVP method (EHVVP; Zhu et al. 2010), and EGBVTD (Liou et al. 2006). Among these existing methods, the use of storm motion as a proxy is the easiest to implement, but it assumes that the mean wind vectors are not a function of height, which is not realistic. The EGBVTD technique can provide a relatively accurate estimation of but requires dual- or multiple-Doppler radar observations of TCs that are rarely available because of the typically long baseline between operational Doppler radars. In comparison, HVVP is attractive because it estimates at different altitudes using only single-Doppler radar data and an empirical modified-Rankine tangential wind profile (referred to as the Rankine profile). The successful applications of HVVP to several real TCs (Harasti 2003; Harasti et al. 2007) have shown its potential for the operational use. Recently, HVVP has been successfully incorporated, together with GBVTD, into the vortex objective radar tracking and circulation (VORTRAC) software for real-time analysis of TCs at the National Hurricane Center of the United States. Despite these encouraging results, the HVVP technique is potentially limited by its fundamental assumption: the modified Rankine-combined vortex exponent is required to separate from the TC circulations using empirical equations. Hence, the HVVP-retrieved may contain large uncertainties resulting from the deviation of the HVVP-estimated and the true . For EHVVP, it is limited by its strict requirement of a Rankine-combined vortex.

In this paper, an improved method, named the modified GBVTD (MGBVTD), is proposed to retrieve the TC mean wind vectors and the primary circulations simultaneously by combining the strength of the HVVP and GBVTD to yield a more realistic TC circulation. Section 2 describes the mathematical formulations and details of MGBVTD. A series of analytical datasets based on a modified Rankine-combined vortex are employed to evaluate the performance of MGBVTD technique in different situations in section 3. In section 4, the MGBVTD is applied to Hurricane Bret (1999) observed simultaneously by two coastal Weather Surveillance Radar-1988 Doppler (WSR-88D) instruments and its retrieved winds are compared with those deduced by EGBVTD. A summary and discussion are given in section 5.

## 2. The MGBVTD method

As MGBVTD combines GBVTD and HVVP, these two methods are summarized below. For simplicity, a unified coordinate system is employed on both methods to make the description more consistent. The acronyms and symbols used in this paper are listed in Tables 1 and 2, respectively.

### a. GBVTD

Only a brief explanation of the original GBVTD formulations is given in this section. Interested readers can refer to Lee et al. (1999) for more details. The same symbols and geometry relationships as in Lee et al. (1999) are adopted in this article except that the TC center is located to the north of the radar (as shown in Fig. 1).

The GBVTD method is proposed to provide an estimate of the horizontal winds of TC circulation relative to the mean wind vector around rings concentric with the circulation center. The mean wind is assumed to be the environmental wind, which only varies with height across the inner core of a TC. For the convenience of later discussions, consists of two components, the along-beam component and the cross-beam component , with respect to north that passes through the circulation center. Least squares curve fitting of the observed Doppler velocity data is performed around the GBVTD rings and the resulting Fourier coefficients can be bridged to various wavenumber components of tangential and radial winds, including . There are many ways to interpret GBVTD solutions since the set of equations is not closed. Lee et al. (1999) proposed the closure assumption in which the asymmetric radial wind is negligible when compared with the corresponding tangential wind. In addition, the maximum wavenumber resolved at each radius varies with the maximum angular data gap; for data having gaps of 30°, 60°, 120°, and 180°, the maximum wavenumbers resolved are 3, 2, 1, and 0, respectively.

GBVTD can only provide an estimation of while the unresolved is aliased into the axisymmetric tangential wind as described in Eq. (20) in Lee et al. (1999):

where *B*_{1} and *B*_{3} are the Fourier coefficients of GBVTD analysis at a given radius [note that the typographical error of the sign before *B*_{3} in Lee et al. (1999) has been corrected], and are the angles for the circulation center relative to the radar and the direction of the mean wind, respectively. We have that is and , where *R* () is the range from the circulation center to the GBVTD ring (radar). The last term on the right-hand side of Eq. (1) was ignored in the original formulation of GBVTD, and it is aliased into .

