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Astron. Astrophys. 337, 43-50 (1998)

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3. Analysis

The data analyzed in this work were taken during the first year of operation of AIROBICC, between March 1992 and March 1993. The mean trigger rate in this period was [FORMULA]16 Hz. The sample contains [FORMULA] events ([FORMULA]800 hours) and after a successful arrival direction reconstruction of those showers with [FORMULA]7 fired stations, about [FORMULA] events ([FORMULA]60%) survive. It has been shown that due to the low counting statistics encountered when dealing with transient phenomena, one has to keep as many events as possible; and so we do not apply the standard cut in the number of fired huts. The angular resolution is then [FORMULA].

The analysis is based on a binned (in time and in space) all-sky search. We define an ON window or source window and an OFF window or background window where events are counted for a certain time interval. The search strategy considers each event as being in the middle (in time) of a burst, so for every single event in the data sample we take a pair of ON-OFF windows with a time interval centered at the time of the event. The ON window is a circle of 0.7° radius centered at the position of the event and the OFF window is an annulus of 1.5° and 10.0° radii concentric to the ON window. The solid angles covered by the ON and OFF windows are denoted by [FORMULA] and [FORMULA] respectively, and their ratio [FORMULA] by [FORMULA]. There is always one event inside the ON window and in order to not overestimate the significance of a possible signal this event is not counted. The size of the ON window is chosen to maximize the S/N according to our resolution (see above) as described in Alexandreas et al. (1993a) and it is expected to contain 75% of the events for a point source. While the size of the OFF window is taken as large as possible to increase statistics, its limit is due to the systematic error introduced because of the non-linear dependence of the counting rate on zenith angle (Alexandreas et al. 1993a). We performed a Monte Carlo (MC) calculation to find the largest radius that keeps the systematic error much lower than the statistical error due to the low counting rate. Results are shown in Fig. 1. The time intervals we chose are 10 seconds, 1 minute and 4 minutes which cover the usual long duration GRBs. Dead time and low counting rate do not allow searches for GRBs with much shorter duration.

[FIGURE] Fig. 1. Systematic error in the estimate of the background with the annulus method as a function of the ratio of the solid angles of the OFF and ON windows. The ON window is a fixed circle of 0.7° radius and the OFF window is an annulus of fixed 1.5° inner radius and variable outer radius. The time interval is 4 minutes which is expected to have the lowest statistical error. The points are the average over several hundred windows. The sample used for this calculation consists of MC events generated as described in the text. The systematic error has a constant value of less than 5% up to a certain size of the OFF window, from which it grows very fast. The arrow points to the value used in this work. This error has to be balanced against the statistical error which in the best case is 20%. Note that the sign of the systematic error is positive, i.e. it is conservative.

The employed procedure has the advantage of being very sensitive but it is also quite time consuming. So in order to look for TeV-emission on longer time scales, a less sensitive search with a 1 hour time window and a classical non-overlapping rectangular grid of 0.5°x0.5° in celestial coordinates has been carried out. In this case the background is estimated by means of a MC method as described in Alexandreas et al. (1993a). For every real event we generate 100 MC events with random directions following the acceptance function of AIROBICC (obtained with the events of a whole run, therefore a different run implies a different acceptance function) but with the same times as that of the original event. The number of MC events that fall within the ON window determines the background.

To obtain the significance for a hypothetical signal we use the probability distribution given in Alexandreas et al. (1993a) which is appropriate for low statistics (Poissonian regime). It is the probability of observing at least [FORMULA] events in the source window, given the observed number of background events [FORMULA], as a result of a background fluctuation:

[EQUATION]

[EQUATION]

where [FORMULA] and [FORMULA] are the number of events in the source and background windows respectively. In case of random distribution of the events (i.e., no GRBs), a cumulative histogram of the number of trials with a chance probability lower than P as a function of -log[FORMULA] should follow a straight line with slope -1, which cuts the Y-axis at the height of the total number of windows (trials). A strong GRB or several weaker GRBs should therefore appear as a deviation from the line. The significance of a deviation can be calculated knowing that theoretically every bin content follows a binomial distribution. Actually this approach is only an approximation because the probability distribution is discrete and there is an oversampling (i.e., search windows overlap and thus trials are not independent) in the search with short time windows. Therefore we estimate the significance repeating the search over a MC sample which is 10 times larger than the real one in the case of short time windows and 20 times larger in the case of 1 hour search window.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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