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

## TL;DR

This paper develops iterative algorithms for a non-linear inverse problem in atmospheric lidar, modeling noise accurately to improve aerosol extinction profile reconstructions from Raman lidar data.

## Contribution

It introduces two novel iterative algorithms based on KKT conditions that better account for Poisson noise in non-linear inverse problems.

## Key findings

- Algorithms outperform standard methods in noisy conditions
- Improved accuracy of aerosol extinction profile retrievals
- Validated with synthetic and experimental data

## Abstract

We consider the inverse problem of retrieving aerosol extinction coefficients from Raman lidar measurements. In this problem the unknown and the data are related through the exponential of a linear operator, the unknown is non-negative and the data follow the Poisson distribution. Standard methods work on the log-transformed data and solve the resulting linear inverse problem, but neglect to take into account the noise statistics. In this study we show that proper modelling of the noise distribution can improve substantially the quality of the reconstructed extinction profiles. To achieve this goal, we consider the non-linear inverse problem with non-negativity constraint, and propose two iterative algorithms derived using the Karush-Kuhn-Tucker conditions. We validate the algorithms with synthetic and experimental data. As expected, the proposed algorithms outperform standard methods in terms of sensitivity to noise and reliability of the estimated profile.