  Current and future imaging surveys will measure cosmic shear with statistical
precision that demands a deeper understanding of potential systematic biases in
galaxy shape measurements than has been achieved to date. We use analytic and
computational techniques to study the impact on shape measurements of two
atmospheric chromatic effects for ground-based surveys such as the Dark Energy
Survey and the Large Synoptic Survey Telescope (LSST): (i) atmospheric
differential chromatic refraction and (ii) wavelength dependence of seeing. We
investigate the effects of using the point spread function (PSF) measured with
stars to determine the shapes of galaxies that have different spectral energy
distributions than the stars. We find that both chromatic effects lead to
significant biases in galaxy shape measurements for current and future surveys,
if not corrected. Using simulated galaxy images, we find a form of chromatic
`model bias' that arises when fitting a galaxy image with a model that has been
convolved with a stellar, instead of galactic, point spread function. We show
that both forms of atmospheric chromatic biases can be predicted (and
corrected) with minimal model bias by applying an ordered set of perturbative
PSF-level corrections based on machine-learning techniques applied to six-band
photometry. Catalog-level corrections do not address the model bias. We
conclude that achieving the ultimate precision for weak lensing from current
and future ground-based imaging surveys requires a detailed understanding of
the wavelength dependence of the PSF from the atmosphere, and from other
sources such as optics and sensors. The source code for this analysis is
available at https://github.com/DarkEnergyScienceCollaboration/chroma .
