## 1. Introduction

Doppler weather radar can provide high-resolution data of precipitation; thus, it can be a great help in the study of severe weather systems, such as rainstorms, hailstorms, and tornadoes. Based on the Doppler effect that there is frequency shift as the object approaches or recedes from an observer, a Doppler radar can determine the frequency shift through measurement of the phase change of electromagnetic waves. Since this frequency shift corresponds to a projected velocity along the line of the object and the radar—that is, the radial direction—a Doppler radar provides only radial speed (receding from the radar or toward the radar). Therefore, algorithms are necessary to retrieve the total wind through the radial speed measured by the Doppler radar.

Many studies have been conducted on wind retrieval algorithms of single-Doppler radar since it was used in meteorology detection. The most frequently used simple-assumption algorithms are velocity–azimuth display (VAD) and velocity volume processing (VVP). Through the assumption of uniform wind at a constant height, Lhermitte and Atlas (1961) proposed VAD to retrieve the uniform wind at the height where a Doppler radar detected radial speed at a certain range and different azimuths. Through the assumption that wind varies linearly with respect to space at a constant height, Browning and Wexler (1968) improved VAD to retrieve wind field parameters by Fourier expansion for radial speeds at different azimuths. Waldteufel and Corbin (1979) proposed a VVP algorithm. VVP separates the radial speed data into many volumes and uses the least squares technique to retrieve the wind field through the assumption that the wind is linear in each volume. Normally, VVP aims at data in an analysis volume and can gain higher-resolution results than VAD. However, because VVP usually produces large errors in practical retrieval for small analysis volumes, it does not exhibit this theoretical superiority.

The support vector machine (SVM) has produced much of the recent progress in many fields (Vapnik 1998; Cristianini and Shawe-Taylor 2000). The SVM is a supervised learning model dealing with classification requirements of an optimization problem. The ultimate purpose of SVMs is to determine a hyperplane to separate points into clusters. The hyperplane is required to maximize the interval from points to the separating hyperplane and to ensure the classification accuracy. Compared with other techniques, the SVM technique has a visual geometric description and it can overcome the high-dimension problem and the nonlinear problem to a certain extent. The outstanding characteristic of SVMs is the solving of the Lagrange dual problem instead of the original problem, which improves optimization models.

Li et al. (2015) introduced an SVM wind retrieval of single-Doppler radar, but they did not give proof and reasons why and in what situation the SVM-based algorithm can give a more accurate retrieval. In this article, a deep discussion is given for the SVM technique in more general situations through theoretical analysis and retrieval experiments. For the equations corresponding to the optimization problems of VVP and SVM-based VVP, coefficient matrices and their condition numbers are analyzed with linear algebra theory to validate the improvement of the new algorithm. Afterward, the two algorithms are implemented to retrieve wind in the simulated cases and real cases to evaluate the practical performance. The cases for retrieval experiments include a simulated large-scale uniform wind field, simulated small-scale mesocyclone wind fields, a dual-Doppler-retrieved typhoon wind field, and single-Doppler observations of a mesocyclone. Based on the retrieval results, the application condition and optimal parameters of the new algorithm are discussed and selected.

## 2. The algorithms

### a. The VVP algorithm

As shown in Fig. 1, VVP aims at radial speeds measured by a Doppler radar in a volume spanning Δ*r*, Δ*θ*, and Δ*ϕ*, where *r*, *θ*, and *ϕ* are the radial range, the azimuth angle, and the elevation angle measured by the Doppler radar.

*V*

_{r}and the wind field parameters is

*u*

_{0},

*υ*

_{0},

*w*

_{0}) are the horizontal and vertical wind components at the center (

*x*

_{0},

*y*

_{0},

*z*

_{0}) of the analysis volume; ∂

*u*/∂

*x*, ∂

*υ*/∂

*y*, ∂

*u*/∂

*y*, and ∂

*υ*/∂

*x*are the horizontal shear of the horizontal wind; ∂

*u*/∂

*z*and ∂

*υ*/∂

*z*are the vertical shear of the horizontal wind; and ∂

*w*/∂

*x*, ∂

*w*/∂

*y*, and ∂

*w*/∂

*z*are the horizontal and vertical shear of the vertical wind.

