Interpretation 2: At fixed \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}

Let's assume that the measured spectrum is made of a line centered around \ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{obs}}}} in the observatory frame. The associated rest frame frequency, i.e., the frequency of the line at rest, will be \ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}}. The observation is naturally set up so that the source frame is also the rest frame. This implies that \ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{obs}}}}, \ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}} and \ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}} are linked through

  $\displaystyle \frac{\ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mat...
...h{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}}{\ensuremath{c}}.
$ (6)

The modeller will use the Doppler effect to interpret the observed frequency difference $\ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}(\ensuremath{i...
...rm{}}}})-\ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{obs}}}}$ as a local velocity difference (projected along the line of sight direction) around the systemic velocity of the source. Let's write \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}}( \ensuremath{i_{\ensuremath{\mathrm{}}}}) the velocity associated to \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}( \ensuremath{i_{\ensuremath{\mathrm{}}}}). The velocity difference will be noted

  $\displaystyle \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensur...
...{}}}})-\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}.
$ (7)
In order to derive this velocity difference, we associate the velocity to each observed frequency through the Doppler effect, i.e.,
  $\displaystyle \frac{\ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{o...
...ath{\mathrm{obs}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}})}{\ensuremath{c}}.
$ (8)
It is easy to deduce that
  $\displaystyle \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensur...
...}})}{\ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}}},
$ (9)
i.e., the local velocity difference is proportional to the observed frequency difference. We can define a linear velocity axis associated to the set of brightnesses, $T(\ensuremath{i_{\ensuremath{\mathrm{}}}})$ as
  $\displaystyle \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}...
...}}}}{\ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}}}.
$ (10)
This velocity scale is only meaningful locally, i.e., as long as the brightnesses can be associated to the line rest frequency under consideration.