Abstract The solution of the lidar equation is an ill-posed problem that requires nonlinear methods to retrieve the atmospheric aerosol optical and microphysical properties. Particularly, in the last decades, the most applied solution for the elastic lidars is through the well known Klett-Fernald-Sasano algorithm for retrieving the backscatter coefficient. To solve this inversion problem, we propose to apply the optimal estimation method to a Vaisala CL51 ceilometer range corrected signals for retrieving under two different frameworks, the particle backscatter coefficient or the ratio and the lidar constant. The optimal estimation is a Bayesian inversion fed by a set of a priori information. In this work, to obtain the suitable prior, we have tested two approaches that involved measurements and synthetic data. The first data set was obtained from previous inversions using the classical Klett-Fernald-Sasano method, and the second one by using Mie simulations fed by aerosol properties from OPAC database. The optimal estimation method used for elastic lidar inversion presents two main advantages compared to the classic approaches. On one hand, there is no need for Rayleigh zone determination and on the other hand, the uncertainty of the retrieved products is directly estimated, therefore the quality of the results is highly dependant on the prior selection. To evaluate the performance of the model, low and high aerosol accumulations scenarios were considered, finding that the backscatter coefficient was oscillating between 5 and 7 (kmsr)⁻¹ in the first 3 km agl with uncertainties lower than 27 $\text{\%}$at degraded spatial resolutions. Additionally, constant and height-dependent priors were tested reaching relative errors in percentage up to 5$\text{\%}$ between them. Besides, relative errors were also analyzed for the prior covariance matrices estimated either from synthetic lidar data and Klett's retrievals, where the errors are lower than 2$\text{\%}$ by using one instead of the another. However, scale factors were applied to the synthetic prior covariance matrices to reach the convergence. The results at retrieving the particle backscattering were compared to those ones estimated from Klett's inversion, considering Klett inversion as the reference. For the extreme scenario of inversions, considering aerosol accumulations at different layers, the bias between the optimized profiles was lower than −0.5 (kmsr)⁻¹ in the first 0.5 km and 0.5 (kmsr)⁻¹ above 1.5 km. Here, we also shown a two-parameter optimization for the lidar constant and lidar ratio, applied to 39 aerosol inversions, finding relative errors lower than 1 $\text{\%}$ and 2.3 $\text{\%}$, respectively, considering those ones from Klett inversion as the reference.

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Highlights • Application of nonlinear regression to solve elastic lidar ill-posed problems for retrieve aerosol properties. • Exploration of different prior scenarios to better constrain the aerosol inversion using the optimal estimation method. • Study of the optimal estimation method capabilities in contrast to the classic elastic Klett approach on lidar inversion.