### b. HVVP

Harasti (2003) proposed the HVVP method to provide estimates of the environmental wind vectors of a TC as a function of height. In contrast to GBVTD, HVVP assumes a modified Rankine-combined vortex model in which the tangential wind profile outside the vortex inner core is described in Eqs. (2) and (3) while inner core exhibits solid body rotation. The HVVP-assumed radial wind profile outside the vortex inner core is also shown in Eqs. (4) and (5):

where *n* represents the wavenumber (*n* = 1, 2, 3), and () is the magnitude (phase) of the asymmetric tangential wind for angular wavenumber *n*. Similarly, for asymmetric radial wind the magnitude (phase) is . Here is tangential (radial) wind at altitude *z* over the radar site, including the axisymmetric and asymmetric components. HVVP uses the axisymmetric solution of the tangential and radial wind [i.e., = = 0 in Eqs. (3) and (5)] to retrieve and at different altitudes over the radar site. Equations (3) and (5) are optionally used by HVVP to analytically assess errors due to asymmetries.

HVVP assumes the observed Doppler velocity () is equal to the sum of the estimated Doppler velocity and the measurement error *ɛ*. It connects the estimated coefficients of a second-order Taylor series expansion of the wind field to the kinematic properties of the analytic datasets in a three-dimension volume:

where are the basis functions and are the predicted parameters that are solved by the least squares curve fitting. HVVP uses a spherical coordinate (*r*, 360 − α, ϕ) where the elevation angle of the radar beam and altitude at each data point are ϕ and *z*, respectively. Here, α is the angle adapted from the GBVTD coordinate that is measured counterclockwise from the radar beam passing through the TC center to the data position on a GBVTD ring. The quantity *z*_{0} represents the altitude of analysis and () and is the total wind component in the cross-beam (along beam) direction at *z*_{0} altitude.

The tangential and radial wind over the radar site can be calculated from retrieved and :

so that (,) can be computed:

Note that Donaldson (1991) proposed a different technique to estimate the kinematic properties of a wind field of a hurricane based also on using the modified Rankine-combined vortex assumption in which the shearing deformation [equivalent to a rearrangement of Eq. (7)] was derived via a different method:

To accurately separate the mean wind, it is imperative to retrieve the TC circulation as close as possible to the truth. However, the HVVP method encounters a potential problem: the in Eq. (7) is calculated using the following empirical equation that is derived from a simplification to the axisymmetric tangential momentum equation [Eq. (2.11)] in Willoughby (1995):

where . As Harasti (2003) suggests the simplifying assumptions in Eq. (12) may not always be valid for different TCs, and thus result in the errors in the estimated mean wind.

### c. MGBVTD method

To reduce the error in estimating the caused by using an empirical *X _{T}* in HVVP, the MGBVTD method is developed by combining the merits of the GBVTD and HVVP methods.

In this framework, the axisymmetric tangential wind and asymmetric TC circulations can be typically expressed as Eqs. (2) and (3). In GBVTD, a guessed () can be provided to estimate the at different radii. Given the radial profile of , the parameter in Eq. (2) can be objectively determined by fitting the GBVTD-derived profile. By substituting this into HVVP, a more accurate () can be retrieved. If converges to , the is considered the “true” . Therefore, MGBVTD is able to search for the “optimal” by examining a reasonable range of .

The procedure of MGBVTD is described as follows. In this study, the magnitude of guessed varies from −20 to 20 m s^{−1} with increments of 0.1 m s^{−1}. In the first step, for an individual , the axisymmetric tangential wind profile can be computed by GBVTD with the correction of in Eq. (1). Based on the GBVTD-retrieved profile, can be calculated by minimizing the objective function f1 derived by taking the logarithm of Eq. (2):

where *i* denotes the *i*th radius whose magnitude is from N1 to N2 km. In our test, N1 is set as and N2 is usually about 70% of to ensure sufficient fitting samples.

In the second step, by substituting into Eqs. (6), (7), and (9), the HVVP-retrieved is obtained. Finally, the difference between and are calculated as

Note that taking the logarithm form amplifies the anomaly. Repeating the first and second steps for all the guessed within the given range and the is considered the optimal when f2 reaches its minimum. Combining the optimal and provided by GBVTD, the MGBVTD method provides an estimate of mean wind without the need for empirical assumptions required by HVVP and hence is likely to improve the accuracy of GBVTD-derived axisymmetric tangential winds.

