the ccd equation back to teaching back to the course start of previous section next page last page of this section help on navigating these pages

To write an equation for SNR we need to know the various noise sources that contribute to an astronomical measurement with a CCD. These are:

Assuming that all of the above noise sources are independent, the total noise, N, is given by the square root of the sum of the squares of the individual errors:

N = (Sobj + Ssky + Sdark + R2)0.5.

Hence the SNR is given by the equation:

SNR = S / N = Sobj / (Sobj + Ssky + Sdark + R2)0.5.

It is important to realise that the above equation applies even after subtracting the mean sky background and dark current levels from each pixel, since the shot noise from these sources will still be present.

To use the above equation correctly, one must be careful with the units used for each of the terms. Typically, when predicting the SNR of an observation, we have:

A few things are worthy of note in the above list. First, Sobj is the total number of photons from the object, which will probably be spread out over a number of pixels, whereas Ssky is the number of sky photons per pixel. Second, Sobj and Ssky are in photon units not electrons. Third, Sobj, Ssky and Sdark will increase with exposure time, but R will not.

So, to clarify the SNR equation given above, we need to account for the exposure time of the CCD image, t, the number of pixels that the object is spread over, npix, and the conversion efficiency of photons to electrons, which is given by the quantum efficiency, QE, of the CCD expressed as a number between 0 and 1. The latter conversion from photons to electrons is essential, as otherwise one would predict a higher signal-to-noise than measured, i.e. the signal in the SNR equation must be the detected signal, not the signal emitted by the source. The resulting equation is sometimes referred to as the CCD equation:

SNR = (Sobj . t . QE) / [ (Sobj . t . QE) + (Ssky . t . QE . npix) + (Sdark . t . npix) + (R2 . npix) ]0.5,

which can be simplified to:

SNR = [Sobj . (t . QE)0.5] / [ Sobj + npix (Ssky + (Sdark / QE ) + (R2 / QE . t)) ]0.5.

Sometimes, Sobj and Ssky are given in counts. In this case, we must also convert them into electron units. This is because Poisson statistics are only applicable when counting independent, random events, which means that the noise is only equal to the square root of the signal if the signal is in units of the detected quantity, i.e. electrons. The conversion from counts to electrons can be performed by replacing QE in the above equation by the CCD gain, g, in units of e-/ADU:

SNR = (Sobj . t . g) / [ (Sobj . t . g) + (Ssky . t . g . npix) + (Sdark . t . npix) + (R2 . npix) ]0.5,

which can be simplified to:

SNR = [Sobj . (t . g)0.5] / [ Sobj + npix (Ssky + (Sdark / g ) + (R2 / g . t)) ]0.5.

Similarly, Sobj and Ssky are sometimes given in flux units, and these must be converted into electrons before the CCD equation can be used. In this case, one must first divide the flux by the energy of a single photon to give the number of photons, and then multiply by the QE to give the number of electrons.

We can use the CCD equation to define three limiting cases: the object-limited case, the background-limited case and the readout-noise limited case.

To maximise SNR, one should always try to expose for long enough to obtain object- or background-limited data, as otherwise one pays a significant penalty for reading out the CCD. However, it isn't always possible to avoid the readout-noise limited regime, particularly when exposure times must be kept short in order to sample short time-scale variability.

Some calculations illustrating how to use the CCD equation are given in the example problems.

©Vik Dhillon, 14th December 2010