a. Benchmark VF

The benchmark VF is configured analytically in appendix A, which is an one-cell vortex intended to resemble a large intense tornadic mesocyclone. As this benchmark VF is configured analytically as a spatially continuous vector field in the vortex-following coordinate system (x′, y′, z′, t′) [see (2.1) and Fig. 1a of Part I], it can be used to evaluate the accuracies of analyzed VFs for differently slanted vortices to any required high spatial resolutions (because the analyzed VF is also a spatially continuous vector field as explained at the end of section 4 of Part I).

For the axisymmetric part of benchmark VF, the scalar field of tangential velocity $V_T^s$ and the vector field of slantwise vertical circulation expressed by ($V_R^s$, w^s) in (R, z′) are plotted by blue contours and black arrows, respectively, in Fig. 1a, where (R, z′) is the vertical cross section for any given β in the slantwise cylindrical coordinate system (R, β, z′) transformed from (x′, y′, z′) [see (2.4) of Part I]. From (A1a), (A2a), and (A2b), it is easy to see that the vertical variations of $V_T^s$, $V_R^s$, and w^s shown in Fig. 1a are governed by 1 + 0.5 tanh(z′/h), [cosh(z′/h)]⁻²/ρ_a and tanh(z′/h)/ρ_a, respectively.

(a) Benchmark $V_T^s$ and ($V_R^s$, w^s) plotted in (R, z′) by blue contours (every 5 m s⁻¹) and black arrows, respectively. (b) Benchmark (u′, υ′) and w′ formulated in (2.4) plotted by black arrows and blue contours, respectively, at z′ = 1 km. (c) As in (b), but at z′ = 4 km. The vector scale is shown below the lower-left corner of each panel. The contours of w′ are plotted every 5 m s⁻¹ in (b), but every 10 m s⁻¹ in (c).

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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(a) Benchmark $V_T^s$ and ($V_R^s$, w^s) plotted in (R, z′) by blue contours (every 5 m s⁻¹) and black arrows, respectively. (b) Benchmark (u′, υ′) and w′ formulated in (2.4) plotted by black arrows and blue contours, respectively, at z′ = 1 km. (c) As in (b), but at z′ = 4 km. The vector scale is shown below the lower-left corner of each panel. The contours of w′ are plotted every 5 m s⁻¹ in (b), but every 10 m s⁻¹ in (c).

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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(a) Benchmark $V_T^s$ and ($V_R^s$, w^s) plotted in (R, z′) by blue contours (every 5 m s⁻¹) and black arrows, respectively. (b) Benchmark (u′, υ′) and w′ formulated in (2.4) plotted by black arrows and blue contours, respectively, at z′ = 1 km. (c) As in (b), but at z′ = 4 km. The vector scale is shown below the lower-left corner of each panel. The contours of w′ are plotted every 5 m s⁻¹ in (b), but every 10 m s⁻¹ in (c).

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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The vector field of (u′, υ′) and the scalar field of w′ in (2.1) are plotted for the benchmark VF by the black arrows and blue contours, respectively, at z′ = 1 km in Fig. 1b and at z′ = 4 km in Fig. 1c. As shown by the black arrows in these two figures, the horizontal velocity is mainly along the tangential direction and its horizontal distribution is close to axisymmetric especially in and around the vortex core (i.e., the circular area of R ≤ R₁, where R₁ is the radius of maximum $V_T^s$). The horizontal flow is thus dominated by the axisymmetric tangential velocity $V_T^s$ and its vertical variation is also dominated by the vertical variation of $V_T^s$ governed by 1 + 0.5 tanh(z′/h).

The blue contours in Figs. 1b and 1c show that the horizontal distribution of w′ is not close to axisymmetric even within the vortex core and becomes increasingly nonaxisymmetric as R increases beyond the vortex core. In particular, the two spiral bands of positive and negative w′ outside the vortex core are distinctly nonaxisymmetric and is thus dominated by w^a—the asymmetric part of w′ formulated in (A4c). According to (A3a) and (A4c), the horizontal distribution of w^a is featured by a pair of spiral bands of positive and negative w^a winding around and toward the vortex center from two opposite directions at each vertical level above the ground. As the vertical level increases from z′ = 0 to D = 5 km, the two paired spiral bands of w^a become increasingly strong [because their amplitudes increase with z′ in proportion to tanh(z′/h)/ρ_a] and their overall patterns rotate around the vortex center cyclonically by π/2 (=90°). Since w′ is dominated by w^a outside the vortex core, the above described structures of w^a also largely describe the structures of w′ outside the vortex core especially in the middle and upper levels of the analysis domain as shown by the blue contours in Fig. 1c.

Note that the density-weighted vertically integrated velocity-potential X formulated in (A3a) and the density-weighted streamfunction Y formulated in (A3b) have the same horizontal distributions but with opposite signs. Note also that the slantwise-vertical velocity for the asymmetric part of VF is related to X by $w^a=-\left({{\partial }^{\prime }}_x^2+{{\partial }^{\prime }}_y^2\right)Y/{\rho }_a=-\left({\partial }_R^2+R^{-1}{\partial }_R+R^{-2}\partial {\beta }^2\right)X/{\rho }_a$ as shown in (A4c) and the slantwise-vertical vorticity for the asymmetric part of VF is defined by ${\zeta }^a\equiv \left({{\partial }^{\prime }}_x^2+{{\partial }^{\prime }}_y^2\right)Y/{\rho }_a=\left({\partial }_R^2+R^{-1}{\partial }_R+R^{-2}\partial {\beta }^2\right)Y/{\rho }_a$. The above relationships imply that the horizontal distribution of ζ^a is also featured by a pair of spiral bands with opposite signs, and the spiral band of positive (or negative) ζ^a is collocated with the spiral band of positive (or negative) w^a at each vertical level above the ground. However, as implied by the formulation of Y in (A3b), the amplitudes of the two paired spiral bands of ζ^a decrease with z′ in proportion to cosh(z′/h)⁻²/ρ_a. Thus, as the vertical level increases from z′ = 0 to D = 5 km, the two paired spiral bands of ζ^a become weak but their overall patterns still rotate around the vortex center cyclonically by π/2 (=90°). The above described structures of ζ^a in connection with w^a are not shown explicitly but they are embedded implicitly and subtly in the vector field of (u′, υ′) plotted by the black arrows in Figs. 1b and 1c. The coupled spiral bands of w^a and ζ^a and their rotations with height are configured to resemble some asymmetric flow structures in observed tornadic mesocyclones (Marquis et al. 2012) so the benchmark VF can have adequate complexities for testing the VF-Var.

