7.10.1 General Principles

  Optimal extractions of spectral information are an important tool when the illumination of a spectrum perpendicular to the dispersion direction is not uniform and there are sources of noise other than statistical fluctuations in photon counts (Horne 1986). While we are operating mostly in the environment where photon counts dominate, the development presented later copes with our unique problem of having substantial variations in photoefficiency. Also, we have some overlap in energy in adjacent orders. We must compensate for the slight contamination of each order on its neighbor and do so in a way that recognizes the increased uncertainties of pixels that have the largest corrections.

The first step in the process is to derive the function that describes illumination profile of a typical order along the vertical (cross dispersion) direction. We have the fortunate situation that the length scales for the vertical profiles are directly proportional to the order separations. Thus, by simply distorting a picture so that all orders have the same separation, chosen to be 10 pixels everywhere, we automatically obtain a universal function for the vertical profiles and the character of their overlap from one order to the next - see Fig. 11. As another convenience, the distortion of the picture was also modified to make the orders perfectly horizontal, so that horizontal slices of entire orders were represented by single rows in the picture. We could thus avoid the complication of dealing with a sloping centerline for each order.

In addition to obtaining the best representation of the signal deposited across an order at some particular wavelength $\lambda$, we must also compensate for interference from adjacent orders that contaminate the upper and lower edges. Either that, or we would have to throw away a good fraction of the signal and make use of only the central part of the order. The mathematical development described in the next section allows one to compensate for the interference, with varying degrees of boldness based on how far the sampling range extends away from the order's core.