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 , 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.