b. Simulated radial-velocity observations

Using (2.4), radial-velocity innovations are generated in the analysis domain by applying idealized Doppler scans and pseudo-operational Doppler scans to an upright (or eastward-slanted) vortex formulated by setting u_c = 10 m s⁻¹ and υ_c = 0 in (2.2) with ∂_zx_c = ∂_zy_c = 0 (or ∂_zx_c = 0.5 and ∂_zy_c = 0). The analysis domain is centered at (x′, y′) = (0, 0) along the z′ coordinate (from z′ = 0 to 5 km) and covers a square area of 20 × 20 km² at each vertical level. The idealized Doppler scans are purely horizontal and unidirectional in parallel along horizontal grid lines of a coarse-resolution grid with ∆x′ = ∆y′ = 0.5 km at 5 vertical levels, every ∆z′ = 1 km, from z′ = 1 to 5 km. In this case, radial-velocity observations and innovations are generated by setting θ = 0° and φ = 270° (or 0°) in (2.3) and (2.4), respectively, on the coarse-resolution grid in the analysis domain. In this case, ${\upsilon }_r^o$ scanned from the east (or south) with φ = 270° (or 0°) is simply −u^o (or υ^o) and the associated ${\upsilon }_r^o$ is simply −uⁱ (or υⁱ). The pseudo-operational Doppler scans mimics the scan mode VCP12 used by operational WSR-88Ds for severe storms. In this mode, a full volume of radial velocity is scanned from each radar on 14 sweeps (at 0.5°, 0.9°, 1.3°, 1.8°, 2.4°, 3.1°, 4.0°, 5.1°, 6.4°, 8.0°, 10.0°, 12.5°, 15.6°, and 19.5°) over the analysis time window from t = 0 to 5 min, while the range resolution is 250 m and the azimuthal beam spacing is 0.5°. The first radar, called radar A (or the second radar, called radar B) is 30 km to the east (or south) of the vortex center on the ground (z = 0) at t = 0. Figure 2 shows the positions of radar A and radar B relative to the vortex-following moving analysis domain at the starting time (t = 0) and ending time (t = 5 min) of the analysis time window.

Positions of radar A and radar B relative to the analysis domain at the starting and ending times of the analysis time window. The square box shows the 20 × 20 km² area of vortex-following analysis domain on the ground level. The plus sign at the center of the square box shows the analysis domain center (also the vortex center) that is moving eastward toward radar A at the speed of 10 m s⁻¹. The black dot labeled with “radar A” (or “radar B”) shows the position of radar A (or radar B), which is 30 km to the east (or south) of the vortex center on the ground (z = 0) at the starting time (t = 0), while the nearby gray dot to the west of the black dot shows the position of radar A (or radar B) at the ending time (t = 5 min) of the analysis time window.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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Positions of radar A and radar B relative to the analysis domain at the starting and ending times of the analysis time window. The square box shows the 20 × 20 km² area of vortex-following analysis domain on the ground level. The plus sign at the center of the square box shows the analysis domain center (also the vortex center) that is moving eastward toward radar A at the speed of 10 m s⁻¹. The black dot labeled with “radar A” (or “radar B”) shows the position of radar A (or radar B), which is 30 km to the east (or south) of the vortex center on the ground (z = 0) at the starting time (t = 0), while the nearby gray dot to the west of the black dot shows the position of radar A (or radar B) at the ending time (t = 5 min) of the analysis time window.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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Positions of radar A and radar B relative to the analysis domain at the starting and ending times of the analysis time window. The square box shows the 20 × 20 km² area of vortex-following analysis domain on the ground level. The plus sign at the center of the square box shows the analysis domain center (also the vortex center) that is moving eastward toward radar A at the speed of 10 m s⁻¹. The black dot labeled with “radar A” (or “radar B”) shows the position of radar A (or radar B), which is 30 km to the east (or south) of the vortex center on the ground (z = 0) at the starting time (t = 0), while the nearby gray dot to the west of the black dot shows the position of radar A (or radar B) at the ending time (t = 5 min) of the analysis time window.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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a. Parameter settings for background error covariance functions

As explained in section 2c of Part I, for the stand-alone applications considered in this paper, the background wind is zero in (x′, y′, z′) so the background wind error is simply the true value of (u′, υ′, w′) in (x′, y′, z′). For the axisymmetric part of benchmark VF shown in Fig. 1a, $V_T^s$ formulated in (A1a) has narrower structures than $V_R^s=-{\partial }_z^{\prime }{\psi }^s/{\rho }_a$ formulated in (A2a), so the background error of $V_T^s$ tends to have narrower correlation structures than the background error of ψ^s. Based on this consideration, the background error decorrelation length factored into r in (3.6) of Part I is set to l = l₁ = 1/2 for $V_T^s$ but set to l = l₂ = 1 for ψ^s, while R_c = 1.5 km is used for the coordinate transformation from R to r in (3.6) of Part I. The background error decorrelation depth factored into h is set to H = H₁ = H₂ = 2 km for $V_T^s$ and ψ^s. Note that the vertical resolution of radial-velocity observations from the idealized (or pseudo-operational) Doppler scans is ∆z′ = 1 km (or coarser than 1 km in upper levels of the analysis domain), so spurious vertical variations can be generated in analyzed VFs if the background error decorrelation depth reduces (from 2 km) to 1 km. This explains why we set H = H₁ = H₂ = 2 km more or less empirically. When the decorrelation length l in r is transformed into the physical radial coordinate R, the decorrelation length in R is a function of R given by L = R_c sinh[(rl + l/2)]–R_c sinh[(rl–l/2)] = 2R_c cosh(rl) sinh (l/2) = 2 (R² + $R_c^2$)^1/2 sinh(l/2). For the above settings of l = l₁ = 1/2 and l = l₂ = 1, the corresponding values of L at R = R_c are $L_1\approx R_c/\sqrt{2}\approx 1\hspace{0.17em}\text{km}$ and $L_2\approx \sqrt{2}{R}_c\approx 2\hspace{0.17em}\text{km}$, respectively.

For the asymmetric part, the background error decorrelation length factored into r is set to l = l₃ = 1 for X but to $l=l_4=1/\sqrt{2}$ for Y, while the background error decorrelation depth factored into h is set to $H=H_3=\sqrt{3}\hspace{0.17em}\left(=1.732\right)\hspace{0.17em}\text{km}$ for X but set to H = H₄ = 2 km for Y. These settings of l and H are selected loosely by considering that the rotational velocity field associated with Y tends to have narrower and deeper structures than the divergent velocity field associated with X. For the above settings of l = l₃ = 1 and $l=l_4=1/\sqrt{2}$, the corresponding values of L at R = R_c are $L_3\approx \sqrt{2}{R}_c\approx 2\hspace{0.17em}\text{km}$ and L₄ ≈ R_c = 1.5 km, respectively.

The root-mean-square values of the divergent (or rotational) part of benchmark (u^a, υ^a) computed from X (or Y) within 5 km from the vortex center in the analysis domain is 4.9 (or 3.2) m s⁻¹, so the background error standard deviation can be set to σ_d = 5 (or σ_r = 3) m s⁻¹ for the divergent (or rotational) part of (u^a, υ^a). The background error standard deviation for ${\partial }_{z^{\prime }}X$ (or Y) can be then estimated by ${\sigma }_d{L}_3/\sqrt{2}$ (or ${\sigma }_r{L}_4/\sqrt{2}$) according to (A13)–(A.14) of Xu et al. (2010), and the background error standard deviation for X can be finally estimated by ${\sigma }_d{L}_3{H}_3/\sqrt{2}$ by using the relationship derived similarly to that in (3.1)–(3.2). Thus, in summary, we set ${\sigma }_3={\sigma }_d{L}_3{H}_3/\sqrt{2}$ with σ_d = 5 m s⁻¹, L₃ = 2 km, l = l₃ = 1 and $H=H_3=\sqrt{3}\left(=1.732\right)\hspace{0.17em}\text{km}$ for the background error covariance function formulated for X in (4.8a) of Part I, and set ${\sigma }_4={\sigma }_r{L}_4/\sqrt{2}$ with σ_r = 3 m s⁻¹, L₄ = 1.5 km, $l=l_4=1/\sqrt{2}$, H = H₄ = 2 km for the background error covariance function formulated for Y in (4.8b) of Part I. For the decorrelation arc defined by Φ ≡ 2 sinh(l/2) + [Φ₀ – 2 sinh(l/2)]R_p/(R + R_p) in (4.9) of Part I, we set Φ₀ = π/2 and R_p = 5 km. The above parameter settings are used for all the experiments designed in the next subsections.

