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In order to evaluate the performance of our three-dimensional variational method proposed in the previous section, we utilize a set of assumed dual-Doppler data from a simulated supercell storm. The Advanced Research Weather Research and Forecasting Model (WRF, Skamarock et al., 2008) is used here to perform a 3-h simulation of a supercell storm. As shown in Fig. 2, the WRF model domain consists of 83 × 83 × 51 grid points with a uniform grid interval of 1 km in the horizontal and model top of 18.5 km in the vertical. The WRF model makes use of the third-order Runge–Kutta explicit time integration scheme with an integration time step of 6 s and turns off the cumulus convection parameterization scheme. Kessler warm cloud microphysical parameterization is used together with a 1.5-order turbulent kinetic energy subgrid parameterization. Open boundary conditions are utilized at the lateral boundaries while a high-level Rayleigh damping layer is introduced to reduce wave reflection from the top of the model.
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).
Except for subtle differences in water mixing ratio above the troposphere, the sounding data used in this case is basically similar to those introduced by Weisman and Klemp (Weisman and Klemp, 1982) (using a surface potential temperature of 300 K and a surface water vapor mixing ratio of 14 g kg−1). For details on the morphology and evolution of supercell storms, readers can refer to Ray (Ray et al., 1981), Klemp et al. (Klemp and Rotunno, 1983), and Schenkman et al. (Schenkman et al., 2011).
The analysis domain shown in Fig. 2 consists of 41 × 41 × 25 grid points with a uniform grid interval of 1 km in the horizontal and 0.5 km in the vertical, in which a supercell is formed and developed. As shown in Fig. 3a, a mature supercell has been formed by 30 min into the simulation. A strong rotating updraft (with maximum vertical velocities exceeding 14 m s−1) near the center of the hook-echo pattern is evident at 2.5 km and is associated with a low-level maximum downdraft of 5.9 m s−1 in the strong echo area. The vertical cross section (Fig. 3b) exhibits a gravitational oscillation pattern, implying that downstream of the overshooting updraft (with maximum vertical velocities reaching 35 m s−1 at 6.5 km) at the tropopause level are downward returning flows (with maximum vertical velocities exceeding 8 m s−1). The structure of simulated supercell storm is similar to that described by Ray et al (Ray et al., 1975), Klemp et al (Klemp and Rotunno, 1983), and Wilhelmson et al (Wilhelmson and Klemp, 1981).
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).
The model-simulated three-dimensional wind field at 30 min is sampled by two assumed radars located at (10, 0) and (80, 0) at ground level in Fig. 2 working in a scanning mode similar to VCP11 with a maximum range of 230 km, gate spacing of 1 km, azimuth resolution of 1°, and elevation angles of 0.5°, 1.0°, 1.5°, 2.0°, 2.5°, 3.0°, 4.0°, 5.0°, 6.0°, 8.0°, 10.0°, 12.0°, 15.0°, and 18.0°, which are selected based on most operational radar configurations. The elapsed times for the volume scans of two pseudo-radars are neglected, and thus the radial winds are observed simultaneously. The simulated wind components are interpolated using a Cressman weighting function (Cressman, 1959) from the grid points to the sampling positions along the radar beams with horizontal and vertical influence radii of 2.4 km and 1.2 km, respectively, and then are synthesized to obtain radial velocities according to Eq. (1). In order to simulate the actual observational errors of radars, random errors with a standard deviation of 1 m s−1 are added to these radial velocities [i.e.,
$ {{V}_{\mathrm{r}}}_{}'=\left(1+\varepsilon \right){V}_{\mathrm{r}} $ is taken as the final observation in the retrieval procedure, where$ \varepsilon $ represents normally distributed random data with an expectation of 0 and a standard deviation of 1]. The wind retrieval is performed in the regions covered by both radars.To give a quantitative assessment of the accuracy of the wind field derived from dual-Doppler radar data, we define the following indices between the retrieved variables and the true counterparts for verification. The mean absolute error (MAE) and correlation coefficient (CC) of wind field are defined as:
the root-mean-square error (RMS) and relative RMS error (RRE) of the horizontal velocity are defined as:
and the RMS error and relative RMS error (RRE) of the vertical velocity are defined as:
where N is the total number of grid points, F stands for any of the wind components, u, v, and w, in the x, y, and z directions, and the subscripts cal and ref represent the retrieved variables and model-simulated variables, respectively.
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In this section, we first evaluate the quality of the retrieved horizontal wind field. It is clearly shown that the distribution of horizontal wind at 2.5 km in Fig. 3c is similar to the simulated field in Fig. 3a. By comparing the retrieved wind fields with the true ones from the WRF model, all the features in the horizontal wind field, including the cyclonic convergence and rotating updrafts around the supercell, are derived well, even though the retrieved wind components are a little stronger than the simulated ones. The vertical cross section (Fig. 3d) reflects the dynamic characteristics of the supercell well, with the retrieved maximum ascending motion of 34 m s−1, which is very close to the simulated one. The downstream oscillations near the top of the main updraft due to gravity waves are also evident in Fig. 3b. As a result, the retrieved three-dimensional wind field agrees well with the true one in Fig. 3. The quality of the retrieved winds can be further quantitatively exhibited by the MAE, CC, RMS, and RRE, comparing the retrieved winds with those from the model. The error statistics of the differences between retrieved and simulated horizontal wind components at each level are given in Table 1. The RMS_V, RRE_V, and MAE errors are all small while the CC is high below 6 km, where most mesoscale convective systems take place, because radars usually work at low elevation angles and the interpolation errors are relatively small with plenty of observation samples. At altitudes above 6 km, the RMS_V error first increases and then decreases with height. Although the RMS_V error is up to 2.5 m s−1 at 9 km and winds at this altitude are rarely used, it is still within an acceptable range. Overall, the general features of the horizontal wind field at all levels are retrieved well, with relatively small RMS error (1.718 m s−1) and high CC (0.916). It is noted that due to radar detection, the coverage of the data decreases with height, which may affect the subsequent estimate of the vertical velocity.
