a. Radar data

Currently the Central Weather Bureau (CWB) and the Air Force Weather Wing in Taiwan operate a total of nine weather radars (Fig. 1): Wu-Fen-Shan (RCWF), Ken-Ting (RCKT), Hua-Lien (RCHL), Chi-Gu (RCCG), Ma-Kung (RCMK), Ching-Chuan-Kang (RCCK), Green Island (RCGI), Lin-Yuan (RCLY), and Nan-Tuan (RCNT). The current study uses data from two S-band radars (Table 1): RCWF (dual polarization) and RCHL (single polarization).

Table 1.

Specifications of radar networks in Taiwan.

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Most of the radars (RCCK, RCMK, RCGI, RCLY, and RCNT) are operated in dual-PRF mode to increase the Nyquist (unambiguous) velocities. In contrast, the Nyquist velocities of those radars operated in single-PRF mode (RCWF, RCHL, RCCG, and RCKT) are less than 30.0 m s⁻¹ (Table 1), and velocity folding is a problem when measuring wind speeds of very intense TCs. As a consequence, an inadequate Doppler velocity dealiasing process would significantly raise the uncertainties of basic interpretations and downstream applications of TCs such as Doppler velocity pattern recognition, single- and dual-Doppler wind retrievals, and data assimilation. To improve the accuracy of velocity dealiasing for TCs for the radar observational domain, a VDVD algorithm is proposed and discussed in the next section.

b. VDVD algorithm

By combining the GBVTD and GBVTD-simplex techniques and profiles of the Rankine-like vortex (e.g., Lemon et al. 1978; Wood and Brown 1992) for TCs, the aliased Doppler velocity can be sequentially dealiased with the iterative procedures. In the VDVD algorithm, the radius of the maximum wind and the maximum velocity can be estimated from the GBVTD technique, and the circulation center can be estimated from the GBVTD-simplex technique. All the estimated information can then be applied to the Rankine-like vortex to extend that wind model to a larger radius.

Based on the original designs of the GBVTD and GBVTD-simplex algorithms, the radar data should first be interpolated from the raw data in polar coordinates [plan position indicator (PPI)] to Cartesian coordinates [constant-altitude plan position indicator (CAPPI)] with the origin at the TC center. However, the effects of the aliasing points of the Doppler velocity will be spread vertically and horizontally during the interpolation procedure. In addition, even though the predealiasing method is used to eliminate the contaminations from the aliased Doppler velocity (Wang et al. 2012), converting the dealiased data from CAPPI coordinates back to polar coordinates is irreversible. Therefore, the VDVD algorithm is calculated on polar coordinates to prevent the data from being contaminated by aliased velocity data when interpolating into CAPPI coordinates.

The inner iterative loop of the VDVD algorithm is implemented to find a set of $(V_{\max },R_{\max })$ to describe the TC structure that is used for reference to dealias the Doppler velocity. However, an improper description of TC structure (unrealistic asymmetric component) can be introduced by a displacement of the TC center (Lee et al. 1999) and results in a failure of dealiasing. Therefore, the outer loop in the VDVD algorithm is designed to reduce the possibility of improper dealiasing induced by a shifted TC center. An overview flowchart of the VDVD algorithm is shown in Fig. 2, and the default values or initial estimates of the empirical parameters and thresholds for the VDVD algorithm are listed in Table 2. The VDVD steps and parameters are described as follows:

1. Initially, estimate the vortex center ($X^0$, $Y^0$) with the origin at the radar site. At any given nth iteration, the vortex center is represented as ($X^n$, $Y^n$).

2. Initially, estimate the radius of the maximum wind (RMW, $R_{\max }^{n,0}$) and the maximum axisymmetric velocity ($V_{\max }^{n,0}$). At any given mth iteration, the RMW and maximum tangential velocity parameters are represented as $R_{\max }^{n,m}$ and $V_{\max }^{n,m}$, respectively.

3. Create a Rankine combined vortex profile via Eqs. (3) and (4) with ($X^n$, $Y^n$), $V_{\max }^{n,m}$, $R_{\max }^{n,m}$, and the predefined wind profile decay ratio ($\alpha$):

   $V_T=V_{\max }^{n,m}{\displaystyle \frac{r}{R_{\max }^{n,m}}}\text{if}\hspace{0.17em}r\le R_{\max }^{n,m},$(3)

   $V_T=V_{\max }^{n,m}{\left({\displaystyle \frac{R_{\max }^{n,m}}{r}}\right)}^{\alpha }\text{if}\hspace{0.17em}r\ge R_{\max }^{n,m},$(4)

   where $V_T$ is the tangential velocity at radius r.