b. Experiment design

The aforementioned upright and slanted vortices are assumed to be stationary in the moving coordinate system (x′, y′, z′) within the analysis time window, so the location of radar A (or radar B) changes with time in (x′, y′, z′) and this affects the calculations of φ and θ at each observation point for the pseudo-operational Doppler scans (but not the idealized Doppler scans). Radial velocities scanned on the two highest sweeps at 15.6° and 19.5° from either radar A or B are located above the analysis domain and thus not used for the analyses, so only 12 sweeps are used. These 12 sweeps are scanned sequentially from each radar every ∆t = 20 s from t = 0 (at 0.5°) to 240 s = 4 min (at 12.5°). When the radial-velocity observations (generated in each radar coordinate system) are transformed into the moving coordinate system (x′, y′, z′), the location of each radar is fixed and calculated for each sweep at the beginning time of the sweep, so the small time differences between azimuthal scans on the same sweep are neglected.

For each type of Doppler scans, experiments are designed in three pairs to test the VF-Var by applying the two-step analysis versus the single-step analysis to three innovation datasets. For the idealized Doppler scans, the three innovation datasets are (i) the dual-Doppler innovations given by (−uⁱ, υⁱ), (ii) the single-Doppler innovations given by −uⁱ only, and (iii) the single-Doppler innovations given by υⁱ only. The three experiments performed by applying the two-step analysis (or single-step analysis) to these three datasets are named E-uv-2, E-u-2, and E-v-2 (or E-uv-1, E-u-1, and E-v-1), respectively. The results of these experiments tested with the upright and slanted vortices will be presented and examined in sections 4a and 4b, respectively. For the pseudo-operational Doppler scans, the three innovation datasets are (i) the dual-Doppler innovations from radar A and radar B, (ii) the single-Doppler innovations from radar A only, and (iii) the single-Doppler innovations from radar B only. The three experiments performed by applying the two-step analysis (or single-step analysis) to these three datasets are named E-AB-2, E-A-2, and E-B-2 (or E-AB-1, E-A-1, and E-B-1), respectively. The results of these experiments tested with the upright and slanted vortices will be presented and examined in sections 5a and 5b, respectively.

As explained at the end of section 4 of Part I, the analyzed VF is a spatially continuous vector field. The benchmark VF formulated in (A2) and (A4) is also a continuous vector field, so the performance of VF-Var can be evaluated on any high-resolution grid. However, the intrinsic spatial resolution of the analyzed VF is limited by the length scale resolvable by the background error correlation functions in connection with spatial distributions of observations. Based on this consideration, the horizontal (or vertical) grid resolution used for evaluating the performance of VF-Var is set to ∆x′ = ∆y′ = 0.25 km (or ∆z′ = 0.5 km) which is finer than the smallest background error decorrelation length (or depth) and roughly matches the highest horizontal (or vertical) resolution of nonuniformly distributed observations from the pseudo-operational Doppler scans. This evaluation grid (with ∆x′ = ∆y′ = 0.25 km and ∆z′ = 0.5 km) will be used in the next two sections to calculate the cylindrical-volume-averaged RMS error (CRE) for each analyzed field centered at (x′, y′) = (0, 0) within R ≤ 5 km through the entire depth (from z′ = 0 to 5 km) of the analysis domain as well as the domain-averaged RMS error (DRE) over the entire analysis domain.

a. Results for upright vortex

For the experiments performed with idealized Doppler scans of the upright vortex, the CRE is listed for each analyzed field of ($V_T^s$, $V_R^s$, w^s) in Table 1 and each analyzed field of (u′, υ′, w′) in Table 2. As shown in row 1 of Table 1 for the two-step analysis in E-uv-2, the CREs of analyzed ($V_T^s$, $V_R^s$) are much smaller than the observation error standard deviation σ_o (=1 m s⁻¹) and the CRE of analyzed w^s is also smaller than σ_o, so the axisymmetric part of VF is analyzed very accurately. The CREs of analyzed (u′, υ′) in E-uv-2 are also smaller than σ_o (see row 1 of Table 2) as they should be because u and υ are observed. However, the CRE of analyzed w′ is larger than σ_o. This is expected because w′ is not observed and has complex structures. Thus, the total VF is also well analyzed in E-uv-2 although the relative CRE (RCRE) of analyzed w′ is as large as 30.2%. On the other hand, the CREs of analyzed ($V_T^s$, $V_R^s$) from the single-step analysis in E-uv-1 are larger than those from the two-step analysis in E-uv-2 (see row 1 vs row 2 of Table 1) but still smaller than σ_o, whereas the CRE of analyzed w^s in E-uv-1 becomes much larger than σ_o. However, the CREs of analyzed (u′, υ′, w′) in E-uv-1 (see row 2 of Table 2) are only slightly larger than those from E-uv-2 (see row 1 of Table 2) and so are their associated RCREs. Thus, in this case, the two-step analysis outperforms the single-step analysis especially in analyzing the axisymmetric part of VF.

Table 1.

Cylindrical-volume-averaged RMS errors (CREs) of analyzed ($V_T^s$, $V_R^s$, w^s) and diagnosed ($V_T^{s+}$, $V_R^{s+}$, w^s+) from experiments performed with idealized Doppler scans of upright vortex in section 4a. The cylindrical-volume average is taken over the volume within R ≤ 5 km through the entire depth (from z′ = 0 to 5 km) of the analysis domain. As described in section 3b, each experiment is named in three parts. The letter “uv,” “u,” or “v” in the middle part means that the experiment is performed with the dual-Doppler innovations given by (−uⁱ, υⁱ), the single-Doppler innovations given by −uⁱ only, or the single-Doppler innovations given by υⁱ only. The number “1” (or “2”) in the last part means that the experiment is performed by applying the two-step analysis (or single-step analysis).

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Table 2.

As in Table 1, but for CREs and RCREs of analyzed (u′, υ′, w′), where RCRE stands for relative CRE defined by the ratio of CRE with respect to the cylindrical-volume-averaged RMS value of the related benchmark field.

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As explained in appendix A, the fields of ($V_T^a$, $V_R^a$, w^a) formulated in (A4) for the asymmetric part of benchmark VF are purely asymmetric. However, the analyzed ($V_T^a$, $V_R^a$, w^a) are not purely asymmetric, so the total axisymmetric part of VF should be diagnosed by $\left(V_T^{s+},V_R^{s+},w^{s+}\right)\equiv \left(V_T^s,V_R^s,w^s\right)+{\left(2\pi \right)}^{-1}{\displaystyle {\int }_{-\pi }^{\pi }\left(V_T^a,V_R^a,w^a\right)\hspace{0.17em}d\beta }$. The CREs of diagnosed ($V_T^{s+}$, $V_R^{s+}$, w^s+) are listed in the last three columns of Table 1. As shown in row 1 (or row 2) of Table 1, the diagnosed ($V_T^{s+}$, $V_R^{s+}$, w^s+) are slightly (or substantially) more accurate than the directly analyzed ($V_T^s$, $V_R^s$, w^s) in E-uv-2 (or E-uv-1), whereas the diagnosed ($V_T^{s+}$, $V_R^{s+}$, w^s+) in E-uv-1 are as accurate as those in E-uv-2.

