## 1. Introduction

*V*in this paper). Nyquist velocities of weather radars are usually 8–32 m s

_{N}^{−1}. The radar can correctly measure velocities within the interval of ±

*V*. Velocities outside the interval, however, cannot be correctly measured resulting in apparent values that reside within the interval of ±

_{N}*V*. In general, the observed velocity values and their true values are related by The true velocity is denoted by

_{N}*V*,

_{T}*V*is the observed velocity,

_{O}*n*is an integer (0, 1, 2, . . .), and

*V*represents the Nyquist velocity.

_{N}The correction of aliased velocities—the so-called dealiasing or unfolding—is a challenging technical task and becomes increasingly difficult with a decreasing Nyquist velocity or increasing noise in the data. Since aliasing is easily identified as abrupt changes in the velocity data field, most of the dealiasing techniques are based on spatial and temporal continuities. These encompass spatial and temporal continuities from one-dimensional (along a radial, e.g., Ray and Ziegler 1977; Bargen and Brown 1980), two-dimensional (along both radial and azimuth, e.g., Merritt 1984; Boren et al. 1986; Bergen and Albers 1988, Eilts and Smith 1990; Gong et al. 2003), to even four-dimensional constraints (along radial, azimuth, elevation, and time, e.g., James and Houze 2001). For the correction of aliased velocities, each velocity observation is compared with a reference velocity and the difference between the two is compared with a prespecified wind-shear threshold based on Nyquist velocity. If the difference exceeds the threshold, then the radial velocity in question is considered aliased and will be subject to correction until it is within the wind shear threshold from the reference velocity.

With many algorithms, the reference velocity is derived from the average of preceding neighbors of the velocity gate in question. The preceding neighbors are assumed to be correctly dealiased. However, at the boundaries, such as the first gates in each radial and first radial/gates in an isolated storm area, no preceding neighbors are available. In this case, independent velocity data, such as those from upper-air soundings (e.g., Eilts and Smith 1990) or velocity azimuth displays (VADs; e.g., Gong et al. 2003), are used. The upper-air sounding data are usually very sparse in space (every 300+ km) and in time (every 12 h). Hence, they are not representative of small-scale wind shears potentially leading to errors within velocity aliasing. VAD winds are available more frequently in time and are spatially consistent with the velocity field to be dealiased. However, the VAD provides little information on horizontal variations in the wind field. In addition, VAD winds are subject to errors caused by the aliased velocities themselves (Gong et al. 2003).

This paper presents a new multiple-pass, 2D (radial and azimuth) technique and the subsequent algorithm to obtain reference velocities more easily than other schemes. The new technique eliminates the dependency on external data sources while facilitating a dealiasing algorithm with a high computational efficiency for operational implementation. Similar to Eilts and Smith (1990), the new technique is based on continuities in velocity fields along radial and azimuth directions. Multiple passes of dealiasing and error checking similar to the concepts in James and Houze (2001) are performed for robust dealiasing. The new multipass velocity dealiasing algorithm first finds a set of reference radials and gates by detecting the weakest wind region. Then from these reference radials and gates, the scheme checks continuities among adjacent gates in radial and azimuth directions and corrects for the velocity values with large differences that exceed the prespecified wind shear threshold(s). Problematic velocity gates that do not have good reference velocities during an initial pass are not processed until a subsequent pass when additional good gates are identified.

The next section, section 2, describes each step in the new dealiasing algorithm. Case studies utilizing the dealiasing algorithm are presented in section 3 followed by a brief summary in section 4.

## 2. Algorithm descriptions

The new velocity dealiasing algorithm includes the following modules:

- the initial radial search,
- the initial three reference radials dealiasing,
- the first round of radial-by-radial dealiasing, and
- the second round of radial-by-radial dealiasing and error check.

These procedures are similar to the “initial dealiasing” and “spatial dealiasing” steps in James and Houze (2001). Figure 1 provides an overview flowchart of the algorithm. Detailed descriptions for each individual module are provided in sections 2a–d.