4. Calculate the radial components of tangential velocities from step 3 in polar coordinates that are used as the references of Doppler velocities. The observed Doppler velocities are dealiased when the differences between the references of Doppler velocities exceed the criterion (1.5 times the Nyquist velocity) for individual radar bins in polar coordinates. The unfolded velocity $V_u$ can be corrected from the observed Doppler velocity $V_d$ with Eq. (2).

5. Calculate the $V_{\max }^{n,m}$ and $R_{\max }^{n,m}$ using the GBVTD algorithm with the unfolded Doppler velocity.

6. Verify whether the values between $V_{\max }^{n,m}$ and $V_{\max }^{n,m-1}$ and between $R_{\max }^{n,m}$ and $R_{\max }^{n,m-1}$ are identical. If yes, return to step 2 to continue the computation.

7. Use ($X^n$, $Y^n$) as the initial guess of the vortex center to calculate the new estimated vortex center ($X^{n+1}$, $Y^{n+1}$) with the GBVTD-simplex algorithm. The radius to start the initial simplex is set to 4 km, and a searching process with a radius of 2 km will be repeated to obtain a higher vortex center precision after finishing the previous search process. Other parameters are the same as those in Lee and Marks (2000).

8. Verify whether the difference between ($X^n$, $Y^n$) and ($X^{n-1}$, $Y^{n-1}$) exceeds the prescribed criterion $C_0$. If yes, return to step 1 to continue the computation. Otherwise, the iterative procedures stop.

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Overview flowchart of the VDVD algorithm.

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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Table 2.

Default values or initial estimates of the VDVD.

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a. Rankine combined vortex

The analytic axisymmetric vortex is located at (0.0, −150.0), which is to the south of the radar at (0, 0). The clear cyclonic flow field and high wind zone around $R_{\max }$ centered at the vortex center are illustrated in Fig. 3a. The $V_{\max }$ and $R_{\max }$ of the Rankine combined vortex and the GBVTD-derived results in each iteration are shown in Table 4 and Figs. 3b–f, respectively. In the initial iteration, the reference vortex (Fig. 3b) is far from the simulated vortex, and only a subset of Doppler velocities is dealiased from the raw Doppler velocities (Fig. 3d). The Doppler velocities near the eyewall are still aliased (Fig. 3e). The $V_{\max }$ and $R_{\max }$ values derived from the GBVTD algorithm are 43.19 m s⁻¹ and 29 km, respectively, which are closer to the given values of the simulated vortex ($V_{\max }$ = 60 m s⁻¹, $R_{\max }$ = 40 km) than the initial estimations ($V_{\max }^0$ = 25 m s⁻¹, $R_{\max }^0$ = 20 km). Thus, the differences in $V_{\max }$ ($\mathrm{\Delta }V$) and $R_{\max }$ ($\mathrm{\Delta }R$) are 18.19 m s⁻¹ and 9 km, respectively. In the second iteration, the GBVTD-derived $V_{\max }$ and $R_{\max }$ ($V_{\max }^1$ and $R_{\max }^1$) values are 60.08 m s⁻¹ and 40 km, respectively, which are very close to those of the simulated vortex. In the third iteration, there are no differences between ($V_{\max }^0,R_{\max }^0$) and ($V_{\max }^2,R_{\max }^2$); thus, the iterations stop. When combining axisymmetric $V_T$ and $V_R$ (Fig. 4a), there is no significant difference in the flow field, but the velocity dipole slightly rotates clockwise (Fig. 4d). From the Doppler velocity field during the convergent process (Figs. 4d–f), the extra $V_R$ component does not raise the number of iterations during the calculation, which is primarily attributed to its relatively small magnitude compared to $V_T$. With this symmetric vortex, the aliased Doppler velocity can be correctly recovered in only three iterations.

Table 4.

GBVTD-derived $V_{\max }$ (m s⁻¹) and $R_{\max }$ (km) at each iteration.

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As in Fig. 3, but for axisymmetric $V_T$ plus $V_R$ flow field.

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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To further investigate the performance of the initial estimates of $V_{\max }^0$ and $R_{\max }^0$ for the initial reference vortex, a $V_{\max }^0$ of 1–80 m s⁻¹ and $R_{\max }^0$ of 1–80 km were used to understand the procedure trigger $V_{\max }^0$ and $R_{\max }^0$ to approach the actual $V_{\max }$ and $R_{\max }$. A convergent feature diagram is constructed to present the convergence path and tendency during the iteration calculations. The differences ($\mathrm{\Delta }V\text{s},\mathrm{\Delta }R\text{s}$) between all the ($V_{\max }^0,R_{\max }^0$) and ($V_{\max }^1,R_{\max }^1$) are calculated and illustrated in the vector form shown in Fig. 5a. It is shown that nearly all vectors are pointing toward the point at $V_{\max }$ = 60 m s⁻¹ and $R_{\max }$ = 40 km and the amount of adjustments are less significant when the initial guesses ($V_{\max }^0$, $R_{\max }^0$) are closer to the true values. This diagram also reveals that most combinations of $V_{\max }^0$ and $R_{\max }^0$ are essentially suitable for an initial estimate. However, the combinations from extreme values of high $V_{\max }^0$ and low $R_{\max }^0$ or high $V_{\max }^0$ and high $R_{\max }^0$ could result in an irreversible unfolding process. When combining axisymmetric $V_T$ and $V_R$ (Fig. 5b), the vectors of convergence paths are very similar to those of $V_T$ that point toward the given values of the simulated vortex ($V_{\max }$ = 60 m s⁻¹ and $R_{\max }$ = 40 km).

