The intermediate file of hours-confirmed sources (in the restricted sense of Section V.E.3 above) accumulated as the satellite data were processed. Cluster analysis, weeks-confirmation, and band-merging were run repeatedly on this intermediate file to optimize the the thresholds in these processors. This section describes how the thresholds were arrived at and discusses the implications of the choices.
It was determined from preliminary analysis that the threshold on the link parameter used in the cluster analysis processor would have to be larger than 2, and that the weeks-confirmation threshold would have to be in the vicinity of 1. It also became clear that in the range of interest, the clustering threshold had the greatest influence on the output. This threshold was therefore optimized with the weeks-confirmation threshold held at 0.8; then the weeks-confirmation threshold was chosen. The final changes in confirmation did not affect the clustering enough to require more tuning.
The goal in optimizing the processor was enhanced reliability; completeness was only a secondary concern. Reliability included the requirement that the source be free of potentially confusing neighbors. Regions of low source density (such as high Galactic latitudes) were the prime areas where the processor was expected to perform well.
As stated earlier (Section V.E.4), cluster analysis was meant to filter out fragments of sources that were larger than 8' and sources that were confused.
Figure V.E.1 This shows that cluster analysis processing does not
greatly affect the number of small extended sources that are
weeks-confirmed at high galactic latitudes. Only at 100 µm
is there a substantial dependence, because of the cirrus.
Neither problem was common at high Galactic latitudes, and while it
was necessary to apply cluster processing in these areas, it was not possible
to select an optimal threshold by studying these areas alone.
Figure V.E.1 illustrates this point clearly by showing
that the number of
weeks-confirmed sources at high Galactic latitudes was essentially independent
of the clustering threshold at 12 µm and 25 µm; and
at 60 µm and 100 µm the number of sources dropped
as the clustering threshold increased indicating the presence of complex
structure at these wavelengths. Figure V.E.1 displays
the results of processing in a region (henceforth Region A) at high Galactic
latitudes, defined in ecliptic coordinates by 0° <
90° and 135° < <
205°; its total area was about 4010°2, about 10% of the
sky. The source density was on the order of 0.02 to 0.05 per sq. deg. too low
for confusion to be a problem.
Figure V.E.2 In contrast with Figure V.E.1, regions of high source
density are heavily affected by cluster analysis.
Figure V.E.2 displays the number of weeks-confirmed sources as a function of clustering threshold for Region B, which includes a crowded portion of the Galactic plane. It was defined in ecliptic coordinates by 25° < < 45°, and 280° < < 300°; its total areawas about 326 sq. deg, which implies a source density of about 0.3 to 6.0 per sq. deg. Both figures were obtained with a weeks-confirmation threshold of 0.8.
The shape of the curves in Figure V.E.2 suggested that a natural choice of clustering threshold could be based on keeping the final source density near the confusion limit. The confusion limit was obtained by requiring a minimum number of 25 beams per source (as in - Section V.H.6), which corresponds to a probability less than 0.1% that two sources will be found in the same beam, and a probability of about 1% that two adjacent beams will both have sources in them, assuming Poisson statistics. To use this criterion, an estimate was needed for beam size. A close upper limit was simply the in-scan width of the largest detector template (about 10') times twice the cross-scan width of a detector (about 10'), leading to an effective "beam size" of 1/36 sq. deg.
In practice, however, and especially at 12 and 25 µm, detections on smaller templates were common, and the effective beam size was found to be smaller than the upper limit by a factor two or more. To estimate the effective beam size, the average density of small extended source detections per survey coverage per band was found in the five most crowded bins on the sky. Each bin was approximately a sq. deg in size. This density was the same at 12 and 25 µm, namely 77 sources per sq. deg. with a population dispersion of 4; at 100 µm, the density was 40 ± 5 per sq. deg At 60 µm, the average density in the five most crowded bins was 78 ± 20; if the highest density bin was thrown out, the average in the next five was 68 ± 10. The result at 100 µm was quite close to the upper limit estimate above, as expected since only the largest template was available at 100 µm.
Adopting as effective beam sizes 1/40, 1/68, 1/77 and 1/77 sq. deg.
respectively in the 100 µm, 60 µm, 25 µm
and 12 µm, the critical densities are 1.6, 2.7, 3.1 and 3.1
sources per sq. deg. To find the corresponding critical clustering threshold,
small heavily populated windows within Region B, with a total area of 9.8
sq. deg were used. The average density of weeks-confirmed sources dropped
quickly as the clustering threshold increased from 2 to 3, and then leveled
off in a way similar to, but steeper than, what was seen in
Fig. V.E.2. The critical source density was reached in
all bands for thresholds
between 3 and 4. A value of 3.5 was chosen for all four bands.
Figure V.E.3 Effect of weeks-confirmation threshold on the number of
sources. The chosen threshold is indicated by the vertical
Figure V.E.3 shows the number of
weeks-confirmed sources as a function
of the weeks-confirmation threshold for Region A. Clearly, almost all
were acquired by a threshold of 2; the slow rise beyond that point was
roughly linear, as expected for false confirmations. The linear rise with
threshold was expected because the search area (rather than the search
radius) scales linearly with the confirmation threshold. Reliability dictated
dictates these false confirmations, while completeness demanded keeping
as many of the better positional matches as possible. A value of 1.4 for
weeks-confirmation threshold was selected because it marked the boundary
between the steep climb due to true confirmations and the gradual climb
due to the false confirmations.
Figure V.E.4 The optimal threshold for band-merging is indicated by the
vertical broken line. It is the same as for weeks-confirmation.