### a. The initial radial search

*V*is radial velocity;

*i*= 1, 2, . . . ,

*N*

_{0}, represents all nonmissing velocity gates; and

*α*is an adaptable parameter with a default value of 0.8.

*N*

_{1}is the number of gates with velocity values that lie between ±

*ßV*interval. Here

_{N}*ß*is another adaptable parameter (default = 0.3) for defining small velocity values. If

*V*

_{M}_{1}changes sign from two consecutive radials to the next two consecutive radials (Fig. 2), then one of the two middle radials that has a larger

*N*

_{1}is selected as the initial radial.

If an initial radial could not be found through the above approach, a second, iterative, search is carried out. In the second search, satisfying the following three criteria identifies a good radial:

- no large wind shears between any adjacent gate pairs in the radial [see Eq. (2)];
- a large number of nonmissing velocity gates (i.e.,
*N*_{0}≥ 40); and - a small mean of all the nonmissing velocities (i.e.,
*V*_{M0}= (1/*N*_{0})Σand_{i}V_{i}*V*_{M}_{0}<*ßV*)._{N}

If more than one such good radial is identified, then the one with the smallest *V _{M}*

_{0}is selected as the initial radial. If no such radials were found in the first-pass search, then criterion 2 is relaxed by 5 (i.e.,

*N*

_{0}≥ 35) and another search for good radials is initiated. This process is repeated until an initial radial is identified or until

*N*

_{0}becomes smaller than 5. If no good radial is found with

*N*

_{0}≥ 5 and

*V*

_{M}_{0}<

*ßV*, then no dealiasing would be performed to the current tilt.

_{N}Within each step of the new dealiasing algorithm, the gates are marked with four different flags to track processes that each point has gone through. The flag values range from 0 to 3 that represent the unprocessed point: 0; the processed point with a high level of confidence: 1; a processed point but with a low level of confidence: 2; and the range-folded or missing value point: 3. Different flags are handled differently during the dealiasing and error checking. Once a velocity gate is flagged as “1,” it is considered a correctly dealiased (i.e., good) gate and will not be processed further saving computational time and improving the overall algorithm efficiency. A similar flagging process was also used in James and Houze (2001).

The initial reference radial only needs to be in the vicinity of the weakest wind region. It does not have to be a perfect radial of zero isodop. False zero isodops may pose challenges to the initial reference radial search process since they can be incorrectly identified as the initial reference radial and consequently fail the dealiasing algorithm. Although this would only occur when radials near the false zero isodop meet the following criteria: 1) having a small mean velocity along the radials and a smaller V_{M0} than have the radials near the true zero isodop, and 2) a low gate-to-gate wind shear along the radials.

### b. Initial three reference radials dealiasing

This module first performs dealiasing for all points in the initial radial using *V _{M}*

_{1}(or

*V*

_{M}_{0}depending on how the initial radial was identified) as the reference velocity. The module then processes the two neighboring radials of the initial radial using the initial radial as the reference. The algorithm dealiases each velocity gate using azimuthal continuity constraints in comparison with the neighboring radials against gates at the same range in the initial radial. If a reference velocity is available in the initial radial, then the corresponding gate in the neighbor radial is processed and flagged as 1. Otherwise, the gate in the neighbor radial is flagged as “0” (“unprocessed”) and is postponed for later passes.

The initial radial and its two neighbor radials (a total of three) will serve as reference data for the next step, which is the first round radial-by-radial dealiasing. Since additional continuity constraints in azimuthal direction are used when obtaining the initial three reference radials, the chances of random errors being in the reference velocities are reduced from using just one radial as the reference.

### c. The first round of radial-by-radial dealiasing

Starting from radials next to the three initial reference radials, the algorithm traverses through the tilt radial-by-radial in two passes: one in the clockwise direction followed by one in the counterclockwise direction. Each pass goes through 180° (Fig. 3). The purpose of employing both clockwise and counterclockwise dealiasing is to improve the algorithm performance around shear zones. This strategy also restricts any potential error propagations inside a 180° sector.