Generally, scholars would simplify the radial speed [Eq. (1)] through neglecting some wind field parameters, since these variables contribute less to the radial speed and thus are difficult to be retrieved and tend to degrade the retrieval of other variables. Waldteufel and Corbin (1979) used 10 variables when they proposed the VVP algorithm, neglecting the horizontal shear of the vertical wind. Koscielny et al. (1982) used eight variables when they proposed a modified VVP, neglecting the vertical wind and its vertical shear, as well as its horizontal shear. Li et al. (2007, 2015) used six variables containing horizontal wind components and their horizontal shear when discussing the calculation process.

*V*

_{rmi}is the measured radial speed at observation point

*i*,

**f**

_{i}is the coefficient vector for observation point

*i*,

*N*is the number of observation points in the analysis volume, and the summation is over

*i*from 1 to

*N*for all observation points in the analysis volume. The derived equations yielding the minimum difference are

### b. The SVM-based VVP algorithm

Regarding the radial speed equation, SVM-based VVP attempts to use SVM regression to retrieve the wind field parameters instead of the least squares technique. Through the analysis of the general support vector regression machine, the least squares support vector regression machine, and the support vector regression machine with the homogeneous decision function described in the references (e.g., Deng and Tian 2009), Li et al. (2015) constructed the least squares support vector regression machine with homogeneous decision function. This type of support vector regression machine is characterized by only one equality constraint in the origin problem and no constraint in the Lagrange dual problem. Using this support vector regression machine, Li et al. (2015) originally proposed the SVM-based VVP algorithm through a six-variable model. By comparison, the authors extend it to a more general situation with *M* variables (2 ≤ *M* ≤ 12) in this article.

**u**represents the slope of the regression hyperplane,

*ξ*

_{i}is the slack variable, and

*c*(

*c*> 0) is a penalization parameter reflecting the balance between the maximized interval and the constraint condition. The Lagrange function can be written as

*α*

_{i}is the Lagrange multiplier. Substituting the optimum conditions of

*L*

*δ*

_{ij}= 1 when

*i*=

*j*and

*δ*

_{ij}= 0 when

*i*≠

*j*. The derived equations yielding the minimum are

**f**

_{1},

**f**

_{2},…,

**f**

_{N})

^{T}· (

**f**

_{1},

**f**

_{2},…,

**f**

_{N}),

**d**= (

*V*

_{rm1},

*V*

_{rm2},…,

*V*

_{rmN})

^{T}, and

**α**= (

*α*

_{1},…,

*α*

_{N})

^{T}is the optimal solution of the Lagrange multipliers. A support vector machine

**f**are just solved wind field variables. The difference is that the wind field parameters are considered variables in the radial speed equation, while the coefficients before the wind field parameters are considered variables in this support vector regression machine.

## 3. Theoretical analysis

In this section, the calculation of VVP and SVM-based VVP is analyzed and compared to see whether and in what situation SVM-based VVP is an improved algorithm that can achieve better retrieval.

**u**=

**b**, the relation between the estimated error of the solution and the error of

**b**can be calculated using

When matrix **f**_{1}, **f**_{2},…,**f**_{N}) is introduced, ^{T} and ^{T} · **+** *c* **=** **+** *c*, where ^{T} · *R*^{N}.

As for symmetric matrices of ^{T}, ^{T} ·

^{T}·

^{−1}·

^{T}·

^{−1}· (

^{T}) ·

^{−1}·

*c*is not introduced, SVM-based VVP cannot change the condition number of the coefficient matrix in VVP.

