#### Abstract

A radar wind analysis system (RWAS) has been developed for nowcast applications. By ingesting real-time wind observations from operational WSR-88D radars and surface mesonet, this system can produce and display real-time vector winds at each selected vertical level or on each conical surface of radar scans superimposed on radar reflectivity or radial-velocity images. An early version of the system has been evaluated and used to provide real-time winds to drive high-resolution emergency response dispersion models. This paper presents the detailed formulations of background error correlation functions used in each of the three steps of vector wind analysis performed in the RWAS and the method of solution used in each step of vector wind analysis. The performances of the RWAS are demonstrated by illustrative examples.

#### 1. Introduction

Built upon the recent successes derived from the Collaborative Radar Acquisition Field Test (Kelleher et al. [1]), a suite of real-time capabilities has been developed at the National Severe Storms Laboratory (NSSL) to process and display NEXRAD level II data through high speed internet (Lakshmanan et al. [2]). In addition, new radar data quality control techniques have been developed (Torres et al. [3], Zhang and Wang [4], Zhang et al. [5], Hubbert et al. [6], Witt et al. [7], and Xu et al. [8, 9]) to deal with various difficult data quality problems (Burgess and Crum [10]), especially those encountered by operationally used radar velocity dealiasing algorithm (Eilts and Smith [11]). With the improved radar wind data quality control, a radar wind analysis system (RWAS) has been also developed for nowcast applications.

The RWAS contains a radial-velocity data quality control (QC) package to preprocess the raw data for the vector wind analysis. The QC package was recently upgraded with the newly developed algorithms to detect and correct aliased velocities over areas threatened by intense mesocyclones and their generated tornados (Xu et al. [9]). The vector wind analysis in the RWAS uses the statistical interpolation (see Lorenc [12] and Daley [13]) with the background error covariance functions derived in Xu et al. [14] to retrieve the horizontal vector wind field from radar radial velocities preprocessed by the QC package.

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 from single-Doppler radial-velocity observations at each selected vertical level or on each conical surface of radar scans superimposed on radar reflectivity or radial-velocity image for nowcast applications. This version of RWAS was evaluated for driving atmospheric dispersion models (Newsom et al. [15]) and implemented for operational test runs with atmospheric dispersion models. To monitor hazardous wind conditions at high spatial and temporal resolutions, surface wind observations from the Oklahoma Mesonet have also been used in addition to radar radial velocities in the RWAS to produce real-time vector wind fields.

The RWAS is upgraded recently with extended capabilities to analyze radial-velocity observations from multiple radars with a model-predicted background wind field. In the upgraded RWAS, high-resolution (250 m in the radial direction and 1° in the azimuthal direction) radial-velocity observations from multiple radars are preprocessed by the upgraded QC package (with the new techniques developed and reported in Xu et al. [9]). The preprocessed high-resolution observations are then combined into two batches of superobservations with the observation resolution coarsened to 30 and 10 km, respectively, based on the criterion that the coarsened superobservations in the first (or second) batch retain and resolve the mesoscale (or storm-scale) structures in the observed radial-velocity field. The purpose is to reduce the observation resolution redundancy and improve the computational efficiency. 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 scale ranges. The background error covariance functions (derived in Xu et al. [14]) for analyzing radial-velocity observations on each tilt of single-radar scan are extended and modified, so the analysis can be applied to radial-velocity observations from multiple radars (see Figure 1) and the analyzed wind field can be computed directly and conveniently in - and -components (as in (11a) and (11b) of this paper) instead of the radial and tangential components (as in of Xu et al. [14]). This paper aims to present the detailed formulations of background error correlation functions and the method of solution used in each step of the vector wind analysis in the upgraded RWAS. The paper is organized as follows. The background error correlation functions used in the RWAS are presented in the next section. The techniques used in the three steps of vector wind analysis are presented in Section 3. The performances of the RWAS are demonstrated by illustrative examples in Section 4. Conclusions follow in Section 5.

#### 2. Background Error Correlation Functions Used in RWAS

##### 2.1. Background Error Correlation Tensor Function

The random horizontal vector fields of background wind errors normalized by their standard deviation are traditionally assumed to be horizontally homogeneous and isotropic over local (or regional) areas in the widely used statistical interpolation and three-dimensional variational techniques for large-scale and synoptic-scale atmospheric data analyses (Daley [13]). This assumption is likely to be less valid at the mesoscale and storm-scale, but it is still adopted commonly and implicitly along with the statistical interpolation and three-dimensional variational techniques in mesoscale data analysis. This assumption has been used to formulate the background wind error correlation functions in the RWAS, while the projection of the background wind error on each conical surface of radar scans was treated approximately as a horizontal component and is denoted by or, simply, by , where denotes the transpose of .

The covariance function of normalized background wind errors or, equivalently, the correlation function of background wind errors involves two points, say and on each vertical level, and is defined by the following second-order tensor function:
where denotes the statistical mean of , and for and 2 and is the background wind error standard deviation (at the concerned vertical level). The velocity vector can be projected onto the -direction along vector (from point to point ) and onto the -direction that is perpendicular to the -direction and is pointing to the left of the -direction (see Figure 1). The resulting components are denoted by and , respectively. These two components are related to by , where is the rotational matrix that rotates the -axis to the *-*direction and is the angle of the rotation, measured positively counterclockwise (see Figure 1).

Under the assumed homogeneity and isotropy, the canonical form of the correlation tensor defined by is diagonal and invariant with respect to translations and rotations of the system of points and (see Figure 5.1 and Section 5.2 of Daley [13]). This implies that is independent of *α*, where , , and are the horizontal distance between the two points. As shown in of Xu et al. [14], the four component equations of yield the following explicit expressions for the four components of :
where and . The two autocorrelation functions satisfy , so and . The above correlation functions will be simplified and extended in the vertical direction to construct the background error covariance matrix not only in the analysis space but also in the observation space of the horizontal vector winds produced by the velocity azimuth display (VAD) analysis (Lhermitte and Atlas [16], Browning and Wexler [17]) from each radar and surface winds from the Oklahoma Mesonet (see Sections 3.1 and 3.2).

##### 2.2. Autocorrelation Function for Radial Velocity

As shown in Figure 1, with respect to radar at , the radial component of ) at is given by , where . Similarly, with respect to radar at , the radial component of ) at is given by , where . Substituting these expressions into that defines the autocorrelation function for gives where , , , , and (2a), (2b), and (2c) are used. Note that (or ) is the angle of vector (or ) with respect to vector (measured positively counterclockwise), while is the angle of vector with respect to vector . The correlation function in (3) will be simplified and extended in the vertical direction to construct the background error covariance matrix in the space of radial-velocity superobservations (see Section 3.3).

##### 2.3. Cross-Correlation Functions

Substituting the expression of and into that defines the cross-correlation function between and gives where (2a), (2b), and (2c) are used. Similarly, substituting the expression of and into that defines the cross-correlation function between and gives The cross-correlation functions in (4a) and (4b) will be simplified and extended in the vertical direction to construct the background error covariance matrix between the analysis and radial-velocity observation spaces in Section 3.3.

##### 2.4. Simplifications

As shown in Xu et al. [18], (3) can be used to compute the autocovariance of radial-velocity innovations (obtained by subtracting the background radial velocity from the radar observed radial velocity at each observation location). From the computed innovation autocovariance, the function forms of and can be extracted and estimated. In particular, according to the results (Figure 4) of Xu et al. [18], the estimated is much larger than the estimated , which implies for mesoscale background errors correlation functions especially those estimated in Xu et al. [18]. As shown in of Xu et al. [18], implies , where and denote the background error covariance functions for the rotational and divergent parts of the horizontal velocity, respectively. This implied result of for the mesoscale background errors covariance functions is consistent with the mesoscale dynamics in which the rotational and divergent parts of the horizontal velocity have about the same order of magnitude. Thus, or, equivalently, has been used for mesoscale vector wind analyses in the RWAS. This simplification is still used in this paper. With this simplification, can be formulated by a Gaussian function and extended in the vertical direction (see (8), (10), (11a), and (11b)).

#### 3. Methods Used in Three Steps of Vector Wind Analysis

Ideally, if the background error covariance is known or can be accurately estimated, then the wind analysis should be performed by analyzing all the observations simultaneously in a single step of statistical interpolation (see Chapters 4-5 of Daley [13]). In practice, however, the background error covariance is often poorly known and cannot be accurately estimated especially if the estimated error covariance needs to cover a wide range of scales or to include some flow-dependences. In this case, it is convenient and effective to formulate the background error covariance with different decorrelation lengths and perform the wind analyses incrementally and successively with different types of observations that cover and resolve different scale ranges (from the synoptic scale to mesoscale). This multistep approach has been used in both the early and current versions of RWAS. The approach is similar to the sequential variational analysis of Xie et al. [19] except that the background error variance is either respecified empirically or reestimated statistically (see (9)) as the background wind field is updated and the background error decorrelation length is reduced or readjusted in each step.

