Removing atmospheric turbulence from ground-based astrometry with fast correlation function measurements

We present a code that removes $\sim 90\%$ of the variance in astrometric measurements caused by atmospheric turbulence, by using Gaussian process regression (GPR) to interpolate the turbulence field from the positions of stars measured by Gaia to the positions of arbitrary targets. This enables robust and routine accuracy of 1-3 milliarcsec on bright sources in single exposures of the Dark Energy Survey (DES) and the upcoming Legacy Survey of Space and Time (LSST). For the kernel of the GPR, we use the anisotropic correlation function of the turbulent displacement field, as measured directly from the Gaia reference stars, which should yield optimal accuracy if the displacement field is Gaussian. We test the code on 26 simulated LSST exposures and 604 real DES exposures in varying conditions. The average correlation function of the astrometric errors for separations $<1^{\prime}$ is used to estimate the variance of turbulence distortions. On average, for DES, the post-GPR variance is $\sim 12 \times$ smaller than the pre-GPR variance. Application of the GPR to LSST is hence equivalent, for brighter stars and asteroids, to having 12 Rubin observatories running simultaneously. The expected improvement in the RMS of turbulence displacement errors is the square root of this value, $\sim 3.5.$ The post-GPR RMS displacement decreases with the density of reference stars as $\sim n_\star^{-0.5}$ for noiseless LSST simulations, and $\sim n_\star^{-0.3}$ for DES data.