*c*, its eigenvalue

*c*,

*c*,

*c*is added to min(

*c*cannot be too small; otherwise it may cause a large condition number. The value of 1/

*c*cannot be too large either, otherwise it may change the properties of the equations. With a proper value of 1/

*c*, the SVM-based VVP algorithm can obtain much more accurate results compared with the VVP algorithm.

Through the introduction of the penalization parameter, the SVM-based algorithm can significantly reduce the condition number of the coefficient matrix and thus can reduce the error. Furthermore, it is noted that the order of matrix

## 4. Retrieval experiments

To verify the improvement of SVM-based algorithm compared with the original algorithm, simulated, Doppler- and dual-Doppler-retrieved wind fields are used to make experiments to evaluate the retrieval performance of the two algorithms. The Doppler radar data were collected by the China New Generation Weather Radar (CINRAD), which is widely used in meteorological services in China. It has technical parameters similar to the WSR-88D in the United States and adopts the volume coverage pattern (VCP) in application. In the following experiments, the frequently used precipitation pattern of VCP 21 is employed with nine elevation angles, including 0.5°, 1.5°, 2.4°, 3.4°, 4.3°, 6.0°, 9.9°, 14.6°, and 19.5°. The maximum detection range for radial speed data is 230 km with a radial resolution of 250 m, and an azimuth resolution about 1°.

In our retrieval experiments, we make the analysis volumes small enough that the wind field is approximately uniform within each volume. Furthermore, the analysis volumes consist of only one elevation angle (i.e., are analysis planes). In addition, the vertical wind component should be neglected in retrieval considering the low elevation scanning of the Doppler weather radar. Therefore, two variables (*u*_{0} and *υ*_{0}) are used in wind retrieval experiments, which is more general and practical.

In the retrieval experiments, the size of an analysis plane is denoted by the number of azimuths and the number of radial gates showing in a square bracket (e.g., [2,12]), and 1/*c* is measured by a relative ratio to the smallest diagonal element in the coefficient matrix, which is denoted in a square bracket (e.g., [0.01]). The absolute value of 1/*c* is the product of the relative ratio and the smallest diagonal element in the coefficient matrix.

### a. A simulated uniform wind

A simulated uniform horizontal wind field is used first to check the performance of the algorithms. The simulated wind field covers the entire Doppler radar domain. It is uniform in horizontal direction and varies linearly in vertical direction from the ground to a maximum height of 12 km. The wind speed at the ground is 10 m s^{−1} and increases to a maximum value of 25 m s^{−1} at 6-km altitude and then decreases to 10 m s^{−1} at 12-km altitude. Random noise is added with no more than 10% of the wind speed. The wind direction is 180° (south wind) at the ground and gradually turns to 270° (west wind) clockwise with the altitude. Random noise is added with no more than 10° difference compared with the wind direction. Radial speed is acquired by projecting the simulated wind in radar radial directions and then smoothed with median filtering (similarly hereinafter), based on which retrieval is implemented for the two algorithms. Tables 1 and 2 and Fig. 2 give the results of the two algorithms during the retrieval.

Mean condition numbers of coefficient matrices corresponding to equations of SVM-based VVP and VVP for the simulated uniform wind field. Two numbers separated by a comma in a square bracket (e.g., [2,12]) denote the analysis plane dimension by the number of azimuths and the number of radial gates, and one number in a square bracket (e.g., [0.01]) denotes the penalization parameter value measured by the reciprocal; this applies to Tables 1–6.

Mean relative error of retrieved wind components by the SVM-based VVP and VVP algorithms for the simulated uniform wind field.

Table 1 gives mean condition numbers of matrices corresponding to equations of SVM-based VVP and VVP for different analysis volumes and different penalization parameters. It is obvious that for SVM-based VVP, the mean condition number is smaller for smaller analysis volumes and for larger 1/*c*, which confirms the theoretical analysis in the previous section. By contrast, for VVP, the mean condition number is smaller for larger analysis volumes, which is consistent with the theoretical analysis in Li et al. (2007).

