XMM-Newton Science Analysis System


bkgfit (ebkgmap-2.10.2) [xmmsas_20230412_1735-21.0.0]

Constraints on the model components:

The fitting procedure will only converge to a unique solution if the components $b_i$ are linearly independent. A measure of this is the quantity

  $$Q_{i,j} = \frac{\sum_{x=1}^{X}\sum_{y=1}^{Y} b_{x,y,i} \, b_{x,y,... ...^{Y} b_{x,y,i}^2 \big) \big(\sum_{x=1}^{X}\sum_{y=1}^{Y} b_{x,y,j}^2 \big) }}.$$ (4)

If one thinks of each image $b_i$ as a vector in an $X \times Y$ dimensional space, then $Q_{i,j}$ is the cosine of the angle between $b_i$ and $b_j$. The task calculates $Q$ for each pair of component images and fails with an error if any $Q$ is greater than 0.99. NOTE that the sums in equation 4 are carried out on unmasked pixels only. Hence components may be linearly independent in masked pixels yet still fail the $Q$ test within the task.

The task also fails if any of the components has only zero-valued pixels. Finally, since the Poisson image values are necessarily all $\ge 0$, negative values are not allowed in the model components.