  While there are no strong observational constraints on the gravitational wave
background across six or more orders of magnitude between 10^{-16} Hz and
10^{-10} Hz and it is difficult to get a constraint below 10^{-12}Hz using
objects in our Galaxy, we suggest that the anisotropy pattern of time variation
of the redshift related to a sample of high redshift objects can be used to
constrain the gravitational wave background around 10^{-12} Hz. Useful
observations for the monitoring of an anisotropy signal in a global redshift
change include spectroscopic observations of the Ly-alpha forest in absorption
towards a sample of quasars, redshifted 21 cm line observations either in
absorption or emission towards a sample of neutral HI regions before or during
reionization, and high frequency (0.1 Hz to 1
  Hz) gravitational wave analysis of a sample of neutron star--neutron star
binaries detected with gravitational wave instruments such as the Decihertz
Interferometer Gravitational Wave Observatory (DECIGO). The low frequency
background can also be constrained by arcsecond-scale anisotropy observations
of the CMB. For reasonable observations in the future involving extragalactic
sources, we find best limits at the level of \Omega_{GW} < 10^{-5} at a
frequency around 10^{-12} Hz while the eventual ultimate limit one cannot beat
is \Omega_{GW} < 10^{-11}.