Each radial is processed in two steps in the clockwise and counterclockwise passes: in the azimuth and the radial direction. Along the azimuth direction, each velocity gate is compared to a reference velocity *V _{R}*, which is the average of three velocities in preceding radials (in the clockwise or the counterclockwise direction) at the same range (Fig. 4). Three instead of one reference gate are used here to avoid incorrect dealiasing due to random errors in preceding velocity gates. To get a valid

*V*, the three preceding gates must satisfy the following criteria:

_{R}- All three gates must have a processing flag of 1 (i.e., they must have been dealiased with high level of confidence); and
- Any two adjacent velocity pairs must satisfy Eq. (2).

*V*is found for any given gate or, if the difference between

_{R}*V*and the velocity at the given gate is larger than

_{R}*αV*, then the gate is flagged as 0, or, unprocessed, and postponed for subsequent dealiasing along the radial direction.

_{N}The radial direction dealiasing starts with a procedure of finding an initial (good) gate. The following criteria are used for initial gate identification (Fig. 4).

- It must have at least 10-km contiguous nonmissing neighbor gates on each side of the radial direction. This criterion is important to assure a reference velocity far away from echo boundaries.
- It must be a processed gate with a flag of 1 and must have at least two neighbor gates on each side with flag of 1 along the radial direction. This will make five contiguous good gates centered at the initial gate and the five gates must satisfy Eq. (2).
- It must have at least three contiguous neighbor gates with flag of 1 in preceding radials at the same range (Fig. 4); and the three velocities must satisfy Eq. (2).

The searching for the initial gate starts from the outermost gate and proceeds inward. Once the initial gate is found, the algorithm performs dealiasing gate-by-gate along the radial direction (both toward and away from the radar) starting from the initial gate. The gates are dealiased using the average of three preceding good velocities (i.e., with flag of 1) in the same radial as a reference. The initial five contiguous good gates ensure that a reference gate can be found at least initially. And the reference gate has good chance to have a correct velocity given the conditions required to identify the initial gate. Note that the dealiasing in the radial direction may overwrite results from the dealiasing in azimuth direction. This strategy helped to correct inappropriate dealiasing along azimuth direction at the far ranges. The inappropriate azimuth dealiasing may occur at far ranges because the adjacent gates in the two neighboring radials are far apart and realistic large wind shears may exist between the gates.

If a break (e.g., a missing-value or range-folded gate) is encountered during the searching/dealiasing, then a new search for another initial gate is performed. The searching/dealiasing are repeated until the ends of the radial are reached. Any gates that get processed are marked with 1 and those that remain unprocessed are marked with a 0 and left for subsequent processes. The radial direction dealiasing process is illustrated in Fig. 4.

The criteria for identifying reference velocities in the first round is relatively strict in that horizontal wind shears around the reference velocity gates must be small and a reference velocity is only allowed to influence a small region in the vicinity. Therefore, the velocities processed during the first round are considered correct with high level of confidence providing a good background for further dealiasing. The unprocessed gates that did not have good reference velocities in their neighborhood during the first round will have a better chance to find valid reference velocities in the second round because of more good velocity gates (with flags of 1) that become available after the first round of dealiasing.

### d. The second round of radial-by-radial dealiasing

The second round of radial-by-radial dealiasing checks and performs dealiasing for all points that were not processed by the first round of radial-by-radial dealiasing. This module is similar to the first round but with a relaxed search radius for finding a reference velocity. For each unprocessed gate in the current radial (i.e., radial 0 in Fig. 5), the algorithm first finds the three neighbor gates in preceding three radials (radials “−1,” “−2,” and “−3,” in Fig. 5). If the three velocity gates satisfy the following conditions

- |
*V*−_{i}*V*_{i−1}| <*αV*, where_{N}*i*= −1 and −2; and, - |
*V*_{0}−*V*| <_{R}*αV*, where_{N}*V*= ⅓Σ_{R}^{−3}_{i=−1}*V*,_{i}

*V*

_{0}gate is dealiased using

*V*as reference. Otherwise, the searching goes one radial further back and checks the three velocities at the same range as

_{R}*V*

_{0}in radials −2, −3, and −4. This procedure is repeated

*M*times (

*M*= 3 as a default) or until a good

*V*is found. If no good

_{R}*V*is found after all the procedures, then the dealiasing is ended.

