Given the location of the photon on the detector, it is clearly not possible to assign a wavelength to it to an accuracy better than the PSF. The uncertainty in the grating period is usually determined by the tolerances in the construction, which are usually quite stringent and far better than the PSF size, and so are quite negligible. The line spread function (LSF) therefore is almost entirely determined by the PSF. This is not the systematic error. Any systematic errors that are present are in addition to these errors.

p.s. I think I’ll understand this post better if I scrutinize later. A first time reading hardly provided me anything.

p.p.s. I came back to read again but another thought came to me. I do not recall that I saw Kalman filter or HMM occasionally from astro-ph but I imagine that the things I learned from nonlinear control and system could help modeling some of astronomical systems with systematic errors or measurement errors in addition to known statistical errors. (Here, known means not quantification but qualification; for example, not estimated σ but Poisson noise). As astrometry is a subject in astronomy, I hope handling errors to be regularized. I’ll appreciate any criticism and comments to my question.