XMM-Newton Science Analysis System


bkgoptrate (bkgfilter-1.8.2) [xmmsas_20230412_1735-21.0.0]

Theory

A possible and good algorithm to achieve the objective, which this task follows, is to search for the point, at which the maximum signal-to-noise (S/N) ratio is achieved for the given background time-series after the bins above a threshold are excluded.

The S/N ratio is described as

  $${\rm S/N} = \frac{{\rm SourceCount}}{\sqrt{ {\rm BackgroundCount}}},$$ (1)

providing the standard Poisson statistics are applicable.

Let us define $Z^*$ as the reference threshold count per time bin, and Exp$^*$ as the exposure time for the bin. When we use $i$ subscripts to represent each time bin, the above formula is rewritten as

	$${\rm S/N}$$	$=$	$$\frac{\Sigma_{i ({\rm Cnt}>Z_i)} {\rm Exp}_i \times {\rm SourceRate}}{\sqrt{ \Sigma_{i ({\rm Cnt}>Z_i)} {\rm Cnt}_i }}$$	(2)
	$$Z_i$$	$\equiv$	$$Z^* \times \frac{{\rm Exp}_i}{{\rm Exp}^*},$$	(3)

where SourceRate, Exp$_i$, Cnt$_i$ and $Z_i$ are the count rate of the source, exposure, count of the given time-series and threshold count in each bin $i$, respectively.

What we here search for is the optimum $Z^*$, which is the background level, with which S/N becomes the maximum. In principle it requires a minimization (or maximization) method for search. We assume SourceRate is constant, then we can ignore SourceRate in comparing S/N ratios for different thresholds ($Z^*$):

	$${\rm S/N}$$	$\propto$	$$\frac{\Sigma_{i ({\rm Cnt}>Z_i)} {\rm Exp}_i}{\sqrt{ \Sigma_{i ({\rm Cnt}>Z_i)} {\rm Cnt}_i }}$$	(4)
	$${\rm S/N}$$	$\propto$	$$\frac{\Sigma_{i ({\rm Cnt}>Z_i)} {\rm Exp}_i}{\sqrt{ \Sigma_{i ({\rm Cnt}>Z_i)} {\rm Rate}_i \times {\rm Exp}_i}},$$	(5)

where the second form represents the one with the rate instead of count of the input time-series, and $Z_i$ is defined in Eq. 3.

If all the time bins have the same exposure, then this formula is further simplified:

	$${\rm S/N}$$	$\propto$	$$\frac{\Sigma_{i ({\rm Cnt}>Z^*)} i}{\sqrt{ \Sigma_{i ({\rm Cnt}>Z^*)} {\rm Cnt}_i }}$$	(6)
	$${\rm S/N}$$	$\propto$	$$\frac{\Sigma_{i ({\rm Rate}>Z^*)} i}{\sqrt{ \Sigma_{i ({\rm Rate}>Z^*)} {\rm Rate}_i }}.$$	(7)