Bivariate polynomial interpolation

Background points are used to approximate the observed background by a polynomial of two variables, sample and line numbers, as:

$$B(x,y)\approx\sum\sum b_{ij}x^i y^j$$ (7.3)

The background of flat field images is usually well modelled by a 2D polynomial of degrees 3 and 4 in variables sample and line respectively. The agreement of the model is typically better than 1% of the background level. For object exposures the signal-to-noise ratio is normally much lower, as is the actual background level. A polynomial of lower degree, for example linear in both dimensions or a constant background should be enough. Because small errors in the determination of the background are carried through the whole rest of the reduction and are even amplified at the edges of the orders, care should be taken in the background fitting.

If no DARK or BIAS frames are available, the background definition might be slightly less accurate because the modelling procedure has to take into account these contributions as well. In some cases the degree of the polynomial has to be increased. As a rule of thumb, one should try to fit the background with a polynomial of the lowest possible degree.

This method gives good results when the main contribution to the background is due to global scattered light.