## 3. Tests using analytic data

### a. Construction of analytic dataset

To quantitatively investigate the performance of the MGBVTD method, a set of idealized vortex flow fields, based on a single-layer (elevation angle ϕ is equal to 0) modified Rankine-combined vortex, is constructed to simulate , following Lee et al. (1999). The mathematical expressions for the axisymmetric tangential wind and radial wind are

where and are set to 50 m s^{−1} and 20 km, respectively. The terms and are scale factors and are assigned 0.1 s^{−1} and 3 m^{0.5} s^{−1}, respectively. Apparently, the outflow (inflow) is inside (outside) according to Eq. (16). The asymmetric tangential wind follows Eq. (3), where (*n* = 1, 2, 3) is the magnitude of each wavenumber and is set to 0.2. Following Lee et al. (1999), there is no asymmetric radial component in the idealized vortex.

A hypothetical Doppler radar is located at the grid origin (0, 0) with a maximum effective range of 150 km and a high effective Doppler velocity where velocity aliasing is not considered. The TC center is set at 80 km north of the radar site at (0, 80). The TC circulation generated by Eqs. (3), (15), and (16) is projected onto the radar beam direction to produce analytic . The mean wind speed and direction are arbitrarily assigned to 10 m s^{−1} and 0–360°. When the is set as 180°(easterly), there is only *V _{M}*

_{⊥}information. The analytic data are used to retrieve the total winds of TCs using GBVTD and MGBVTD for comparison against the analytic modified Rankine-combined vortex. As a quantitative measure of the accuracy of GBVTD and MGBVTD retrievals, the root-mean-square error (RMSE) of the total winds between retrieved and the true values is calculated as

Here *V* and are the quantities to be verified and true value, respectively. The quantity *N* indicates the total number of data points of the valid values. Besides, the correlation coefficient (CC) between the retrieved and true value is calculated in the idealized experiments.

### b. Results of retrieved MGBVTD winds

A series of experiments was designed to examine the performance of MGBVTD-retrieved mean wind in the presence of 1) different direction of , 2) different , 3) various , 4) tangential wind asymmetry, and 5) a misplaced center. A description of these experiments is given in Table 3. Without specific description, is generated from the same idealized axisymmetric vortex with = 50 m s^{−1}, = 20 km, and = 1.0 using Eqs. (13) and (14). In this study, the TC center is defined as the circulation center and its location is known for the analytical series.

#### 1) Sensitivity to the direction of (GM1)

To examine the impact of different directions of mean wind on the MGBVTD method, GM1 is conducted in which is equal to 10 m s^{−1} while varies from 0° to 360°. The retrieved and are nearly identical to their true counterparts (Fig. 2a). The magnitude of retrieved mean wind is within 0.1 m s^{−1} to its true value in every run and the diagonal line of retrieved direction of mean wind indicates the error is negligible. Thus it is concluded that MGBVTD is insensitive to the direction of mean wind.

To better understand the searching process of for MGBVTD, an example (Fig. 2b) is given when is equal to 180° and | |= 10.0 m s^{−1}. The search range of is set from −20 to 15 m s^{−1}. When is equal to the true (−10.0), the fitting error f2 reaches its minimum (near −30.0) and the retrieved is treated as the true value of the Rankine profile. The corresponding GBVTD- and MGBVTD-retrieved total winds (axisymmetric tangential and radial winds plus a mean wind) are shown in Figs. 3b and 3c, as compared with the analytic wind in Fig. 3a. Clearly, we see better agreement between the MGBVTD-retrieved magnitudes in Fig. 3c and the analytical wind magnitudes in Fig. 3a. Despite a slight underestimation of the total wind in the south part of the TC at ~20 km radius, the general wind pattern (Fig. 3c) indicates a coherent wavenumber 1 pattern with similar magnitude and phase to the analytical wind field (Fig. 3a), which is consistent with the characteristics of the Rankine-combined vortex with an easterly mean wind as shown in Fig. 5 of Lee et al. (1999). On the contrary, as GBVTD cannot retrieve the cross-beam component of mean wind, there is no wavenumber 1 signal in the wind pattern retrieved by GBVTD (Fig. 3b). The corresponding error statistics, RMSE (CC) of GBVTD- and MGBVTD-retrieved total wind field, are 8.5 m s^{−1} (0.82) and 0.2 m s^{−1} (1.0), respectively (not shown), which quantitatively proves that MGBVTD can retrieve a more accurate wind field than GBVTD with the presence of .

