Low and dense cloud cover is often encountered in daily observing experiments with lidar. In such scenarios, the laser cannot penetrate the clouds and it is not possible to use the clean atmosphere to initially calibrate the calibration heights required for the actual inversion, which can pose a significant challenge to the inversion of aerosols under clouds. This paper proposes an iterative proximal calibration algorithm based on slope method and Fernald's backward integral equation, combined with the inherent observational mode characteristics of the micro infrared lidar (mIRLidar). By conditioning the iterative process, the relative error between the obtained extinction coefficient value and the true value becomes smaller and smaller as the number of iterations increases. According to the inversion results of the actual lidar observation data, it can be tentatively concluded that the use of the proximal calibration iterative algorithm eliminates the limitations of the traditional Fernald height calibration method and improves the accuracy of the under-cloud aerosol inversion.
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