Here is a derivation of the variance of a WIRE pixel in a single subframe: (Fig. 1) .
Here is a derivation of the variance of a WIRE pixel averaged over many subframes: (Fig. 2) . (This is NOT how the current pipeline works, but is presented as an example.)
For the actual WIRE pipeline, first I define variables and show that our exposure-based weighting scheme is approximately optimal: (Fig. 3) .
Here I derive the variance for a WIRE pixel averaged over many subframes and reduced to DN per second. This is how the pipeline currently work, but WITHOUT any upsampling (NREP=1): (Fig. 4) .
Here is the variance corrected for upsampling: (Fig. 5) . This is the correct variance if the source extractor program calculates stars by summing over many of the upsampled pixels.
If we set SUM=1 and AVE=1 in wdaophot, then for aperture photometry in the upsampled coadd image, the effective inverse gain and readout noise are:
(coadd readout noise) = (original WIRE readout noise in DN) * sqrt(total number of subframes) / (upsampling factor "NREP") / (total exposure time of all frames)
Note that the variance for an upsampled pixel presented above is NOT equal to the square of the RMS (or standard deviation) per pixel of the background in the coadd image. Because the pixels are correlated, the RMS will be smaller by a factor of the upsampling factor (u=NREP) than the the appropriate computed variance. Deriving the apparent RMS: (Fig. 6) .
Also note that the measured RMS will be slightly lower than the derived apparent RMS due to smoothing caused by pixel rebinning and interpolation. On the other hand, the RMS may be slightly higher if the background is not flat, and will also be increased by confusion noise of faint, undetected sources.
Here I show that the ratio of the Poisson and readout variance is the same for a single raw frame and the coadd image: (Fig. 7) .
Old email which was modified from an email from Joe C.: joe.email .