Source confusion, the contribution to noise in an image due to the superimposed signals from faint unresolved sources clustering on the scale of the observing beam (Condon, 1974; Scheuer, 1974), is a significant problem for observations in the submm waveband (Blain et al., 1998; Eales et al., 2000; Hogg, 2001). This is due to the relatively coarse ( 10 arcsec) spatial resolution currently available. In fact, a significant fraction of the noise in the deepest 850-µm SCUBA image of the HDF-N (Hughes et al., 1998) can be attributed to confusion (Peacock et al., 2000). At present, the practical confusion limit for galaxy detection in SCUBA observations at the atmospherically favored 850-µm wavelength is about 2 mJy. This limit makes it difficult to determine accurate sub-arcsec positions for the centroids of the submm emission from faint SCUBA-selected galaxies, rendering follow-up observations more challenging. Unfortunately, experience has shown that many known high-redshift galaxies, especially optically-selected LBGs, are typically fainter than the confusion limit, and so are difficult to study using SCUBA.
The variety of count data for dusty galaxies shown in Figs. 9 and 10 can be used to estimate the effect of source confusion in observations made at a wide range of frequencies and angular scales.
The distribution of flux density values from pixel to pixel in an image due to confusion noise depends on the underlying counts of detected galaxies. Confusion noise is always an important factor when the surface density of sources exceeds about 0.03 beam-1 (Condon, 1974). The results of a confusion simulation for the 14-arcsec SCUBA 850-µm beam are shown in Fig. 12: see also Hogg (2001), Eales et al. (2000) and Scott et al. (2002). Note that Eales et al. assume a very steep count and obtain larger values of confusion noise than those shown here. Observations made with finer beams at the same frequency suffer reduced confusion noise, while for those made in coarser beams the effects are more severe: compare the results for the much larger 5-arcmin beam in the three highest frequency submm-wave observing bands of the planned Planck Surveyor space mission all-sky survey shown in Blain (2001a).
Figure 12. Histograms showing the simulated effects of confusion noise in deep SCUBA integrations at 850 µm. Left: the expected distribution of pixel flux densities when the telescope samples the sky in a standard (-0.5,1,-0.5) chopping scheme, with no additional noise terms present. The flux distribution is non-Gaussian, with enhanced high- and low-flux tails as compared with the overplotted Gaussian, which has the width predicted by simple calculations (Fig. 13). Right: the same confusion noise distribution is shown in the right-hand panel, but convolved with Gaussian instrument and sky noise with an RMS value of 1.7 mJy, which is typical of the noise level in the SCUBA Lens Survey (Smail et al., 2002). At this noise level, the additional effect of confusion noise is small.
The simulated confusion noise distribution is non-Gaussian (Fig. 12), but can be represented quite accurately by a log-normal distribution, leading to many more high-flux-density peaks in an image than expected assuming a Gaussian distribution of the same width. The width of the central peak of the distribution in flux density is approximately the same as the flux density at which the count of sources exceeds 1 beam-1. This provides a useful indication of the angular scales and frequencies for which confusion noise is likely to be significant, and of the limit imposed to the effective depth of surveys by confusion for specific instruments: see Fig. 13.
Figure 13. An approximate measure of the 1- confusion noise expected as a function of both observing frequency and angular scale from the mm to mid-IR waveband (updated from Blain et al., 1998). The contributions from extragalactic and Galactic sources are shown in the left and right panels respectively. Radio-loud AGN may make a significant contribution to the top left of the jagged solid line (Toffolatti et al., 1998). A Galactic cirrus surface brightness of B0 = 1 MJy sr-1 at 100 µm is assumed. The ISM confusion noise is expected to scale as B01.5 (Helou and Beichman, 1990; Kiss et al., 2001). The bands and beamsizes of existing and future experiments (see Tables 1 and 2) are shown by: circles - Planck Surveyor; squares - BOOMERANG; empty stars - the SuZIE mm-wave Sunyaev-Zeldovich instrument; triangles - BOLOCAM, as fitted to CSO (upper 3 points) and the 50-m Large Millimeter Telescope (LMT; lower 3 points); filled stars - SCUBA (and SCUBA-II); diamonds - Herschel; asterixes - Stratospheric Observatory for Infrared Astronomy (SOFIA); crosses - SIRTF. The resolution limits of the interferometric experiments ALMA and SPECS lie far below the bottom of the panels. The confusion performance of the 2.5-m aperture BLAST balloon-borne instrument is similar to that of SOFIA. Confusion from extragalactic sources is expected to dominate over that from the Milky Way ISM for almost all of these instruments.
3.1.1. Confusion and follow-up observations of submm galaxies
The real problem of confusion for identifying and conducting multiwaveband studies of submm-selected galaxies is illustrated by the results of the first generation of surveys. The very deepest optical image that matches a submm-wave survey is the HDF-N, in which there are several tens of faint optical galaxies (at R > 26) that could be the counterpart to each SCUBA detection (Hughes et al., 1998; Downes et al., 1999a). It is thus impossible to be certain that a correct identification has been made from the submm detection image and optical data alone: compare the identifications in Smail et al. (1998a, 2002). In some cases, EROs and faint non-AGN radio galaxies (Smail et al., 1999, 2000; Gear et al., 2000; Lutz et al., 2001) can be associated with submm galaxies, especially after higher-resolution mm-wave interferometry observations have provided more accurate astrometry for the submm detection (Downes et al., 1999; Frayer et al., 2000; Gear et al., 2000; Lutz et al., 2001), to reduce the effects of submm confusion, with the investment of significant amounts of observing time. The surface density of both EROs and faint non-AGN radio galaxies is less than that of the faintest optical galaxies, and so the probability of a chance coincidence between one and a submm galaxy is reduced. A very red color and detectable radio emission from a high-redshift galaxy are both likely to indicate significant star-formation/AGN activity and/or dust extinction, making such galaxies better candidate counterparts even in the presence of confusion-induced positional uncertainties.