Although the analytical formal solution of the transfer equation allows this, such inversions often produce numerical instabilities, especially, in the method of short characteristics. Algorithms used for interpolating the source function and the opacity either to facets of grid voxels or along the short characteristic, have inherent numerical errors, which might produce at the same time opposite signs of the source function and optical depth. Such an unphysical situation destroys the convergence of the M-ALI iteration scheme.

We tried the method of , who proposed to branch the integration of the transfer equation in terms of emissivities and opacities instead of source functions and optical depths, but found this to be unstable too. Finally, we simply removed the upper levels of the transitions that had population inversions from the model atom.

For  we used a four-level plus continuum model atom of @2013ApJ...772...89L, where we treat only the  k line in PRD and  h in CRD to lower the computational costs.

Finally, for  we used the five-level plus continuum model atom of where we similarly treat only  K in PRD and  H in CRD.

We tested using the 1D FAL-C model atmosphere [@1993ApJ...406..319F] to see whether the neglection of PRD in 