Because both spatial resolution and source properties changed with wavelength, an astronomical object could appear extended in one band and point-like in others. In view of that, band-merging was carried out after confirmation, in contrast to point source processing.
Figure V.E.4 shows the output of the band-merge
processor as a function of the band-merging link parameter threshold in
Region B how Galactic latitudes). As anticipated most sources turned out
to be single-band sources. Past a threshold of 1.4 very little new
band-merging took place. A threshold of 1.4, the same as for
weeks-confirmation, was adopted.
It was evident that the performance of the small extended source processor at high Galactic latitudes varied slowly as a function of the cluster processing threshold. In contrast, crowded regions provided the testing ground for selecting an optimal clustering threshold. With a first determination of 3.5 as the clustering threshold high Galactic latitudes provided the optimal choice of 1.4 as the weeks-confirmation threshold (Fig. V.E.2). A value of 1.4 was also chosen as the optimal threshold for band-merging.
Figure V.E.5 A final check on the optimal thresholds: the
weeks-confirmation threshold used here is the final one (1.4);
the effects of cluster analysis are quite drastic, as expected.
The tick mark on each curve indicates the critical source density;
in all cases this density is obtained at thresholds greater than
the optimal choice of 3.5.
The final iteration was to repeat clustering optimization using the final choice for confirmation threshold. This was done using Fig. V.E.5, where the density of weeks-confirmed sources is shown as a function of the clustering threshold in the three crowded regions mentioned in Section V.E.7.a. The confusion limits were 3.1, 3.1, 2.7, and 1.6 sources per sq. deg in the 12, 25, 60, and 100 µm bands, respectively. This critical density was reached for all bands between clustering thresholds of 3 and 4; as expected a value of 3.5 was still the optimal common choice for all bands.
To assess the significance of this choice, one can estimate the size of the area searched for close neighbors by the cluster analysis processor both in relative and absolute terms.
If an extended source is thought of as a square-wave in one dimension with total width W, then its corresponding rms size is W/(2 x 3½). The template used for detecting this source would have been itself square-wave shaped with a width W, and baseline segments W/2 on each side. A clustering threshold of 3.5 implies that 2 sources are considered close neighbors as soon as the baseline segments of their respective detection templates star to overlap. This was clearly a reasonable, though somewhat conservative, way of guarding against confusion.
To estimate the angular distances involved in cluster analysis, the mean size of a sample of 111 sources in each band was calculated after clustering and weeks-confirmation; these mean sizes (always close to the medians as well) were 1.5', 1.5', 1.8', and 2.2' at 12, 25, 60, and 100 µm. The largest size in any band was 3'. On average, therefore, cluster analysis treated as "close neighbors" two sources within 10' of each other at 12 and 25 µm, 12' at 60 µm, and 15' at 100 µm.
Cluster analysis fulfilled its objective in recognizing and setting aside large structures that were fragmented into small extended sources; this was the reason for the decrease in 100 µm and 60 µm source counts with increasing clustering threshold in Region A (Fig. V.E.1): cirrus was integrated into larger structures and dropped from further processing. It should be stressed, however, that cirrus is not absent from the small extended source catalog.
Cluster processing also fulfilled its objective as a confusion processor, as shown by the reduction by an order of magnitude of the source density in crowded areas (Fig. V.E.4) as the clustering threshold was varied from 1 to greater than 3.5. The sources that survived in densely populated areas were either very isolated or locally dominant. isolated sources had no neighbors within the search window. Dominant sources were so much brighter than their neighbors that when the latter were combined with them the source parameters were barely altered so that the size in particular did not grow beyond the maximum cutoff value. In confused areas most sources were dropped because they combined with a neighbor within the search radius, but far enough away that the combined structure exceeded the size limit. Such occurrences were recognized by the rejected source having an axial ratio much larger than unity.
It should therefore, be stressed that the absence of a small extended source where one was expected, in crowded or uncrowded regions, may be due to the presence of a neighbor; the two sources may have combined into too large a source.
Table V.E.1 traces the number of small extended sources that were processed through clustering analysis and weeks-confirmation with the final choice for the thresholds in Regions A and B. The fraction of sources that survived cluster analysis and went on to weeks-confirmation was much higher in Region A (high latitude) than in Region B (low latitude). At 12 and 25 µm, about 90% of the sources surviving cluster analysis did not pass weeks- confirmation and were therefore discarded this percentage decreased at longer wavelengths but remained substantial. The main reason for this high failure rate was the lack of a rigorous requirement for hours-confirmation, such as was required for point sources. Detector noise or other transients could trigger detections which seconds-confirmed, and then were used to construct a source that was discarded only at weeks-confirmation.
The excess of 25 µm detections in Region A was a direct result of the lack of hours-confirmation: the dead detectors in this band relaxed the seconds-confirmation filter, and therefore allowed many more. stray detections than in other bands. The problem was hardly noticeable in Region B because most detections there were triggered by real but complex structure on the sky. That difference just reflects the contrasting definitions of Regions A and B: A has a low surface density of sources at the survey sensitivity, and the noise was dominated by the detector noise; B was dominated by confusion noise, in the sense that it was densely packed with detectable sources. The result was that most detections were discarded by cluster processing in Region B, and by weeks-confirmation in Region A. When all bands were combined it turned out that in both regions about one out of every seven detections ended up contributing to a source in the catalog; this fraction was remarkably similar for Regions A and B.
Region B was surveyed three times by the satellite, but only about a quarter of all sources were detected on all three passes; this was mostly due to "shadowing" by the Galactic plane, and hysteresis in the detectors (see Section VIII.D). The coverage of Region A was more complex including areas with