1. Introduction
Detecting and tracking mesocyclones from Doppler radial-velocity fields are very important processes for tornado-related severe weather warning operations, but the tasks involved often present enormous difficulties especially when mesocyclones are poorly resolved in the far radial ranges or confused with other signatures or data artifacts (such as noisy or improperly dealiased velocities) in radial-velocity fields. To overcome the difficulties, various automated mesocyclone detection methods and algorithms have been developed by many investigators (Stumpf et al. 1998; Smith and Elmore 2004; Liu et al. 2007; Newman et al. 2013; Miller et al. 2013). These methods rely on the assumption that a mesocyclone is behaving as a Rankine vortex and identify it as an object with no attempt to diagnose the detailed vortex wind field. By using a modified Rankine vortex model in combination with a uniform flow, a linear shear flow, and a linear divergence flow, Potvin et al. (2009, 2011) developed a technique for detecting mesocyclones and other convective vortices from multiple-Doppler observations and retrieving their size, strength, and translational velocity, but not the detailed vortex wind fields. To diagnose the full storm wind field, Gao et al. (2013) adapted a real-time three-dimensional variational data assimilation (3DVAR) system and showed the value of the wind field assimilated from multiple-Doppler radar data. This 3DVAR system compares favorably with the methods described above with regard to identifying storm-scale midlevel circulations, but the circulation may not be fully resolved because of the isotropic univariant background covariance used for each velocity component in the cost function. It is possible to improve the mesocyclone wind analysis by formulating vortex-flow-dependent background error correlation functions in cylindrical coordinates cocentered with the mesocyclone. This approach will be explored in this paper to develop a new method for mesocyclone wind analyses. The method can be used either in a stand-alone fashion or can be incorporated into the radar wind analysis system (RWAS; Xu et al. 2015) and performed as an additional step of targeted finescale wind analysis in the RWAS for nowcast applications.
The paper is organized as follows. The RWAS is briefly reviewed in the next section. The method for mesocyclone wind analyses is developed in section 3. The effectiveness and satisfactory performances of the method are demonstrated by illustrative examples in section 4. Conclusions follow in section 5.
2. Review of the RWAS
The initial version of RWAS was developed as a stand-alone system (without using any model-predicted background wind field) to retrieve real-time vector wind field data from single-Doppler radial-velocity observations at each selected vertical level or each selected tilt of radar scan superimposed on the radar reflectivity or radial-velocity image for nowcasting applications. This version of RWAS was evaluated for driving atmospheric dispersion models (Fast et al. 2008; Newsom et al. 2014) and implemented for operational test runs with atmospheric dispersion models. To monitor hazardous wind conditions, surface wind observations from the Oklahoma Mesonet have also been used in addition to radar radial velocities in the RWAS.
The RWAS contains a radial-velocity data quality control (QC) package to preprocess the raw data before the vector wind analysis. The QC package was recently upgraded with the newly developed algorithms to correct aliased velocities over areas threatened by intense mesocyclones and their generated tornados (Xu et al. 2013). The vector wind analysis in the RWAS uses the statistical interpolation (Daley 1991) to retrieve the horizontal vector wind field from radar radial velocities after QC. The vector wind analysis was also upgraded recently with extended capabilities to analyze radial-velocity observations from multiple radars with a model-predicted background wind field. In particular, high-resolution radial-velocity observations from multiple radars are combined into two (or three) batches of superobservations with the observation resolution coarsened to match the effective resolution of the analysis (i.e., about one-third of the decorrelation length of the background error correlation function used in the analysis) for each batch, so the observation resolution redundancy can be reduced to improve the computational efficiency (Xu 2011; Xu and Wei 2011). After this, the analysis is performed incrementally in multiple steps for different types of observations (from coarse to fine resolution) to cover and resolve different scales (from the synoptic to storm scale). For the mesocyclone vortex wind analysis presented in section 4a of this paper, the upgraded RWAS will be used to produce the mesoscale wind field by performing the following three steps:
A vertical profile of vector wind v = (zonal component u, meridional component υ) is produced by the velocity–azimuth display (VAD) method as a by-product of the VAD-based dealiasing (Xu et al. 2011, 2013) for each radar, and then the VAD winds are analyzed into the background wind field using the method of statistic interpolation described in section 3.1 of Xu et al. (2015). The background wind field is extracted from the nearest forecasts from the operational Rapid Refresh (RAP) model (http://rapidrefresh.noaa.gov/) by interpolating the predicted wind fields in time and space onto the analysis grid in a 800 × 800 × 10 km3 domain centered at the Twin Lakes, Oklahoma, Weather Surveillance Radar-1988 Doppler (KTLX). The analysis grid has a horizontal grid spacing of 10 km and contains 41 levels from the surface level (z = 10 m) to z = 10 km above the ground.
