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We consider the inverse problem of retrieving aerosol extinction coefficientsfrom Raman lidar measurements. In this problem the unknown and the data arerelated through the exponential of a linear operator, the unknown isnon-negative and the data follow the Poisson distribution. Standard methodswork on the log-transformed data and solve the resulting linear inverseproblem, but neglect to take into account the noise statistics. In this studywe show that proper modelling of the noise distribution can improvesubstantially the quality of the reconstructed extinction profiles. To achievethis goal, we consider the non-linear inverse problem with non-negativityconstraint, and propose two iterative algorithms derived using theKarush-Kuhn-Tucker conditions. We validate the algorithms with synthetic andexperimental data. As expected, the proposed algorithms outperform standardmethods in terms of sensitivity to noise and reliability of the estimatedprofile.

Iterative algorithms for a non-linear inverse problem in atmospheric lidar