A Study of the Point Spread Function in SDSS Images.

We use SDSS imaging data in $ugriz$ passbands to study the shape of the point spread function (PSF) profile and the variation of its width with wavelength and time. We find that the PSF profile is well described by theoretical predictions based on von K\'{a}rm\'{a}n's turbulence theory. The observed PSF radial profile can be parametrized by only two parameters, the profile's full width at half maximum (FWHM) and a normalization of the contribution of an empirically determined "instrumental" PSF. The profile shape is very similar to the "double gaussian plus power-law wing" decomposition used by SDSS image processing pipeline, but here it is successfully modeled with two free model parameters, rather than six as in SDSS pipeline. The FWHM variation with wavelength follows the $\lambda^{\alpha}$ power law, where $\alpha \approx-0.3$ and is correlated with the FWHM itself. The observed behavior is much better described by von K\'{a}rm\'{a}n's turbulence theory with the outer scale parameter in the range 5$-$100 m, than by the Kolmogorov's turbulence theory. We also measure the temporal and angular structure functions for FWHM and compare them to simulations and results from literature. The angular structure function saturates at scales beyond 0.5$-$1.0 degree. The power spectrum of the temporal behavior is found to be broadly consistent with a damped random walk model with characteristic timescale in the range $\sim5-30$ minutes, though data show a shallower high-frequency behavior. The latter is well fit by a single power law with index in the range $-1.5$ to $-1.0$. A hybrid model is likely needed to fully capture both the low-frequency and high-frequency behavior of the temporal variations of atmospheric seeing.