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The convergent feature diagram for axisymmetric (a) $V_T$ and (b) $V_T$ plus $V_R$. The arrows in black show the iteration path. The Rankine combined wind profiles used in consecutive iterations are indicated by solid lines and iteration numbers based on an initial estimate. The color scale indicates the square root of the sum of the differences between the estimated and true value of $R_{\max }$ and $V_{\max }$ during the calculations (see the text for details).

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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b. Effect of $V_{\text{T}1}$

Because the conditions of the wavenumber-1 component of $V_T$ ($V_{\text{T}1}$) on different azimuths have similar influences on testing the asymmetric effects between 0° (90°) and 180° (270°) relative to the vortex center, $V_{\text{T}1}$ values located at azimuths 0° and 90° superimposed on the basic flow (Rankine combined vortex) are chosen to examine the effects of asymmetry. Figure 6a shows that the Doppler velocity pattern for $V_{\text{T}1}$ at 0° is antisymmetric with respect to the y axis, while the peak magnitude in the Doppler velocity dipole is approximately 5–10 m s⁻¹ higher than that in the basic flow. The final $V_{\max }$ and $R_{\max }$ values derived from the GBVTD algorithm are 61.64 m s⁻¹ and 40 km, respectively, which are very close to the idealized vortex values (Fig. 6b). The convergent feature diagram also shows that nearly all vectors are pointing toward the point at $V_{\max }$ = 60 m s⁻¹ and $R_{\max }$ = 40 km (Fig. 6c). However, it also shows a potentially irreversible unfolding process under conditions of a higher $V_{\max }$ with a smaller $R_{\max }$ that could result in inadequate dealiased Doppler velocity. The Doppler velocity pattern for $V_{\text{T}1}$ at 90° is shown in Fig. 6d. An obvious asymmetric pattern for the Doppler velocity dipole is observed, while the difference is approximately 20 m s⁻¹ between the peak magnitudes of inbound and outbound Doppler velocities. Similar to the results shown in Fig. 6b, the aliased Doppler velocity can be recovered by the inner loop of the VDVD algorithm (Fig. 6e), while the number of iterations is 4 (Fig. 6f), which is slightly higher than that in Fig. 6c because of its asymmetry on the Doppler velocity field.

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(a) Doppler velocity fields of the original, (b) final iteration, and (c) the convergent feature diagram for axisymmetric $V_T$ plus $V_{\text{T}1}$ at 0°. (d)–(f) As in (a)–(c), but for axisymmetric $V_T$ plus $V_{\text{T}1}$ at 90°.

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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c. Effect of $V_M$

The Doppler velocity patterns are characterized by the directions of $V_M$. Figure 7 shows the Doppler velocity patterns of 10 m s⁻¹ along-beam northerly and cross-beam easterly $V_M$ components superimposed on the axisymmetric $V_T$ flow. The definitions of the along and cross beam are the same as those in Lee et al. (1999). When combining the along-beam $V_M$ (Fig. 7a), the zero isodop of Doppler velocity and the peak magnitude of the Doppler velocity dipole show remarkable change compared to those from the cross-beam $V_M$ (Fig. 7d). The easterly (westerly) $V_M$ enhances (reduces, not shown) the Doppler velocity of the axisymmetric vortex because the effect is a function of the beam angle, as documented in Lee et al. (1999).

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As in Fig. 5, but for (a)–(c) axisymmetric $V_T$ plus northerly $V_M$, and (d)–(f) axisymmetric $V_T$ plus easterly $V_M$.