The wind field produced in step 1 is used as background to analyze surface wind observations (at z = 10 m) from the Oklahoma Mesonet employing the method described in section 3.2 of Xu et al. (2015).
The wind field produced in step 2 is used as background to analyze radar radial-velocity superobservations generated [using the method in section 3.2 of Lu et al. (2011)] in three batches with the observation resolutions coarsened to 30, 21, and 13 km (in both the radial and azimuthal directions), respectively, over the far radial range (r > 80 km), the middle radial range (40 < r ≤ 80 km), and the near radial range (r ≤ 40 km) from each radar. The analysis method is the same as described in section 3.3 of Xu et al. (2015) but is applied serially to the above three batches of superobservations. The background error decorrelation length (or depth) is reduced consecutively to 25 (or 2), 18 (or 1), and 11 (or 0.3) km when the analysis is performed with the first, second, and third batch, respectively.
Figure 1a shows the background wind field from the operational RAP forecast. Figure 1b shows the analyzed wind field produced by the RWAS using radial velocities scanned from five operational radars plus Oklahoma Mesonet wind data around 2211 UTC for the tornadic storm on 24 May 2011. In comparison with the background winds in Fig. 1a, the analyzed winds in Fig. 1b are adjusted toward radar-observed radial winds in and around the areas covered by radar radial-velocity observations, but the adjustments are too coarse and too smooth to resolve the mesocyclone (marked by the small yellow circle).
(a) RAP forecast wind field plotted by color-scaled arrows on the 800 × 800 km2 horizontal domain at z = 4 km superimposed on the reflectivity image from five radars for the tornadic storm at 2211 UTC 24 May 2011. (b) As in (a), but for the RWAS-analyzed wind field superimposed on the dealiased radial-velocity images at 4.0° tilt from KTLX and the Vance Air Force Base, Oklahoma (KVNX) radar; 0.9° tilt from KFDR; and 0.5° tilt from the Tulsa, Oklahoma (KINX), and the Fort Smith, Arkansas (KSRX) radars. In (b), the image from KTLX covers the image from KVNX, while the images from the remaining three radars are largely isolated around their respective radars. Positive (negative) values shown by the image from any one of the radars indicate horizontal flow away from (toward) that radar, while zero or near-zero values indicate flow perpendicular to the viewing direction from that radar. Each radar site is marked by a purple dot with the radar name in (b). The small yellow circle in (b) marks the tornadic mesocyclone. The thin green lines plot the state boundaries in (a) and (b), and the county boundaries in Oklahoma only in (b).
Citation: Weather and Forecasting 30, 5; 10.1175/WAF-D-15-0046.1
The existing RWAS is clearly unable to resolve the mesocyclone. This inability is tied up with the following two limitations. First, the effective resolution of the RWAS-produced incremental wind field is limited by the superobservation resolution and the decorrelation length of the background error correlation function used in the analysis. As shown in Fig. 1b, the mesocyclone is about 100 km away from KTLX, so the decorrelation length is 18 km and the superobservation resolution is coarsened to 21 km around the mesocyclone. The effective analysis resolution is thus limited by 21 km, which is obviously insufficient to resolve the mesocyclone. In addition, the ability of the existing RWAS to retrieve the unobserved wind component tangential to the radar beam is limited by the homogeneous and isotropic background error correlation functions [see (2) in Xu et al. (2015)] used for the vector wind analysis. This is the second limitation.