Figure 2 shows the performance of the two algorithms at 1.5-km altitude for analysis plane [2,12] and 1/*c* [0.01] and [0.5]. There is a uniform south-southwest wind at this altitude (Fig. 2a). SVM-based VVP generally gives the consistent wind direction and wind speed (Fig. 2c). Because the radar beams are perpendicular to the wind direction, yielding very small radial speeds (Fig. 2b), the retrieved wind in areas around the east–west direction has relative larger error (Fig. 2c). By comparison, VVP gives an unsuccessful retrieval for both wind speed and wind direction (Fig. 2d).

Table 2 gives the mean relative error of the *u* and *υ* components retrieved by the two algorithms for different analysis volumes and different penalization parameters compared with the simulated wind components. Since a very small wind speed could induce a very large relative error, the simulated *u* and *υ* components, whose speed is less than 1 m s^{−1}, are disregarded in error calculation, and 1 m s^{−1} is also approximately the sensitivity of the wind measurement instruments. It can be seen that for SVM-based VVP, the error first decreases and then increases with the increasing of 1/*c*. Although the condition number may be largely reduced for a large 1/*c* as shown in Table 1, the corresponding equation in retrieval may change its property, and the retrieved wind direction tends to be thoroughly along the radial as shown in Fig. 2e. Furthermore, although the condition number is smaller for smaller analysis volumes, the error is smaller for larger analysis volumes given the same 1/*c*. This is because the error is also related to the norm ratio of the detection error to the real values as shown in Eq. (10). When larger analysis volumes could be used, the norm of the vector **d** for SVM-based VVP becomes larger and the norm ratio becomes smaller, leading to a smaller error. For VVP, it gives better results for a larger analysis volume due to a smaller condition number and a smaller norm ratio. Compared with SVM-based VVP, it gives a larger error for smaller analysis volumes but gives a smaller error for larger analysis volumes.

### b. A simulated mesocyclone wind

Since the condition number of the coefficient matrix in SVM-based VVP is smaller for smaller analysis volumes, contrary to that of VVP, it has more superiority for wind retrieval of mesoscale and small-scale precipitation. To verify this point, a simulated mesocyclone is constructed for the wind retrieval experiment. The vertical structure of the mesocyclone uses the wind field pattern proposed by Brown and Wood (1983, 1991), and presents convergent rotation near the ground, pure rotation at lower midlevels, divergent rotation at upper midlevels, and pure divergence near the storm top. The center of the mesocyclone is positioned at (70 km, −70 km) relative to the Doppler radar. The simulated wind speed and direction of the mesocyclone follows the Rankine model (Inoue et al. 2011; Tanamachi et al. 2013). The wind speed first increases linearly with the distance from the center to 5 km and then decreases inversely with the distance from 5 to 15 km. Random noise is added with no more than 10% of the wind speed, and with no more than 10° difference compared with the wind direction.

It should be noted that the uniform wind assumption may be less justified when larger analysis volumes are used for the mesocyclone. In these situations, additional variables could be retained and the linear wind assumption with six variables (*u*_{0}, *υ*_{0}, ∂*u*/∂*x*, ∂*υ*/∂*y*, ∂*υ*/∂*x*, ∂*u*/∂*y*) is tested by repeating the retrievals besides the uniform wind assumption with two variables (*u*_{0}, *υ*_{0}).

Table 3 gives the mean condition numbers of matrices corresponding to equations of SVM-based VVP and VVP under the uniform wind assumption within the analysis volume for different analysis volumes and different 1/*c*. The variation of the condition number with the size of the analysis volume and 1/*c* is similar to that for the simulated uniform wind. It is noted that for VVP there appears the condition number of “infinite” for several analysis volumes because in these analysis volumes the elevation and azimuth angles of the points are all the same and thus the coefficient matrix is singular. This is more likely when small analysis volumes are used.

Mean condition numbers of coefficient matrices corresponding to equations of the SVM-based VVP and VVP algorithms under the uniform wind assumption for the simulated mesocyclone with a radius of 15 km. The term *infinite* indicates that the condition number equals infinity for an analysis volume and that the corresponding coefficient matrix is singular.

