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:
- Random fluctuations in the detected photons
from the source,
i.e. the shot noise on the source
signal, (S_{obj})^{0.5}.
- Random fluctuations in the detected photons from the sky background,
i.e. the shot noise on the sky
signal, (S_{sky})^{0.5}.
- Random fluctuations in the thermally-generated electrons produced in
the CCD, ie. the shot noise on the dark current, (S_{dark})^{0.5}.
- Time-independent detector noise, i.e.
readout noise,
R. Note that no square root is required here as there is no
signal associated with this noise source and hence Poissonian statistics
do not apply.
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 = (S_{obj} + S_{sky} +
S_{dark} + R^{2})^{0.5}.
Hence the SNR is given by the equation:
SNR = S / N = S_{obj} / (S_{obj} + S_{sky} +
S_{dark} + R^{2})^{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:
- S_{obj} in units of photons per second.
- S_{sky} in units of photons per second per pixel.
- S_{dark} in units of electrons per second per
pixel.
- R in units of electrons per pixel.
A few things are worthy of note in the above list. First,
S_{obj} is the total number of photons from
the object, which will probably be spread out over a number of pixels,
whereas S_{sky} is the number of sky photons per
pixel. Second, S_{obj} and
S_{sky} are in photon units not electrons. Third,
S_{obj}, S_{sky} and
S_{dark} 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, n_{pix}, 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 = (S_{obj} . t . QE) /
[ (S_{obj} . t . QE) + (S_{sky} . t . QE . n_{pix})
+ (S_{dark} . t . n_{pix}) +
(R^{2} . n_{pix}) ]^{0.5},
which can be simplified to:
SNR = [S_{obj} . (t . QE)^{0.5}] /
[ S_{obj} + n_{pix} (S_{sky}
+ (S_{dark} / QE ) + (R^{2} / QE . t)) ]^{0.5}.
Sometimes, S_{obj} and S_{sky} 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 = (S_{obj} . t . g) /
[ (S_{obj} . t . g) + (S_{sky} . t . g . n_{pix})
+ (S_{dark} . t . n_{pix}) +
(R^{2} . n_{pix}) ]^{0.5},
which can be simplified to:
SNR = [S_{obj} . (t . g)^{0.5}] /
[ S_{obj} + n_{pix} (S_{sky}
+ (S_{dark} / g ) + (R^{2} / g . t)) ]^{0.5}.
Similarly, S_{obj} and S_{sky} 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.
- Object limited: In this case, the object signal per pixel is
much larger than the sky signal, dark current or readout noise per pixel.
Hence,
SNR = S_{obj} / (S_{obj} + S_{sky} +
S_{dark} + R^{2})^{0.5}
~ (S_{obj})^{0.5}.
Hence the SNR increases as the square root of the object signal. Since
the object signal is proportional to the exposure time, this means
that the SNR is proportional to the square root of the exposure
time. If the exposure time is doubled, the SNR will increase by
2^{0.5} ~ 1.4. The object signal is also proportional to the
area of the telescope aperture, which is proportional to the square of
the diameter. Hence the SNR is proportional to the diameter of the
telescope aperture - if the diameter is doubled, the SNR will double.
- Background limited: In this case, the sky signal per pixel is
much larger than the object signal, dark current or readout noise per pixel.
Hence,
SNR = S_{obj} / (S_{obj} + S_{sky} +
S_{dark} + R^{2})^{0.5}
~ S_{obj} / (S_{sky})^{0.5}.
This is similar to the object-limited case. Both the object and sky
signal increase linearly with exposure time, hence the SNR is
proportional to the square root of the exposure time. Both the object
and sky signal also increase linearly with telescope area, hence the
SNR is proportional to the diameter of the telescope aperture.
For a given sky signal, the SNR will increase linearly with the object
signal. For a given object signal, however, the SNR decreases as the
square root of the increasing background level. This is why it is so
important to minimize light pollution and observe faint objects when the
Moon is new.
- Readout-noise limited: In this case, the readout noise per pixel
is much larger than the object signal, sky signal or dark current per pixel.
Hence,
SNR = S_{obj} / (S_{obj} + S_{sky} +
S_{dark} + R^{2})^{0.5}
~ S_{obj} / R.
Since the readout noise is independent of integration time or telescope
aperture, the SNR will now increase linearly with exposure time and as
the square of the telescope aperture diameter.
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