3. Vortex wind analysis
To resolve the mesocyclone, it is necessary to overcome the aforementioned two limitations. To this end, a new method is developed with the following three key components: 1) an algorithm for estimating the vortex center of the mesocyclone on a selected tilt of a radar radial-velocity scan, 2) a vortex-flow-dependent background error correlation function formulated for the vortex wind analysis over the mesocyclone area on the selected tilt, and 3) the square root of the vortex-flow-dependent background error covariance matrix derived analytically to precondition the cost function and thus enhance the computational efficiency. The method can be used as an additional (fourth) step of targeted finescale analysis after the third step is performed in the RWAS. It can be also used in a stand-alone fashion. In the latter case, it is necessary to estimate the environmental mean wind. The detailed techniques in the three components are presented in the following subsections.
a. Estimating vortex center location and environmental mean wind
The mesocyclone area is identified as a by-product of the automated velocity dealiasing [see the appendix in Xu et al. (2013)] on a selected tilt of low-elevation radar scan, and this is done by applying four cyclonic-rotation conditions to an 11 × 41 data window (11 beams and 41 range gates) centered at each flagged special data point that fails to pass a tightened continuity condition. The vortex center location is then estimated, also as a by-product of the automated velocity dealiasing, and is used here as the first guess. From this first guess, the vortex center location is further estimated in the mesocyclone area on the selected tilt by applying the following two-step algorithm to the data field of dealiased radial-velocity observations, denoted by 
- Find
Denote by
- Denote by
where
If the vortex wind analysis is performed in a stand-alone fashion with zero 
b. Cost function formulated with vortex-flow-dependent background error covariance
In (16b), the Gaussian correlation function is modified by subtracting its mirror image obtained by mirror reflecting the corrected point 
(a) Plot of 
Citation: Weather and Forecasting 30, 5; 10.1175/WAF-D-15-0046.1
Since the radial decorrelation length equals 1 in ρ, the associated radial decorrelation length in the physical space can be estimated by 
As shown above, l and Φ control the decorrelation lengths in the physical space along the radial and azimuthal directions, respectively. These scaling factors are specified empirically in this paper. The background wind error correlation structures and associated decorrelation lengths in the radial and azimuthal directions may be estimated statistically from time series of radar radial-velocity observations of mesocyclones by modifying the innovation method of Xu et al. (2007) with the covariance model used in this paper. This approach needs to be explored beyond this paper.
c. Square root of background error covariance and preconditioned cost function
For the selected values of l = ½, Φ = 1, and 
Since the nested domain is small, 
For the illustrative examples presented in the next section, the error standard deviation for the dealiased radial-velocity observations is set to σo = 2 m s−1 in the cost function, and this setting is the same as that used to estimate the superobservation error standard deviation [see section 3.3 of Xu et al. (2015)] for the vector wind analysis in step 3 of section 2. As we have seen in Fig. 1b, the RWAS-produced background winds are too smooth to capture the mesocyclone, so the background wind errors can be as large as the true vortex winds. Since the maximum of the true vortex winds estimated from 
Comparisons between the single- and dual-Doppler vortex wind analyses that use the RWAS-produced background wind field for the Oklahoma tornadic storm at 2211 UTC 24 May 2011. Here, VM (or RM) is the max tangential velocity of the vortex (or the radial distance of the max tangential velocity from the vortex center) estimated by (3) or (4) from single-Doppler radial-velocity observations, Vmax (or Rmax) is the max tangential velocity (or the radial distance of the max tangential velocity from the vortex center) in the analyzed vortex wind field, and RMSd (or RMSd5) is the RMS deviation of the single-Doppler-analyzed vortex winds from the dual-Doppler-analyzed vortex winds over the nested domain (or within R ≤ 5 km from the vortex center).