Figure 3 gives the results retrieved by the two algorithms. At low height, the mesocyclone presents a superimposed wind field of an anticlockwise vortex and a convergence (Fig. 3a). For the retrieved wind field by SVM-based VVP, the anticlockwise trend of the wind direction—the primary characteristic of the mesocyclone—is acquired in Fig. 3e. Because the radar beams are perpendicular to the wind direction around the southeastern region of the cyclone, the retrieved wind direction in this area has a relatively larger error. In addition, the retrieved wind speed in the center of the mesocyclone has a large error because the wind speed and the radial speed are very small. The center and the magnitude of the derived convergence and the vorticity in Figs. 3f and 3g, respectively, correspond to those in Figs. 3b and 3c, respectively, but there are some inconsistencies. By contrast, VVP provides a depressing retrieval containing remarkable error because of the large condition number. The retrieved wind vectors and the calculated divergence and the vorticity entirely fail to fit the simulated wind field (Figs. 3h–j, respectively). Figures 3k–m give the retrieved wind field by SVM-based VVP under the linear wind assumption with six variables. Affected by the additional 1/*c* in the equation, the retrieved wind vectors are all oriented to radial directions, but the distributions and values of the retrieved divergence and vorticity agree with those in Figs. 3b and 3c, respectively. Table 4 gives the specific error value of the retrieved wind components for the uniform wind model. It is noticeable that SVM-based VVP gives a better retrieval than VVP. It is worth pointing out that when 1/*c* increases, the error decreases but the retrieved wind direction tends to be along the radial direction. This is a sign that the equation property is changing, similar to the situation of the simulated uniform wind. When the retrieved wind direction is completely along the radial direction, retrieval becomes meaningless. Therefore, 1/*c* should not be too large.

Mean relative error of retrieved wind components and the calculated divergence and vorticity by the SVM-based VVP and VVP algorithms under the uniform wind assumption for the simulated mesocyclone with a radius of 15 km.

The radius of 15 km seems a little big for actual mesocyclones. The experiments are repeated for a smaller simulated mesocyclone with a radius of 7.5 km that has exactly the same structures as the bigger mesocyclone. The retrieval error of the wind vectors by the uniform wind model is given in Table 5. The variation tendency of the error with different 1/*c* and different analysis plane dimensions is similar to that of the bigger mesocyclone. Meanwhile, similar to the situation of the bigger mesocyclone, the retrieved wind vectors by the linear wind model are along radial directions.

Mean relative error of retrieved wind components and the calculated divergence and vorticity by the SVM-based VVP and VVP algorithms under the uniform wind assumption for the simulated mesocyclone with a radius of 7.5 km.

Based on the standard of balance between small retrieval error and small equation property changing, appropriate penalization parameters could be selected in actual retrieval. Table 6 gives the appropriate 1/*c* for different analysis volumes summarized from the retrieval results of the simulated uniform wind and the two mesocyclones’ wind. It can be seen that 1/*c* should be relatively smaller for large-scale precipitation and for small analysis volumes and relatively larger for small-scale precipitation and for large analysis volumes. It should be noted that for large-scale uniform wind field, the retrieval of SVM-based VVP has no remarkable advantage compared with VVP for analysis volumes [5,32] and [3,20]. The SVM-based algorithm is more suitable for high-resolution retrieval with small analysis volumes.

Appropriate 1/*c* for different analysis volumes summarized from the retrieval results of the simulated uniform wind and the wind of the two mesocyclones.

In practice, wind fields typically contain multiple scales, and therefore the chosen volume size and 1/*c* should work well for both larger- and smaller-scale wind fields. According to Table 6, analysis volume [2,12] and 1/*c* [0.01] are chosen for general scale precipitation, and analysis volume [2,12] and 1/*c* [0.05] are chosen for very small-scale precipitation, like mesocyclones.