_{R}Along the radial direction, the search for unprocessed gates starts at an arbitrary midrange (default = 20 km) and proceeds both toward and away from the radar. The midrange beginning is preferred over the first-gate beginning because random noises often exist near the radar due to residual clutter. For any unprocessed gate, the algorithm searches for the nearest three contiguous dealiased gates along the radial direction both toward and away from the radar. The search is continued until three contiguous dealiased velocities are found or a range limit Δ*r* (i.e., the distance between the velocity gate in question and the closest gate in the three contiguous gates; default = 5 km) is exceeded.

The second round radial-by-radial dealiasing is repeated 2 more times with relaxed *M* (search radius in azimuth direction, default = 6° and 10°, respectively, in the two additional passes) and Δ*r* (search radius in radial direction, default = 20 and 50 km, respectively, in the two additional passes). The additional two passes are needed for dealiasing velocities in isolated storm regions away from the radar.

## 3. Case studies

The new 2D multipass velocity dealiasing algorithm has been tested on more than 1000 volume scans of data from the Weather Surveillance Radar-1988 Dopplers (WSR-88Ds) located in Taiwan (RCWF, Wu-Fen-Shan, Taipei, Taiwan) and in the United States (KTLX, Oklahoma City, Oklahoma, and KMHX, Morehead City, North Carolina). The processed volumes encompassed events of tornadoes, typhoons and hurricanes. The dealiasing algorithm performs very well in more than 99% of the aliased velocity observations (the aliased velocities were identified by meteorological experts). There are less than 1% of the cases where the algorithm failed to recover the correct velocities. Those velocities not correctly dealiased were primarily confined to small areas near data holes or near range-folded velocity data. The following sections include several case examples of the dealiasing results for several cases.

The operational WSR-88D velocity dealiasing algorithm was used to provide an independent dealiasing scheme to compare performance results with the automated 2D multipass velocity dealiasing algorithm. For each case dataset evaluated, the WSR-88D algorithm was run twice: the first run was used to initialize the algorithm’s environmental wind table with a representative vertical wind profile obtained by processing the velocity data through the VAD algorithm; the second run performed the final velocity dealiasing using the updated environmental wind table. Comparisons of the results from the two schemes are shown in sections 3a and 3b.

### a. Continuous velocity fields with vertical wind shear and tornadic storms

When the velocity field is continuous, both the automated 2D multipass dealiasing algorithm and the operational WSR-88D algorithm performed very well. Figure 6 shows radial velocity fields before and after using the new dealiasing algorithm for Hurricane Isabel as observed by KMHX radar at 1537 UTC 18 September 2003. The operational WSR-88D results (not shown here) are the same as the results from the automated 2D multipass dealiasing algorithm. Note that the zero radial velocity spirals around the radar in this case (Fig. 6a) indicating high wind shears in vertical direction. There are two false zero velocity regions in the tilt: one to the northeast and one to the southeast of the radar between the 50- and 100-km range (Fig. 6a). The algorithm successfully chose the initial reference radials near the true zero isodop to the southwest of the radar because the false zero velocity regions did not meet the weak gate-to-gate wind shear condition due to aliasing errors.

Figure 7 provides another example of continuous velocity field using a tornado case that occurred in Oklahoma. Even though this is an isolated supercell storm with small spatial scale (∼50 km × 50 km; Fig. 7d), the velocity field was very continuous given the fine spatial resolution of 250 m × 1° (Fig. 7a). The automated 2D multipass velocity dealiasing algorithm and the operational scheme produced similar results for this case (Figs. 7b,c), although the operational scheme used the VAD velocities as references while the automated 2D multipass scheme did not use any external data for references.

