Atmospheric Lidar takes advantage of Rayleigh backscattering to create a relative density profile of the atmosphere. The method for temperature derivation is based on the work of Chanin and Hauchecorne (CH). Beginning with an initial temperature, and utilizing the ideal gas law, a downward integration procedure is applied to create a temperature profile from the density profile down to forty- five kilometers. Since this initial temperature is only a best guess, the temperatures towards the top of the profile may not be accurate. However, so long as the guess is reasonable, within perhaps a fifty Kelvin margin (though hopefully not so much), multiple guesses seem to converge after working down fifteen or so kilometers.
Hobbs, Jaren, "Forward Model for Temperature Derivation from Atmospheric Lidar" (2013). Physics Capstone Projects. Paper 1.