Effects of misestimated far-end boundary values on two common lidar inversion solutions

Solutions of the lidar equation in both the Klett [Appl. Opt. 20, 211 (1981)] and the Fernald [Appl. Opt. 23, 652 (1984)] approaches include the effect of errors in the estimated boundary value at the far end. In the present study an attempt is made to formulate the effect of the error in the boundary value on the solution of the lidar equation. Using a modified extinction coefficient, we can simplify and unify the error expression of the lidar inversion solution. From the unified error expression and numerical experiments, we found that (a) in the case of overestimation of the boundary value, the discrepancy between the estimated value and the real value decreases near the lidar more rapidly than in the case of underestimation; and (b) the error for the Fernald solution converges to zero more rapidly than the error for the Klett solution, but the convergence of the Fernald solution sometimes shows oscillatory behavior, whereas the convergence of the Klett solution is always monotonic.