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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The final $V_{\max }$ and $R_{\max }$ values derived from the GBVTD algorithm are 60.08 (62.78) m s⁻¹ and 40 (40) km, respectively, for northerly (easterly) $V_M$ components (Fig. 7b and Fig. 7e). The final $V_{\max }$ is slightly greater than the analytic vortex because of the uncertainty in retrieving the $V_{\max }$ from the GBVTD algorithm when the cross-beam $V_M$ component is superimposed with basic flow (Lee et al. 1999). Even though the $V_{\max }$ is slightly overestimated, the VDVD still performs well because the tolerance of the difference (1.5 times the Nyquist velocity) between the reference and aliased velocities is relatively large [Eq. (2)].

d. Combined vortex

To further evaluate the performance and limitations of the VDVD algorithm, all the axisymmetric and asymmetric components are superimposed on the Rankine combined vortex to provide the analytic dataset. Figure 8 shows two representative examples with extreme asymmetry of the Doppler velocity pattern combined $V_T$ with 1) 20% $V_T$ as $V_{\text{T}1}$ at azimuth 0°, $V_R$, and 10 m s⁻¹ northerly $V_M$ and 2) 20% $V_T$ as $V_{\text{T}1}$ at azimuth 90°, $V_R$, and 10 m s⁻¹ southerly $V_M$. For the former test (Fig. 8a), the Doppler velocity field is characterized by a bending zero isodop and asymmetric dipole structure. The final $V_{\max }$ and $R_{\max }$ values derived from the GBVTD algorithm are 61.68 m s⁻¹ and 40 km, respectively (Fig. 8b). Although the $V_{\max }$ and $R_{\max }$ are approximations of the true value, the aliased Doppler velocities in a limited area cannot be well recovered near the northwestern direction at a range of approximately 70 km from the vortex center. The unsuccessful result is due to the high asymmetry of the combined vortex contributed by $V_{\text{T}1}$ in the northern vortex and the northerly $V_M$ superimposed on symmetric $V_T$ flow. The convergent feature diagram (Fig. 8c) shows the conversion paths for vectors in a fashion similar to those shown in Figs. 7 and 8, but the number of iterations increases to 8. The asymmetric Doppler velocity structure for the latter test is shown in Fig. 8d. The magnitude difference between peak inbound and outbound Doppler velocities reaches approximately 30 m s⁻¹. Even in such an extreme condition, the VDVD is still able to recover the aliased Doppler velocities. The final $V_{\max }$ and $R_{\max }$ values derived from the GBVTD algorithm are 60.07 m s⁻¹ and 40 km, respectively (Fig. 8e), values that are closer to the simulated vortex values than the values provided by the former test. However, the number of iterations increases to 12 (Fig. 8f), being considerably higher than the numbers in the previous tests. This finding suggests that the more asymmetric the Doppler velocity dipole is, the greater the number of convergence iterations required.

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As in Fig. 5, but for (a)–(c) axisymmetric $V_T$ plus $V_R$, $V_{\text{T}1}$ at 0°, and northerly $V_M$, and (d)–(f) axisymmetric $V_T$ plus $V_R$, $V_{\text{T}1}$ at 90°, and southerly $V_M$.

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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Table 3 summarizes the sensitivity tests for various combinations from the wavenumber 1 of $V_T$ ($V_{\text{T}1}$) given with 20% of $V_T$ and $V_M$ given with a value of 5 and 10 m s⁻¹ with various directions from 0° to 360°. The results show that the number of iterations is less than five and generally increases with the vortex asymmetry. This finding indicates that the aliased velocities from almost all the tests can be recovered by the VDVD algorithm except the first experiment in test A09 shown in Fig. 8b, which is combined with symmetric $V_T$ and $V_R$ and asymmetric $V_{\text{T}1}$ at azimuth 0° and northerly $V_M$. The unsuccessful recovery of aliased Doppler velocity in this test occurs because the simulated vortex is far from the Rankine vortex assumption when the direction of $V_M$ is parallel to the radar beam. The aliased Doppler velocity can be successfully corrected with 5 m s⁻¹ $V_M$ for all directions because of the reduced asymmetry in the analytic vortex. In practice, this uncertainty in the dealiasing Doppler velocity will increase with higher $V_M$ values, which could be mitigated by including the short-term motion of TCs or previous analysis of $V_M$ from the GBVTD algorithm.

e. Effect of TC center displacement

The subsections above focus on the sensitivities of the inner loop in the VDVD algorithm with the correct vortex center. The sensitivity for the effect of vortex center displacement is discussed in this subsection. From the overview flowchart of the VDVD algorithm (Fig. 2), the necessary inputs of the initial estimation are the vortex center location ($X^0,Y^0$), $V_{\max }^0$, and $R_{\max }^0$. In practice, typhoon centers can be provided by the best track analysis, which is issued every hour by the CWB in Taiwan, for example. However, this track information usually has a 20–30-min timing delay because it takes time to integrate the information from all available data such as radar, satellite, and surface observations. Alternatively, the approximate center position can be estimated using the current TC center location, and the track can be forecasted with a linear extrapolation.

