  The absolute measurement of the Earth angular rotation rate with ground-based
instruments becomes challenging if the 1 part in $10^9$ of precision has to be
obtained. This threshold is important for fundamental physics and for geodesy,
to investigate effects of General Relativity and Lorentz violation in the
gravity sector and to provide the fast variation of the Earth rotation rate.
  High sensitivity Ring Laser Gyroscopes (RLG) are currently the only promising
technique to achieve this task in the near future, but their precision has been
so far limited by systematics related to the laser operation.
  In this paper we analyze two different sets of observations, each of them
three days long. They were obtained from the G ring laser at the Geodetic
Observatory Wettzell. The applied method has been developed for the GINGERINO
ring laser in order to identify and extract the laser systematics. For the
available data sets the residuals show mostly white noise behavior and the
Allan deviation drops below 1 part in $10^9$ after an integration time of about
$10^4$~s.
