  The measurement of shape parameters of sources in astronomical images is
usually performed by assuming that the underlying noise is uncorrelated.
Spatial noise correlation is however present in practice due to various
observational effects and can affect source shape parameters. This effect is
particularly important for measurements of weak gravitational lensing, for
which the sought image distortions are typically of the order of only 1%. We
compute the effect of correlated noise on two-dimensional gaussian fits in full
generality. The noise properties are naturally quantified by the noise
autocorrelation function (ACF), which is easily measured in practice. We
compute the resulting bias on the mean, variance and covariance of the source
parameters, and the induced correlation between the shapes of neighboring
sources. We show that these biases are of second order in the inverse
signal-to-noise ratio of the source, and could thus be overlooked if bright
stars are used to monitor systematic distortions. Radio interferometric surveys
are particularly prone to this effect because of the long-range pixel
correlations produced by the Fourier inversion involved in their image
construction. As a concrete application, we consider the search for weak
lensing by large-scale structure with the FIRST radio survey. We measure the
noise ACF for a FIRST coadded field, and compute the resulting ellipticity
correlation function induced by the noise. In comparison with the weak-lensing
signal expected in CDM models, the noise correlation effect is important on
small angular scales, but is negligible for source separations greater than
about 1 arcmin. We also discuss how noise correlation can affect weak-lensing
studies with optical surveys.