The vortex asymmetry, especially for the wavenumber-1 structure that is computationally generated during the GBVTD calculation, exhibits a displacement of the TC center of an axisymmetric vortex, as documented by Lee and Marks (2000). To simulate the effect of a shifted vortex center in the inner loop of the VDVD algorithm, four experiments are conducted with a displacement of 8 km (20% of simulated RMW) in four directions (north, east, south, and west) from the true value based on the analytics vortex shown in section 3a and Fig. 3a. All the experiments reveal obvious convergence patterns, and the convergence approximately points to the given values of the idealized vortex $(V_{\max }=60\hspace{0.17em}{\text{m}\hspace{0.17em}\text{s}}^{-1},R_{\max }=40\hspace{0.17em}\text{km})$ (Fig. 9). The aliased Doppler velocities can be recovered by the reference wind field based on those approximated $V_{\max }$ and $R_{\max }$ values even at a weaker $V_{\max }$ and larger $R_{\max }$. It is shown that the $R_{\max }^m$ and $V_{\max }^m$ gradually increase toward the true values. The $\mathrm{\Delta }V\text{s}$ are greater than $\mathrm{\Delta }R\text{s}$ in magnitude initially but are contrary at the end of iteration. An obvious curved convergence path is presented that is slightly different from that in section 3a (Fig. 5). Generally, a displacement of the vortex center makes the convergence pattern less significant and increases the number of iterations during the structure-finding procedure in this sensitivity test of the VDVD algorithm. However, the uncertainty of the VDVD algorithm will generally increase with the large center displacement and result in the failure of the convergence (discussed in the discussion section).

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As in Fig. 5, but for the centers at (a) (0, −142), (b) (8, −150), (c) (0, −158), and (d) (−8, −150).

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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f. Mitigation of the TC center displacement effects

To mitigate the effects of center displacement in the VDVD algorithm described in the previous subsection, the GBVTD-simplex center-finding algorithm is combined with the GBVTD-based unfolding algorithm. The GBVTD-simplex technique (Lee and Marks 2000) was proposed to determine the TC center and can provide a proper center position for well-organized TCs when compared with the reflectivity-based algorithm (Chang et al. 2009a). The GBVTD-simplex algorithm assumes that the vortex center is located where the local maximum of $V_{\max }$ can be derived from the GBVTD algorithm. Therefore, the simplex method used to approximate the location of the local maximum is combined with GBVTD to provide a proper vortex center in the GBVTD-simplex algorithm.

Accordingly, there are two loops in the VDVD algorithm, which are designed to mitigate the effect of a shifted TC center. The purpose of the inner loop (GBVTD iteration) of the VDVD algorithm (Fig. 2) is to dealias the Doppler velocities as discussed above; the purpose of the outer loop (GBVTD-simplex iteration) is to determine the vortex center based on the Doppler velocity field from the inner loop. To examine the mutual influence between the outer and inner loops, the same symmetric $V_T$ superimposed with 20% of $V_T$ ($V_{\text{T}1}$) at 90°, $V_R$, and 5 m s⁻¹ southerly $V_M$ with an eastward 8-km displacement (20% of the simulated RMW) centered at the vortex center $\left(X^0,Y^0\right)=\left(8,-150\right)$ are applied. For the initial outer iterative loop, the inner loop converges within three iterations to a set of $\left(V_{\max }^{0,3},R_{\max }^{0,3}\right)=\left(22.18\hspace{0.17em}{\text{m}\hspace{0.17em}\text{s}}^{-1},49\hspace{0.17em}\text{km}\right)$, which is substantially different from the structure of the analytic vortex because of the improperly dealiased velocity field (Fig. 10a). Subsequently, a first apparent center $\left(X^1,Y^1\right)=\left(-9.25,-134.65\right)$ is determined from the GBVTD-simplex algorithm in the outer loop. With the new TC center, a set of $\left(V_{\max }^{1,4},R_{\max }^{1,4}\right)=\left(48.54\hspace{0.17em}{\text{m}\hspace{0.17em}\text{s}}^{-1},44\hspace{0.17em}\text{km}\right)$ can be calculated by using the VDVD algorithm with a better dealiased Doppler velocity field (Fig. 10b). The second apparent center determined from the GBVTD-simplex technique is $\left(X^2,Y^2\right)=\left(-2.12,-147.79\right)$, which is very close to the center location of the analytic vortex. Combined with the outer loop (GBVTD-simplex iteration), the Doppler velocities can be subsequently fully recovered in two iterations (Fig. 10c) and find the result $\left(V_{\max }^{2,4},R_{\max }^{2,4}\right)=\left(58.38\hspace{0.17em}{\text{m}\hspace{0.17em}\text{s}}^{-1},39\hspace{0.17em}\text{km}\right)$ of the inner loop.