### c. Dual-Doppler analysis of typhoon wind field

Using the chosen analysis volume [2,12] and penalization parameter [0.01], practical performance of SVM-based VVP and VVP is evaluated through the comparison with the dual-Doppler radar retrieval. The two Doppler radars (both are S-band CINRAD) are located in Wenzhou and Taizhou, respectively, of Zhejiang Province in southeastern China. From 0001 to 0054 LST 9 August 2015 when Typhoon Soudelor made landfall in Fujian Province southwest of Zhejiang Province, the Wenzhou radar and the Taizhou radar collected 10 volume scans of the typhoon rainbands. These data were used in the dual-Doppler retrieval.

*u*and

*υ*can be derived in Cartesian coordinates:

*R*

_{1}and

*R*

_{2}are the distance of the observation point from Wenzhou radar and Taizhou radar, respectively;

*V*

_{1}and

*V*

_{2}are the radial speed of the observation point detected by Wenzhou radar and Taizhou radar, respectively; (

*x*,

*y*) is the coordinate of the observation point; (

*x*

_{1},

*y*

_{1}) is the coordinate of the Taizhou radar (

*x*

_{1}= 76.40,

*y*

_{1}= 80.28); and

*d*is the baseline length of the two radars (

*d*= 110.82). The appropriate area of dual-Doppler analysis was discussed in Lhermitte and Miller (1970) and Miller and Strauch (1974). Considering that the radial speed resolution is 0.5 m s

^{−1}for the two radars and that the corresponding calculation error of

*u*and

*υ*should be smaller than 1 m s

^{−1}, the range parameter

*β*is set to 30°, which determines the minimum difference between the two azimuth angles of the two radars. The appropriate analysis area is the nonoverlapping parts of the two circles with a radius of 0.5

*d*× csc(

*β*) and centers on the perpendicular bisector of the baseline, whose distance from the baseline is 0.5

*d*× ctg(

*β*) on both sides.

Figure 4 gives the retrieval of the two algorithms for Wenzhou radar data at 1.5 km and the dual-Doppler retrieval at 0001 LST 9 August 2015. There is an apparent cyclonic motion retrieved by the dual-Doppler method (Fig. 4c). The wind speed and direction retrieved by SVM-based VVP have good agreement with those retrieved by the dual-Doppler method, except the wind direction in the southeastern region (Fig. 4d). By comparison, the wind speed and direction retrieved by VVP have significant differences from those retrieved by the dual-Doppler method (Fig. 4e). Table 7 gives the mean relative difference of wind components for Wenzhou radar and Taizhou radar compared with the dual-Doppler retrieval for all 10 times. It is more evident that SVM-based VVP gives a much better retrieval compared with VVP.

Mean relative difference of wind components for the Wenzhou radar and the Taizhou radar on 9 Aug 2015 compared with the dual-Doppler retrieval for analysis plane [2,12] and 1/*c* [0.01].

### d. Doppler observations of mesocyclone wind field

On the evening of 1 June 2015, a strong storm occurred in Jianli in Hubei Province in central China. The precipitation system was related to a mesocyclone that overturned the ship *Eastern Star*, which was traveling along the Yangtze River with 454 passengers at about 2120 LST but only 12 survived, attracting worldwide attention. The Jingzhou Doppler weather radar is in the northwestern region of Hubei Province and about 100 km away from the incident location. Figure 5a shows the radial speed detected by the Jingzhou radar at an elevation of 0.5° at 2113 LST 1 June 2015, on which the radial speed pattern of the mesocyclone can be discriminated. For this small-scale system, high-resolution retrieval is required, otherwise details of the wind field are difficult to acquire.

Figures 5b–d give retrieval results by SVM-based VVP using the chosen analysis volume [2,12] and penalization parameter [0.05]. The anticlockwise flow of the mesocyclone is acquired, and the negative maximum and the positive maximum for the divergence and the vorticity are also obtained. Compared with the simulated mesocyclone and the corresponding retrieval results shown in Fig. 3, SVM-based VVP has achieved significant characteristics of the mesocyclone’s wind field and has acquired satisfactory results. By contrast, VVP gives a retrieved wind field far from reasonable (figures omitted).

