CCD noise budget ($\mathrm{Var}=S+B+dt+M{r}^2$)

For source electrons $S$, sky electrons $B$, dark-current rate $d$, total exposure $t$, $M$ reads and rms read noise $r$, independent variances add to $S+B+dt+M{r}^2$. calibration can remove reproducible means but cannot remove the science exposure's random photon or dark-current fluctuations. A finite background estimate and imperfect calibration add further uncertainty.

For total source electrons $S$, $n$ aperture pixels, sky-plus-dark rate $b+d$, time $t$, $M$ reads with rms $r$, and an independent mean-background estimate of variance ${\sigma }_{\hat{B}}^2$ per pixel, the aperture variance is $S+n(b+d)t+nM{r}^2+n^2{\sigma }_{\hat{B}}^2$ before additional calibration terms. The last factor is $n^2$ because one uncertain background mean is subtracted from all aperture pixels. Pixel correlations require covariance terms.