For further inspection, GBVTD- and MGBVTD-retrieved axisymmetric tangential wind profiles are shown as gray lines in Fig. 3d. Compared to the Rankine profile (black solid line), the GBVTD retrieval underestimates . The discrepancy between them becomes larger at greater radii, which can be easily understood since a larger fraction of is aliased into mean tangential wind when is larger as shown in Eq. (1). Whereas, MGBVTD performs very well because of its ability to retrieve , as indicated by the overlap between its retrieved profile (gray dashed line) and Rankine profile.

#### 2) Sensitivity to Rankine (GM2)

The multistorm statistical study of Gray and Shea (1973) indicates that the mean value found for was 0.5 and was close to the expected theoretical value, but with relatively large standard deviations (0.3). It is believed that the stage of TC life cycle accounts for much of the variability in the tangential wind profile shape (e.g., Weatherford 1989). Considering this fact, the testing range for is set from 0.3 to 1.0 in GM2 run. The larger is, the faster the profile drops down with radius.

A dimensionless parameter Par is introduced and is defined as

where represents the radius of HVVP analysis domain centered at the radar. When the HVVP analysis domain extends to , Par = 1.0. The lower bound of Par is set to 0.3 in this sensitivity test to ensure enough data points for analysis. The upper bound of Par is set to 0.7 as larger value of Par tends to cause large estimation error of the deformation term of HVVP (), thus degrading the retrieved . For diverse and Par, RMSE of the retrieved axisymmetric tangential wind profile to their true counterparts is computed (Fig. 4a). It is evident (Fig. 4a) that RMSE is proportional to Par with the same while RMSE is inversely proportional to with fixed Par. The similar correlation can be inferred with the retrieved (Fig. 4b). The largest error of the retrieved is ~1.3 m s^{−1} to the true value when = 0.3 and Par = 0.7. To quantitatively measure the variation of the retrieved for different Par, the standard deviation (STD) of for each is shown in Table 4. The maximum STD of 0.4 occurs when = 0.3, which is consistent with the large space between the five lines shown in Fig. 4b. However, the low values for both the RMSE of the retrieved profile (<1.3 m s^{−1}) and the STD of indicate MGBVTD can reliably retrieve and the TC primary circulations on a wide range of wind profiles (i.e., ).

To illustrate the advantage of MGBVTD over HVVP in the estimation of TC winds, the retrieved and using HVVP are shown in Figs. 4c and 4d. Apparently, the estimated from HVVP are nearly constant despite the variation of the true due to the application of the empirical Eq. (12), thus leading to a larger error in the estimate of (more than 9 m s^{−1} when > 0.7). However, HVVP is not expected to perform well with a radial wind model such as Eq. (16) since it is not related to the expected radial wind profile derived from the axisymmetric momentum equation from which Eq. (12) result. Additional tests using the model proposed in HVVP [i.e., Eq. (4)] were also performed. However, even in this situation, HVVP can only retrieve a comparable result to MGBVTD when inflow exists outside and (not shown), which is expected since it is exactly the empirical equation in Eq. (12). This fact supports MGBVTD could be applied more generally.

#### 3) Sensitivity to (GM3)

As the size of TC's eye changes considerably from case to case, it is indispensable to test the performance of MGBVTD to TCs with different . A dimensionless parameter ρ is introduced as . The errors for the retrieved and shown in Fig. 5 remain considerably small generally but become larger when ρ is greater than 0.5 (i.e., the radar is within twice the of the TC center). When the of a storm approaches a radar, there are fewer radii available for the GBVTD analysis to deduce leading to a less stable fitting of the radial wind profile. However, even in this situation, the largest retrieved error of () is 0.2 m s^{−1} (0.01) that can be essentially neglected.

#### 4) Sensitivity to asymmetry (AV series)

Based on GM1 to GM3 tests aforementioned, three experiments are conducted to examine the impact of asymmetric circulation, including wavenumber 1, 2, and 3 (AV1, AV2, and AV3) embedded within the axisymmetric vortex plus a mean flow. The asymmetric structure is generated using Eq. (3) and the parameters are listed in the Table 3.