## 5. Conclusions

Doppler weather radars can measure only the radial speed of the wind based on the Doppler effect; therefore, retrieval algorithms are required to retrieve the wind field. VVP is a frequently used algorithm under a simple assumption of linear wind, and it uses the least squares technique to calculate wind parameters. However, the coefficient matrix of equations corresponding to the optimization problem of the least squares technique has a very large condition number, making retrievals highly error prone.

An SVM can be used to replace the least squares technique. In this article, the authors first extend the SVM-based algorithm to a more general situation with a random number of *M* variables (2 ≤ *M* ≤ 12). Consequently, coefficient matrices of equations corresponding to VVP and SVM-based VVP are analyzed and compared using the linear algebra theory. It can be found that the new algorithm can significantly reduce condition numbers of matrices due to a penalization parameter in the SVM and that a smaller analysis volume can lead to a smaller condition number. Therefore, the SVM-based algorithm has the ability to acquire more accurate wind fields for high-resolution retrieval.

While wind fields are often highly spatially nonlinear, especially within convection, they approach linearity and then uniformity when viewed at smaller scales. Therefore, a uniform wind model with two variables of horizontal wind components (*u*_{0}, *υ*_{0}) is adopted in retrieval experiments to find out the appropriate analysis volume and penalization parameter in actual wind retrieval. Simulated and dual-Doppler-retrieved wind fields were used to compare and test the two retrieval techniques for different sizes of analysis volumes. The retrievals show that VVP gives better results for a larger analysis volume compared with a smaller analysis volume. SVM-based VVP gives better results for a larger analysis volume given a large-scale uniform wind field compared with a smaller analysis volume while for a smaller analysis volume given a small-scale local wind field compared with a larger analysis volume. In addition, SVM-based VVP gives better results for a smaller 1/*c* given a large-scale wind field and a smaller analysis volume, while for a larger 1/*c* given a small-scale wind field and a larger analysis volume. Compared with VVP, the new algorithm has definite superiority with the selected appropriate penalization parameters when smaller analysis volumes are used. In actual wind retrieval, because both large-scale and small-scale winds should be retrieved usually at the same time, a small analysis volume should be adopted and thus the new algorithm should be used instead of the traditional algorithm. The new algorithm can provide acceptable high-resolution results of wind vectors, as well as the divergence and the vorticity.

We therefore recommend using the new algorithm to replace the traditional algorithm. It has the capability to analyze the dynamical structures of mesoscale and small-scale weather systems that usually produce severe weather. The wind retrievals obtained by the algorithm also have the potential to be assimilated into numerical models to produce still higher-quality research analyses or to improve the initialization of forecasts. However, the quantitative accuracy of the new algorithm still needs to be improved to acquire more satisfied results. There is concern that the tangential wind is difficult to acquire when the number of radials in an analysis volume adjoining the radar is very few, because in this situation the real variation of the radial speed between radials in such a small area is too obscure to achieve. In the current scheme of the algorithm, the wind retrieval is independent in each analysis volume and the relationships between the analysis volumes are ignored. It is necessary to make use of observations not only within but also outside an analysis volume to obtain more information and to keep the consistency of the wind field. A possible effective approach is to take into account the background wind by the variational method or the statistic interpolation (Xu et al. 2015).

## Acknowledgments

The authors are grateful to the reviewers and the editors for their constructive suggestions regarding earlier versions of this article. This study was supported by the China Commonwealth Industry Research Project (GYHY201306078), the Young Scientists Fund of the National Natural Science Foundation of China (41305031), and the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institution.

## REFERENCES

Armijo, L., 1969: A theory for the determination of wind and precipitation velocities with Doppler radars.

