## Introduction

Although Doppler radar can scan the atmosphere with a high spatial and temporal resolution, its major limitation is that only the radial component *V*_{rad} of a three-dimensional wind vector can be detected. The unobserved cross-beam components, that is, the azimuthal *V*_{azi} and polar *V*_{θ} velocities, need to be retrieved by other means. The deployment of multiple Doppler radars is one solution for determining the complete flow structure (Armijo 1969; Doviak and Zrnić 1984). Unfortunately, difficulties such as the economic cost of more radar sites, accurate antenna calibration, and synchronous observations by all radars make this scenario unlikely to be common in daily operations. Over the past decade, the radar-meteorology community has made a considerable effort to develop techniques whereby the unobserved transverse wind components can be estimated directly from single-Doppler radar measurements without sacrificing data resolution (e.g., Liou et al. 1991; Sun et al. 1991; Qiu and Xu 1992; Laroche and Zawadzki 1994; Shapiro et al. 1995). This technique is called the single-Doppler velocity retrieval (SDVR). Because of the discreteness of radar measurements, estimates of local temporal derivatives by the finite differencing of two observations from consecutive volume scans may lead to large errors. To mitigate this kind of error, Gal-Chen (1982) suggested the use of a moving frame of reference for which the time tendency would be as small as possible. Zhang and Gal-Chen (1996) adopted this concept to perform single-Doppler wind retrievals for a microburst case during the Phoenix II field program and for a Florida sea-breeze case during the Convection and Precipitation/Electrification Experiment. The behavior and sensitivity of this retrieval scheme in comparison with different kinds of internal and external factors, based on idealized flow and reflectivity patterns, have been examined by Lazarus et al. (1999). Liou (1999) has demonstrated that a significant improvement in the original moving-frame method could be achieved by taking into account the continuity equation and a weak constraint for the vertical vorticity in the retrieval formulation. This modified method is referred to hereinafter as L99.

Usually, the SDVR techniques are validated using the results obtained from either a numerical model, or from multiple-Doppler syntheses. In other words, the “correct” flow structure is available as a reference. However, when retrieval schemes are actually applied in the real world, the most commonly seen scenario is that there is no prior knowledge available about the true wind fields. Under this condition, when a wind field is finally recovered, it is very natural to ask the following questions. Can one trust this result? Specifically speaking, if the performance and scanning strategy of all radars are presumably the same, then is there an “optimal” location for the SDVR? How would the SDVR be affected by the radar's viewing angle? Suppose the radar happens to be at a location for which most of the wind vectors are parallel (perpendicular) to the radar beams. Would this condition be favorable (unfavorable) for the SDVR? Last, a very practical issue is that, when the results of SDVR turn out to be bad, can this fact be immediately recognized from the SDVR information alone?

In this research, Doppler radar data collected during the 1987 Taiwan Area Mesoscale Experiment (TAMEX) intensive observation period (IOP) 2 are first used to explore the performance of L99. In addition, the focus is on the study of how the SDVR results vary with respect to the radar's viewing angle.

The paper is organized as follows. In the next section the L99 method is reviewed. Section 3 briefly introduces the TAMEX IOP-2 datasets and the weather phenomena associated with them. Section 4 designs several indices for inter comparisons of the results. Section 5 demonstrates the usefulness of L99. In section 6, experiments are conducted to study the impact of the radar location on the retrievals, followed by conclusions in section 7.

## The modified moving-frame-of-reference SDVR technique

In this section we briefly review L99, the modified moving-frame-of-reference method for SDVR.

### The mean wind

*U,*

*V,*and

*W*can be obtained by minimizing the local temporal derivative of a scalar field (or reflectivity

*η*in the radar observation) in a moving frame. If the formulation is written in a fixed coordinate with respect to the ground, one getswhere

*W*

_{t}represents the terminal velocity of the precipitation particles and can be estimated empirically from the reflectivity data, Ω covers the entire retrieval domain and extends in time from the first to the last radar scans, and

*α*is a weighting coefficient that balances the magnitude of each penalty term in the cost function. The

*U,*

*V,*and

*W*can be solved by setting the derivatives of

*J*with respect to the unknowns to vanish in (1). After obtaining the optimal moving speed, a new reference frame is formed byHere

*t*

_{0}is a reference time. All the radar data need to be redefined within this moving frame.