The retrieved () in all AV series oscillate in a wavelike behavior around the true value consistent with the corresponding asymmetric structures (i.e., wavenumbers) (Figs. 6a,b). The retrieved errors of () are less than 0.05 (0.5 m s^{−1}) in AV1–AV3 experiments where the errors are less than 5% of the specified values. The performances of GBVTD and MGBVTD for asymmetric TCs are compared in all AV1–AV3 experiments (Fig. 7). Similar to that in Fig. 2c, the GBVTD cannot accurately deduce total wind structure in AV1–AV3 tests, which is mainly due to the inability of GBVTD to retrieve . In comparison, MGBVTD reproduces all major features of the wavenumber 1–3 structures well, especially the amplitude and phase of asymmetry. Nevertheless, the retrieved total winds of MGBVTD do suffer pronounced distortion in higher wavenumber asymmetry similar to that of GBVTD. For example, the peak amplitude of wavenumber 3 is significantly reduced on the far side of the TC in both Figs. 7b3 and 7c3, even though MGBVTD has contained the information. This is mainly due to the geometric distortion inherent in GBVTD nonlinear coordinate, consistent with the description in Lee et al. (1999). The corresponding error statistics (Fig. 8) show that the experiments with asymmetric tangential wind component tend to have slightly larger errors than that of GM1–3, especially for higher wavenumbers. The RMSEs of GBVTD- (MGBVTD) retrieved total winds in the AV1–AV3 experiments are 8.8 (0.2), 8.4 (0.7), and 10.6 (3.2) m s^{−1}, respectively. The MGBVTD-retrieved total winds in the AV1–AV3 experiments also show higher values of CCs (Fig. 8). The main reason for the large RMSE errors and the relatively low CCs in the GBVTD-retrieved total wind is its inability to retrieve at 10 m s^{−1} amplitude. In contrast, MGBVTD can retrieve accurate and is also quite robust when the TC circulation is asymmetric.

In contrast to the asymmetric tangential winds, the asymmetric radial winds cannot be resolved in the MGBVTD framework. When significant wavenumber 2 components of () exist as shown in Eq. (1), the estimated axisymmetric tangential wind and may contain large error, and thus lead to the bias in the estimate of . In this situation, some extra wind measurements are required to retrieve the asymmetric radial flow (Liou et al. 2006), which is beyond the scope of this paper.

#### 5) Misplaced centers (GC)

Previous studies (e.g., Roux and Marks 1996; Lee and Marks 2000) have shown the quality of GBVTD-retrieved winds is sensitive to the center uncertainties. Lee and Marks (2000) noted that a 5-km deviation of the TC center can produces 20% error in GBVTD-retrieved . To examine the impact of the center uncertainty on MGBVTD retrievals, we calculated the errors of the MGBVTD-retrieved and for various center displacements. As shown in Fig. 9, the error is proportional to the center displacement, and the error is more sensitive to the center displacement in the *x* axis (perpendicular to the beam through the TC center) than that in the *y* axis. In general, MGBVTD performs very well when the misplaced center is within 3 km, and the maximum error of the retrieved () is about 0.08 (1 m s^{−1}). Lee and Marks (2000) developed a “Simplex” algorithm, which can estimate the TC center within 0.34 km (2 km) of the true center for analytical (real) TCs. This suggests that MGBVTD has an ability to retrieve accurate for a real TC by using the GBVTD-Simplex-estimated TC center.

## 4. Testing of MGBVTD with Hurricane Bret

In this section, Hurricane Bret (1999) is selected to test the performance of MGBVTD. Bret was a category-4 hurricane before it weakened to a category-3 hurricane a few hours before landfall along the coast of Texas. Two WSR-88D coastal instruments located at Corpus Christi (KCRP) and Brownsville (KBRO) made simultaneous observations as Bret made landfall midway between them.

A constant-altitude PPI (CAPPI) mosaic of reflectivity from KCRP and KBRO at 0000 UTC August 23 1999 is shown in Fig. 10a and indicates that KCRP was located in a region of mostly convective precipitation while KBRO was located in a region of mostly stratiform precipitation. The corresponding 2-km CAPPI image of Doppler velocity from KCRP at 2357 UTC (KBRO at 0000 UTC) is illustrated in Fig. 10b (Fig. 10c). The coverage of Doppler radar data in the real TC is not as complete as that in the analytic TCs. Note that HVVP cannot perform well with a large data gap (e.g., large gap of Doppler velocity data south of KBRO as shown in Fig. 10c), which will affect the accuracy of retrieved 16 variables in Eq. (6) and further degrade the accuracy of .

