  # Three-Dimensional Wind Field Retrieved from Dual-Doppler Radar Based on a Variational Method: Refinement of Vertical Velocity Estimates

• In this paper, a scheme of dual-Doppler radar wind analysis based on a three-dimensional variational method is proposed and performed in two steps. First, the horizontal wind field is simultaneously recovered through minimizing a cost function defined as a radial observation term with the standard conjugate gradient method, avoiding a weighting parameter specification step. Compared with conventional dual-Doppler wind synthesis approaches, this variational method minimizes errors caused by interpolation from radar observation to analysis grid in the iterative solution process, which is one of the main sources of errors. Then, through the accelerated Liebmann method, the vertical velocity is further re-estimated as an extra step by solving the Poisson equation with impermeable conditions imposed at the ground and near the tropopause. The Poisson equation defined by the second derivative of the vertical velocity is derived from the mass continuity equation. Compared with the method proposed by O'Brien, this method is less sensitive to the uncertainty of the boundary conditions and has better stability and reliability. Furthermore, the method proposed in this paper is applied to Doppler radar observation of a squall line process. It is shown that the retrieved vertical wind profile agrees well with the vertical profile obtained with the velocity–azimuth display (VAD) method, and the retrieved radial velocity as well as the analyzed positive and negative velocity centers and horizontal wind shear of the squall line are in accord with radar observations. There is a good correspondence between the divergence field of the derived wind field and the vertical velocity. And, the horizontal and vertical circulations within and around the squall line, as well as strong updrafts, the associated downdrafts, and associated rear inflow of the bow echo, are analyzed well. It is worth mentioning that the variational method in this paper can be applied to simultaneously synthesize the three-dimensional wind field from multiple-Doppler radar observations.
• • Figure 1.  A schematic view of the geometry of a ground-based radar in the Cartesian coordinate system (r is the distance between the target and the radar, $\theta$ is the azimuth angle, $\varphi$ is the elevation angle, and $u$ , v, and ${w}'$ are the velocity components in the x, y, and z directions, respectively).

Figure 2.  WRF model domain (solid box), dual-Doppler analysis domain (dashed box), and positions of two assumed radars (marked by hollow five-pointed star).

Figure 3.  The WRF model-simulated supercell storm (a, b) at 30 min and retrieved field (c, d) from two virtual Doppler radars denoted in Fig. 2. (a, c) Horizontal wind vectors (arrow, m s−1; the scale is in the bottom-left corner), vertical velocity (contours every 3 m s−1) and simulated reflectivity (shaded, dBZ) at 2.5 km above ground level (AGL). (b, d) Vertical cross section of synthesized wind field (horizontal and vertical wind vectors projected onto the cross section, arrow, m s−1), vertical velocity (contours, solid line for upward every 3 m s−1 and dashed line for downward every 2 m s−1) and simulated reflectivity (shaded, dBZ) along line A–B in (a, c).

Figure 4.  Error distribution of vertical velocity obtained from model-simulated horizontal wind by scheme SO, DO, SP, DP, and DPE compared with simulated vertical velocity at each level. (a) Mean absolute error MAE (m s−1). (b) Correlation coefficient CC. (c) Root-mean-square error RMS_W (m s−1). (d) Relative root-mean-square error RRE_W.

Figure 5.  As in Fig. 4, but for vertical velocity derived from two virtual Doppler radars denoted in Fig. 2.

Figure 6.  Locations of Nanchang and Fuzhou radar stations (solid five-pointed star) that observed the squall line on 11 May 2017 and the dual-Doppler analysis domain (solid box) with terrain heights (shaded and contoured every 250 m). The dotted circles denote the 150 km range of measurement while it is suitable to perform wind synthesis in the non-overlapping domain of two solid cycles. The grid point marked with an asterisk represents the position 47 km away from the Nanchang radar. The red dots indicate the mosaic of composite radar reflectivity greater than or equal to 45 dBZ at 1242 UTC 11 May 2017.

Figure 7.  Hodographs showing the wind profile (black solid line) retrieved from dual-Doppler radar observations and the VAD wind profile (grey solid line) of the Nanchang radar at 1242 UTC 11 May 2017. The black dashed and gray dashed lines represent the vertical 1–3.5 km wind shear based on the retrieved wind profile and the VAD wind profile, respectively. The position of the retrieved wind profile is shown in Fig. 6.

Figure 8.  The observed radial velocity (a, c, shaded, m s−1, positive away from and negative toward the radar) at 1.45° elevation angle from the Nanchang (a, b) and Fuzhou (c, d) radars and the retrieved radial velocity (b, d, shaded, m s−1) based on the derived wind from dual-Doppler radar data according to Eq. (1) at 1242 UTC 11 May 2017. The locations of the Nanchang and Fuzhou radars are shown in Fig. 6. The black arrow represents the heading of the squall line while the red ellipse indicates the positions of the positive and negative velocity centers and the horizontal wind shear. The negative values of x- and y-axis represent the west and south sides of the radar.

Figure 9.  Scattergraph of the observed–retrieved radial velocity comparisons from all elevation angles of the Nanchang (a) and Fuzhou (b) radars, with the concentration (0–1) represented by different colors of the scatters.