### The perturbation wind

*u,*

*υ,*and

*w*are decomposed intowhere

*U*and

*V*are the optimal advection velocities obtained in (1) and

*u*′,

*υ*′, and

*w*′ are the unknown perturbation velocities. A slightly revised cost function from L99 is formulated as follows:where

*J*

_{1}–

*J*

_{6}, the prime denotes that these variables are obtained in the moving frame of reference. In

*J*

_{4},

**V**′ is the three-dimensional perturbation wind, and ‖‖ denotes the magnitude of the vector. In

*J*

_{6}, we haveNote that

*p*

^{′}

_{1}

*p*

^{′}

_{2}

*p*

^{′}

_{3}

*r*′ is the distance between the radar and the grid point, and

*V*

^{′}

_{rad}

In (4), *J*_{1} denotes the conservation law for reflectivity in the moving frame; *J*_{2} is the geometric relation between the observed radial wind and the retrieved *u*′, *υ*′, and *w*′. Constraint *J*_{3} is implemented to reduce the divergence; in *J*_{4}, a weak vorticity condition is added to serve as a smoothness constraint to prevent ill conditioning and to reduce the impact of noise on the retrievals (Sasaki 1970; Qiu and Xu 1996; Xu et al. 1995). In *J*_{5}, the overbar represents the spatial average throughout a horizontal plane. When information about the horizontal mean winds is available from other analyses, such as the Velocity Azimuth Display (VAD; Browning and Wexler 1968), this constraint minimizes the difference between the horizontally averaged *u*′ and *υ*′ and the VAD products. As discussed in the next section, the Doppler radar data collected for this research were obtained from radar sector scans. A VAD analysis was therefore not performed during IOP-2. Nevertheless, *J*_{5} remains as an option in the retrieval formulations and can be easily turned on/off by giving it a nonzero/zero weighting coefficient. Constraint *J*_{6} provides higher-order spatial filters to remove noise. Its weighting coefficient *α*_{6} is estimated in a manner similar to Testud and Chong (1983).

After the gradients of cost function *J* with respect to the unknown variables *u*′, *υ*′, and *w*′ are calculated, (4) is minimized using a limited-memory, quasi-Newton, conjugate-gradient algorithm known as “VA15AD” (Liu and Nocedal 1988). Readers can refer to Liou (1999) for more details about the derivations. The resulting wind perturbations *u*′, *υ*′, and *w*′ are added along with the mean wind *U* and *V* in (3) to obtain the complete flow field.

## TAMEX IOP-2 and the dual-Doppler observations

TAMEX IOP-2 took place on 16–17 May 1987. This particular case represented a long-lasting subtropical squall line and has been discussed in detail by many researchers. For example, an investigation of the squall line's kinematic structure was made by Wang et al. (1990), a look at its thermodynamic structure and momentum budgets was taken by Lin et al. (1990), and the kinetic energy budgets were studied by Lin et al. (1991). Chen (1991) explored the squall line's behavior using a two-dimensional cloud model, and the study of the orographic effects of Taiwan's terrain on this squall line can be found in Teng et al. (2000).