When dual-Doppler innovations (−uⁱ, υⁱ) reduce to single-Doppler innovations −uⁱ, the analyzed ($V_T^s$, $V_R^s$, w^s) and diagnosed ($V_T^{s+}$, $V_R^{s+}$, w^s+) become slightly less accurate (as seen from row 3 vs row 1 in Table 1), the analyzed u′ (or w′) become slightly less (or more) accurate but the analyzed υ′ becomes much less accurate (as seen from row 3 vs row 1 in Table 2). The reduced accuracy in υ′ is apparently caused by the absence of υⁱ in the single-Doppler innovations. On the other hand, as shown in row 4 of Table 1 for the single-step analysis applied to −uⁱ in E-u-1, the CREs of analyzed ($V_T^s$, $V_R^s$) are larger than those from the two-step analysis in E-u-2 (see row 3 of Table 1) but still smaller than σ_o whereas the CRE of analyzed w^s becomes larger than σ_o. In this case, the diagnosed $V_T^{s+}$ (or $V_R^{s+}$) is slightly (or substantially) more accurate than the analyzed $V_T^s$ (or $V_R^s$) and the diagnosed w^s+ is much more accurate than the analyzed w^s. Thus, the axisymmetric part of VF is not well analyzed directly but well diagnosed indirectly in E-u-1. As shown in row 4 versus row 3 of Table 2, the CREs of analyzed (u′, υ′, w′) in E-u-1 are nearly the same as those from E-u-2 and so are their associated RCREs.

For the two-step analysis applied to υⁱ in E-v-2, the CREs of analyzed ($V_T^s$, $V_R^s$, w^s) and diagnosed ($V_T^{s+}$, $V_R^{s+}$, w⁺) in row 5 of Table 1 are comparable to those from E-u-2 in row 3 of Table 1, and the CREs of analyzed u′ and (υ′, w′) in row 5 of Table 2 are comparable to those of analyzed υ′ and (u′, w′) from E-u-2 in row 3 of Table 2. The increased CRE of analyzed u′ in E-v-2 is similar to the increased CRE of analyzed υ′ in E-u-2 but is now caused by the absence of uⁱ in the single-Doppler innovations. For the single-step analysis applied to -υⁱ in E-v-1, the CREs of analyzed ($V_T^s$, $V_R^s$, w^s) and diagnosed ($V_R^{s+}$, w⁺) (or $V_T^{s+}$) in row 6 of Table 1 are larger (or slightly smaller) than those from the two-step analysis in E-v-2 (see row 5 of Table 1), but the CREs of analyzed (u′, υ′, w′) in row 6 of Table 2 are nearly the same as those from the two-step analysis in E-u-2 (see row 5 of Table 2) and so are their associated RCREs.

DREs of analyzed (u′, υ′, w′) are also computed (on a grid with ∆x′ = ∆y′ = 0.25 km and ∆z′ = 0.5 km) over the entire analysis domain (rather than within R ≤ 5 km). The computed DRE is roughly a half of CRE for each analyzed field of (u′, υ′, w′) while the domain-averaged RMS value is also roughly a half of the cylindrical-volume-averaged RMS value for each benchmark field. Thus, the relative DRE (RDRE; defined by the ratio of DRE with respect to the domain-averaged RMS value of the related benchmark field) is roughly the same as the RCRE listed in Table 2 for each analyzed field of (u′, υ′, w′) in each experiment. For each analyzed field of (u′, υ′, w′), the DRE is a vertical average of the area-averaged RMS error (ARE), and the ARE is calculated (on a grid with ∆x′ = ∆y′ = 0.25 km) over the entire area of the analysis domain at each vertical level. The AREs of analyzed (u′, υ′, w′) from the above three pairs of experiments are shown as functions of z′ in Figs. 3a–c. As shown, the AREs of single-step analyzed (u′, υ′, w′) have nearly the same vertical distributions as those of their respective two-step analyzed (u′, υ′, w′). As shown by the red (or green) dashed curve in Fig. 3b (or Fig. 3c), the ARE of analyzed u′ (or υ′) in E-u-2 (or E-v-2) is similar to that from E-uv-2 shown in Fig. 3a, which is nearly constant (around σ_o/4) for z′ ≥ 1 km but increases nearly 4 times (to about σ_o = 1 m s⁻¹) as z′ decreases from 1 to 0 km. This increase of ARE is due to the absence of observation below z′ = 1 km. On the other hand, the ARE of analyzed υ′ (or u′) in E-u-2 (or E-v-2) is substantially larger than that from E-uv-2 shown in Fig. 3a, and this increase of ARE is due to the absence of υⁱ (or uⁱ) in the single-Doppler innovations as explained earlier. However, the absence of υⁱ (or uⁱ) causes only a moderate increase of ARE for the analyzed w′ in E-u-2 (or E-v-2) from that in E-uv-2. Note that the analyzed w′ is ensured to satisfy the boundary condition of w′ = 0 at z′ = 0 (see section 4 of Part I) and the area-averaged RMS value of benchmark w′ increase with z′. This explains why the ARE of analyzed w′ is zero at z′ = 0 and increases with z′ as shown by the blue curve for each experiment in Fig. 3.

