http://www.aanda.org/articles/aa/pdf/2013/09/aa21551-13.pdf
The prescription of Bourrier et al.. is to calculate (at each pixel) the average optical depth \tau_i for range of velocity bins with spacing \Delta v (or frequency bins with size \Delta \nu, so that
\sum \tau_i \Delta \nu = \tau
So we have
\tau_i \Delta \nu = \tau_i \frac{\Delta v}{\lambda_0} = \displaystyle \int_{v_i-\frac{1}{2}\Delta v}^{v_i + \frac{1}{2}\Delta v} \tau (v) dv
and
\tau (v) = \sigma_{v0} \int N(v') \phi ( v-v' ) dv'
Now to get the line profile in terms of velocities, we can start with the line profile in terms of frequencies:
\phi ( \nu - \nu' ) = \frac{\Gamma}{4 \pi^2 (\nu - \nu')^2 + \left ( \Gamma / 2 \right )^2}
and realizing that this is a distribution, we can use
\phi ( v - v') dv' = \phi ( \nu - \nu') d \nu
where
\nu = \nu_0 + \nu_0 \frac{v}{c}
so
\nu - \nu' = \frac{v - v'}{\lambda_0}
and
d \nu = \frac{dv}{\lambda_0}
to get
\phi ( v-v' ) =\frac{\Gamma \lambda_0}{4 \pi^2 (v - v')^2 + \left ( \Gamma \lambda_0 / 2 \right )^2}
Putting this all together we have…
\tau_i = \frac{\lambda_0}{\Delta v}\displaystyle \int_{v_i-\frac{1}{2}\Delta v}^{v_i + \frac{1}{2}\Delta v} \sigma_{v0} \int N(v') \frac{\Gamma \lambda_0}{4 \pi^2 (v - v')^2 + \left ( \Gamma \lambda_0 / 2 \right )^2} dv' dv
Now if we bin the column densities into velocity bins, we have
N_j = \displaystyle \int_{v_j-\frac{1}{2}\Delta v}^{v_j + \frac{1}{2}\Delta v} N (v') dv'
and
\int N(v') \frac{\Gamma \lambda_0}{4 \pi^2 (v - v')^2 + \left ( \Gamma \lambda_0 / 2 \right )^2} dv' = \sum N_j \frac{\Gamma \lambda_0}{4 \pi^2 (v - v_j)^2 + \left ( \Gamma \lambda_0 / 2 \right )^2}
and we have (wave arms in air for a bit)
\tau_i = \frac{\lambda_0}{\Delta v} \sigma_{v0} \sum N_j \displaystyle \int_{v_i-\frac{1}{2}\Delta v}^{v_i + \frac{1}{2}\Delta v} \sigma_{v0} \frac{\Gamma \lambda_0}{4 \pi^2 (v - v_j)^2 + \left ( \Gamma \lambda_0 / 2 \right )^2} dv
or
\tau_i = \frac{\lambda_0}{\Delta v} \sigma_{v0} \sum N_j C_{ij}
where
C_{ij}=\displaystyle \int_{v_i-\frac{1}{2}\Delta v}^{v_i + \frac{1}{2}\Delta v} \sigma_{v0} \frac{\Gamma \lambda_0}{4 \pi^2 (v - v_j)^2 + \left ( \Gamma \lambda_0 / 2 \right )^2} dv
Now \Delta v is of order 20 km/s and \Gamma \lambda_0 is 76 m/s… so if i == j, then the integral may as well go to infinity and we have
C_{ii} = 1
and for i \ne j
C_{ij} = \frac{\Gamma \lambda_0}{4 \pi^2 (v_i-v_j)^2} \Delta v
so we have
\tau_i = \frac{\sigma_{v0} N_i \lambda_0}{\Delta v} + \displaystyle \sum_{i \ne j} N_j \sigma_{v0} \frac{\Gamma \lambda_0^2}{4 \pi^2 (v_i-v_j)^2}
Or at least that's how I get agreement with their equation 11…
So to implement this approach, I plan on creating projections (integrations along LOS) of \kappa_i where
\kappa_i(n,v) = \delta_{ij}\frac{\sigma_{v0} n \lambda_0}{\Delta v} + \left ( 1 - \delta_{ij} \right ) n \sigma_{v0} \frac{\Gamma \lambda_0^2}{4 \pi^2 (v_i-v_j)^2}
where j corresponds to the bin that the LOS cell velocity v lies within