During IOP-2, two 5-cm Doppler radars [National Center for Atmospheric Research (NCAR) CP-4 and Tropical Ocean and Global Atmosphere (TOGA)] were operated simultaneously in the early morning of 17 May to observe the northern portion of the squall line while it was still over the ocean. Figure 1 shows the dual-Doppler analysis domain (enclosed by an inner rectangle), of which CP-4 and TOGA conducted intensive sector scans. The shaded contour lines denote the radar reflectivity at the height of 2 km. A strong north–south reflectivity line is easily identified. To the east of the radar sites reflectivity data were not available because of the sector scanning strategy. The coordinates of CP-4 and TOGA are (0.0, 0.0, 0.009) km, and (−14.66, −41.25, 0.206) km, respectively. The height is measured as above ground level. The datasets utilized for this research, with approximately 3 min of separation, were collected by CP-4 at 0038:56–0041:33, 0041:38–0044:14, and 0044:19–0046:55 and by TOGA at 0038:29–0041:41, 0041:48–0044:59, and 0045:29–0048:39 local standard time (LST), respectively. Dual-Doppler syntheses were performed using the Custom Editing and Display of Reduced Information in Cartesian Space (CEDRIC) software, developed by NCAR (Mohr et al. 1986), to generate the “true” low-level (2 km) wind field at 0042:00 LST, shown in Fig. 2. The shaded area represents the region having radar reflectivity greater than 30 dB*Z.* Although it is somewhat difficult to identify, the wind vectors seem to converge along a north–south-oriented line, mainly because of changes in their magnitudes. This line also agrees well with the high-reflectivity zone. According to Wang et al. (1990), the squall line moved from 250° at a nearly constant velocity of 16.5 m s^{−1}. Therefore, by subtracting the system's velocity, one can obtain the storm-relative wind field, displayed in Fig. 3. The north–south-oriented low-level convergence line mentioned above is now even more evident, especially over the northern portion of the domain. Based on these wind distributions, Fig. 4 illustrates the negative horizontal divergence (convergence) field in a more quantitative manner. To the east of the convergence line, the warm, moist environmental air moves toward the squall line, feeding the convective updrafts. This mechanism results in a wide area of high reflectivity along the convergence zone. Figures 2, 3, and 4 are used as references to validate the SDVR results.

## Indices for verification and comparison

*A.*” In this paper its root-mean-square (rms) amplitude is defined bywhere

*N*is the number of points in the retrieval domain. Let the subscripts

*t*and

*r*stand for the dual-Doppler-derived true field and the “retrieved” result from SDVR, respectively; the directional difference between the true (

*u*

_{t},

*υ*

_{t}) and the retrieved (

*u*

_{r},

*υ*

_{r}) wind vectors, ranging from 0° to 180°, is evaluated at each grid point. Then, the averaged directional difference over the entire area is computed by the following formula:

Equation (10) is applied only to the ground-relative wind fields. One can easily show that the angle between two vectors can change significantly before and after being subtracted by a third common vector, that is, using the storm-relative coordinate. For example, if at a certain point, the observed wind vector is **V**_{1} = (*a,* *a*), its retrieved counterpart is **V**_{2} = (*b,* *b*). In this case, their directional difference is 0°, which means the wind direction has been perfectly recovered. Now, suppose the storm's speed of movement is **U** = (*b,* *a*); then the storm-relative vectors for the observed and retrieved winds become **V**_{1} − **U** = (*b* − *a,* 0) and **V**_{2} − **U** = (0, *b* − *a*), respectively, and the directional difference turns out to be 90°. In real applications, including the case presented in this research, the moving speed of a weather system is sometimes determined subjectively, and this example demonstrates that one has to be careful when interpreting the results on a storm-relative coordinate. Thus, the directional difference is estimated using the ground-relative wind fields.

## Results of SDVR

### Using CP-4 data

Using the CP-4 radar observations as the input data, the optimal moving speed obtained from (1) is *U* of about 11.20 m s^{−1} and *V* of about 9.10 m s^{−1}. Figure 5 displays the resulting SDVR low-level wind structure from (4), and Fig. 6 shows the storm-relative flow pattern. As discussed in section 4, comparisons of two storm-relative flow fields are sometimes very sensitive to the storm's velocity. Thus, the good agreement between Figs. 2 and 5 and the similarity between Figs. 3 and 6 are strong evidence that the unknown total velocity vectors, which involve both the magnitude and the direction, have been satisfactorily recovered by the SDVR scheme. Based on the SDVR wind vectors, the negative horizontal divergence (convergence) field is plotted in Fig. 7. When compared with the dual-Doppler solutions shown in Fig. 4, it can be seen that the L99 SDVR method has successfully reproduced the convergence line. Table 1 lists the values of the quantitative indices introduced in section 4, along with the true values from the dual-Doppler analyses. We note that the radial wind is slightly underestimated by about 0.5 m s^{−1}, but the underestimation along the tangential component is large (∼2.2 m s^{−1}). A further discussion of Table 1 is given in section 5c.

