Correct for scattered light in the PFS spectrographs

It has become clear (e.g. PIPE2D-1212) that there is serious scattered light in the PFS spectrographs. Dealing with this is a little tricky, as there is no clean separation between "scattered light" and "the wings of the profiles". Nethertheless, it is possible to think of ways of handling the scattering which don't involve inverting dense 600^2 matrices for every row of the data.

Let us arbitrarily decompose the full profile P into a core and a scattering component P = C + S. Given some data D which is a realisation of a true (2-D) spectrum M we have D = M(C + S) + epsilon where the product is to be interpreted in Fourier space (or a convolution in pixel space), and epsilon is a noise term. We may then
trivially write CM = D - SM = D - SD/(C + S).

If we assume that we can find a model for S which has a lower amplitude than C and is more compact in Fourier space, then we may approximate D - SD/(C + S)) ~ D - SD/C. D/C is an estimate of M ignoring the scattered lights contribution to D. This allows us to subtract the scattered light, leaving an estimate of CM, which we may use to solve for M using only the core profile. It is clear that one could iterate, if desired, using our estimate of M to refine the scattered light correction.

This procedure is self-consistent, treating the scattering as a part of the total profile (you can think of this procedure as an iterative approach to solving the dense matrix mentioned above on the assumption that it is diagonal dominant).

Some preliminary results are discussed in scattering.pdf