6.7.2 Extended Sources

In general, the signal-to-noise level for extended sources can be computed by comparing the expected signal, S, in each pixel, determined from Equations 6.8 through 6.13, to the noise N=(S+B)^1/2, where B is the equivalent background, determined in a manner similar to that for point sources. Unlike for point sources, the calculation does not, in a first approximation, involve the sharpness of the point spread function. For example, let us consider the observation of a source with a V surface brightness of 24 mag arcsec^-2, assuming the F569W filter, WFC camera, and sky background V=23.3 mag arcsec^-2. The signal-to-noise estimate goes as follows. The signal in each WFC pixel is 24.0+5.0 = 29.0 magnitude. By Equation 6.11, the total signal collected from the source in a 3000 second integration is S = 44.1 electrons, neglecting the small AB color correction. The sky signal per pixel is 84.1 electrons. The dark current is ~12 electrons. The total equivalent background is thus B = 84.1+5.32+12 = 124.2 electrons, larger than the signal detected, thus the noise is background-dominated. The noise is N=(S+B)^1/2= 13.0 electrons, and the signal-to-noise per pixel expected in this case is 3.4. Similar calculations can be carried out for other filters; for observations in narrow-band filters and in the UV, the sky background signal will usually be unimportant. For very long observations of faint objects, other noise terms, such as flat field uncertainty, and errors in dark (and possibly bias) subtraction, must be considered more carefully.

If the scale of features in the target is larger than one pixel, the signal-to-noise can sometimes be improved by smoothing the observed image or - if read noise is a significant contributor - by reading the image out in AREA mode (see CCD Orientation and Readout).