Distortion Model

The deviations of the spectra from a straight line is parameterized as

$$\Delta y = b_0 + b_1r + b_2r^2 + b_3r^3$$ (2)

where

$$r = \sqrt{(x-x_{obj})^2 + (y-x_{obj})^2}$$ (3)

is the distance of a pixel (x,y) from the object coordinates (x_obj, y_obj) in pixels and $\Delta y$is the deviation in pixels of the spectrum from a horizontal line.

In addition, the rotation of the spectrum relative to an exact alignment along a row is specified by a rotation angle. The angle is measured clockwise in units of degrees.

The alignment of the spectrum is taken into account by rotating the grism image around the object position (x_obj, y_obj) prior to the extraction. The distortions in the spectra are taken into account by introducing a corresponding distortion in the weights used for the extraction.