Area-averaged RMS errors (AREs) plotted as functions of z′ for analyzed (u′, υ′, w′) from three pairs of experiments performed with idealized Doppler scans of upright vortex: (a) E-uv-1 vs E-uv-2 for dual-Doppler analyses, (b) E-u-1 vs E-u-2 for single-Doppler analyses with u-component observations only, and (c) E-v-1 vs E-v-2 for single-Doppler analyses with υ-component observations only. The area average is taken over the entire 20 × 20 km² area of the analysis domain at each vertical level.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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Area-averaged RMS errors (AREs) plotted as functions of z′ for analyzed (u′, υ′, w′) from three pairs of experiments performed with idealized Doppler scans of upright vortex: (a) E-uv-1 vs E-uv-2 for dual-Doppler analyses, (b) E-u-1 vs E-u-2 for single-Doppler analyses with u-component observations only, and (c) E-v-1 vs E-v-2 for single-Doppler analyses with υ-component observations only. The area average is taken over the entire 20 × 20 km² area of the analysis domain at each vertical level.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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Area-averaged RMS errors (AREs) plotted as functions of z′ for analyzed (u′, υ′, w′) from three pairs of experiments performed with idealized Doppler scans of upright vortex: (a) E-uv-1 vs E-uv-2 for dual-Doppler analyses, (b) E-u-1 vs E-u-2 for single-Doppler analyses with u-component observations only, and (c) E-v-1 vs E-v-2 for single-Doppler analyses with υ-component observations only. The area average is taken over the entire 20 × 20 km² area of the analysis domain at each vertical level.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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The analyzed $V_T^s$ and ($V_R^s$, w^s) in E-u-2 are shown in Fig. 4a against their respective benchmark fields. As shown, even with single-Doppler innovations, the analyzed $V_T^s$ and ($V_R^s$, w^s) match their respective benchmark fields quite closely, which is consistent with their small CREs listed in row 3 of Table 1. The analyzed (u′, υ′) and w′ in E-u-2 are shown against their respective benchmark fields at z′ = 1 km in Fig. 4b and z′ = 4 km in Fig. 4c. As shown, the analyzed (u′, υ′) is very close to the benchmark field and the difference from the benchmark is small (< or ≪5 m s⁻¹) and mainly in the direction of unobserved υ component, while the analyzed w′ captures the dominant updraft and downdraft structures. However, largest errors are seen for the analyzed w′ inside the vortex core, especially in the upper levels. These large errors are caused by the rapid horizontal variations of w′ inside the vortex core that cannot be well resolved by the VF-dependent background error correlation functions in connection with the very limited number of observations inside the vortex core. When the innovations are given by (−uⁱ, υⁱ) instead of −uⁱ, the analyzed (u′, υ′) and w′ (not shown) become, as expected, much closer to their respective benchmark fields than those in Figs. 4b and 4c. Thus, when the vortex is upright, the two-step analysis performs very well with the idealized dual-Doppler innovations and reasonably well with the idealized single-Doppler innovations, whereas the single-step analysis underperforms the two-step analysis slightly in analyzing (u′, υ′, w′) and substantially in analyzing ($V_T^s$, $V_R^s$, w^s).

(a) As in Fig. 1a, but overlapped by analyzed $V_T^s$ and ($V_R^s$, w^s) from E-u-2 plotted by red contours and green arrows, respectively. (b),(c) As in Figs. 1b and 1c, but overlapped by analyzed (u′, υ′) and w′ from E-u-2 plotted by green arrows and red contours, respectively. E-u-2 is performed with idealized Doppler scans of upright vortex.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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(a) As in Fig. 1a, but overlapped by analyzed $V_T^s$ and ($V_R^s$, w^s) from E-u-2 plotted by red contours and green arrows, respectively. (b),(c) As in Figs. 1b and 1c, but overlapped by analyzed (u′, υ′) and w′ from E-u-2 plotted by green arrows and red contours, respectively. E-u-2 is performed with idealized Doppler scans of upright vortex.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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(a) As in Fig. 1a, but overlapped by analyzed $V_T^s$ and ($V_R^s$, w^s) from E-u-2 plotted by red contours and green arrows, respectively. (b),(c) As in Figs. 1b and 1c, but overlapped by analyzed (u′, υ′) and w′ from E-u-2 plotted by green arrows and red contours, respectively. E-u-2 is performed with idealized Doppler scans of upright vortex.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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b. Results for slantwise vortex

For the experiments performed with the eastward-slanted vortex [formulated by setting ∂_zx_c = 0.5 and ∂_zy_c = 0 in (2.2)–(2.4)], the resulting CREs are listed in Tables 3 and 4 . As shown in row 1 of Table 3 for the two-step analysis applied to (−uⁱ, υⁱ) in E-uv-2, the CREs of analyzed ($V_T^s$, $V_R^s$, w^s) become much larger and the CREs of diagnosed ($V_T^{s+}$, $V_R^{s+}$, w^s+) also become larger when compared to the upright vortex (see row 1 of Table 1) but the latter CREs are still smaller than σ_o. As shown in row 1 of Table 4, the CREs of analyzed (u′, υ′) are slightly larger but the CRE of analyzed w′ is substantially smaller when compared to the upright vortex (see row 1 of Table 2). Thus, when the vortex becomes slanted, the axisymmetric part of VF becomes difficult to analyze directly but still can be diagnosed indirectly with less reduced accuracies. In this case, the analyzed (u′, υ′) become slightly less accurate but the analyzed w′ becomes substantially more accurate for the two-step analysis in E-uv-2. On the other hand, as shown in row 2 of Table 3 for the single-step analysis applied to (−uⁱ, υⁱ) in E-uv-1, the analyzed ($V_T^s$, $V_R^s$, w^s) and diagnosed ($V_T^{s+}$, $V_R^{s+}$, w^s+) are more accurate than those from the two-step analysis in E-uv-2 (see row 1 of Table 3). Also, as seen by comparing row 2 with row 1 in Table 4, the analyzed (u′, υ′, w′) in E-uv-1 are slightly more accurate than those from E-uv-2. Thus, the single-step analysis outperforms the two-step analysis in this case.

Table 3.

As in Table 1, but from experiments performed with idealized Doppler scans of eastward-slanted vortex in section 4b.

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Table 4.

As in Table 2, but from experiments performed with idealized Doppler scans of eastward-slanted vortex in section 4b.

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When only considering single-Doppler innovations with (−uⁱ, υⁱ) reduced to −uⁱ, the analyzed ($V_T^s$, $V_R^s$, w^s) and diagnosed ($V_T^{s+}$, $V_R^{s+}$, w^s+) become less accurate especially for the two-step analysis [as seen by comparing rows 3 with row 1 in Table 3] while the analyzed (u′, w′) also become less accurate and the analyzed υ′ becomes much less accurate [as seen by comparing row 3 (or row 4) with row 1 (or row 2) in Table 4]. In this case, the CREs in row 3 of Table 3 (or Table 4) for the two-step analysis are still larger than those from the single-step analysis shown in row 4 of Table 3 (or Table 4). Thus, when (−uⁱ, υⁱ) reduce to −uⁱ, the axisymmetric part of VF and (u′, w′) become more difficult to analyze, the analyzed υ′ becomes much less accurate (due again to the absence of υⁱ in the single-Doppler innovations), and the single-step analysis still outperforms the two-step analysis.

When only considering single-Doppler innovations with (−uⁱ, υⁱ) reduced to υⁱ, the innovations υⁱ used in E-v-2 and E-v-1 are not affected by the vortex eastward slanting because υⁱ is given by υ′ + ε for θ = 0° and φ = 0° with ∂_zy_c = 0 in (2.4) although ∂_zx_c = 0.5 ≠ 0. In this case, the innovations υⁱ remain the same as those used in section 4a for the upright vortex with ∂_zx_c = ∂_zy_c = 0 except that they are now located in the slantwise coordinates (x′, y′, z′) with ∂_zx_c = 0.5. This explains why the CREs in rows 5 and 6 of Table 3 (or Table 4) are the same as those for the upright vortex shown in row 5 and 6 of Table 1 (or Table 2), respectively.