The vector wind analysis in the current RWAS performs three steps. The first step analyzes the VAD vector winds produced at each radar site in the analysis domain into the background wind field. In this step, the nearest predictions from the NCEP operational rapid refresh (RAP) model are mapped using linear interpolations in time and space to the analysis grid to generate the background wind field. The analysis grid covers an area of 800 × 800 km^{2} centered at the KTLX radar (or any other selected location). The horizontal grid resolution is 10 km. The vertical grid contains 41 levels from the surface level to = 10 km above the ground with a resolution of = 250 m (except for the first vertical layer from = 10 m to 250 m). The wind field produced by the above first step is used as the background in the second step to analyze surface wind observations (at = 10 m) from Oklahoma Mesonet. The wind field produced from the second step is then used as a new background in the third step to analyze serially two batches of radial-velocity superobservations in two substeps. The first (or second) batch of superobservations is generated from dealiased radial velocities from each radar (by using the method described in Section 3.2 of Lu et al. [20]) with the observation resolution coarsened to 30 (or 10) km. In particular, the data area on each tilt of radar scan is partitioned into small sector areas of 30 (or 10) km radial length and azimuthal arc length for the first (or second) batch, and superobservations are generated by averaging the dealiased radial-velocity observations over each small sector area, as shown by the examples in Figure 4 of Lu et al. [20]. The wind field produced by analyzing the first batch in the first substep is used as a new background to analyze the second batch in the second substep. The methods used in the three steps of analyses are described in the following subsections.

##### 3.1. Method for Analyzing VAD Winds in the First Step

A vertical profile of VAD vector wind is produced above each radar site as a byproduct of the VAD-based dealiasing (Xu et al. [8]). The VAD winds are then used as observations and analyzed into the background wind field on the analysis grid by minimizing the following cost function: where , is the state vector of the analyzed wind field, is the state vector of the background wind field, is the background error covariance matrix, is the observation error covariance matrix, is the innovation vector, is the state vector of the VAD winds, and is the observation operator for the VAD winds that transforms the state vectors from the analysis space to the observation space via linear interpolations.

The minimizer of is obtained by solving the linear system of algebraic equations derived from . By using the matrix equality in (7B.6) of Jazwinski [21], the solution of the derived linear system can be cast into the following form:
The solution in (6) can be computed in the following two steps:
Here, the observation errors (for the VAD winds) are assumed to be unbiased and uncorrelated in space, so , where (≈3^{2} m^{2} s^{−2}) is the VAD wind error variance and is the identity matrix in the observation space. The and are computed by using (2a)–(2c) with formulated by a Gaussian function form extended in the vertical direction (see defined in (8)). Since is used here according to the simplification explained in Section 2.4, in (2c). This implies that the two components of VAD vector winds can be analyzed separately, so the formulations in (5)–(7a) and (7b) can be applied simply to each component wind field.

With the above simplification, the th element of can be computed directly for each component wind field by where , , , and (or ) denotes the decorrelation length (or depth) used in the vertical extended . The index (or ) goes from 1 to in the observation space for each component wind field, where is the total number of VAD vector winds.

The th element of is computed similarly as in (8) except that the index goes from 1 to in the analysis space for the analyzed -component (or -component) winds, where is the total number of analysis grid points. The above formulations will be used for analyzing the VAD winds in the first step by setting = 3 m s^{−1}, = 3 m s^{−1}, = 150 km, and = 1 km. The selection of = 150 km and = 1 km can be justified by the need of retaining and resolving mainly the subsynoptic-scale structures in the analysis, so the analyzed wind field can smoothly fill the gaps between sparsely distributed VAD winds.