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The dealiased Doppler velocity in the VDVD outer loop (a) first, (b) second, and (c) third iteration used the condition that symmetric $V_T$ superimposed 20% $V_T$ as $V_{\text{T}1}$at 90°, $V_R$, and 5 m s⁻¹ southerly $V_M$ with eastward 8-km displacement (20% of simulated RMW).

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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A series of experiments are conducted to understand the improvement of the VDVD compared to the VDVD without the GBVTD-simplex technique (Table 5). The combinations of $V_{\text{T}1}$ with 10% or 20% of $V_T$ at 90° and $V_M$ with 5 or 10 m s⁻¹ from 180° are selected to induce the greatest asymmetry of a TC structure, and an 8-km (20% of the simulated RMW) displacement of the TC center is also applied. The success rate is determined according to the ratio of the experiments that can be totally dealiased. For the experiments with 10% $V_T$ as $V_{\text{T}1}$, the success rate of the VDVD algorithm without GBVTD simplex is less than 75%. The VDVD algorithm provides a success rate 25%–50% higher than that of the VDVD without the GBVTD-simplex technique except in experiment B03, which has extreme conditions of 20% of $V_T$ as $V_{\text{T}1}$ and 10 m s⁻¹ $V_M$. The outer loop of the VDVD successfully approximates the vortex center to the true value. However, the higher $V_{\text{T}1}$ and $V_M$ could increase the uncertainties, which would result in a decreased success rate of the VDVD algorithm.

Table 5.

Sensitivity tests of the VDVD algorithm for various combinations of $V_T$, $V_{\text{T}1}$, $V_R$, and $V_M$ and displacement of the TC center. A simplex center difference in italic format indicates that the test is not fully dealiased by the VDVD algorithm, and one in bold format indicates that the test is not fully dealiased by the VDVD without GBVTD simplex. The information in the second and third row of V_T1 and V_M are the same as that in Table 3.

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a. Typhoon Fitow (2013)

Figures 11 and 12 show the extreme examples of ZW06 and VDVD for Typhoon Fitow (2013) as observed by the RCWF radar, respectively. The center of Typhoon Fitow was located over the ocean northeast of Taiwan at a distance of ~240 km from the radar site at 2311 UTC 5 October 2013. Typhoon Fitow exhibited a weakly circular-symmetric eyewall structure with a radius of ~50 km (Fig. 11a). The raw Doppler velocities around eyewall areas are mostly aliased (Fig. 11b) because the wind speeds exceed the Nyquist velocity. With the ZW06 algorithm, it is found that the unreasonable Doppler velocities were observed near the northwestern to southwestern quadrants of the TC center, which occasionally occurred during operation because of relatively noisy data around the eyewall regions (Fig. 11c). The case from the ZW06 algorithm has the lowest success rate, 74.5%, compared to that of the subjective dealiasing analysis (Fig. 11e). In contrast, the VDVD algorithm has a success rate of more than 99% at that time (Fig. 11d). Approximately 9.5 h later, Typhoon Fitow moved to the ocean north of Taiwan at a distance of ~150 km from the radar site (Fig. 12a). The reflectivity structure showed a more intense circular-symmetric eyewall compared to that at 2311 UTC 5 October (Fig. 11a). Figure 12c shows the dealiasing result of Doppler velocity using the VDVD algorithm. The low-velocity region caused by sea clutter is overly unfolded near the east side of the radar site. Similar to the results of the VDVD algorithm, the overly unfolded phenomenon of the Doppler velocity is also found in the same region using the ZW06 algorithm (Fig. 12b). Hence, the sea clutter signals should be adequately removed before the dealiasing algorithms are applied, which can potentially improve the dealiased velocity performance.

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Radar observations of Typhoon Fitow (2013) at 0.5° from RCWF at 2311 UTC 5 Oct 2013. (a) Base reflectivity (dBZ). Doppler velocities (m s⁻¹) of (b) raw data, (c) ZW06, (d) VDVD, and (e) subjective analysis. Two range rings are indicated at 150 and 300 km from the radar site.

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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As in Fig. 11, but at 0837 UTC 6 Oct 2013.

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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To quantitatively evaluate the VDVD algorithm performance, comparisons are conducted with a subjective dealiasing analysis based on PPI pixels. Approximately 21.5% of the pixels were aliased from Doppler velocity data in 23 h (1907 UTC 5 October to 1757 UTC 6 October), and on average, the VDVD and ZW06 algorithms can recover 99.4% and 98.3% of the aliased velocity data, respectively. Table 6 shows the success rate from a total of 472 elevation sweeps. A total of 87% and 70% of the elevation sweeps exhibit success rates exceeding 99% for the VDVD and ZW06 algorithms, respectively. The statistical results show that the VDVD has a more stable performance for providing high quality data. Moreover, only approximately 2% and 14% of the elevation sweeps have a success rate under 97% for the VDVD and ZW06 algorithms, respectively.

