The purpose of this work is to use data from the Atmospheric Infrared Sounder (AIRS) and the Moderate Resolution Imaging Spectroradiometer (MODIS) to refine our knowledge of post-launch AIRS point spread functions (PSFs), including suspected changes over the mission. We develop methodology, by deriving mathematical optimization formulation based on variational principles and Sobolev gradient descent, for reconstruction of AIRS spatial response functions. We use the data over the ocean, collected for the duration of a day, to reconstruct a single PSF. We examine the repeatability of our reconstructions by computing PSFs based on data collected during two consecutive days, and also investigating the change in the reconstructions by comparing the reconstructed PSF based on data collected in the beginning and the middle of the mission. We also quantify uncertainties in our reconstruction results.