Signal-to-noise considerations for sky-subtracted CCD data

The standard equation for calculating the uncertainty of photometry obtained from CCDs does not correctly consider the random errors, or 'noise', introduced into observations by procedures used in reducing the data. This paper presents a thorough derivation of the theoretical error equation that considers the contributions from all internal noise sources in the signal-to-noise ratio (S/N) of a sky-subtracted image. A simplified version used for estimating the internal errors from empirical data is also derived. The propagation of noise through the data-reduction process is illustrated through a series of equations for the change in S/N that results from a variety of different operations performed on a CCD frame. Comparing these effects with the results expected for an observation made with an ideal detector suggests a number of ways to improve the precision of photometry through the practices employed in obtaining and reducing the observations.