On the other hand, −uⁱ is given by −(u′ + w′∂_zx_c + ε) for θ = 0° and φ = 270° with ∂_zx_c = 0.5 in (2.4) and thus contains a projection of the slantwise vertical velocity w′. When the VF is expressed by (u′, υ′, w′) in terms of the covariant basis vectors in (x′, y′, z′), w′∂_zx_c is not a part of u′ and thus can be partitioned into w^s∂_zx_c and w^a∂_zx_c. However, when the VF is expressed by (u_υ, υ_υ, w) in (2.4) and observed in (x, y, z), w′∂_zx_c becomes a part of u_υ, so w^s∂_zx_c (or w^a∂_zx_c) is no longer purely axisymmetric (or asymmetric) as a part of (u_υ, υ_υ). For this reason, as explained in section 2b of Part I, the VF cannot be analyzed in a properly partitioned form by a 2DVar at a given vertical level unless ∂_zx_c = ∂_zy_c = 0. For the same reason, when the axisymmetric part of VF is analyzed alone in the first step in E-uv-2 or E-u-2, the attribution of w^s∂_zx_c (or w^a∂_zx_c) to the asymmetric (or axisymmetric) part of (u_υ, υ_υ) is not counted. This explains why the analyzed ($V_T^s$, $V_R^s$, w^s) in E-uv-2 and E-u-2 become less accurate as the vortex becomes slanted eastward (with ∂_zx_c increased from 0 to 0.5). On the other hand, when the axisymmetric part is analyzed jointly with the asymmetric part in E-uv-1 (or E-u-1), the attributions of w^s∂_zx_c and w^a∂_zx_c to the asymmetric and axisymmetric parts of (u_υ, v_υ) are all counted. This explains why ($V_T^s$, $V_R^s$, w^s) are analyzed more accurately in E-uv-1 (or E-u-1) than in E-uv-2 (or E-u-2). Besides, since the analyzed (u^a, υ^a, w^a) diminish rapidly toward the vortex center in the vortex core (see appendix C of Part I), errors in ($V_T^s$, $V_R^s$, w^s) produced in the first-step analysis cannot be corrected (or corrected effectively) at (or around) the vortex center by the second-step analysis in E-uv-2 or E-u-2. This explains why the diagnosed ($V_T^{s+}$, $V_R^{s+}$, w^s+) in E-uv-2 (or E-u-2) are still less accurate than those from the single-step analysis in E-uv-1 (or E-u-1). Furthermore, by noting that w′ is a part of −uⁱ = −(u′ + w′∂_zx_c + ε) and thus is partially observed for ∂_zx_c = 0.5, one can see why w′ is analyzed more accurately in E-uv-2 (or E-uv-1) when the vortex becomes slanted eastward [as seen by comparing row 1 (or row 2) in Table 4 with that in Table 2].

The AREs of analyzed (u′, υ′, w′) from the above two pairs of experiments are shown as functions of z′ in Figs. 5a and 5b. As shown, the single-step analyzed (u′, υ′, w′) are more accurate than their respective two-step analyzed (u′, υ′, w′), and the analyzed u′ (or υ) in E-u-1 or E-u-2 is less (or much less) accurate than that in E-uv-1 or E-uv-2. These comparisons are consistent with the results in Table 4. In addition, as shown by the solid curves in Fig. 5 versus those in Fig. 3, when the vortex becomes slanted eastward, the dual-Doppler analyzed u′ (or w′) in E-uv-1 becomes less (or more) accurate and the single-Doppler analyzed u′ (or w′) in E-u-1 becomes substantially less (or slightly more) accurate, and these changes of accuracies (caused by the increase of ∂_zx_c from 0 to 0.5) are seen mainly for z′ > 1 km and become increasingly large as z′ increases above 1 km. Similar changes of accuracies have been seen and explained in terms of CRE when the results in Table 4 were compared with those in Table 2.

As in Figs. 3a and 3b, but for eastward-slanted vortex (with ∂_zx_c = 0.5 and ∂_zy_c = 0).

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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As in Figs. 3a and 3b, but for eastward-slanted vortex (with ∂_zx_c = 0.5 and ∂_zy_c = 0).

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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As in Figs. 3a and 3b, but for eastward-slanted vortex (with ∂_zx_c = 0.5 and ∂_zy_c = 0).

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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Figure 6a shows that the diagnosed $V_T^{s+}$ and ($V_R^{s+}$, w^s+) in E-u-1 match their respective benchmark fields reasonably well. Figures 6b and 6c show that the analyzed (u′, υ′) (or w′) in E-u-1 matches its benchmark field very closely (or reasonably well) at z′ = 1 and 4 km, respectively. When the innovations are given by (−uⁱ, υⁱ) instead of −uⁱ, the diagnosed $V_T^{s+}$ and ($V_R^{s+}$, w^s+) and analyzed (u′, υ′) and w′ (not shown) are much closer to their respective benchmark fields than those in Figs. 6a–c. Thus, when the vortex becomes slanted eastward, the single-step analysis performs very well with the idealized dual-Doppler innovations and reasonably well with the idealized single-Doppler innovations, whereas the two-step analysis underperforms the single-step analysis (except for single-Doppler scans perpendicular to the vortex slanting direction).

(a) As in Fig. 1a, but overlapped by diagnosed $V_T^{s+}$ and ($V_R^{s+}$, w^s+) from E-u-1 plotted by red contours and green arrows, respectively. (b),(c) As in Figs. 4b and 4c, but from E-u-1. E-u-1 is performed with idealized Doppler scans of eastward-slanted vortex.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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(a) As in Fig. 1a, but overlapped by diagnosed $V_T^{s+}$ and ($V_R^{s+}$, w^s+) from E-u-1 plotted by red contours and green arrows, respectively. (b),(c) As in Figs. 4b and 4c, but from E-u-1. E-u-1 is performed with idealized Doppler scans of eastward-slanted vortex.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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(a) As in Fig. 1a, but overlapped by diagnosed $V_T^{s+}$ and ($V_R^{s+}$, w^s+) from E-u-1 plotted by red contours and green arrows, respectively. (b),(c) As in Figs. 4b and 4c, but from E-u-1. E-u-1 is performed with idealized Doppler scans of eastward-slanted vortex.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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a. Results for upright vortex

For the experiments performed with pseudo-operational Doppler scans of the upright vortex, the resulting CREs are listed in Tables 5 and 6 . As shown by comparing row 1 with row 2 in Table 5, the CREs of analyzed ($V_T^s$, $V_R^s$) in E-AB-2 are smaller than σ_o but larger than those in E-AB-1, the CRE of analyzed w^s in E-AB-2 is larger than σ_o but smaller than that in E-AB-1, and the CREs of diagnosed ($V_T^{s+}$, $V_R^{s+}$, w^s+) in E-AB-2 are all further reduced below σ_o but slightly larger than those in E-AB-1. Thus, the axisymmetric part of VF is diagnosed slightly more accurately in E-AB-1 than in E-AB-2. As shown by comparing row 1 with row 2 in Table 6, the CREs of analyzed (u′, υ′) in E-AB-2 are smaller than σ_o but slightly larger than those in E-AB-1 whereas the CRE of analyzed w′ in E-AB-2 is smaller than that in E-AB-1. Thus, in this case, the two-step analysis is more accurate (or slightly less accurate) than the single-step analysis in analyzing w′ [or analyzing (u′, υ′) and diagnosing ($V_T^{s+}$, $V_R^{s+}$, w^s+)].

Table 5.

As in Table 1, but from experiments performed with pseudo-operational Doppler scans of upright vortex in section 5a. As described in section 3b, each experiment is named in three parts. The letter “AB,” “A,” or “B” in the middle part means that the experiment is performed with the dual-Doppler innovations from radar A and radar B, the single-Doppler innovations from radar A only, or the single-Doppler innovations from radar B only. The number “1” (or “2”) in the last part means that the experiment is performed by applying the two-step analysis (or single-step analysis).

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Table 6.