##### 3.2. Method for Analyzing Mesonet Surface Wind Observations in the Second Step

The wind field produced by the previous step is used as a new background to analyze surface wind observations (at = 10 m from the Oklahoma Mesonet). The analysis is performed by minimizing the similar form of cost function as in (5) except that the observation state vector** y**, the observation operator , and the innovation vector are for the surface wind observations. The minimizer of the cost function is obtained by solving a similar form of linear system as in (6) with the solution computed in two steps similar to those in (7a) and (7b), but the decorrelation length (or depth) is reduced to = 60 km (or = 0.2 km) although we still set (but around a reduced value of about 2 m s^{−1}). The selection of = 60 km and = 0.2 km can be justified by the need of retaining and resolving mesoscale boundary-layer structures in the analysis, so the analyzed wind field can not only smoothly fill the gaps between the surface observations from the Oklahoma Mesonet but also extend the impact of the surface observations upward (up to = 0.5 km) within the boundary layer.

##### 3.3. Method for Analyzing Radial-Velocity Superobservations in the Third Step

The wind field produced from the previous step is used as a new background to analyze the first batch of radial-velocity superobservations (with the observation resolution coarsened to 30 km to retain and resolve the mesoscale structures, as explained in the introduction section) in the first substep of the third step. The wind field produced from the first substep is then used as a new background to analyze the second batch of radial-velocity superobservations (with the observation resolution coarsened to 10 km to retain and resolve the storm-scale structures) in the second substep of the third step. The analysis of each batch of superobservations is performed by minimizing a similar form of cost function as in (5) except that the observation state vector** y**, the observation operator , and the innovation vector are for the radial-velocity superobservations in the first (or second) batch. The minimizer of the cost function is obtained by solving a similar form of linear system as in (6) with the solution computed in two steps similar to those in (7a) and (7b), but the observation variance is estimated for each individual superobservation as described below and background error covariance matrices are computed by using (9)–(11a) and (11b) formulated below.

The observation errors (for the radial-velocity superobservations) are also assumed to be unbiased and uncorrelated in space, but the error variance is nonconstant as it is estimated for each individual superobservation by , where is the number of dealiased radial velocities used to produce the concerned radial-velocity superobservation, = 2 m s^{−1} is the error standard deviation estimated for dealiased radial velocities, and = 1 m s^{−1} accounts for bias errors (Salonen et al. [22]) in the dealiased radial velocities that are not considered when the superobservation error variance is estimated using the local averaging. Thus,** O** is a diagonal matrix with nonconstant diagonal elements.

After the background is updated by VAD winds in the first step, the new background errors should still be largely independent of the radial-velocity superobservations errors. Thus, the background error variance can be estimated by subtracting the superobservations error variance from the superobservation-generated innovation variance, as shown by the diagonal part of the innovation covariance matrix formulated in of Xu et al. [18]. When the innovation variance in this relationship is computed approximately by a local spatial averaging (instead of statistic averaging), the background error variance can be estimated as a smooth function of by
where denotes the summation over index from 1 to* M*—the number of radial-velocity superobservations in the batch to be analyzed, is the innovation (radial-velocity superobservation minus background radial velocity) at the th superobservation point, is the error variance estimated for the th superobservation, and (or ) is the decorrelation length (or depth) in the vertical extended used below in (10), (11a), and (11b).

By using the vertically extended (3) with and the above second refinement, the th element of can be computed directly by where is the observation space for the radial-velocity superobservations, , , and (or ) is the radar location for the radial-velocity superobservation at point (or ). Similarly, by using the vertically extended (4a) and (4b) with and the above second refinement, the th element of can be computed directly by where (or ) denotes the analysis subspace for the analyzed -component (or -component) winds. We set = 100 km (or 50 km) and = 0.8 km in (9)–(11a) and (11b) for analyzing the first (or second) batch of radial-velocity superobservations in the first (or second) substep of the third step.

