  The cross-correlation function and template matching techniques have
dominated the world of precision radial velocities for many years. Recently, a
new technique, named line-by-line, has been developed as an outlier resistant
way to efficiently extract radial velocity content from high resolution
spectra. We apply this new method to archival HARPS and CARMENES datasets of
the K2-18 system. After reprocessing the HARPS dataset with the line-by-line
framework, we are able to replicate the findings of previous studies.
Furthermore, by splitting the full wavelength range into sub-domains, we were
able to identify a systematic chromatic correlation of the radial velocities in
the reprocessed CARMENES dataset. After post-processing the radial velocities
to remove this correlation, as well as rejecting some outlier nights, we
robustly uncover the signal of both K2-18b and K2-18c, with masses that agree
with those found from our analysis of the HARPS dataset. We then combine both
the HARPS and CARMENES velocities to refine the parameters of both planets,
notably resulting in a revised mass and period for K2-18c of
$6.99^{+0.96}_{-0.99}$M$_{Earth}$ and $9.2072\pm0.0065$d, respectively. Our
work thoroughly demonstrates the power of the line-by-line technique for the
extraction of precision radial velocity information.