At a specified site, the radar observes each portion of the retrieval domain from several different viewing angles. Therefore, it is interesting to examine how the quality of the retrievals varies spatially. This issue is dealt with in the following.

*V*

_{azi}by its radial part

*V*

_{rad}to show the weight of the former. However, in those regions in which the flow is nearly perpendicular to the radar beam (

*V*

_{rad}∼ 0), the resulting ratio may become extremely large and is difficult to interpret. Thus, another parameter

*R*

_{at}, which measures the ratio between the azimuthal wind and the total wind at each point is given byWhen |

*V*

_{azi}| is much larger (smaller) than |

*V*

_{rad}|,

*R*

_{at}approaches 1 (0). When the two components are comparable,

*R*

_{at}is approximately 1/

*V*

_{rad}(retv) or

*V*

_{azi}(retv)] from SDVR, at each grid point, are compared with their dual-Doppler-synthesized true counterparts [

*V*

_{rad}(true) or

*V*

_{azi}(true)] by computing the following two quantities:

As viewed from the CP-4 site, Fig. 8 displays the spatial distributions of the true *R*_{at}, using wind data from dual-Doppler analyses. The dark area indicates the ratio where *R*_{at} is less than 0.05, and the light-shaded area denotes *R*_{at} greater than 0.95. This plot reveals that, for the IOP-2 flow field, the total winds near the southern boundary of the domain are mostly parallel to the CP-4 radar beam (i.e., |*V*_{rad}| ≫ |*V*_{azi}|), but they gradually become more perpendicular to the radar beam (i.e., |*V*_{rad}| ≪ |*V*_{azi}|) as one moves toward the northeastern corner of the domain. The above-mentioned relation between *V*_{rad} and *V*_{azi} is found to be indicated well by the retrieved *R*_{at} field, which is based on the radial and azimuthal winds deduced from the SDVR algorithm and the CP-4 data, as shown in Fig. 9.

To study how the retrievals vary spatially, the parameters *R*(rad) and *R*(azi) defined in (13) and (14) are illustrated in Figs. 10 and 11, respectively. In these two plots, “bad” retrievals can be identified easily by their negative or large values (∼1). In Fig. 10, this occurs in the northeastern part of the domain, which implies that the radial winds have been improperly retrieved in this region. The same situation takes place for the azimuthal winds near the southern border of the domain, as revealed in Fig. 11. By a comparison with Figs. 8 or 9, it fortunately can be found that the location where a component (either radial or azimuthal) is poorly recovered coincides with the region where this particular wind component is also the weaker one. As a result, the SDVR-deduced total winds would not be hampered too seriously by the bad retrievals.

### The SDVR using CP-4 data over a larger domain

In the previous experiment, the dual-Doppler analysis covered a relatively small domain, with a size of 50 km × 60 km. In Wang et al. (1990) and Lin et al. (1990), the analysis field was even smaller (45 km × 25 km). In the next experiment, the CP-4 datasets, centered at 0041:38 LST, are used to recover the wind field for a more extensive 80 km × 100 km area. Note the size of this domain already exceeds that for which dual-Doppler syntheses can be performed properly. The distance between the lower-left-hand corner of this large domain and the CP-4 site almost reaches the maximum detectable range of the CP-4 (120 km). Figure 12 depicts the SDVR storm-relative winds. The verification by the dual-Doppler syntheses over the limited domain suggests that SDVR wind estimates using CP-4 data should be reliable over a larger domain. In addition, Fig. 13 shows the derived negative horizontal divergence (convergence) field. It can be seen that the north–south-oriented low-level convergence line stretches over a longer range and that its position agrees very well with the high-reflectivity zone, especially outside the dual-Doppler domain. This result implies that one should be able to trust the SDVR flow field for the large domain. The experiment discussed in this section demonstrates the usefulness of the L99 method, especially when dual-Doppler observations are limited to a small domain or even are not available at all.