#### 4. Illustrative Examples

##### 4.1. Illustrative Examples from Stand-Alone RWAS

The stand-alone version of RWAS has been applied to single-Doppler radial-velocity observations (quality-controlled but without superobbing) from the operational KTLX radar in Oklahoma, surface wind observations from the Oklahoma Mesonet, and upper-level wind observations from the NOAA Profiler Network (see Figure 2). No model-predicted background wind field is needed or used by this stand-alone system. Instead, a single vertical profile of VAD winds is produced in the first step as a byproduct from the VAD-based dealiasing (Xu et al. [8]), and this vertical profile of VAD winds provides a vertical varying but horizontally uniform background wind field for the analysis in the next step, that is, the second step (or third step without superobbing) described in Section 3.2 (or Section 3.3) if surface mesonet winds are available and used (or not available or not used). This stand-alone RWAS can run very efficiently on a single workstation to produce real-time vector wind fields in a mesoscale domain of 160 × 160 × 5 km^{3} centered at the radar site. The real-time wind products can be displayed and superimposed on radar reflectivity and radial-velocity images with NSSL/WDSS-II (Lakshmanan et al. [2]) to monitor low-level wind conditions at high spatial and temporal resolution for nowcast applications. As shown by the example in Figure 3, the vector wind fields produced by the stand-alone RWAS can capture strong horizontal convergences and/or shears associated with mesoscale fronts. This capability has been examined in detail and demonstrated for both idealized and real radar observations of mesoscale fronts in Xu et al. [14]. As mentioned in the introduction, the early stand-alone RWAS (without using surface mesonet winds and profiler winds) was also used and evaluated for driving atmospheric dispersion models (Newsom et al. [15]) and implemented for operational test runs with atmospheric dispersion models.

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The stand-alone RWAS has been also used for monitoring radar data quality and wind product quality since its real-time run started in May 2007. From the real-time runs, occasional failures were detected from the wind products during severe ice storms in 2008 and January 2009. The failures were found to be rooted in falsely dealiased radial velocities. The VAD-based dealiasing (Xu et al. [8]) was found to be unsuitable for raw radial-velocity data scanned from winter ice storms by the operation WSR-88D radars using VCP31 mode with the Nyquist velocity reduced below 12 m s^{−1}. A new dealiasing scheme was then developed by using the alias-robust variational (AR-Var) analysis to replace the alias-robust VAD analysis for the reference check (see Section 2 of Xu et al. [9]). This AR-Var-based dealiasing upgraded the VAD-based dealiasing adaptively for scan modes VCP31 (used by the operational WSR-88D radars) and Mod80 (used by the FAA Terminal Doppler Weather Radar (TDWR)) with reduced Nyquist velocities (<18 m s^{−1}). The superior performance of this adaptively upgraded dealiasing is shown by the example in Figure 4 for severely aliased raw radial velocities scanned at 0.5° tilt by the KTLX radar using VCP31 for the Oklahoma ice storm on January 27, 2009 (examples for severely aliased raw radial velocities scanned at higher tilts and different times from winter ice storms were shown in Figure 1 of Xu et al. [9]).

Figure 5 shows the vertical profiles of zonal and meridional components, , of VAD wind (produced as byproducts from the AR-Var-based dealiasing) for the winter ice storm case in Figure 4. As shown, there was a very strong vertical shear around = 1 km above the ground. Above and below this shear layer, the horizontal winds are both strong and quite uniform but in the opposite directions. Around and across the shear layer, the incremental wind field produced in the third step tends to be adversely affected by the excessively specified vertical correlation depth ( = 0.8 km) in the background error covariance matrices (constructed in (10), (11a), and (11b)). This can cause spurious horizontal variations (associated with the nonuniform coverage and distributions of the radar volumetric data) in and/or near the shear layer. This problem is shown by the wind field produced by the stand-alone RWAS at = 0.7 km in Figure 6(a). To solve the problem, a new shear-dependent factor is introduced to the vertically extended (with = 0 in (3)-(4a) and (4b)) to reduce the vertical correlation adaptively across a strong vertical-shear layer. This shear-dependent factor is formulated by where is the difference between the background horizontal velocities at the two corrected points (i.e., points (, ) and ) and is the decorrelation scale in the new parameter dimension of . This shear-dependent factor is multiplied to the background error covariance formulation in (10)-(11a) and (11b) for constructing and . The vertical correlation in the background error covariance is thus refined adaptively (for winter ice storms). As explained at the beginning of this section for the stand-alone RWAS, the background wind field is given by a single vertical profile of VAD winds produced in the first step. In this case, in (12) becomes independent of and thus is simply a function of . The shear-dependent factor formulated in (12) remains applicable when the background wind field becomes horizontally nonuniform either due to the use of mesonet winds in the second step in the stand-alone RWAS or due to the use of a model-predicted background wind field in the upgraded RWAS. This shear-dependent factor is used adaptively for winter ice storms or stratiform precipitations in which the horizontal winds are strongly sheared in the vertical around inversion layers.