As in Table 2, but from experiments performed with pseudo-operational Doppler scans of upright vortex in section 5a.

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Note that the CREs of analyzed (u′, υ′, w′) from E-AB-2 (or E-AB-1) in Table 6 are substantially smaller than those from E-uv-2 (or E-uv-1) in Table 2 and so are the associated RCREs. These CRE reductions can be expected because the pseudo-operational Doppler scans have much higher data density and data coverage than the idealized Doppler scans. Note also that w′ is not directly observed from the idealized dual-Doppler scans of the upright vortex but now becomes partially observed from the pseudo-operational Doppler scans because θ is now not only nonzero but also can be large and so is the projection of w′ (=w for upright vortex) on ${\upsilon }_r^o$ [see (2.3)]. This provides an additional explanation for the CRE reduction of analyzed w′ in E-AB-2 (or E-AB-1) versus that in E-uv-2 (or E-uv-1).

When the dual-Doppler innovations from radars A and B reduce to single-Doppler innovations from radar A only, the analyzed ($V_T^s$, $V_R^s$, w^s) and diagnosed ($V_T^{s+}$, $V_R^{s+}$, w^s+) become less accurate especially for the two-step analysis [as seen from row 3 vs row 1 in Table 5]. The analyzed (u′, w′) also become less accurate and the analyzed υ′ becomes much less accurate [as seen from row 3 (or row 4) vs row 1 (or row 2) in Table 6]. The reduced accuracy in analyzed υ′ can be explained by the largely diminished projections of υ′ [=υ_υ for the upright vortex as shown in (2.4)] on ${\upsilon }_r^i$ for single-Doppler innovations from radar A. In this case, the diagnosed ($V_T^s$, $V_R^s$, w^s) and analyzed (u′, υ′, w′) in E-A-1 are more accurate than those in E-A-2, so the single-step analysis outperforms the two-step analysis.

The same behavior that is seen for radar A is replicated when radar B is considered alone (see row 5 and row 6 in Tables 5 and 6), except in this scenario, the analyzed u′ (instead of υ′) becomes much less accurate [as seen from row 5 (or row 6) vs row 1 (or row 2) in Table 6]. The reduced accuracy in analyzed u′ can be explained by the largely diminished projections of u′ (= u_υ for the upright vortex) on ${\upsilon }_r^i$ for single-Doppler innovations from radar B. In this case, the diagnosed ($V_T^s$, $V_R^s$, w^s) and analyzed (u′, υ′, w′) in E-B-1 are more accurate than those in E-B-2, so the single-step analysis still outperforms the two-step analysis.

The AREs of analyzed (u′, υ′, w′) from the above three pairs of experiments are shown in Figs. 7a–c. As shown in Fig. 7a, the AREs of analyzed (u′, υ′) from E-AB-2 are very close to those from E-AB-1 and the CRE of analyzed w′ from E-AB-2 is slightly smaller than that from E-AB-1. These comparisons are consistent with those seen for CREs in row 1 versus row 2 in Table 6. The AREs of analyzed (u′, υ′, w′) in Fig. 7a vary with z′ similarly to those in Fig. 3a and these vertical variations can be explained similarly (as explained for Fig. 3a in section 4a) except for the following differences: (i) The AREs of analyzed (u′, υ′) in Fig. 7a start to increase not as z′ decreases to 1 km (as seen in Fig. 3a) but as z′ further decreases below 0.5 km, and this difference can be explained by the fact that observations are now available between z′ = 0.5 and 1 km from the pseudo-operational Doppler scans. (ii) The AREs of analyzed (u′, υ′) in Fig. 7a do not stay nearly constant (as seen in Fig. 3a) but increase slowly (or substantially) as z′ increases above 2.5 (or 4.5) km, and this difference can be explained by the fact that the projection of (u′, υ′) [=(u_υ, υ_υ) for the upright vortex] on (uⁱ, υⁱ) is independent of z′ but its projection on ${\upsilon }_r^i$ is affected by z′. In particular, as z′ increases above 2.5 (or 4.5) km, cosθ decreases slowly (or substantially) and so does the projection of (u′, υ′) on ${\upsilon }_r^i$ according to (2.3)–(2.4). The increased elevation angle required for analyzing the flow at and above z′ = 2.5 and 4.5 km implies that (u′, υ′) are being less detected but the detection of w′ is increasing. (iii) The AREs of analyzed (u′, υ′, w′) in Fig. 7a are substantially smaller than those in Fig. 3a [except for (u′, υ′) above z′ = 4.5 km], and this is consistent with the earlier explained difference of CREs in the first two rows of Table 6 versus those in Table 2. As shown by the green solid (or dashed) curve in Fig. 7b versus that in Fig. 7a, the ARE of analyzed υ′ from E-A-1 (or E-A-2) is much larger than that from E-AB-1 (or E-AB-2), and this is consistent with the earlier explained increase of CRE in row 4 (or row 3) from that in row 2 (or row 1) of Table 6. As shown by the red solid (or dashed) curve in Fig. 7c versus that in Fig. 7a, the ARE of analyzed u′ from E-B-1 (or E-B-2) is much larger than that from E-AB-1 (or E-AB-2), and this is consistent with the earlier explained increase of CRE in row 6 (or row 5) from that in row 2 (or row 1) of Table 6.

As in Fig. 3, but from three paired experiments performed with pseudo-operational Doppler scans of upright vortex: (a) E-AB-1 vs E-AB-2 for dual-Doppler analyses, (b) E-A-1 vs E-A-2 for single-Doppler analyses with observations from radar A only, and (c) E-B-1 vs E-B-2 for single-Doppler analyses with observations from radar B only.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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As in Fig. 3, but from three paired experiments performed with pseudo-operational Doppler scans of upright vortex: (a) E-AB-1 vs E-AB-2 for dual-Doppler analyses, (b) E-A-1 vs E-A-2 for single-Doppler analyses with observations from radar A only, and (c) E-B-1 vs E-B-2 for single-Doppler analyses with observations from radar B only.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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As in Fig. 3, but from three paired experiments performed with pseudo-operational Doppler scans of upright vortex: (a) E-AB-1 vs E-AB-2 for dual-Doppler analyses, (b) E-A-1 vs E-A-2 for single-Doppler analyses with observations from radar A only, and (c) E-B-1 vs E-B-2 for single-Doppler analyses with observations from radar B only.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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Figure 8a shows that the diagnosed $V_T^{s+}$ and ($V_R^{s+}$, w^s+) in E-A-1 match their respective benchmark fields quite closely. Figures 8b and 8c show that the analyzed (u′, υ′) (or w′) in E-A-1 matches its benchmark field very closely (or reasonably well) at z′ = 1 and 4 km, respectively. When the innovations are given from both radar A and radar B (instead of radar A only), the diagnosed $V_T^{s+}$ and ($V_R^{s+}$, w^s+) and analyzed (u′, υ′) and w′ (not shown) become much closer to their respective benchmark fields than those in Figs. 8a–c. Thus, when the vortex is upright, the single-step analysis performs very well with the dual-Doppler innovations from radars A and B and reasonably well with single-Doppler innovations from either radar A or B. In this case, the two-step analysis underperforms the single-step analysis in general except for analyzing w^s and w′ with the dual-Doppler innovations.

