Title: A Hierarchical PSF Reconstruction Method

Abstract: Reconstruction of the point spread function (PSF) plays an important role in many areas of astronomy, including photometry, astrometry, galaxy morphology, and shear measurement. The atmospheric and instrumental effects are the two main contributors to the PSF, both of which may exhibit complex spatial features. Current PSF reconstruction schemes typically rely on individual exposures, and its ability of reproducing the complicated features of the PSF distribution is therefore limited by the number of stars. Interestingly, in conventional methods, after stacking the model residuals of the PSF ellipticities and (relative) sizes from a large number of exposures, one can often observe some stable and nontrivial spatial patterns on the entire focal plane, which could be quite detrimental to, e.g., weak lensing measurements. These PSF residual patterns are caused by instrumental effects as they consistently appear in different exposures. Taking this as an advantage, we propose a multi-layer PSF reconstruction method to remove such PSF residuals, the second and third layers of which make use of all available exposures together. We test our method on the i-band data of the second release of Hyper Suprime-Cam. Our method successfully eliminates most of the PSF residuals. Using the Fourier\_Quad shear measurement method, we further test the performance of the resulting PSF fields on shear recovery using the field distortion effect. The PSF residuals have strong correlations with the shear residuals, and our new multi-layer PSF reconstruction method can remove most of such systematic errors related to PSF, leading to much smaller shear biases.