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 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.
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To examine the reliability of the retrieved vertical velocity in more detail, we design the following five experiments listed in Table 2.
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 2. List of experiments for vertical velocity estimation.
On the vertical velocity retrieval, experiments SO (SP) and DO (DP) are designed to compare the differences between the shallow and deep convective continuity equations with the O'Brien (Poisson) method, and experiments DP and DPE are used to evaluate the sensitivity of different air density distributions. Since the solution of the numerical model satisfies the basic equations of atmospheric motion, we take the vertical velocities from the WRF model as the true values for evaluation. Firstly, the simulated horizontal wind fields are used to estimate the vertical velocity, and the errors of each experiment are verified.
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The advantage of using a simulated horizontal wind field is that the data at each grid point is valid, which can eliminate the uncertainties caused by insufficient data in the retrievals. Height profiles of the statistical deviation of the differences between the vertical velocities retrieved from the simulated horizontal wind components by each scheme and those obtained directly from the WRF model are shown in Fig. 4. On the whole, the errors of retrieved–simulated vertical velocity comparisons derived with the deep convective continuity equation (experiments DO, DP, and DPE) are reasonably smaller than those derived with the shallow convective continuity equation (experiments SO and SP). The RMS errors of experiments SO and SP are larger than the other three schemes at each layer in Fig. 4c. At altitudes below 11 km, the correlation coefficients in experiments DO, DP, and DPE remain relatively high, varying from 0.95 to 0.98 (Fig. 4b), while the RMS error in Fig. 4c and the RRE in Fig. 4d remains smaller than 1 m s−1 and 0.3 at each layer, respectively. As a result, the vertical wind field considering the density change with height by means of the deep convective continuity equation is closer to the true state (i.e., the air density should not be ignored since the supercell storm develops in a deep and strong convective environment). In addition, there seems to be little difference in the vertical velocity error between the simulated air density experiment (DP) and the empirical air density experiment (DPE). The above results lay the experimental foundation for the three-dimensional wind field derived from dual- or multiple-Doppler radar observations for a real case, as presented in the next section.
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.
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Invalid data inevitably exist at upper levels due to the noncontinuous radar elevation angles. This problem may have a certain impact on the wind field retrieval, especially on the estimation of the vertical velocity. As shown in Fig. 5, the errors of two experiments (SO and DO) performed with the O’Brien method increase faster with height than the other three experiments above 5.5 km. One of the main reasons is that the data at the highest level is used to adjust the vertical velocity from top to bottom in the O'Brien method [see Eq. (12)], thus it will be impossible to modify the data at a lower layer if the upper data is missing. The data coverages at 12 km and 12.5 km are only 86% and 64% respectively, which is common in practice, so the O’Brien method has certain defects. In contrast, the errors of the three experiments (SP, DP, and DPE) performed with the Poisson method remain relatively smaller and more stable. The RMS error reaches its maximum (about 2.5 m s−1) near 9 km and then decreases with height. The errors of experiments DP and DPE are smaller than those of experiment SP at each layer (Fig. 5), indicating that the vertical velocities derived with the deep convective continuity equation are more accurate than those derived with the shallow convective continuity equation. These features are basically consistent with the results outlined in the previous paragraph. Although the errors of experiment DPE at the upper level (>7 km) are slightly larger than those of experiment DP, the differences are negligible. Approximate error distributions of experiments DP and DPE are also presented in Fig. 5, demonstrating that the true air density can be replaced by empirical air density. This conclusion has important practical significance, because the true air density of the atmosphere is difficult to obtain. Moreover, the Poisson method is rather stable and robust, as concluded from the results of experiments DP and DPE.
The mean error statistics of retrieved velocities at all levels for the above five experiments are listed in Table 3. As shown, both the RMS error (<1.78 m s−1) and the RRE (0.609) remain reasonably small, and the CC is as high as 0.78 in experiments DP and DPE, indicating that the vertical wind is recovered with plausible accuracy. The above results are surprisingly similar to the conclusions of Gao et al. (Gao et al., 1999) (refer to experiment CNTL in the literature, with RMS error of 1.937 m s−1, RRE of 0.609, and CC of 0.825), proving that the method described in this paper is basically consistent with the variational solution with the mass continuity equation imposed as a weak constraint in the cost function (Gao et al., 1999, 2001) for the calculation of vertical velocity.
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 Table 3. List of experiments with mean error statistics of retrieved vertical wind compared with simulated wind.