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XMM-Newton Science Analysis System


bkgfit (ebkgmap-1.2) [xmmsas_20070708_1801-7.1.0]

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Description

Although the task is non-XMM specific, it is primarily intended to generate maps of the background in XMM EPIC images. The task does this by fitting a linear combination of background-model component images. The task takes three main inputs (see section 6 for details): (i) the Poissonian FITS image which is to be fitted; (ii) a list of $N$ model component FITS images; (iii) (optionally) a FITS mask image. The output is a single FITS image which represents the best-fit background model. The best-fit amplitudes $\mathbf{a}_{\rm {opt}}$ and the names of the component datasets are recorded in this output dataset in a binary table extension.

Clearly all the input images must have the same dimensions: call this $X \times Y$ pixels.

The fitting is done by minimizes the maximum-likelihood estimator $L$ defined as follows:

$$L(\mathbf{a}) = -2\sum_{x=1}^{X}\sum_{y=1}^{Y} ln[P_{x,y}(\mathbf{a})]$$ (1)

where $\mathbf{a}$ is the vector of $N$ component amplitudes, and the sums are understood to be over all unmasked image pixels. Suppressing the $x,y$ subscript for the sake of brevity, the Poissonian probability $P$ is given by

$$P(\mathbf{a}) = \frac{B^{I}(\mathbf{a}) \, exp[-B(\mathbf{a})]} {I!}$$ (2)

where the total background model $B$ is the linear combination of the $N$ components $b_i$, viz

$$B(\mathbf{a}) = \sum_{i}^{N} a_i b_i,$$

and $I$ represents the value at that pixel of the Poissonian image. Inserting equation 5 into 1 gives

$$L(\mathbf{a}) = -2\sum_{x=1}^{X} \sum_{y=1}^{Y} \big\{ I_{x,... ...x,y}(\mathbf{a})] - B_{x,y}(\mathbf{a}) - ln(I_{x,y}!) \big\}.$$