4. Illustrative examples
a. Vortex wind analyses performed with RWAS
The vortex wind field for the tornadic mesocyclone (marked by the small yellow circle in Fig. 1b) on 24 May 2011 is analyzed in this section as an incremental wind field with respect to the RWAS-produced background wind field. The analyses are performed by using single-Doppler wind observations (dealiased 
(a) Single-Doppler-analyzed vortex winds plotted by black arrows superimposed on color contours of the radial-velocity innovation field 
Citation: Weather and Forecasting 30, 5; 10.1175/WAF-D-15-0046.1
For the vortex winds in Fig. 3a (or Fig. 3b), the maximum velocity is Vmax ≡ max|(Δu, Δυ)| = 38.7 (or 34.1) m s−1 at the radial distance of Rmax = 1.06 (or 0.56) km from the vortex center, as listed in the first (or second) row of Table 1. For KFDR, the listed value of Vmax = 38.7 m s−1 is slightly larger than the maximum tangential velocity VM = 32.6 m s−1 estimated by (3), but the listed value of Rmax = 1.06 km is much smaller than the RM = 3.55 km estimated by (4) because RM is bounded below by the azimuthal resolution (
The radial-velocity innovation fields in Figs. 3a and 3b are separated in time (or height) by merely about 2 min (or 0.15 km), so these two data fields can be used together to analyze the averaged vortex wind field over the time interval between 2210:15 and 2212:23 UTC and in the vertical layer between 4.29 ≤ z ≤ 4.44 km. The dual-Doppler-analyzed vortex wind field is plotted by the black arrows in Fig. 3c (or Fig. 3d) against the green arrows that replot the single-Doppler-analyzed vortex wind field in Fig. 3a (or Fig. 3b). As shown in Fig. 3c (or Fig. 3d), the KFDR (or KTLX) single-Doppler-analyzed vortex winds match the dual-Doppler-analyzed vortex winds with the absolute value of their vector difference below 10 m s−1 in most areas (outside the purple contour loops). As listed in the last two columns of Table 1, the root-mean-square (RMS) deviation of the KFDR (or KTLX) single-Doppler-analyzed vortex winds from the dual-Doppler-analyzed vortex winds is RMSd = 7.1 (or 8.6) m s−1 over the nested domain and is RMSd5 = 7.4 (or 8.6) m s−1 within R ≤ 5 km from the vortex center. These RMSd and RMSd5 values are much smaller (by about 5 times) than the Vmax listed in the third column of Table 1.
The radial-velocity innovation field in Fig. 3a (or Fig. 3b) reveals 
As in the last four columns of Table 1, but for vortex wind analyses using the adjusted radial-velocity innovation data [see (10)]. In the first column, 
As in Fig. 3, but for adjusted radial-velocity innovation data [see (10)] in the vortex wind analyses.
Citation: Weather and Forecasting 30, 5; 10.1175/WAF-D-15-0046.1
The above comparisons suggest that the dual-Doppler analysis in Fig. 4 is slightly more accurate than that in Fig. 3, although the domain-averaged RMS difference between the two dual-Doppler-analyzed total wind fields is merely 0.2 m s−1. For the dual-Doppler analysis in Fig. 4, the total wind field obtained by adding the adjusted background wind field 
The total wind field produced by the KFDR (or KTLX) single-Doppler analysis in Fig. 4a (or Fig. 4b) is plotted in Fig. 5a (or Fig. 5b) by the color-scaled arrows superimposed on the KFDR (or KTLX) radial-velocity image. The total wind field produced by the dual-Doppler analysis in Fig. 4a is plotted in Fig. 5c by the color-scaled arrows superimposed on the KTLX radial-velocity image. The total wind field in Fig. 5c is replotted by the black arrows in Fig. 5d against the sparse brown arrows for the background wind field 
(a) As in Fig. 4a, but for the total wind field plotted by color-scaled arrows superimposed on the dealiased radial-velocity image from KFDR. (b) As in (a), but for KTLX. (c) As in (b), but for the dual-Doppler-analyzed total wind field. (d) As in (b), but for black arrows plotted together with the background winds 
Citation: Weather and Forecasting 30, 5; 10.1175/WAF-D-15-0046.1
b. Vortex wind analyses performed in stand-alone fashion
As explained at the end of section 3a, when the vortex wind analysis is performed in a stand-alone fashion, it is necessary to estimate 
As in the first three columns of Table 2, but for vortex wind analyses using the proxy background winds (listed in the first column). In the fourth (or fifth) column, RMSE (or RMSE5) is the RMS error of the analyzed total wind field evaluated against the benchmark truth total wind field in Fig. 4c over the nested domain (or within R ≤ 5 km from the vortex center).
