  In very crowded fields, the modulation of the background by the sea of
unresolved faint sources induces centroid shifts. The errors increase with the
number of sources per beam. Even the most optimistic simulations of imaging
data show that position errors can become severe (on the order of the beam
size) at flux levels at which images contain 1/50 to 1/15 sources per beam,
depending on the slope of the number-flux relation $d\log N/d\log S$. These
problems are expected to be significant for recent observations of faint
submillimeter sources and may be the reason that some sources appear to lack
optical counterparts.