### The SDVR using TOGA radar data

In section 5a, the L99 scheme has demonstrated its ability to obtain satisfactory retrievals. However, one is not certain whether the conclusions reached by using CP-4 datasets still remain valid for other radars. Thus, in this section, the data measured by TOGA radar are substituted into the L99 method. The SDVR products are examined along with the CP-4 results and the dual-Doppler synthesized winds. The strength and weakness of the L99 method can be further studied by performing this three-way cross comparison.

Using the TOGA data, the optimal moving speed obtained from (1) is *U* of about 12.72 m s^{−1} and *V* of about 9.46 m s^{−1}. Figure 14 illustrates the ground-relative wind field, and it agrees well with the dual-Doppler and the CP-4 results. However, the storm-relative wind field, as displayed in Fig. 15, does show apparent differences from the dual-Doppler and CP-4 analyses, especially along the convergence line and throughout the southwestern portion of the domain. Although the example in section 4 has demonstrated the sensitivity of the storm-relative wind with respect to the moving speed of the system, Fig. 15 still implies that the quality of the SDVR using TOGA data is inferior to its CP-4 counterpart. In this particular case, it is somewhat difficult to identify the convergence line from the winds. Therefore, the negative divergence (convergence) field is calculated and is displayed in Fig. 16. From this, one does indeed find that the retrieved convergence by SDVR is weaker and smaller in scope, although its location still agrees with the high-reflectivity zone.

Table 1 shows a quantitative comparison between the CP-4 and TOGA results. The smaller AOR (0.84) for CP-4 reveals that, because of its relatively “good” position, it can extract more information from the total winds than the TOGA radar does. The TOGA radar is located at a disadvantageous site, with a much higher AOR (2.57), so that most of the wind vectors are perpendicular to the radar beam, thus becoming unobservable. By comparing the rms magnitudes of the true radial wind with the retrieved one, this component is found to be underestimated, but only slightly (<0.5 m s^{−1}). However, the underestimation turns out to be severe for the azimuthal winds (2.2 m s^{−1} for CP-4 and 3.2 m s^{−1} for TOGA). This finding may indicate that the underestimation of *V*_{azi} is unavoidable, which is probably acceptable when |*V*_{azi}| is smaller than |*V*_{rad}|. When the cross-beam wind happens to be the principal component of the total wind (because of the geographic location of the radar site), however, a full recovery of the missing component faces greater difficulties. The MVE value also suggests that the CP-4 retrievals are better than those of TOGA retrievals. However, information about total wind directions, even from the relatively bad viewing angle of TOGA radar, probably still is useful, as revealed by the small *θ*_{diff}.

Figure 17 shows the spatial distributions of the dual-Doppler-derived true *R*_{at} from the TOGA radar site. As explained in (12), this variable represents the relative strength of the azimuthal winds with respect to the total winds. In this picture, a northwest–southeast-oriented light-shaded area where *R*_{at} is greater than 0.95 is very prominent. Apart from this region, *R*_{at} gradually decreases outward to approximately 0.7 near the boundaries. In other words, from the TOGA radar site, the unobserved azimuthal winds are either greater than, or comparable to, the observed radial winds for the entire scanning domain. It is very encouraging to find that the above-mentioned structure has been reproduced in the retrieved *R*_{at} from SDVR, as depicted by Fig. 18.

