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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.