As in Fig. 3, but for the stand-alone analyses that use the proxy background winds listed in the first column of Table 3 in place of the RWAS-produced background wind field.
Citation: Weather and Forecasting 30, 5; 10.1175/WAF-D-15-0046.1
The vortex wind field analyzed from the dual-Doppler proxy radial-velocity innovation data is plotted by the black arrows in Fig. 6c (or Fig. 6d) against the green arrows that replot the vortex wind field in Fig. 6a (or Fig. 6b). As shown in Fig. 6c (or Fig. 6d), the single-Doppler-analyzed vortex winds match the dual-Doppler-analyzed vortex winds as closely as those in Fig. 4c (or Fig. 4d) over the area of R ≤ 3 km around the vortex center (or over the entire nested domain), so the single-Doppler analysis can be as useful as the dual-Doppler analysis in terms of extracting the vortex wind field from radar radial-velocity observations. However, as shown in the first column of Table 3, the vector value of 
As indicated by the results in Fig. 6 and Table 3, the stand-alone single-Doppler analysis can reliably extract the vortex wind field but cannot reliably capture the high-wind areas in the total wind field if 
The stand-alone single-Doppler analysis has also been applied to the KTLX radial-velocity scans of the tornadic mesocyclone and its produced tornado, which was rated as category 5 on the enhanced Fujita scale (EF5). The tornado struck the cities of Newcastle and Moore, Oklahoma, in the afternoon (between 1445 and 1535 local time) on 20 May 2013. As an example, the analyzed vortex wind field is plotted in Fig. 7a by the black arrows superimposed on the color contours of the proxy radial-velocity innovation field computed from the KTLX radial-velocity observations at 0.5° tilt (around z = 0.67 km) at 2008:42 UTC. The estimated vortex center is at (rc, φc) = (28.875 km, 266°) in the KTLX coordinates, and this estimated vortex center is very close to the solid green zero-
(a) Stand-alone single-Doppler-analyzed vortex winds plotted by black arrows superimposed on the color contours of the proxy radial-velocity innovation field from KTLX at 0.5° tilt (around z = 0.67 km) for the tornadic mesocyclone at 2008:42 UTC 20 May 2013. (b) Single-Doppler-analyzed total wind field plotted by color-scaled arrows superimposed on the radial-velocity image from KTLX for the same case as in (a). The thin green lines in (b) show the county boundaries and the streets of Moore. KTLX is located at (x, y) = (28.80, −2.01) km outside the analysis domain.
Citation: Weather and Forecasting 30, 5; 10.1175/WAF-D-15-0046.1
As in Table 3, but for the single-Doppler vortex wind analysis of the tornadic mesocyclone scanned by KTLX at 0.5° tilt at 2008:42 UTC 20 May 2013. Here, RMSE (or RMSE5) is the RMS error of the total wind field in Fig. 7b evaluated over the nested domain (or within R ≤ 5 km from the vortex center) against the benchmark truth total wind field obtained by the same single-Doppler vortex wind analysis but using 
The total wind field [i.e., the vortex wind field in Fig. 7a plus the proxy background wind 
5. Conclusions
In this paper, a new method is developed for analyzing the vortex wind fields from radar-observed mesocyclones. The method contains three key components. The first component is an automated algorithm for estimating the vortex center of the mesocyclone (detected as a by-product of Doppler velocity dealiasing) on a selected tilt of a radar scan. The second component is a vortex-flow-dependent background error correlation function formulated in the cylindrical coordinates cocentered with the mesocyclone for the vortex wind analysis on the selected tilt. The third component is the square root of the vortex-flow-dependent background error covariance matrix derived analytically to precondition the cost function and enhance the computational efficiency.
