  Context. Stellar activity and instrumental signals are the main limitations
to the detection of Earth-like planets using the radial velocity (RV)
technique. Recent studies show that the key to mitigating those perturbing
effects might reside in analysing the spectra themselves, rather than the RV
time series and a few activity proxies. Aims. The goal of this paper is to
demonstrate that we can reach further improvement in RV precision by performing
a principal component analysis (PCA) decomposition of the shell time series,
with the shell as the projection of a spectrum onto the spacenormalised flux
versus flux gradient. Methods. By performing a PCA decomposition of shell time
series, it is possible to obtain a basis of first-order spectral variations
that are not related to Keplerian motion. The time coeffcients associated with
this basis can then be used to correct for non-Dopplerian signatures in RVs.
Results. We applied this new method on the YARARA post-processed spectra time
series of HD10700 and HD128621. On HD10700, we demonstrate, thanks to planetary
signal injections, that this new approach can successfully disentangle real
Dopplerian signals from instrumental systematics. The application of this new
methodology on HD128621 shows that the strong stellar activity signal seen at
the stellar rotational period and one-year aliases becomes insignificant in a
periodogram analysis. The RV root mean square on the five-year data is reduced
from 2.44 m/s down to 1.73 m/s. This new approach allows us to strongly
mitigate stellar activity, however, noise injections tests indicate that rather
high signal-to-noise ratio (S/N>250) is required to correct for the observed
activity signal on HD128621.