,*J. Atmos. Sci.***26**, 570–573, doi:10.1175/1520-0469(1969)026<0570:ATFTDO>2.0.CO;2.Brown, R. A., and V. T. Wood, 1983: Improved severe storm warnings using Doppler radar.

,*Natl. Wea. Dig.***8**(3), 17–27.Brown, R. A., and V. T. Wood, 1991: On the interpretation of single-Doppler velocity patterns within severe thunderstorms.

,*Wea. Forecasting***6**, 32–48, doi:10.1175/1520-0434(1991)006<0032:OTIOSD>2.0.CO;2.Browning, K., and R. Wexler, 1968: The determination of kinematic properties of a wind field using Doppler radar.

,*J. Appl. Meteor.***7**, 105–113, doi:10.1175/1520-0450(1968)007<0105:TDOKPO>2.0.CO;2.Cristianini, N., and J. Shawe-Taylor, 2000:

*An Introduction to Support Vector Machines and Other Kernel-Based Learning Method*s. Cambridge University Press, 204 pp.Deng, N., and Y. Tian, 2009:

*Support Vector Machine—Theory, Algorithm and Expansion*. Science Press, 244 pp.Inoue, H. Y., and Coauthors, 2011: Finescale Doppler radar observations of a tornado and low-level misocyclones within a winter storm in the Japan Sea coastal region.

,*Mon. Wea. Rev.***139**, 351–369, doi:10.1175/2010MWR3247.1.Koscielny, A., R. Doviak, and R. Rabin, 1982: Statistical considerations in the estimation of divergence from single-Doppler radar and application to prestorm boundary-layer observation.

,*J. Appl. Meteor.***21**, 197–210, doi:10.1175/1520-0450(1982)021<0197:SCITEO>2.0.CO;2.Lhermitte, R. M., and D. Atlas, 1961: Precipitation motion by pulse Doppler radar. Preprints,

*Ninth Conf. on Radar Meteorology*, Kansas City, MO, Amer. Meteor. Soc., 218–223.Lhermitte, R. M., and L. J. Miller, 1970: Doppler radar methodology for the observation of convective storms. Preprints,

*14th Conf. on Radar Meteorology*, Tucson, AZ, Amer. Meteor. Soc., 133–138.Li, N., M. Wei, X. Tang, and Y. Pan, 2007: An improved velocity volume processing method.

,*Adv. Atmos. Sci.***24**, 893–906, doi:10.1007/s00376-007-0893-0.Li, N., M. Wei, X. Mu, and C. Zhao, 2015: A support vector machine-based VVP wind retrieval method.

,*Atmos. Sci. Lett.***16**, 331–337, doi:10.1002/asl2.564.Miller, L. J., and R. G. Strauch, 1974: A dual Doppler radar method for the determination of wind velocities within precipitating weather systems.

,*Remote Sens. Environ.***3**, 219–235, doi:10.1016/0034-4257(74)90044-3.Tanamachi, R. L., H. B. Bluestein, M. Xue, W.-C. Lee, K. A. Orzel, S. J. Frasier, and R. M. Wakimoto, 2013: Near-surface vortex structure in a tornado and in a sub-tornado-strength convective-storm vortex observed by a mobile, W-band radar during VORTEX2.

,*Mon. Wea. Rev.***141**, 3661–3690, doi:10.1175/MWR-D-12-00331.1.Vapnik, V., 1998:

*Statistical Learning Theory.*John Wiley & Sons, 768 pp.Waldteufel, P., and H. Corbin, 1979: On the analysis of single-Doppler radar data.

,*J. Appl. Meteor.***18**, 532–542, doi:10.1175/1520-0450(1979)018<0532:OTAOSD>2.0.CO;2.Xu, Q., L. Wei, and K. Nai, 2015: Analyzing vortex winds in radar-observed tornadic mesocyclones for nowcast applications.

,*Wea. Forecasting***30**, 1140–1157, doi:10.1175/WAF-D-15-0046.1.