If the conclusions obtained from the CP-4 results in section 5a are correct, then, according to the plots of *R*_{at}, either the observed or the retrieved ones, we may expect to see poorly retrieved radial winds throughout the northwest–southeast-oriented region. To confirm this expectation, parameter *R*(rad), defined in (13), is displayed in Fig. 19. One does see a narrow band of bad values stretching from the northwest corner to the southeast corner. As for the tangential winds, they should be recovered reasonably well everywhere in the domain, because ratio *R*_{at} is found to be large enough (>0.5) for the entire area (see Figs. 17 and 18). This is also confirmed by *R*(azi), shown in Fig. 20. It should be pointed out that no negative *R*(azi) is spotted in Fig. 20, which implies that the wind directions at all points are correctly retrieved. However, in the northwest–southeast-oriented region in which the radial winds are improperly retrieved, the magnitude of *R*(azi) ranges from approximately 0.60 to 0.65. By definition, from (14), this is equivalent to saying that at these points only 75%–85% of the true azimuthal winds are reproduced. This explains the underestimation of the azimuthal winds shown in Table 1.

The above discussion of the CP-4 and TOGA results does shed some light on the performance of L99. However, this comparison may not be appropriate because the CP-4 radar is distinct from the TOGA radar, not only in terms of the geographic locations but also hardware configuration. As indicated by Wang et al. (1990), for example, the CP-4 radar has a narrower beam width (1.02°) and a stronger peak power (400 kW) and is more sensitive to echoes (the minimum detectable reflectivity is −16 dB*Z*). Thus, the performance and quality of their measurements must be different, and the conclusions reached in the previous sections should be applied with caution. In the next section, another approach is used to investigate how the radar location affects the retrievals.

## SDVR versus radar location

In this section, the factors associated with the radar viewing angle are isolated so that one can discuss their influence on the retrievals. Lazarus et al. (1999) reported that the quality of the SDVR retrieval did depend on the radar location. However, this conclusion was based on a highly idealized divergent flow, and thus its applicability to real datasets such as in the current study is not certain.

*u,*

*υ,*and

*w*are projected onto a specified site at which a “virtual” Doppler radar is positioned. Now, assuming that the location of this virtual radar is the point of origin (0.0, 0.0, 0.0), its observed radial wind

*V*

_{rad}can be calculated using the following formula:where (

*x,*

*y,*

*z*) is the location of each data point and

*r*is the distance to the virtual radar. Through this process, the quality of the datasets collected by any virtual radar can be placed on an equal basis and any comparisons become more meaningful.

With the above experimental designs in mind, the CP-4 and TOGA radar datasets are synthesized to produce three-dimensional flow fields at three times, 0040:00, 0043:00 and 0046:00 LST, respectively. A total of 11 virtual radars, to which the L99 algorithm is applied, are created using the newly projected radial winds to recover the complete velocity field at 0043:00 LST. Figure 21 illustrates the relative positions of these 11 virtual radars with respect to the retrieval domain. From due north, they are equally spaced 18° apart. As one can see, for the same area and flow pattern, each radar actually observes a different portion of the wind vectors. For some radars, for example, radar 4 or radar 5, most of the wind vectors are parallel to the antenna beam and can thus be observed well. By contrast, radar 8 and radar 9 contain wind vectors in the unobservable cross-beam components. How the radar position affects the SDVR is the main focus of this section.

Quantitative SDVR results using the L99 technique for the 11 virtual radars are listed in Table 2. For convenience of discussion, the dual-Doppler- and SDVR-generated AOR for each virtual radar is characterized somewhat subjectively into three groups. As discussed later, the group-averaged statistics do exhibit systematic variations. In group I, the AOR is smaller than 0.7. This condition is considered to be the one in which the unobserved azimuthal wind is smaller than the radial wind. In group II, the AOR ranges from 0.7 to 1/0.7 (∼1.4), indicating that the two components are approximately comparable. In group III, the AOR is greater than 1.4, the situation whereby the unknown azimuthal component dominates the radial wind. The interpretation of Table 2 is discussed in detail in the following.