At 2357 UTC, the circulation center of Bret is located to the south of KCRP; the coordinates require a clockwise rotation from the true north to the azimuthal angle of the circulation center for later MGBVTD analysis. The circulation center is identified by the “Simplex” algorithm (Lee and Marks 2000). Table 5 shows the sensitivity of the MGBVTD retrievals using KCRP data with respect to different Par at 2-km height where Par is set from 0.3 to 0.7 similar to GM2. The final is chosen from the optimal results when the RMSE for MGBVTD-retrieved profile to the Rankine profile with the fitted is minimized. The STD of for different values of Par is 1.6 m s^{−1}, larger than that in GM2 (Table 4), but still within 2.0 m s^{−1}. The RMSE for MGBVTD-retrieved profile increases when Par increases, indicating a degradation of the retrieved due to the increasing importance of the deformation with increasing Par (i.e., getting close to the eyewall). The retrieved mean STD of for KBRO is 4.6 m s^{−1} (not shown), which is more than 10 times of that in GM2. This discrepancy is most likely a result from the large gaps in the KBRO data affecting the accuracy of the HVVP analyses. Therefore, the mean wind retrieved from KCRP will be used for KBRO analyses in this study.

The vertical profile of mean wind over KCRP experiences anticyclonic shear above 2-km altitude (Fig. 11a). To test the validity of retrieved mean wind, the original GBVTD axisymmetric tangential wind profiles and their corresponding counterparts with the correction of by MGBVTD for two radars at 2-km altitude are shown in Fig. 11b. Before the correction, the difference of tangential wind for KCRP and KBRO at is about 3 m s^{−1} and it is larger at farther radii. This can be understood since the difference of the radar-viewing angle from the two radar sites toward the TC center is nearly 180° and the effect of on the retrieved axisymmetric tangential wind as shown in Eq. (1) is in opposite sign. After the correction with from either MGBVTD, the profiles for two radars are nearly coincident, which hints at the accuracy of retrieved . Similarly, the axisymmetric tangential wind profiles for KCRP and KBRO at 3-km altitude are also shown in Fig. 11c, in which the mean difference of amplitude between KCRP and KBRO has dropped from 5.28 m s^{−1} (GBVTD) to 1.63 m s^{−1} (MGBVTD). To quantitatively evaluate the accuracy of MGBVTD-retrieved , the EGBVTD-retrieved mean wind vector is projected onto the cross-beam direction of both KBRO and KCRP for comparison (Table 6). The MGBVTD-retrieved proves to be reliable since it is only 1 m s^{−1} (2 m s^{−1}) larger than its counterpart of EGBVTD at 2-km (3 km) altitude on average. Meanwhile, the profiles with the correction of the EGBVTD-finding cross-beam component for two radars show consistent results with those from MGBVTD (Figs. 11b,c).

Similar to the idealized case, the retrieved total winds from GBVTD and MGBVTD are also shown for comparison in Fig. 12. Without retrieving , the GBVTD-retrieved total winds at 2-km height from KCRP and KBRO are shown in Figs. 12a,b. The total wind pattern for KCRP clearly indicates a wavenumber-1 structure while a more asymmetric structure for KBRO. In addition, the magnitude of the total wind for KBRO is generally much smaller than that for KCRP. The difference is about 5 m s^{−1} in the southeast of the radius of maximum tangential wind (RMW). When including the retrieved , MGBVTD-retrieved total winds for both KCRP and KBRO shows a consistent wavenumber-1 structure whose maxima winds are greater than 56 m s^{−1} and located in the northwest quadrant (as shown in Figs. 12c,d). To investigate how the large data gaps could affect the accuracy of MGBVTD retrievals, the total winds retrieved for KBRO using MGBVTD-retrieved from KBRO data are shown in Fig. 12e. As the derived from KBRO data is nearly 6 m s^{−1} larger than that of EGBVTD, there is a distinct overestimation (Fig. 12e) in the wind magnitude at the northern part of TC, as compared with Figs. 12c and 12d. This suggests that the large data gap would degrade the accuracy of MGBVTD-retrieved total wind speed.

