## 2 Materials and Methods
### Retrieval Method
#### 2.3.2 Integrated Retrieval Technique Using Multimode Doppler Spectra
For retrieval of vertical air velocity from an observed Doppler spectrum, without the knowledge of \(\sigma_{\mathrm{DSD}}^{2}\), the latter three terms are used to correct for the effects of the broadening of the left edge of the spectrum due to non-hydrometeor size distribution. Figure 4: Schematic of the small-particle-traced idea. The blue dashed line is the estimated radar mean noise, the solid red line is the max noise, the \(\omega_{\mathrm{initial}}\) represents the initial retrieval of vertical air velocity. For a symmetric Gaussian antenna, \(\sigma_{\rm B}^{2}\) can be estimated by using the horizontal wind speed, U (m\(\cdot\)s\({}^{-1}\), obtained by radiosonde), and the radar beamwidth (\(\theta\)) [17] as follows:  
\[\sigma_{\rm B}^{2}=\frac{\rm U^{2}\theta^{2}}{2.76^{2}} \tag{2}\]  
\(\sigma_{\rm S}^{2}\) is caused by vertical (k\({}_{\rm h}\)) and horizontal (k\({}_{\rm v}\)) shear in the vertical winds (both in s\({}^{-1}\)), and this term can be estimated from Equation (3) [17; 25; 26]. The first term represents transverse shear and the second term represents radial shear. R is the distance from the target to the radar, \(\Delta\) is the radar gate length, and both k\({}_{\rm h}\) and k\({}_{\rm v}\) can be calculated by the MV across three adjacent points. \[\sigma_{\rm S}^{2}=\frac{\rm k_{\rm h}^{2}R^{2}\theta^{2}}{2.766^{2}}+\frac{ \rm k_{\rm v}^{2}\Delta^{2}}{12} \tag{3}\]  
Turbulent broadening of the spectrum variance can be derived by using the temporal variance of the measured mean Doppler velocity, \(\sigma_{\rm vm}^{2}\), with the assumption that the length scales of the turbulent eddies observed by the radar are within the inertial subrange of the turbulence spectrum [33; 34]. The \(\sigma_{\rm vm}^{2}\) can be calculated as follows:  
\[\sigma_{\rm vm}^{2}=\frac{\sum_{i=-n/2}^{i=n/2}\left({\rm MV_{i}-\overline{MV} }\right)^{2}}{n} \tag{4}\]  
Figure 5: Flow diagram of the integrated technique for retrieval of vertical air velocity using multimode Doppler spectra. where n is the adjacent gate number (here set to 7, as the turbulence length scales are assumed to be mostly within 60 s of the radar observational time), \(\mathrm{MV_{i}}\) represents the mean Doppler velocity of each gate, and \(\overline{\mathrm{MV}}\) is the overall average. [PERSON] et al. (2005) showed that the relationship between \(\sigma_{\mathrm{T}}^{2}\) and \(\sigma_{\mathrm{vm}}^{2}\) can be described by a simple form:  
\[\frac{\sigma_{\mathrm{T}}^{2}}{\sigma_{\mathrm{vm}}^{2}}=\frac{\mathrm{L}_{ \mathrm{s}}^{2/3}}{\mathrm{L}_{\mathrm{I}}^{2/3}-\mathrm{L}_{\mathrm{s}}^{2/3}} \tag{5}\]  
In Equation (5), \(\mathrm{L_{s}}\) is the horizontal length of the radar sampling volume for a 2 s dwell time and \(\mathrm{L_{I}}\) represents the larger eddies passing through the effective sampling volume that results from an average of the radar observations over 60 s ([PERSON] et al., 2017). \(\mathrm{L_{s}}\) and \(\mathrm{L_{I}}\) can both be given as \(\mathrm{L}=\mathrm{Ut}+2\mathrm{Resin}(\theta/2)\), where t is the observation time. After estimating these three spectral broadening components, a correction factor \(\delta\) (m\(\cdot\)s\({}^{-1}\)) can be calculated, by which the initial air velocity \(\omega_{\mathrm{initial}}\) is increased in the direction of the arrow in Figure 4. \(\delta\) can be written as  
\[\delta=\sigma_{\mathrm{v}}-\sqrt{\sigma_{\mathrm{v}}^{2}-\left(\sigma_{ \mathrm{T}}^{2}+\sigma_{\mathrm{S}}^{2}+\sigma_{\mathrm{B}}^{2}\right)} \tag{6}\]  
For relatively strong convective clouds or light precipitation, the smallest traced targets in the radar sampling volume may possibly have non-negligible sizes, corresponding to terminal speeds that affect the accuracy of the derived air velocities. However, these retrieval biases are difficult to estimate because the particles in convective clouds on the TP have unresolved phases or habits.