### The underestimation of the wind speed

From radars 1–11, whatever the AOR is, the magnitudes of the retrieved radial winds are always smaller than but comparable to their dual-Doppler counterparts, as shown in column 1. The difference is less than 0.5 m s^{−1} for all radars. However, this is not the case for the azimuthal winds. The numbers displayed in column 2 indicate that, the SDVR method tends to underestimate the azimuthal wind components. In addition, when the azimuthal component gradually begins to dominate the radial wind, from the group I radars to the group III radars, the underestimation increases accordingly. It can be seen that the underestimation is less than 1 m s^{−1} for radars 3 or 4 in group I but exceeds 3 m s^{−1} for the group-III radars.

### The sequence of AOR

The column 3 of Table 2 gives the dual-Doppler-analyzed and SDVR-derived values of AOR. As defined in (15), this parameter estimates the ratio of the azimuthal to the radial component. Although none of the SDVR-retrieved AORs are identical to their dual-Doppler counterparts, the statistics do show that the L99 SDVR technique is capable of placing the retrieved AOR into the correct category. The most encouraging finding is that the order of the retrieved AOR, from small to large, is exactly the same as that obtained using the dual-Doppler AOR. From the operational point of view, this implies that even lacking any prior information about the true flow pattern, one can still determine whether the unknown portion of the total wind vectors would be smaller than, comparable to, or larger than the observable component.

### The value of MVE and the directional difference θ diff

The magnitude of MVE, as defined in (11), is a quantitative parameter that describes the quality of the retrievals. When MVE is equal to zero, the retrieval is perfect. Column 4 lists the MVE parameters for all 11 radars; the average MVE for each group is depicted in column 5. It can be determined easily that the group-I radars produce the best results (average MVE = 2.34 m s^{−1}), and the largest average MVE (=3.57 m s^{−1}) occurs in group III. This finding indicates that the L99 SDVR method does depend on the radar's position. When a radar happens to be at a bad location at which most of the velocity vectors are perpendicular to the radar beams, thus becoming undetectable to the radar, the quality of the SDVR results deteriorates accordingly. Nevertheless, the small averaged directional differences *θ*_{diff}, estimated using (10) and shown in column 6, demonstrate that for the group-III radars the SDVR information regarding wind directions is still reliable, even though the unknown portion is much stronger.

### The spatial distribution of R_{at}, R(rad), and R(azi)

The spatial distributions of *R*_{at}, *R*(rad), and *R*(azi) are examined for all 11 radars. The conclusions obtained from studying the CP-4 and TOGA results still hold. That is, the SDVR-retrieved *R*_{at} field agrees well with its dual-Doppler counterpart and is a reliable source for determining the relative magnitudes of the azimuthal winds related to the total winds throughout the entire domain. In addition, the relatively bad retrievals of a particular wind component always take place in the region in which this component is weaker. To demonstrate this finding, Fig. 22 illustrates the SDVR-retrieved *R*_{at} for radar 4 (with the smallest AOR). The northeast–southwest-oriented dark region denotes *R*_{at} of less than 0.05, which implies that the azimuthal winds would be poorly retrieved along this area. This is confirmed by showing the *R*(azi) in Fig. 23. By contrast, because Fig. 22 does not show any *R*_{at} with values near 1, one would expect that all the retrieved radial winds for the whole domain should be of good quality. This expectation is found to be true by examining *R*(rad) in Fig. 24. Note that the contour with a value of 0.7 implies nearly perfect retrievals, as defined by (13).

### The horizontal convergence fields

The negative horizontal divergence (convergence) fields are studied using the dual-Doppler-synthesized wind field and all 11 single-Doppler-retrieved results. It is found that the strength of the convergence, represented by its rms amplitude, is always underestimated by the SDVR scheme. Furthermore, similar to the tangential winds, the underestimation increases from the group-I to the group-III radars. For example, the intensity of the dual-Doppler-synthesized convergence zone is about 0.91 × 10^{−3} s^{−1}. Using radar-4 (in group I) data, which have the smallest AOR, the intensity of the convergence line is reproduced as 0.63 × 10^{−3} s^{−1}; for radar 9 (in group III), with the largest AOR, the intensity is recovered to be only 0.48 × 10^{−3} s^{−1}. Note that by reexamining the CP-4 and TOGA results, one also finds a similar feature. The CP-4 radar (AOR = 0.84) deduced convergence zone is 0.74 × 10^{−3} s^{−1}, as compared with 0.43 × 10^{−3} s^{−1} for the TOGA radar (AOR = 2.57), whereas the true value obtained from dual-Doppler analyses is equal to 0.89 × 10^{−3} s^{−1}.

