## Signal and Noise Effects from Flat Fielding

Flat frames are maps of the sensitivity of the detector. The flat frame records dust donuts, optical vignetting, and nonuniformities in the detector. After combining raw images and dark frames during calibration, the image is divided by the flat frame. In this process, noise in the flat frame will be transferred to the final calibrated image.

To avoid adding significant noise from the flat-field frame, all you need to do is make flats with a very high signal-to-noise ratio. Doing that turns out to be quite easy.

To make a flat frame, your telescope points at a bright, uniformly illuminated surface. Some astronomers prefer the twilight sky; others prefer a screen mounted near the telescope or an internally illuminated light box. In all cases, the source is bright enough to produce a solidly exposed image in 2 to 15 seconds.

In a good flat frame, the signal should be about half the full-well capacity of the detector, or about 32,000 electrons for a typical astronomical CCD in a 10-sec-ond exposure. From Equations 2.12 and 2.13, we find the output signal to be:

Sraw= + + 100= 12904 [ADUS]" <EqU' 2'29>

The noise in the raw flat-frame signal is:

araw= ^V(V32000)2 + (VT0)2 + 82= 71.6 [ADUs]. (Equ.2.30)

At the same time you make the flat-frame exposures, you make dark frames to accompany them.

The output signal is:

Sdark= Y5 + f~5 + 100= 104 [ADUS]' (EqU" 2"31)

The noise in the output signal is:

Odark= ^V(VO)2 + (a/TO)2 + 82= 3.44 [ADUs]. (Equ. 2.32)

To be sure of attaining a high signal-to-noise ratio, a conscientious observer might make 10 flat frames and 10 flat dark frames. Since your are making an averaged image, you can apply Equations 2.24 and 2.25:

Snat= Sraw-Sdark = 12904- 104 = 12800 [ADUs] (Equ. 2.33)

The resulting signal-to-noise ratio is over 4000. Since it takes just a few minutes to make the flat frames, there is no reason to settle for less.

Dividing a dark-subtracted image by a flat frame will add noise to the image. The following equation gives the resulting signal-to-noise ratio when an image is divided by a flat frame: