XMM-Newton Science Analysis System


imweightadd (tools-1.68.1) [xmmsas_20230412_1735-21.0.0]

Estimation of the optimal weights

In order to estimate the optimal weights, imweightadd needs two lists: a list of background maps and a list of source relative expectation values $\sigma_i$.

The first job is to reduce each background map to a single representative value $\beta_i$ of background counts per pixel. This is done by making a histogram of all the non-zero values in each map and selecting the value which falls nearest the 90% mark on the histogram. The rationale behind this is as follows. Source detection is likely to be most sensitive near the centre of the image; this is also the place where one would expect the maximum to be in the background values. Hence it makes sense to choose a value which is nearer to the maximum value than to the minimum. However there is also the possibility of local increases in background due to out-of-time events or such like. Because of this possibility it was thought undesirable to pluck the background value right from the top of the tree so to speak: hence the 90% figure was arrived at as a compromise.

The next thing is to normalize the $\sigma$s to 1. Naturally at least one of them must be non-zero.

imweightadd then performs a minimisation via a simplex algorithm. The quantity to be minimized is the detection sensitivity as defined in section 3.3. The analogue of equation 2 in the present case is

  $$L_{\rm {cutoff}} = -\ln[1 - Q(ky_{\rm {cutoff}},k\mu)]$$ (6)

where

$$y_{\rm {cutoff}} = \sum_{i=1}^N w_i (\beta_i + S_{\rm {cutoff}} \sigma_i),$$

$$\mu = \sum_{i=1}^N w_i \beta_i,$$

$$v = \sum_{i=1}^N w_i^2 \beta_i,$$

and

$$k = \mu / v$$

as before. At each step, equation 6 is inverted numerically via a Ridders-method algorithm to yield the sensitivity $S_{\rm {cutoff}}$. The minimization procedure therefore arrives at the set of weights which yield the minimum value (within convergence limits) of $S_{\rm {cutoff}}$. These weights are then applied to generate weighted sums of the input images and also the input background maps and exposure maps.