### b. Aliased velocities near data voids and range-folded areas

When significant data voids and discontinuities exist in the velocity field, dealiasing schemes often fail. Figures 8 and 9 show examples of velocity fields observed during the Typhoon Nari event on 15 September 2001. As a result of many range-folded regions and data voids, the operational WSR-88D dealiasing scheme incorrectly modified radial velocities in a couple of areas (see white bold circles in Figs. 8b and 9b). In contrast, the new 2D multipass dealiasing algorithm was able to dealiase the velocity field correctly (Figs. 8c and 9c). The multipass and azimuth direction dealiasing procedure helped to improve the results for this case. Furthermore, multiple error checks along both azimuth and radial directions helped minimize any conflicts in the final velocity field.

Figure 10 shows examples of the velocity fields observed during the Hurricane Isabel event on 18 September 2003. As a result of many range-folded regions and data voids, the operational WSR-88D dealiasing scheme incorrectly modified radial velocities in a region west-northwest of the radar (see the white bold circle in Fig. 10b). In contrast, the new 2D multipass dealiasing algorithm was able to dealiase the velocity field correctly (Fig. 10c) except for a much smaller area surrounded by the range-folded gates.

## 4. Summary

A new 2D multipass velocity dealiasing algorithm has been developed. The algorithm is an automated scheme that finds reference velocities in the input velocity field and does not require external data sources. A multipass dealiasing procedure is employed in the algorithm that uses very strict criteria in earlier passes and relaxed criteria in later passes. The strict criteria for finding reference velocities in the earlier passes ensure relatively high-quality dealiased velocities. These dealiased velocities then provide a good reference for dealiasing in the later passes. The initial research presented here found that multipass neighborhood checking and iterative dealiasing procedures improve the stability and robustness of the dealiasing algorithm. The dealiasing algorithm was tested across ∼1000 volume scans of radar data from domestic and international WSR-88Ds encompassing events of tornadoes, typhoons, and hurricanes. The algorithm can correct for aliased Doppler velocity data properly in more than 99% of the cases tested and examined. The automated 2D multipass dealiasing algorithm showed improvements over the current operational WSR-88D dealiasing scheme in regions near data voids and range-folded observations. The algorithm dealiases radial velocity fields tilt by tilt without using external data or VAD velocities as references. Thus, it may facilitate a rapid update of dealiased radial velocity fields for use in severe weather applications on a tilt-by-tilt basis.

The new automated 2D multipass dealiasing algorithm is simple, stand-alone, and computationally efficient (average CPU is 5 s for processing one volume scan data on a Dell PC with 700-MHz processor and 3.6 GB of RAM). The current algorithm has been tested mainly on hurricane, typhoon, and tornado events. Further testing on a variety of dealiasing situations (including precipitation events in complex terrain environments, high shear environments, or isolated severe thunderstorm events) will be conducted in the future.

The authors thank Dr. Jen-Tsin Deng and Dr. Pao-Liang Chang of the Central Weather Bureau of Taiwan for their input during the initial development of the scheme. Mr. Hoyt Burcham and Mr. Mike Jain of the National Severe Storms Laboratory provided results from the operational dealiasing algorithm and their help is greatly appreciated. The authors would like to thank Mr. Ken Howard and Mr. Kurt Hondl for their reviews that helped immensely in the preparation of this manuscript. Major funding for this research was provided under the Federal Aviation Administration (FAA) Aviation Weather Research Program Advanced Weather Radar Technologies Product Development Team MOU and partial funding was provided under NOAA–University of Oklahoma Cooperative Agreement NA17RJ1227, U.S. Department of Commerce, and through the collaboration with the Central Weather Bureau of Taiwan.

This research is in response to requirements and funding by the FAA. The views expressed are those of the authors and do not necessarily represent the official policy or position of the FAA.

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