## 3 Methodology
### Retrieval of the RMRL  
In this subsection, two methods are used to retrieve the optical properties of aerosols. Generally, by assuming a constant lidar ratio between \(\alpha(r)\) and \(\beta(r)\), the inversion algorithm proposed by [PERSON] (KF method) can be used to retrieve \(\alpha(r)\) and \(\beta(r)\)[38; 39]. This method can be used to invert the extinction and backscatter with only one elastic backscatter lidar, with the assumption of the lidar ratio and boundary value. Different assumptions of boundary value may lead to various results [40; 41]. In general, the boundary value is chosen at a reference altitude where the atmosphere is relatively pure and the signal-to-noise ratio (SNR) is high enough. Sometimes, it may be difficult to find the reference height, and soft targets such as clouds are used to estimate the boundary value [42]. In addition, the Raman backscatter signals can be used alone to retrieve aerosol extinction profiles [43]. The independent aerosol extinction coefficient can be expressed as: [44; 45]  
\[\alpha_{aer}(\lambda_{L},r)=\frac{\frac{\mathrm{d}}{\mathrm{d}r}\left[\ln \frac{n_{x}(r)}{rP(\lambda_{x},\lambda_{L},r)}\right]-\alpha_{\mathrm{mol}}( \lambda_{L},r)-\alpha_{\mathrm{mol}}(\lambda_{x},r)}{1+\left(\frac{\lambda_{ L}}{\lambda_{x}}\right)^{k}}, \tag{4}\]  
where the subscripts \(aer\) and \(mol\) stand for aerosol and atmospheric molecules, respectively. \(n_{x}(r)\) is the number density of the Raman molecule; \(\lambda_{X}\) and \(\lambda_{L}\) are the wavelength of the Raman-shifted laser and the output laser, respectively. \(P(\lambda_{X},\lambda_{L},r)\) is the backscatter power of the Raman-shifted signal, and \(k\) is the Angstrom exponent. With the combination of Mie scattering data, aerosol backscatter coefficient profiles can be obtained by:  
\[\beta_{\mathrm{aer}}(\lambda_{L},r)=-\beta_{\mathrm{mol}}(\lambda_{L},r)+ \beta_{\mathrm{mol}}(\lambda_{L},r_{0})\cdot\frac{P(\lambda_{L},r)\cdot P( \lambda_{X},\lambda_{L},r_{0})\cdot n_{x}(r)}{P(\lambda_{L},r_{0})\cdot P( \lambda_{X},\lambda_{L},r)\cdot n_{x}(r_{0})}\cdot\frac{\exp[-\int_{r_{0}}^{ r}\alpha_{\mathrm{aer}}(\lambda_{X},r)+\alpha_{\mathrm{mol}}(\lambda_{X},r) \mathrm{d}z]}{\exp[-\int_{r_{0}}^{r}\alpha_{\mathrm{aer}}(\lambda_{L},r)+ \alpha_{\mathrm{mol}}(\lambda_{L},r)\mathrm{d}z]}, \tag{5}\]  
where \(r_{0}\) is the reference height and \(P(\lambda_{L},r)\) is the backscatter power of the Mie lidar signal. \(\beta_{\mathrm{mol}}\) can be calculated by the atmospheric temperature and pressure from the radiosonde measurement or the atmosphere model. The two approaches present different advantages and disadvantages; for example, the KF method relies on the accurate assumption of the lidar ratio and boundary value, and the Raman method relies on the accurate assumption of the Angstrom exponent. Figure 2 depicts some retrieval results of the experiments, and the two algorithms show good consistency under correct assumptions. To compare the data retrieval of the RMRL and CDWL, the backscatter and extinction coefficients of aerosol at the wavelengths of 355, 532, and 1064 nm had to be converted to the wavelength of 1548 nm. The extinction coefficient retrieved with the Raman method only uses the backscatter signal at the Raman wavelength of 386 nm, which may bring errors to the data conversion process. In this work, the two methods were combined. First, the aerosol extinction coefficient was obtained at the reference height with the Raman method using the data from 386 nm and then applied to the KF method as the boundary value. Next, with the KF method, the extinction coefficient at the wavelengths of 355, 532, and 1064 nm could be obtained using the data from the different elastic lidars.