Flux vs. Velocity representation of spectra?
I think the velocity scale is calculated something like this:
$$v_r = c\left( \frac{\lambda - \lambda_0}{\lambda_0}\right),$$
where $\lambda$ is the observed wavelength and $\lambda_0$ is wavelength of the Mg II line, corrected for the redshift of the quasar ($z=0.868)$ it is observed towards.
The confusing thing is, for Fig.~3 in the cited paper, that the Mg II feature is actually doublet with lines at rest wavelengths of $\lambda_r =2796.35$ and $2803.53$ Angstroms. Therefore
$$\lambda_0 = \lambda_r (1+z) = 5223.58\ \ {\rm or} \ \ 5236.99\ \ {\rm Angstroms}$$
corresponding to the thick and thin solid lines on Fig.3 respectively.
In other words, zero on the velocity scale would correspond to a Mg II absorber at the redshift of the quasar (an intrinsic absorber), whereas negative velocities correspond to absorbers at lower redshift, by the velocities indicated.
For a check on this - calculate the velocity of feature &quot;b&quot;, which appears to be at about 5221 Angstroms in the observed spectrum for the redder doublet component (the thin solid line in Fig.3). Using the above definition, this would be a velocity of -915 km/s.
Looking at Fig.3, there seems to be a major problem. Feature &quot;b&quot; is at -680 km/s.  However, if we look further on at Figs. 4 and 5, we see feature &quot;b&quot; plotted at -850 km/s in the Mg II spectrum, which is much closer, and would perhaps be in exact agreement if feature &quot;b&quot; was actually at 5222 Angstroms (which looks unlikely) or that a 4th, unspecified decimal place had been used in the redshift. e.g. if $z=0.8676$ then $\lambda_0 = 5235.87$ Angstroms, and the velocity of a feature measured at 5221 Angstroms would then be -852 km/s in perfect agreement with Figs. 4 and 5.
Either way, there is major disagreement between Fig. 3 (which I suspect is in error) and Figs. 4 and 5 (which could be reconciled with the above analysis if the redshift was 0.8676).
I'm sorry this is not very conclusive and I certainly wouldn't accept it without someone else verifying that the x-axis of Fig.3 appears to be problematic.