## 3 Methodology
### Retrieval Algorithm of the CDWL  
In this subsection, we prefer to discuss the retrieval of aerosol optical properties rather than wind profiles, for wind measurement, as discussed in many works [21; 34; 35], is not this paper's focus. In the CDWL measurements, the photocurrent in response to the beat signal of the atmospheric backscatter and the local oscillator can be expressed as [35]  
\[i_{d}=RP_{Lo}+RP_{s}(r)+2R[\eta_{h}(r)P_{Lo}P_{s}(r)]^{0.5}\cos(\omega_{c}t+ \theta)+i_{n}, \tag{1}\]  
where \(R\) is the response of the detector (\(R=\eta_{q}q/h\
u\), \(\eta_{q}\) is the quantum efficiency, \(q\) is the elementary charge, \(h\) is the Planck constant, and \(v\) is the photon frequency), \(r\) is the range, and \(P_{Lo}\), and \(P_{s}(r)\) are the power of the signal from the local oscillator and the atmospheric backscatter, respectively. \(\eta_{h}(r)\) is the heterodyne efficiency, which represents the fraction of the total signal power matched with the local oscillator field. \(\omega_{c}\) and \(\theta\) are the frequency and phase of the IF (intermediate frequency) signal. The first two terms of Equation (1) are filtered through AC (alternating current) coupling, and the third and fourth terms are \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Parameter** & **CDWL** & **RMRL** \\ \hline Laser & & & \\  & Wavelength & 1548 nm & 355/532/1064 nm \\  & Frequency offset & 80 MHz & / \\  & Pulse energy & 110 \(\mu\)J & 250/350/350 mJ \\  & Repetition rate & 10 kHz & 20 Hz \\  & Pulse width & 200 ns & 8 ns \\ Telescope & & & \\  & Diameter & 100 mm & 450 mm \\  & Elevation angle & 70\({}^{\circ}\) & 90\({}^{\circ}\) \\  & Detection type & BD & PMT \\ Data acquisition & Noise bandwidth & 200 MHz & 0.3 nm \\  & Sampling rate & 250 MHz & 20 MHz \\  & Temporal resolution & 1 s & 60 s \\  & Spatial resolution & 30, 60, 150 m & 30 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Key parameters of CDWL and RMRL systems. signal current \(i_{s}\) and noise current \(i_{n}\), respectively. The wide band CNR (carrier-to-noise ratio) is defined as the ratio of signal power and noise power and can be expressed as  
\[CNR(r)=\frac{\langle i_{s}^{2}\rangle}{\langle i_{n}^{2}\rangle}=\frac{\eta_{q} \eta_{h}(r)P_{\mathrm{s}}(r)}{h\upsilon B}, \tag{2}\]  
where \(B\) is the receiver noise equivalent bandwidth. Combined with the lidar equation [36], the lidar equation of coherent lidar can be expressed as:  
\[\frac{CNR(r)r^{2}}{\eta_{h}(r)}=\frac{\eta_{q}CE_{0}}{h\upsilon B}\beta(r) \exp\left[-\int_{0}^{r}2\alpha(r)\mathrm{d}z\right], \tag{3}\]  
where \(C\) represents the lidar constant, \(E_{0}\) is the pulse energy of the outgoing light, and \(\beta(r)\) and \(\alpha(r)\) are the backscatter and extinction coefficient, respectively. For a monostatic pulsed coherent lidar, due to the heterodyne efficiency caused by the telescope focus function, a distance-dependent modulation is applied to the backscatter signal, resulting in the distortion of the near-field signal. This may have little impact on wind measurements, but it will cause false values in the attenuated backscatter coefficient profiles. Using the estimation or calibration of the heterodyne efficiency, the calibrated backscatter signal can be obtained, and the backscatter and extinction coefficient of atmospheric aerosol can be further retrieved. Similar to the processing method of overlap factor correction [37], the modulation of the heterodyne efficiency over the backscatter signal can be estimated from the horizontal measurements. To calibrate the heterodyne efficiency in the near field, an advanced horizontal measurement is performed, and the average result of the CNR is applied and fitted, as shown in the subfigure of Figure 1a. Due to the different range resolutions, the density of numerical points varies with the range. Using the ratio of the range-correct CNR and the linear fit result, the heterodyne efficiency can be obtained and used to calibrate the CNR profiles in the experiment.