The mean wind

To reduce the errors associated with the discrete nature of radar measurements, Gal-Chen (1982) suggested that a set of optimal advection velocities 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 gets

where 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 by

Here t₀ is a reference time. All the radar data need to be redefined within this moving frame.

The perturbation wind

With the assumption that the frame of reference only moves horizontally, the true wind fields u, υ, and w are decomposed into

where 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

In the expressions of the penalty terms J₁–J₆, the prime denotes that these variables are obtained in the moving frame of reference. In J₄, V′ is the three-dimensional perturbation wind, and ‖‖ denotes the magnitude of the vector. In J₆, we have

Note that p^′₁, p^′₂, and p^′₃ are the Cartesian coordinates of the radar in the moving frame, represented by

Furthermore, r′ is the distance between the radar and the grid point, and V^′_rad stands for the radial winds observed by the radar, defined by

In (4), J₁ denotes the conservation law for reflectivity in the moving frame; J₂ is the geometric relation between the observed radial wind and the retrieved u′, υ′, and w′. Constraint J₃ is implemented to reduce the divergence; in J₄, 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 J₅, 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, J₅ remains as an option in the retrieval formulations and can be easily turned on/off by giving it a nonzero/zero weighting coefficient. Constraint J₆ provides higher-order spatial filters to remove noise. Its weighting coefficient α₆ 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.

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.

For a given wind vector, one can divide its azimuthal part 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 by

When |V_azi| is much larger (smaller) than |V_rad|, R_at approaches 1 (0). When the two components are comparable, R_at is approximately 1/2(∼0.7).

To examine how the L99's capability varies spatially, the retrieved winds [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:

If the radar site is selected as the origin, the radial wind is said to be positive (negative) when the air is blowing away from (toward) the radar, and the azimuthal wind is positive (negative) when the flow motion is in the clockwise (counterclockwise) direction. The expression sgn[] on the rhs of (13) and (14) is used to determine the correctness of the SDVR wind direction. It gives the values of 1 (−1) when the SDVR-generated wind direction is identical (opposite) to the dual-Doppler one. Based on (13) and (14), one gets the following information from which to determine the accuracy of the SDVR winds:

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.

The CP-4 and TOGA radars are located at two different geographic locations; therefore, for a given wind vector, they observe different portions. In other words, the ratio of the missing component to the observable part varies between CP-4 and TOGA. In this research, it is desirable to describe this difference by defining an area-wide-averaged parameter AOR:

A large (small) AOR simply means that, from this radar site, the unknown azimuthal winds are generally stronger (weaker) than the observed radial winds.

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 dBZ). 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.

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 θdiff

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).