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As shown in Figure 6(b), when the shear-dependent factor in (12) is used with = 1 m s^{−1}, the spurious horizontal variations are eliminated from the wind field produced by the stand-alone RWAS at = 0.7 km. The wind field produced by the stand-alone RWAS at = 1 km is plotted in Figure 6(c). As shown, the wind intensity and horizontal variability increased significantly and the wind directions largely reversed as increased from 0.7 to 1 km. The horizontal variations around the radar at = 1 km in Figure 6(c) are consistent with the dealiased radial velocities around = 1 km on each and every different tilt (data not shown), so they are more realistic and much less spurious than those in Figure 6(a). Thus, the stand-alone RWAS is able to capture and resolve not only strong horizontal convergences and/or shears associated with mesoscale fronts (as shown in Figure 3) but also sharp vertical shears in winter ice storms.

##### 4.2. Illustrative Examples from Upgraded RWAS

The upgraded RWAS has been applied to radial-velocity observations from five operational WSR-88D radars (KTLX, KFDR, KINX, KVNX, and KSRX) and one TDWR radar (TOKC) in combination with surface wind observations from Oklahoma Mesonet to produce vector wind fields for the Oklahoma Moore tornadic storm case on May 20, 2013. As explained at beginning of Section 3, the background wind field is from the nearest RAP model predictions interpolated in time and space to the analysis grid in a 800 × 800 × 10 km^{3} domain centered at the KTLX radar. As an example, Figures 7(a) and 7(b) plot the background wind fields at the vertical levels of = 2 km and 4 km, respectively, from the RAP 1-hour prediction valid at 1900 UTC on May 20, 2013. The background wind field at = 2 km (or 4 km) is superimposed on the dealiased radial-velocity images (reflectivity images) scanned at 0.5° tilt from the six radars. The plotted background winds show the prestorm wind conditions on the subsynoptic scale around 1900 UTC, which was about 1 hour before the small emerging storm cell marked by the white letter E in Figure 7(b) developed into a supercell and generated a devastating tornado over the Oklahoma Moore area on May 20, 2013. In particular, the prevailing environmental winds were southwesterly and increased rapidly with height in the lower and middle troposphere (from = 1 km to 5 km) over the entire Oklahoma except for the panhandle area.

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Over and to the northwest of the Oklahoma panhandle area, as shown in Figure 7(a) (or Figure 7(b)), the environmental winds changed to northwesterly (or westerly) around = 2 km (or 4 km), indicating that a cold front had moved into the northwest corner of the analysis domain around 1900 UTC. The movement of this cold front reflected by the RAP 1-hour prediction in Figure 7(a) is slower than that implied by the radial-velocity observations from the KVNX and TOKC radars. The radial velocities scanned at 0.5° tilt from KVNX and TOKC were confined mostly within the radial range of 100 km from each radar and thus below = 2 km. The color images of the dealiased radial velocities from these two radars in Figure 7(a) indicate that the lower-level winds had already changed to northeasterly around KVNX and northwesterly around TOKC at 1900 UTC. The radial-velocity images from these two radars revealed that the lower-level frontal shear and convergence zone was much sharper and located further southeastward (by nearly 200 km) than that suggested by the RAP 1-hour prediction as shown by the predicted background winds at = 0.25 km in Figure 8(a). This lower-level frontal shear and convergence zone is well captured by the upgraded RWAS as shown by the analyzed winds at = 0.25 km in Figure 8(b).

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To show the overall performance of the upgraded RWAS, the incremental winds produced by the upgraded RWAS are plotted at = 2 km in Figure 9(a) and at = 4 km in Figure 9(b). Note that the incremental winds are plotted with enlarged arrow lengths and narrowed color scales in Figure 9 than those for the background winds in Figure 7, so the analyzed winds (i.e., the incremental winds plus the background winds) are not too far from the background winds. The incremental winds reveal major and significant corrections made to the background winds in and around areas covered by the radial-velocity observations from the six radars. In particular, the incremental winds in Figure 9(a) (or Figure 9(b)) show how the background winds are corrected at = 2 km (or 4 km) over the areas covered by the radial-velocity observations from the KVNX and TOKC radars as upward extensions to those shown in Figure 8 and discussed in the previous paragraph. The incremental winds in Figure 9 also reveal a major correction made to the background winds with enhanced cyclonical rotation and horizontal shear along and across the dry line (along the west edge of the series of storm cells in Figure 9(b)) in the lower and middle tropospheres.

