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4. Data analysis software

The telemetry sent by the satellite to the groundstation comprises housekeeping and scientific data. The housekeeping data is used to monitor the overall condition of the satellite and the instruments, and to evaluate the scientific data.

The data from the instruments consists of a list of events labelled with detector position, energy channel and time of detection. From this raw data one can determine fluxes and positions for all detected celestial X-ray sources (in case of extended sources also morphology). This will all be done on ground.

Figure 17: Point Spread Function in y-direction measured at 8 keV through a 0.1 mm pinhole, the FWHM is 0.6 pixel which is well within the required 1 pixel. The broadened distribution including the two side lobes is acceptably low (tex2html_wrap_inline1876), and is caused by pile-up events

However, first the instrument and satellite housekeeping must be processed in order to correctly analyse the data. For this purpose software was developed to read the Final Observation Tapes (FOT) and produce cleaned eventlists in FITS format, the so called stage I software. Using these cleaned eventlists, Stage II software creates data products such as images, fluxes, and spectra (Jager et al. 1996).

4.1. Stage I, FOT processing

The following extraction and processing activities are carried out on FOTS, in principle without the intervention of the operator:

4.2. Stage II, image reconstruction

  The stage I software produces a list of events (photons) that were measured in the detector. Without further analysis, it is impossible to tell whether a photon came from a certain X-ray source, it could have reached the detector through any of the transparent mask elements from almost any part of the sky. Unlike most imaging instruments, the data of a coded mask camera needs first be deconvolved.

The main objective of the stage II software is to reconstruct the observed sky in an optimal time and energy channel interval in order to obtain information on X-ray sources. This information typically consists of images, lightcurves and spectra for each source in the field of view. A number of reconstruction methods have been described in the literature, like correlation of the detector image with the mask pattern, the Maximum Entropy Method and Maximum Likelihood Fitting (Willingale et al. 1984). We briefly describe here the method of cross-correlation between detector image and mask pattern. We use this method because it is numerically relatively fast and it has produced satisfactorily results with the COMIS instrument. A full description can be found in in 't Zand (1992). Investigation of other methods is ongoing.

4.2.1. The correlation method

  Each point source in the large field of view projects the shadow of the mask pattern onto the detector. Extracting the locations and strenghts of these sources from the sum of their projections, means solving an inverse problem. This can be done by deconvolving or cross-correlating the total image on the detector with the mask pattern as follows.

If we represent the detector image of the sky by a two-dimensional matrix D, then D is given by the convolution tex2html_wrap_inline1884 of the mask pattern M and the true sky image S:
Reconstruction of the sky image S' can be performed by means of a cross-correlation tex2html_wrap_inline1892 of D with a certain reconstruction matrix R, which has to be specified.
The reconstruction is perfect if R satisfies the condition
where I is the identity matrix. If M is based on a pseudo-random URA pattern, R can be found and is uniquely given by M. The pattern of R is then equal to the pattern of M.

If D contains complete uncorrupted shadows of M, Eq. (3 (click here)) is always exactly matched and S' = S. However, in a detector-mask configuration such as employed in the BeppoSAX-WFCs (i.e. with equally sized mask and detector, the ``simple" camera), D only contains a complete shadow of M for the on-axis position and so-called ``coding noise" emerges. The reconstruction of a single sky pixel k can then be represented by
where s'k is the reconstructed intensity at sky pixel k, sk the true intensity, and fi,k a factor representing the cross talk between different sky pixels introduced by the coding noise.

Equation (4 (click here)) illustrates that coding noise is in fact deterministic: if the configuration of sky sources is known, the cross talk term may be evaluated and its influence on the estimate of sk eliminated. In practice, suppression of coding noise may be accomplished by an iterative method. Such a method is Iterative Removal Of Sources or IROS (Hammersley 1986; in 't Zand 1992).

In IROS, each step involves the subtraction of simulated point source exposures in the detector plane, where the point sources are those which are significant with respect to the noise in the cross-correlation of that detector image. A cross-correlation on the resulting detector image contains less coding noise and provides the means to evaluate weaker sources for a subsequent iteration. The final product of IROS is a sky image with almost no coding noise and no sources. The subtracted sources can then be displayed in this image.

A problem which arises when using IROS, is the accuracy of the point source location. Due to the coding noise in the image, the Point Spread Functions of the sources are perturbed in a non-statistical manner so that the localisation by means of fitting with the PSF can be slightly off. Subtraction of sources with errors in the positions of only a few tens of a pixel then introduces residues in the image that can not be recovered later. One solution to this problem is to make use of the catalog position of well known sources. After subtraction of the known X-ray sources, any new sources will remain in the sky image and can be analysed. Other possibilities are being studied.

Apart from IROS there are more ways to optimize the signal-to-noise ratio of the reconstruction as determined by the cross-correlation given by Eq. (2 (click here)), e.g.,

  1. Weighing.

    The Poisson noise may be suppressed by differently weighing subsections of the detector plane during cross-correlation. The idea behind this technique may be thought of as follows: consider the case when one strong and one weak point source are in the observed sky whose exposures have a small overlap in the detector plane. We may omit counts in this overlap during the reconstruction of the weak source, thus eliminating the strong contribution in Poisson noise from the strong source. This may improve the signal-to-noise of the weak source several tens of percents. Generally, this improvement may be achieved by weighing detector subsections in stead of omission.

  2. Transparency correction and detector background subtraction.

    The value of the reconstruction matrix R (uniquely given by M, see in 't Zand 1992) does not account for possible disturbing influences on the imaging due to the detector window support structure. Part of this disturbance may be eliminated by redefining R. The remainder may be eliminated by subtracting an estimate of the contributions by the sky and detector background in the detector plane. This must be done before the cross-correlation. The total countrate of the background was found to be about 190  c s-1 in the full energy passband (cf. the countrate for the Crab X-ray source of 300  c s-1).

The result of the complete reconstruction consists of two images, one representing estimates of the intensity per sky pixel and the other containing estimates of the Poisson noise standard deviation per pixel. Furthermore a list of sources and upper limits is produced for the selected time and energy windows. This can subsequently be used to do a more detailed analysis, i.e. determination of spectra and lightcurves.

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