For a visual comparison, the negative divergence (convergence) field itself, rather than the wind field, is shown in Figs. 25 and 26 for radars 4 and 9, respectively. Qualitatively speaking, it is very encouraging to see that in both cases the low-level convergence line is successfully placed at the correct position. In fact, this is also true for the other radars (not shown).

## Conclusions and future work

The validity of the L99 SDVR technique is investigated using TAMEX IOP-2 Doppler radar data. The value of L99 is demonstrated using a case over a larger domain on which dual-Doppler synthesis is not possible. In addition, this research focuses on the impact of the radar viewing angle on the retrievals. Several conclusions, along with plans for future work, are given as follows:

- The magnitude of the SDVR-retrieved azimuthal wind is always underestimated by the L99 scheme. This underestimation becomes serious when the azimuthal wind is the principal portion of the total wind vector.
- The ratio AOR is used as an index to show the relative importance of radar-unobservable azimuthal winds to the observed radial winds. Experiments indicate that, when the AOR is small, one obtains the best retrievals. However, for those radars with a large AOR, the retrievals turn out to be less reliable. In this regard, it is concluded that the quality of the winds retrieved by L99 SDVR does depend on the radar location.
- Substituting the observational data from 11 virtual radars into the L99 scheme, it can also provide a set of retrieved AOR indices and is capable of placing them into the correct category. In other words, the relative magnitudes between the azimuthal and the radial winds would not be mistaken.
- In the scenario in which the unobservable azimuthal wind is much stronger than the observable radial component, our experiments show that the deduced wind direction is robust to the radar position.
- For a certain domain, the L99 scheme correctly describes spatial variations in the ratio between
*V*_{azi}and*V*_{rad}. From this information one can predict the locations at which the relatively bad radial (or azimuthal) wind retrievals would take place. - The convergence zone may be positioned qualitatively correctly by the L99 method.

The above conclusions deliver a simple but important message: the performance of the L99 technique depends on the radar viewing angle relative to the total wind vectors. However, without any prior knowledge about the true wind structure, which is usually the case in real applications, one can still refer to the SDVR information to determine whether the missing portion of the total wind is less than, comparable to, or greater than the known radial component. When the last situation occurs, it implies that for this particular case the radar happens to be at a disadvantageous location for executing the radar observations and SDVR calculations. The users then can be alerted that the radar data and the single-Doppler-retrieved products must be used cautiously.

In this research, emphasis is placed on investigating how SDVR results vary with radar location. It should be realized that, for the same radar location and the same scanning strategy, the accuracy of the retrievals might depend on the complexity of the flow structure. In other words, does the SDVR favor (disfavor) certain types of wind fields? In future work, more real case studies involving various kinds of weather phenomena are planned to explore this problem.

The dual-Doppler data are synthesized using the CEDRIC software developed by NCAR. Three anonymous reviewers' comments helped to improve this manuscript substantially. The authors appreciate the discussions with Prof. Tai-Chi Chen Wang. This research is sponsored by the National Science Council of Taiwan under Grants NSC88-2111-M-008-027-A10, NSC89-2111-M-008-012-A10, and NSC89-2111-M-008-062.

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Statistical results from SDVR using the L99 method and CP-4 and TOGA data. “Dual” stands for dual-Doppler syntheses. “Single” means the SDVR results

Quantitative comparisons of SDVR results from 11 virtual radars. “Dual” stands for dual-Doppler syntheses. “Single” means the SDVR results