Table 6.

Numbers of elevation sweeps falling between the criteria of raw data for VDVD and the algorithm in ZW06. The numbers in parentheses are the percentages of the 472 total sweeps.

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As mentioned previously, the VDVD algorithm can dealias the Doppler velocity and simultaneously provide a reasonable center and inner-core wind structure for TCs. Figure 13 shows the track of Typhoon Fitow derived automatically from GBVTD simplex, which is generally consistent with the official track issued by the CWB and indicates that the track information from the GBVTD-simplex algorithm is potentially usable for real-time operation. For the wind structure of Typhoon Fitow (2013), Fig. 14 shows the time series of $V_{\max }$ and $R_{\max }$ from the first elevation angle data derived from the VDVD algorithm. The $V_{\max }$ and $R_{\max }$ at 1907 UTC 5 October were 40 m s⁻¹ and 100 km, respectively. Sequentially, the $V_{\max }$ changed slightly (40–50 m s⁻¹) before 1300 UTC 6 October and then dramatically weakened to approximately 35 m s⁻¹. The $R_{\max }$ decreased from 100 to 60 km from 0000 UTC 5 October to 0900 UTC 6 October. A subsequent $R_{\max }$ contraction from 90 to 60 km occurred at approximately 1100 UTC 6 October. Generally, $V_{\max }$ and $R_{\max }$ are out of phase, which is consistent with previous eyewall contraction studies (Marks and Houze 1987).

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Track of Typhoon Fitow (2013) derived from GBVTD simplex (blue line) and observations (black line). The numbers above the tracks are the dates and times in UTC.

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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Time series of $V_{\max }$ (blue line) and $R_{\max }$ (red line) derived from the GBVTD-simplex technique with the first elevation data of RCWF.

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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b. Typhoon Nesat (2017)

Figure 15a shows the reflectivity field of Typhoon Nesat observed from RCHL at 0622 UTC 29 July. An obvious asymmetric eyewall structure with strong reflectivity was located in the southern quadrant of the typhoon center. The center was approximately 130 km northeast of the radar site, and the eyewall radius was approximately 55 km. Notably, there are no valid data on the mountainside west of the RCHL because the electromagnetic wave emissions were turned off, as mentioned previously. It is also found that a narrow beam blockage wedge to the northeast of the radar is the result of a tall building near the radar site. The aliased Doppler velocities from raw data occurred over the eyewall and surrounding area (Fig. 15b). With the ZW06 algorithm, part of the aliased Doppler velocities is not well dealiased in the inbound region near the eyewall and surrounding area (Fig. 15c). In contrast, the aliased Doppler velocities can be successfully recovered by the VDVD algorithm (Fig. 15d).

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Radar observations of Typhoon Nesat (2017) at 0.5° from RCHL at 0622 UTC 29 Jul 2017. (a) Base reflectivity. Doppler velocities of (b) raw data, (c) VDVD, and (d) VDVD with radial verification procedure.

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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When Typhoon Nesat moved toward the east coast of Taiwan, most of the aliased velocity data are recovered near eyewall regions (Fig. 16a). However, there are some discontinuities in the Doppler velocities that are overly dealiased near the offshore regions of southeastern Taiwan. It results from the flow direction of terrain-induced local circulation that deviates from that of the Rankine combined vortex (e.g., Wu and Kuo 1999; Wu et al. 2002; Chang and Lin 2011). As a result, the Rankine combined vortex assumption of the VDVD algorithm cannot be adequately applied because of the local circulations. To avoid this situation and extend the VDVD algorithm usage, a procedure of radial-by-radial verification similar to the beam checking in ZW06 is used to recover the discontinuities resulting from inadequate unfolding. The concept is illustrated in Fig. 17. Two radials passing through two tangential points of RMW are chosen as initial references. The radial-by-radial verification procedure begins counterclockwise from the inbound site of the Doppler velocity and clockwise from the outbound site and is also applied to the regions inside the RMW with opposite directions to the regions outside the RMW. The procedure is stopped at the azimuth of circulation center determined by the VDVD algorithm.

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Radar observations of Typhoon Nesat (2017) at 0.5° from RCHL at 0922 UTC 29 Jul 2017. Doppler velocities of (a) VDVD and (b) VDVD with radial verification procedure.

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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Schematic diagram of radial-by-radial verification based on the VDVD algorithm. The black circle indicates the maximum wind radius centered on the typhoon symbol. The thick black lines denote the initial references from the inflow and outflow parts. The gray arrows represent the beam verification in the clockwise and anticlockwise directions.

