  The wavelength dependence of atmospheric refraction causes elongation of
finite-bandwidth images along the elevation vector, which produces spurious
signals in weak gravitational lensing shear measurements unless this
atmospheric dispersion is calibrated and removed to high precision. Because
astrometric solutions and PSF characteristics are typically calibrated from
stellar images, differences between the reference stars' spectra and the
galaxies' spectra will leave residual errors in both the astrometric positions
(dr) and in the second moment (width) of the wavelength-averaged PSF (dv) for
galaxies.We estimate the level of dv that will induce spurious weak lensing
signals in PSF-corrected galaxy shapes that exceed the statistical errors of
the DES and the LSST cosmic-shear experiments. We also estimate the dr signals
that will produce unacceptable spurious distortions after stacking of exposures
taken at different airmasses and hour angles. We also calculate the errors in
the griz bands, and find that dispersion systematics, uncorrected, are up to 6
and 2 times larger in g and r bands,respectively, than the requirements for the
DES error budget, but can be safely ignored in i and z bands. For the LSST
requirements, the factors are about 30, 10, and 3 in g, r, and i
bands,respectively. We find that a simple correction linear in galaxy color is
accurate enough to reduce dispersion shear systematics to insignificant levels
in the r band for DES and i band for LSST,but still as much as 5 times than the
requirements for LSST r-band observations. More complex corrections will likely
be able to reduce the systematic cosmic-shear errors below statistical errors
for LSST r band. But g-band effects remain large enough that it seems likely
that induced systematics will dominate the statistical errors of both surveys,
and cosmic-shear measurements should rely on the redder bands.
