Broad lines are, well, broad. This creates a problem because the interpolation gets really bad because the gradient of the solution is so big that any case of extrapolation destroys the solution. 

I think the solution is to pull back the broadening of the lines, i.e., artificially reduce the Lorentzian component and create new solutions.  Of course, in that case, the solution for the particular broadened HgAr lines will be wrong.... This is a more general problem that we have to solve - how we should treat different lines differently (I will raise this issue in Slack in order to see how to best approach the problem - FYI hassan )