Figure 10.  Evolution of the squall line together with radar reflectivity, horizontal storm-relative wind field, horizontal divergence, and vertical velocity retrieved from dual-Doppler radar observations at (a–c) 1224 UTC 11 May, (d–f) 1242 UTC 11 May, and (g–i) 1300 UTC 11 May. (a), (d), (g) Horizontal storm-relative wind vectors (arrow, m s−1; the scale is in the bottom-right corner) and radar reflectivity (shaded, dBZ) at 2.5 km AGL. The blue solid lines describe horizontal storm-relative winds of 20 m s−1 and 24 m s−1. (b), (e), (h) Vertical velocity (shaded, m s−1; positive for upward and negative for downward) and horizontal divergence (contours, 10−4 s−1; dashed line for convergence, solid line for divergence) at 2.5 km AGL. (c), (f), (i) Vertical cross section of the synthesized wind field (storm-relative wind vectors projected onto the cross section, arrow, m s−1), vertical velocity (contours, solid line for upward and dashed line for downward; every 5 m s−1), and radar reflectivity (shaded, dBZ) along line A–B in (a), respectively. The thick gray solid line represents the updraft and downdraft.

Table 1.  Error statistics for the retrieved horizontal winds compared to the simulated winds. The bold values at the bottom of the table indicate the averages of each indice.

 Height (km) MAE (m s−1) CC RMS_V (m s−1) RRE_V Data cove-rage (%) u v u v 1.0 1.391 0.999 0.934 0.956 1.449 0.286 100.000 2.0 1.432 0.546 0.963 0.957 1.292 0.335 100.000 3.0 1.230 0.619 0.972 0.951 1.216 0.239 100.000 4.0 1.209 0.677 0.975 0.958 1.238 0.152 100.000 5.0 1.558 1.022 0.945 0.936 1.740 0.148 100.000 6.0 1.556 1.129 0.929 0.932 1.824 0.117 100.000 7.0 1.004 1.028 0.958 0.901 1.710 0.091 100.000 8.0 1.205 0.886 0.910 0.931 1.927 0.103 99.941 9.0 1.437 1.126 0.809 0.899 2.536 0.135 99.762 10.0 1.217 1.167 0.871 0.915 2.052 0.105 99.108 11.0 1.107 0.974 0.908 0.877 1.822 0.091 96.074 12.0 0.957 1.017 0.869 0.781 1.811 0.092 86.437 Mean 1.275 0.933 0.920 0.916 1.718 0.158 98.443

Table 2.  List of experiments for vertical velocity estimation.

 Experiments Mass continuity equation Solution Air density ($\rho$) SO Shallow convective continuity equation [Eq. (11)] O'Brien method − DO Deep convective continuity equation [Eq. (10)] O'Brien method From model SP Shallow convective continuity equation [Eq. (11)] Poisson method − DP Deep convective continuity equation [Eq. (10)] Poisson method From model DPE Deep convective continuity equation [Eq. (10)] Poisson method From Eq. (3)

Table 3.  List of experiments with mean error statistics of retrieved vertical wind compared with simulated wind.

 Experiments MAE (m s−1) CC RMS_W (m s−1) RRE_W SO 1.283 0.708 2.072 0.724 DO 1.474 0.722 2.360 0.844 SP 1.058 0.764 1.884 0.650 DP 1.045 0.780 1.767 0.609 DPE 1.058 0.781 1.770 0.609
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

## Three-Dimensional Wind Field Retrieved from Dual-Doppler Radar Based on a Variational Method: Refinement of Vertical Velocity Estimates

###### Corresponding author: Zhiying DING, dingzhiying@nuist.edu.cn;
• 1. Key Laboratory of Meteorological Disaster, Ministry of Education/Joint International Research Laboratory of Climate and Environment Change/Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science and Technology, Nanjing 210044, China
• 2. Jiangxi Meteorological Observatory, Nanchang 330096, China
• 3. Guangdong Province Key Laboratory of Regional Numerical Weather Prediction, Institute of Tropical and Marine Meteorology, CMA, Guangzhou 510080, China
• 4. Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai), Zhuhai 519082, China
• 5. Jiangxi Institute of Land and Space Survey and Planning/Jiangxi Geomatics Center, Nanchang 330000, China

Abstract: In this paper, a scheme of dual-Doppler radar wind analysis based on a three-dimensional variational method is proposed and performed in two steps. First, the horizontal wind field is simultaneously recovered through minimizing a cost function defined as a radial observation term with the standard conjugate gradient method, avoiding a weighting parameter specification step. Compared with conventional dual-Doppler wind synthesis approaches, this variational method minimizes errors caused by interpolation from radar observation to analysis grid in the iterative solution process, which is one of the main sources of errors. Then, through the accelerated Liebmann method, the vertical velocity is further re-estimated as an extra step by solving the Poisson equation with impermeable conditions imposed at the ground and near the tropopause. The Poisson equation defined by the second derivative of the vertical velocity is derived from the mass continuity equation. Compared with the method proposed by O'Brien, this method is less sensitive to the uncertainty of the boundary conditions and has better stability and reliability. Furthermore, the method proposed in this paper is applied to Doppler radar observation of a squall line process. It is shown that the retrieved vertical wind profile agrees well with the vertical profile obtained with the velocity–azimuth display (VAD) method, and the retrieved radial velocity as well as the analyzed positive and negative velocity centers and horizontal wind shear of the squall line are in accord with radar observations. There is a good correspondence between the divergence field of the derived wind field and the vertical velocity. And, the horizontal and vertical circulations within and around the squall line, as well as strong updrafts, the associated downdrafts, and associated rear inflow of the bow echo, are analyzed well. It is worth mentioning that the variational method in this paper can be applied to simultaneously synthesize the three-dimensional wind field from multiple-Doppler radar observations. followshare  DownLoad:  Full-Size Img  PowerPoint