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With the enhanced cyclonical rotation and horizontal shear along and across the dry line as discussed above, the analyzed wind field by the upgraded RWAS contains a lower-middle-level jet to the southeast of the supercell marked by the white letter S, as shown in Figure 10(b). This lower-middle-level jet is quite distinctive in the analyzed wind field as shown by the red arrows in Figure 10(b) but is not so for the RAP model-predicted background wind field in Figure 10(a). As an important feature in the prestorm environment wind field; this jet and its associated cyclonic shear (as revealed by the strong incremental winds marked by the red arrows in Figure 9(b)) were conducive for the further intensification of the supercell (marked by the white letter S) that moved to the jet-headed convergence area (marked by the red letter S in Figure 9(b)) during the next two hours. As shown by the above example, the upgraded RWAS is useful for studying and nowcasting the environment wind conditions prior to and during severe storms.

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#### 5. Conclusions

The early stand-alone version of radar wind analysis system (RWAS) developed at NSSL is upgraded with extended capabilities to analyze radial-velocity observations from multiple radars, while the initial background wind field is no longer zero but extracted from the nearest operational model predictions by interpolating the predicted wind fields in time and space to the analysis grid in an enlarged mesoscale domain (800 × 800 × 10 km^{3}). The background error correlation functions used in the upgraded RWAS are formulated in Section 2 with detailed derivations. The methods of solution used in the three steps of the vector wind analysis in the upgraded RWAS are presented in Section 3 with detailed descriptions and explanations concerning why and how the analysis is performed in multiple steps with the background field updated in each step. It is also described and explained how the observation and background error variances are estimated either statistically or empirically in each step and how the radial-velocity superobservations are generated in two batches with their error variances estimated individually for the analyses performed serially with the two batches in the third step. The performances and related utilities of the early stand-alone RWAS and upgraded RWAS are demonstrated by illustrative examples in Section 4.

The formulations and methods of solution used in the RWAS have several important merits not discussed in the previous sections. These merits are described and explained below. Note that the analyzed incremental winds are computed directly from the analytical formulation of the background error covariance matrix (i.e., in (7b)), so the analyzed wind field is essentially a continuous field infinitely differentiable. This has the following two implications. (i) The vorticity and divergence fields can be also computed directly with no numerical discretization and related truncation error from the analytical formulations (not shown in this paper) derived by applying the corresponding differential operators to the background error covariance function in the analysis space. (ii) The effective horizontal (or vertical) resolution of the analyzed wind field is a fraction (about 1/3) of the decorrelation length (or depth) of the background error correlation function used in the analysis. The effective resolutions of the analyzed wind field in each step further determines the optimal analysis grid spacing and the optimal observation resolution for each batch of radial-velocity superobservations generated by superobbing the dealiased high-resolution radial velocities in the third step. Optimizing the observation resolution by superobbing with minimized loss of information content from observations can reduce the computational cost substantially without deteriorating the analysis accuracy (Purser et al. [23], Liu et al. [24], Alpert and Kumar [25], Salonen et al. [26], and Simonin et al. [27]). The use of superobbing in the upgraded RWAS is thus computationally critical for real-time wind analyses and related nowcast applications.

Although the stand-alone RWAS and upgraded RWAS are both useful for studying and nowcasting the environment wind conditions prior to and during severe storms, their analyzed wind fields are still too smooth to resolve fine interior structures of storm winds especially mesocyclones imbedded within supercells. To overcome or alleviate this limitation, an additional step of targeted fine-scale analysis without superobbing needs to be designed and performed in a nested domain (to cover just the concerned storm) after the third step in the upgraded RWAS. Along this direction, two approaches can be explored and developed. The first approach is to use properly simplified model equations to constrain the wind analysis in a way similar to that of Gao et al. [28]. The second approach is to formulate the background error correlation function adaptively with certain desired flow-dependences. This approach is being explored for analyzing radar observed mesocyclone using vortex-flow-dependent background covariance. The above approaches will be developed to further upgrade the RWAS for detecting tornadic and nontornadic mesocyclones and nowcasting their associated storm winds from radar radial-velocity observations.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors are thankful to Dr. Jidong Gao of NSSL and the anonymous reviewer for their comments and suggestions that improved the presentation of the results. This research was supported by the ONR Grant N000141410281 to the University of Oklahoma (OU). Funding was also provided by DOC/NOAA/OAR under NOAA-OU Cooperative Agreement no. NA17RJ1227.