As in Fig. 6, but from E-A-1 performed with operational Doppler scans of upright vortex.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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As in Fig. 6, but from E-A-1 performed with operational Doppler scans of upright vortex.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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As in Fig. 6, but from E-A-1 performed with operational Doppler scans of upright vortex.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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b. Results for slantwise vortex

For the experiments performed with pseudo-operational Doppler scans of the eastward-slanted vortex, the resulting CREs are listed in Tables 7 and 8 . As shown in row 1 of Table 7 for the two-step analysis in E-AB-2, the CREs of analyzed ($V_T^s$, $V_R^s$, w^s) and diagnosed ($V_T^{s+}$, $V_R^{s+}$, w^s+) become larger when compared to the upright vortex (see row 1 of Table 5). As shown in row 1 of Table 8, the CREs of analyzed (u′, υ′) also become larger but the CRE of analyzed w′ becomes smaller when compared to the upright vortex (see row 1 of Table 6). Thus, when the vortex becomes slanted, the axisymmetric part of VF becomes more difficult to analyze and diagnose, and the analyzed (u′, υ′) become less accurate but the analyzed w′ becomes more accurate for the two-step analysis in E-AB-2, and these changes can be explained by the decreased detection of (u′, υ′) and increased detection of w′ for the slanted vortex. On the other hand, as shown in row 2 of Table 7 for the single-step analysis in E-AB-1, the analyzed ($V_T^s$, $V_R^s$, w^s) and diagnosed ($V_T^{s+}$, $V_R^{s+}$, w^s+) are substantially more accurate than those from the two-step analysis in E-AB-2 (see row 1 of Table 7). As shown in row 2 versus row 1 of Table 8, the analyzed (u′, υ′) in E-AB-1 are more accurate than those from the two-step analysis in E-AB-2 but the analyzed w′ in E-AB-1 is slightly less accurate than that from the two-step analysis in E-AB-2. Thus, in this case, the single-step analysis is substantially more accurate than the two-step analysis in analyzing ($V_T^s$, $V_R^s$, w^s) and diagnosing ($V_T^{s+}$, $V_R^{s+}$, w^s+) but slightly more (or less) accurate than the two-step analysis in analyzing (u′, υ′) (or w′).

Table 7.

As in Table 1, but from experiments performed with operational Doppler scans of eastward-slanted vortex in section 5b.

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Table 8.

As in Table 2, but from experiments performed with pseudo-operational Doppler scans of eastward-slanted vortex in section 5b.

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When the dual-Doppler innovations from radars A and B reduce to single-Doppler innovations from radar A, the analyzed ($V_T^s$, $V_R^s$, w^s) and diagnosed ($V_T^{s+}$, $V_R^{s+}$) become less accurate (as seen from rows 3 and 4 vs rows 1 and 2, respectively, in Table 7) but the diagnosed w^s+ become slightly more (or less) accurate for the single-step (or two-step) analysis [as seen from row 3 (or row 4) vs rows 1 (or row 2) in Table 7]. In this case, the analyzed (u′, υ′, w′) also become less accurate (as seen from rows 3 and 4 vs rows 1 and 2, respectively, in Table 8). Thus, when the dual-Doppler innovations reduce to single-Doppler innovations from radar A, the axisymmetric part of VF becomes more difficult to analyze and diagnose, the analyzed (u′, w′) become less accurate, and the analyzed υ′ becomes much less accurate (due to the reduced projection of υ′ on ${\upsilon }_r^i$ for single-Doppler innovations from radar A). In this case, the single-step analysis outperforms the two-step analysis.

When the dual-Doppler innovations (from radars A and B) reduce to single-Doppler innovations from radar B, the axisymmetric part of VF also becomes more difficult to diagnose especially for the two-step analysis (as seen from row 5 versus row 1 in Table 7), the analyzed (υ′, w′) become less accurate and the analyzed u′ becomes much less accurate (due to the reduced projection of u on ${\upsilon }_r^o$ for each single-Doppler observation from radar B) as seen from rows 5 and 6 versus rows 1 and 2, respectively, in Table 8. In this case, the single-step analysis outperforms the two-step analysis.

Figure 9a shows that the AREs of analyzed (u′, υ′, w′) from the paired E-AB-1 and E-AB-2 vary with z′ similarly to those in Fig. 7a, and their vertical variations can be explained similarly (to those given for Fig. 7a in section 5a). As z′ increases above 0.5 km, however, the AREs of analyzed (u′, υ′) in Fig. 9a increase slightly more rapidly but the analyzed w′ in Fig. 9a increases less rapidly than those in Fig. 7a, and this difference can be explained by the fact that the projection of (u′, υ′) [=(u_υ–w′∂_zx_c, υ_υ) in this case] on ${\upsilon }_r^o$ decreases but the projection of w′ on ${\upsilon }_r^o$ increases as ∂_zx_c increases from 0 to 0.5, as shown in (2.3). Figure 9b shows that the ARE of analyzed υ′ from E-A-1 (or E-A-2) is much larger than that from E-AB-1 (or E-AB-2), and this is consistent with the earlier explained increase of CRE seen in row 4 (or row 3) versus row 2 (or row 1) in Table 8. Figure 9c shows that the ARE of analyzed u′ from E-B-1 (or E-B-2) is much larger than that from E-AB-1 (or E-AB-2), and this is consistent with the earlier explained increase of CRE seen in row 6 (or row 5) versus row 2 (or row 1) in Table 8. The difference between Figs. 9b and 7b is significant because the single-Doppler innovations from radar A are strongly affected by the eastward slanting of the vortex (with ∂_zx_c increased from 0 to 0.5), but the difference between Figs. 9c and 7c is much less significant because the single-Doppler innovations from radar B (to the south of the vertex) are not much affected by the eastward slanting of the vortex.

As in Fig. 7, but for eastward-slanted vortex.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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As in Fig. 7, but for eastward-slanted vortex.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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As in Fig. 7, but for eastward-slanted vortex.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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Figure 10a shows that the diagnosed $V_T^{s+}$ and ($V_R^{s+}$, w^s+) in E-A-1 match their respective benchmark fields quite closely. Figures 10b and 10c show that the analyzed (u′, υ′) (or w′) in E-A-1 matches its benchmark field very closely (or reasonably well) at z′ = 1 and 4 km, respectively. When the innovations are from both radar A and radar B (instead of radar A only), the diagnosed $V_T^{s+}$ and ($V_R^{s+}$, w^s+) and analyzed (u′, υ′) and w′ become much closer to their respective benchmark fields than those in Figs. 10a–c. Thus, when the vortex becomes slanted eastward, the single-step analysis performs very well with the dual-Doppler innovations from radars A and B and reasonably well with single-Doppler innovations from radar A or B, whereas the two-step analysis underperforms the single-step analysis for single-Doppler scans but not for dual-Doppler scans. However, the single-step analysis as well as the two-step analysis performed with single-Doppler innovations from radar B are not much affected by the vortex eastward slanting (as seen and explained above when Fig. 9c is compared with Fig. 7c).

As in Fig. 8, but for eastward-slanted vortex.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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As in Fig. 8, but for eastward-slanted vortex.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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As in Fig. 8, but for eastward-slanted vortex.

Citation: Journal of the Atmospheric Sciences 78, 3; 10.1175/JAS-D-20-0159.1

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