To further quantitatively assess the performance of GBVTD and MGBVTD, the corresponding RMSE (CC) of the retrieved total winds from the two methods (Table 7) are 6.3 (0.91) and 2.0 (0.96), respectively. Clearly, MGBVTD retrieves better TC circulation, which is a clear advantage, and it is necessary to include information in deducing accurate TC circulation. We also applied the MGBVTD to other heights, and MGBVTD still obtained a better TC circulation than GBVTD. For example, the RMSE (CC) of the retrieved total winds from GBVTD and MGBVTD at 3-km height are 6.1 (0.90) and 2.0 (0.98) (not shown).

## 5. Summary and discussion

In this paper, the modified GBVTD (MGBVTD) method is developed based on the GBVTD (Lee et al. 1999) and HVVP (Harasti 2003) techniques. The individual weaknesses inherited in GBVTD and HVVP are well known. GBVTD can retrieve reasonable TC circulations but is hindered by its inability to retrieve . HVVP is ideally suited to deduce but it requires an underlying empirical Rankine profile assumption. By combining GBVTD and HVVP algorithms, MGBVTD retains the strength in both algorithms but avoids their weakness. The GBVTD-retrieved TC circulation is used to anchor the TC wind profile for HVVP to deduce , which in turn is included in the GBVTD analysis to reduce the biases in the retrieved TC circulation. A better TC circulation is obtained via an iterative process in MGBVTD.

When tested with a series of analytic TC datasets, the difference of MGBVTD-retrieved compared to the true value is within 1 m s^{−1} (~10%) in most cases and the retrieved wind structure is close to the given one after the correction with . In addition, the sensitivities of MGBVTD to several parameters are examined using analytical TCs. It has been demonstrated that the MGBVTD algorithm is not sensitive to the mean wind vector (direction and magnitude), the TC axisymmetric wind structure modeled by the modified Rankine-combined vortex (i.e., ), the asymmetric winds of the TC as well as misplaced centers. MGBVTD is, however, sensitive to the size of the HVVP analysis domain especially when it includes a portion of eyewall circulation where deformation is significant.

When applied to Hurricane Bret, MGBVTD also shows its ability to retrieve a more consistent TC structures when using data from KBRO and KCRP than the GBVTD-retrieved structures because of its ability to deduce an accurate . The closeness of the individually retrieved axisymmetric winds at constant heights from KCRP and KBRO demonstrates the strength of the MGBVTD technique over the GBVTD technique and its potential to be included in real-time TC wind retrieval packages like VORTRAC and for research use. Note that GBVTD has been also successfully applied to retrieving tornado wind fields in recent years (Lee and Wurman 2005; Tanamachi et al. 2007). It is recommended that MGBVTD be applied to tornado research in the future to investigate its performance compared to GBVTD for smaller-scale vortices as well.

The uncertainties of the MGBVTD method may result from several factors that need to be noted. The first one is the deviation of modified Rankine-combined vortex from real TCs. However, this may be a secondary effect because the modified Rankine-combined vortex approximation is adequate to represent the major circulation characteristics of real TCs. Second, data coverage (e.g., missing data between TC rainbands or in weaker TCs) may contain large gaps as shown in the KBRO data in Hurricane Bret, which can pose great challenge for HVVP to find accurate mean wind information. Under this circumstance, it would be better to verify the MGBVTD-estimated mean wind with the mean wind estimated from other sources, if possible. Finally, the asymmetric radial flow is unresolved in MGBVTD and is aliased into the tangential wind and along-beam mean wind. As shown in Eq. (1), when the significant wavenumber 2 radial wind exists, the retrieved can contain a large error. In this situation, some extra wind measurements are required to retrieve the asymmetric radial flow.

## Acknowledgments

This work was primarily supported by the Social Commonwealth Research Program (GYHY201006007), National Fundamental Research 973 Program of China (2009CB421502 and 2013CB430101), and the National Natural Science Foundation of China (Grants 40975011, 41275031, and 40921160381).

## REFERENCES

*31st Conf. on Radar Meteorology,*Seattle, WA, Amer. Meteor. Soc., 14B.1. [Available online at https://ams.confex.com/ams/32BC31R5C/techprogram/paper_64625.htm.]

*33rd Conf. on Radar Meteorology,*Cairns, Australia, Amer. Meteor. Soc., P11A.6. [Available online at https://ams.confex.com/ams/33Radar/techprogram/paper_123747.htm.]

*Global Perspectives on Tropical Cyclones,*R. L. Elsberry, Ed., WMO/TD 693, 21–62.

## Footnotes

The National Center for Atmospheric Research is sponsored by the National Science Foundation.