Following the successful completion of Step 1, an attempt is made to refine the order shifts by determining systematic differential shifts across the image, e.g., due to an expansion/contraction term. A failure in the Step 2 tests described below results in the adoption of the global fit determined from Step 1. Differential shifts cannot be computed for these cases.
Order centroid positions and weights found above are used in Step 2. This step differs from Step 1 in two important respects. First, a sufficient number of orders containing flux must be found both in the short- and long-wavelength (spatial) ends of the camera. If this distribution test fails, the global shift from Step 1 is adopted and applied to all lines in the image. A second difference from Step 1 is that Step 2 computes order spacings from the found centroids. A least-squares solution is then determined from the differences of the logarithms of these spacings versus echelle order number (because of the expected 1/m2-dependence in order separation) and the logarithms of the corresponding order spacings for the fiducial image. A quadratic least-squares solution is attempted for SWP images because a curvature term is sometimes necessary. For LWP images the least-squares solution is linear, while for LWR images the solution is two joined line segments across the camera. As a quality-control check, ORDERG makes a test on the derived shifts at the end of Step 2. If any of the shifts exceed a threshold value of 4.0 pixels, Step 2 reverts to the solution from Step 1.
The found shifts resulting from Step 2 are defined literally only for the order centroid positions. As a practical matter, the final shift for a given order is applied uniformly to all lines associated with that order, including the adjacent lines containing background fluxes. Note that this can produce small shift discontinuities for lines located midway between the orders.