The noise sources are:

• the dark current noise, Od, in r.m.s. electrons; and

The total noise in the raw image, Graw , converted into units of ADUs, is:

At first glance, it looks like we've just done something sneaky. Whereas we simply summed the signals; when we added the noise sources, we squared each noise, summed them, and took the square root of the sum. However, recall that the noise originates as the random variation in an event count. By squaring the noise, we recover the number of events. To get the total number of events, we add them, and then take the square root to find the expected random variation in the sum of events. Another way of looking at this is that because noise is random, the variations in a collection of noise sources will partially cancel each other.

It is important to recognize that because noise sources represent counts of events, then even when we subtract a signal, we must add its noise! Why? The number of events under consideration has increased, and that's where the noise comes from, not whether we've added the events or subtracted them. In the next section, you'll see why this matters.

As a reality check, let's compute the signal and noise in a raw image using realistic values for a good-quality CCD camera. In a 60-second exposure under fairly good suburban-quality skies, you can expect roughly 1000 photons per pixel, and therefore roughly 600 electrons of signal.

The dark current in a cooled CCD camera will be around 1 electron per second per pixel; so in 60 seconds, the dark current signal will be about 60 electrons. We'll assume a gain of 2.5 electrons per ADU, a readout noise of 8 root-mean-square electrons, and a bias (typical of many astronomical CCD cameras) set at the convenient value of 100.

The output signal is:

The noise in the output signal is:

Graw= ^ V(V600)2 + (V60)2 + 82= 10.76 [ADUs]. (Equ. 2.15)

Although we could compute a signal-to-noise ratio for the raw image by dividing the signal we care about (the 240 ADUs of sky) by the total noise, it would not be a meaningful calculation. We've treated the dark current as uniform, but in fact it usually varies considerably from pixel to pixel. The dark current signal adds a highly distracting noise pattern to the image plus a scattering of hot pixels to it. (Technically speaking, we remind our readers that a noise pattern is not noise, but an unwanted signal.) To see the image properly, we will make a dark frame by making an exposure having the same duration as the raw image, but with the camera shutter closed. The dark frame will capture a record of the noise pattern and hot pixels, which we then subtract to produce a clean image.

2.4.2 Signal and Noise in a Dark Frame

Making a dark frame is exactly the same as making an exposure except that the camera shutter is kept closed so that no light reaches the detector. Most CCD cameras and digital cameras support making dark frames so that the user can remove the dark current pattern.

During a 60-second exposure to make a dark frame, the photon signal is zero photons per pixel. In 60 seconds, the dark current signal will be about 60 electrons, as it was for the raw image. The gain, readout noise, and bias are all the same as before.

The output signal is:

The dark frame is a map of the dark current from the sensor. Note that the useful portion of the signal—the 24 ADUs of dark current—is uncertain with a standard deviation of 4.45 ADUs; so the dark frame is not a perfect map of the dark current, but a sample of the dark current with an accompanying error.

2.4.3 Signal and Noise in a Dark-Subtracted Image

We are finally ready to investigate signal and noise in a dark-subtracted image. Dark subtraction removes both the bias and the dark current pattern from the raw image, but adds noise from the dark frame to the noise already present in the raw frame. If this somehow seems a bit "unfair," remember that to remove the dark current pattern, we've been forced to count more electrons, which, by the inexorable statistics of event counting, leads to greater uncertainty. The photon signal in the dark-subtracted image is:

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