  High precision spectrographs might exhibit temporal variations of their
reference velocity or nightly zero point (NZP). One way to monitor the NZP is
to measure bright stars, which are assumed to have an intrinsic radial velocity
variation much smaller than the instrument's precision. While this method is
effective in most cases, it does not fully propagate the uncertainty arising
from NZP variations. We present a new method to correct for NZP variations in
radial-velocity time series. This method uses Gaussian Processes based on
ancillary information to model these systematic effects. It enables us to
propagate the uncertainties of this correction into the overall error budget.
Another advantage of this approach is that it relies on ancillary data
collected simultaneously with the spectra rather than solely on dedicated
observations of constant stars. We applied this method to the SOPHIE
spectrograph at the Haute-Provence Observatory using a few instrument's
housekeeping data, such as the internal pressure and temperature variations.
Our results demonstrate that this method effectively models the red noise of
constant stars, even with a limited amount of housekeeping data, while
preserving the signals of exoplanets. Using both simulations with mock planets
and real data, we found that this method improves the false-alarm probability
of detections by several orders of magnitude. By simulating numerous planetary
signals, we were able to detect up to 10 percent more planets with small
amplitude radial velocity signals. We used this new correction to reanalysed
the planetary system around HD158259 and improved the detection of the
outermost planets. We also suggest decreasing the observing cadence of the
constant stars to optimise telescope time for scientific targets.
