Computation of Contamination

For the regions of overlap, the spectral point F_j,k contains the correct flux from object j, but in addition it contains flux from spectral point F_i,l, where l= k- x_max,i-x_min,j and vice versa. Let's call F_i,l the spectrum as normally extracted, and F_i,l(true) the true spectrum of object i at wavelength l, and c_ij,l the contamination factor, then:

F_i,l= F_i,l(true) + c_ij,l F_j,k(true)

were k= l + x_max,i-x_min,j and

F_j,k= F_j,k(true) + c_ji,k F_i,l(true)

were l= k - x_max,i-x_min,j. Now we want to estimate the 2 contamination factors c_ji,k and c_ij,l: Normally the spectra are computed

$$F_{i,l} = \sum_{\rm region\ of\ object\ i} g_i(y) w_i(y)$$ (14)

where g is the flux on the grism image and w is the weight. The sum goes perpendicular to the dispersion direction in the region of object i.

Now in order to estimate the contamination from spectrum j, we just add the weights for object j at the position of object i:

$$c_{ij}= \sum w_j(y)$$ (15)

The sum is along y in the region of object i. Note that the sum of the weights of object j is in region of object i!

and similarly:

$$c_{ji}= \sum_{\rm along\ y\ in\ region\ of\ object\ j} w_i(y).$$ (16)

 

Figure 3.5: Two examples of situations where the spectra can be deblended at the wavelength marked with the dashed lines. a) even though there is no region of spectrum 1 which is not contaminated at the dashed line, both the spectrum 2 and 3 are partially uncontaminated and can therefore be estimated. This estimate in turn can be used to correct spectrum 1. b) Again, spectrum 1 can be estimated from the uncontaminated part, which in turn allows to extract spectrum 2.  

 

Figure 3.6: Example of a complex situation with the overlap of several spectra.  

 

Figure 3.7: Two examples of situations where the spectra cannot be deblended at the wavelength marked with the dashed lines. In both situations, there is no region of spectrum 1 which is not contaminated by other spectra. Therefore, there is no information in that image which could be used to decide to which spectrum a particular feature belongs.