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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Figure 16b shows the dealiasing velocity field from the VDVD algorithm with the radial-by-radial verification procedure. The discontinuous regions are significantly reduced, indicating that the radial-by-radial verification procedures can recover the inadequately dealiased Doppler velocities, once the local circulations outside the TC inner-core regions are far from the Rankine combined profile assumption.

a. GBVTD retrievals

As described in section 2, the VDVD algorithm is based on the GBVTD and GBVTD-simplex techniques, which can simultaneously provide the mean flow, axisymmetric tangential and radial winds, and asymmetric tangential winds from single-Doppler radar data after the VDVD algorithm dealiasing procedures are completed.

Figure 18 shows comparisons of the temporal evolutions of the mean tangential winds derived from the GBVTD based on the dealiasing data of the ZW06 and VDVD algorithms from the RCWF and RCHL radars for Typhoon Nesat (2017). It presents a very evident RMW contraction from 57 to 35 km (0903 to 1101 UTC) before Typhoon Nesat made landfall on Taiwan based on the dealiased data processed by the VDVD algorithm and observed from the RCWF radar (Fig. 18a). The RMW contraction rate is ~10 km h⁻¹, which is comparable to that found by Chang et al. (2009a). In contrast, using the ZW06 dealiased data, three remarkable, discontinuous gaps from the mean tangential wind distributions were found, where the wind speeds were dramatically smaller than those of proximate times (Fig. 18b). In addition, the retrieved wind with ZW06 dealiased data (Fig. 18b) is generally slightly weaker than those with VDVD dealiased data (Fig. 18a), which reflects the effects of inadequate dealiasing by the ZW06 algorithm. However, the GBVTD algorithm is not very sensitive because it is constructed in a harmonic analysis when the inadequately dealiased data do not spread into large regions.

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(a) Temporal composite of the Typhoon Nesat mean tangential wind (m s⁻¹) derived with the GBVTD technique using dealiased data from the VDVD algorithm from the RCWF radar between 0503 and 1101 UTC 29 Jul 2017. The shading intervals are every 1 m s⁻¹, and the solid lines indicate the RMW. (b) As in (a), but with ZW06 dealiased data. (c) As in (a), but from the RCHL between 0500 and 1100 UTC 29 Jul 2017. (d) As in (c), but with ZW06 dealiased data.

Citation: Journal of Atmospheric and Oceanic Technology 36, 8; 10.1175/JTECH-D-18-0139.1

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For the data observed from the RCHL radar, the regions of wind retrievals are remarkably smaller than those from the RCWF radar (Figs. 18a and 18c), which resulted from noisier data being removed by the RCHL radar signal processor. A similar decreasing trend is observed for the RMW when using the dealiased data from the VDVD algorithm (Fig. 18c), and there are systematic biases in the RMWs between the data from RCWF and RCHL, with a difference of 5–20 km. The obvious differences are possibly contributed by effects including the asymmetric circulation structure of Typhoon Nesat and a relatively shorter observational range of 190 km from RCHL compared to 300 km from RCWF (Table 1), which could cause differences in the uncertainties of the center determinations of the GBVTD-simplex algorithm and the wind retrievals of the GBVTD algorithm (Lee et al. 1999; Lee and Marks 2000). In addition, because the TC centers are close to the RCHL during most of the analysis time, the outward tilting of the vertical eyewall (e.g., Marks and Houze 1987; Hazelton and Hart 2013) may partially contribute the biases of GBVTD-derived RMWs because the analyses are examined in the polar coordinate system. For the difference in the results of the RCWF using the dealiased data from ZW06 (Fig. 18d), the extremely discontinuous gaps in the spatial and temporal distributions of the mean tangential wind indicated that a large region of data was not adequately dealiased.

b. Dual-Doppler wind retrievals

The dual-Doppler wind retrievals were processed using 3D Cartesian grid data. The u and υ components at each grid point were retrieved using the approach of Ray et al. (1975). Furthermore, the horizontal wind components were variationally adjusted to minimize the mean difference between horizontal and vertical divergences, and then the horizontal divergence and vertical velocity were derived. Notably, there are uncertainties in the vertical velocity calculations in regions with incomplete coverage of radar observations in the boundary layer. However, the horizontal wind adjustments are relatively small compared to the original derivations, especially in weather systems with strong winds, such as those produced by TCs.

Figure 19 shows examples of the dual-Doppler wind retrievals by using RCWF and RCHL radar data from Typhoon Nesat at an altitude of 4 km at 0712 UTC 29 July 2017. The retrieved winds are only shown near the northern part of the eyewall regions because of the incomplete coverage at this altitude. Additionally, the cyclonic circulation is quite obvious with a maximum wind speed greater than 55 m s⁻¹ based on the VDVD data (Fig. 19a). In contrast, unreasonable retrieval winds are found with irregular wind directions shown in the northern and northeastern quadrants of the typhoon using the ZW06 data, which partially results from the failure of the Doppler velocity dealiasing (Fig. 19b).