Is your aerosol backscatter retrieval afflicted by a sign error?

Precise knowledge about the prevailing aerosol content in the atmosphere is very important for several reasons, as aerosols are involved in multiple important processes that not only have a direct impact on air quality, but also influence cloud formation and the earth's radiation budget. Besides that, continuous aerosol observations provide valuable information on atmospheric transport dynamics.
Aerosol backscatter coefficient measurements with elastic backscatter lidars are conducted since multiple decades [1], while the implemented retrieval algorithms predominantly refer to the seminal publications by Klett 1985, Fernald 1984 and Sasano 1985 [2,3,4]. The respective inversion algorithm is often simply called the 'Klett inversion', being a main reason why this algorithm is most often adapted. While more sophisticated aerosol lidars (e.g. Raman lidars, HSRL, ...) have been developed since, simple elastic backscatter lidar measurements are still very frequently conducted as they are technically easy to implement, often as a byproduct. In most cases, the corresponding retrieval algorithms still refer to the 'Klett inversion'.
Unfortunately, the inversion algorithm by Klett 1985 is afflicted by a sign error. In his publication, the sign error is hidden within a substitute, making it very hard to be recognized, representing a major pitfall. A comprehensive literature review revealed, that large parts of the aerosol lidar community are aware of this problem and have tacitly corrected it or, to a much smaller amount, even referred to an erratum which was published by Kaestner in 1986 [5].
However, at the same time and up to this date, a considerable error propagation can be found in literature as well, using and referring to the incorrect algorithm with the sign error included.
Therefore, we want to renew the awareness towards this sign error and show a corrected and slightly improved Klett inversion algorithm. In addition, we present the overall implication resulting from the uncorrected inversion algorithm by using exemplary case studies. Depending on the lidar location and prevailing atmospheric conditions, potential errors reach from marginal to major, often preventing error detection solely based on the magnitude of the calculated results. Simple a posteriori corrections are not possible, as the error magnitude depends on multiple factors.

[1] T. Trickl, H. Giehl, H. Jäger, and H. Vogelmann. 35 yr of stratospheric aerosol measurements at Garmisch-Partenkirchen: From Fuego to Eyjafjalla-   jökull, and beyond. Atmospheric Chemistry and Physics, 13(10):5205–5225, 2013.
[2] James D. Klett. Lidar inversion with variable backscatter/extinction ratios. Appl. Opt., 24(11):1638–1643, June 1985.
[3] Frederick G. Fernald. Analysis of atmospheric lidar observations: Some comments. Appl. Opt., 1984.
[4] Yasuhiro Sasano, Edward V. Browell, and Syed Ismail. Error caused by using a constant extinction/backscattering ratio in the lidar solution. Appl. Opt., 24(22):3929–3932, November 1985.
[5] Martina Kaestner. Lidar inversion with variable backscatter/extinction ratios: Comment. Applied Optics, 25(6):833–835, March 1986.