Ring laser gyroscopes are top sensitivity inertial sensors used in the measurement of angular rotation rates. It is well known that the response of such remarkable instruments can in principle access the very low frequency band, but the occurrence of nonlinear effects in the laser dynamics imposes severe limitations in terms of sensitivity and stability. We report here general relationships aimed at evaluating corrections able to effectively account for nonlinear laser dynamics. The so-derived corrections are applied to analyse thirty days of continuous operation of the large area ring laser gyroscope GINGERINO leading to duly reconstruct the Sagnac frequency $omega_S$. The analysis shows that, on the average, the evaluated corrections affect the measurement of the Earth rotation rate $Omega_E$ at the level of 1 part in $1.5times10^{3}$. Among the identified corrections, the null shift term $omega_{NS}$ is the dominant one. It turns out proportional to the optical losses $mu$ of the ring cavity, which are changing in time at the level of $10%$ within the considered period of thirty days. The time behaviour is reconstructed based on available signals (interferogram and mono-beam intensities), and the Allan deviation of the estimated $Omega_E$ shows a remarkable long term stability, leading to a sensitivity better than $10^{-10}$rad/s with more than $10$s of integration time, and approaching $(8.5pm 0.5)times 10^{-12}$rad/s with $4.5times10^{5}$s of integration time.
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