The method is incorporated into the existing radar wind analysis system (RWAS; Xu et al. 2015) as an additional (fourth) step of targeted finescale analysis after the third step is performed in the RWAS. The method can be also used in a stand-alone fashion, but this stand-alone application requires the environmental mean wind to be estimated around the vortex and used as a proxy background wind.
The effectiveness and performance of the method are demonstrated by examples of analyzed vortex wind fields and total wind fields for the Oklahoma tornadic mesocyclones observed by KFDR and KTLX on 24 May 2011 and by KTLX on 20 May 2013. The results are summarized below.
When the method is used with the RWAS-produced background wind field, the single-Doppler-analyzed vortex wind field can match the gross pattern of the dual-Doppler-analyzed vortex wind field and the RMS difference between the two analyzed fields ranges from 7.1 to 8.6 m s−1 (see Fig. 3 and Table 1). Estimating the environmental mean wind around the vortex atop the RWAS-produced background wind field [see (8) and (9)] and using it to adjust the background wind field and thus the radial-velocity innovation data [see (10)] may only slightly improve the vortex wind analyses (see Fig. 4 and Table 2).
When the RWAS-produced background wind field or the adjusted background wind field is used, the total wind field produced by either single-Doppler or dual-Doppler analysis can be useful or, at least, much more useful than the RWAS-produced background wind field for nowcasting the tornadic mesocyclone and associated high-wind areas (see Fig. 5).
When the method is used in stand-alone fashion, it is necessary to use the estimated environmental mean wind around the vortex as a proxy background wind for the vortex wind analysis. In this case, although the analyzed vortex wind field is not very sensitive to the estimated environmental mean wind (see Fig. 6), the accuracy of the analyzed total wind field depends on the accuracy of the estimated environmental mean wind (see Tables 3 and 4).
The environmental mean wind can be estimated reliably from dual-Doppler radial-velocity observations [see (8)] but not reliably from single-Doppler radial-velocity observations [see (9)]. For the single-Doppler case, the vortex center moving velocity (estimated by the time change of the vortex center location from the previous to the current volume scan) can be used as a proxy background wind for the stand-alone single-Doppler vortex wind analysis. With a reliably estimated proxy background wind, the total wind field produced by the stand-alone single-Doppler analysis can be useful for nowcasting the tornadic mesocyclone and associated high-wind areas (see Fig. 7).
The consistency and stability of the method can be further examined and have been well verified by applying the method to consecutive data volumes for the two cases considered in this paper, but the detailed results are omitted. In summary, the method is computationally very efficient and it can retrieve the vortex part of the mesocyclone winds from merely single-Doppler observations. These are the strengths of the method. The method is expected to work best when the mesocyclone is intense and not too far (within 100 km) from the radar, and this is another strength of the method. On the other hand, the method may not work well when the mesocyclone is small and far from the radar where the radar beam becomes wider than the vortex core diameter. In addition, the method may not accurately retrieve the total wind field when the environmental mean wind cannot be reliably estimated from a single-Doppler volume scan in the stand-alone application. These are the weaknesses of the method. The vortex moving velocity may be used as a proxy background wind for the stand-alone application, but how to accurately estimate the vortex center moving velocity requires further investigation.
Since the multifunction phased-array radar (PAR) technology is under consideration for future operational weather radars (Zrnic et al. 2007), we will also use the PAR data to evaluate the performances of the method developed in this paper. The method can be extended to analyze three-dimensional vortex winds in Cartesian coordinates from either single- or multi-Doppler scans of mesocyclones with the background wind error correction functions formulated in a slantwise cylindrical coordinate system cocentered with the mesocyclone at each vertical level. This capability is currently being developed.
Acknowledgments
The authors are thankful to the anonymous reviewers, Vincent Wood, and Travis Smith of NSSL for their comments and suggestions that improved the presentation of the paper. The research was supported by ONR Grant N000141410281 to the University of Oklahoma (OU). Funding was also provided to CIMMS by NOAA/Office of Oceanic and Atmospheric Research under NOAA–OU Cooperative Agreement NA11OAR4320072, U.S. Department of Commerce.
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