Effective gain and read noise in the MegaPipe stacks
Summary: This page gives a derivation of how approximate values for the effective gain and read noise may be calculated for the MegaPipe stacks. Values for the NGVS data are given.
The noise properties of a single image can be described by two parameters: the gain and the read noise. Gain is the number of electrons for a given data number of particular image pixel. The read noise is the noise caused by reading out the CCD.
Neither concept is particularly well defined for the stacked MegaPipe output images. In the stacking process, each input image has a sky level subtracted and is then scaled. The scaled, sky-subtracted images are then combined. These operations change the noise properties enough that they can no longer be described by two simple parameters. This is why MegaPipe produces weight images. However some astronomical software packages require a value for gain and read noise in order to function properly. Therefore, this page attempts to produce a value for effective gain and effective read noise for the MegaPipe stacks.
In calculating the effective gain and read noise, the following assumptions are made:
- The gain is the same for all the chips in the MegaCam mosaic. In fact the gain varies from 1.2 to 1.6 e/ADU. A value 1.5 is adopted.
- The sky background levels are the same for all image. If the images were taken on the same night, this is often true to within 10%. The average sky level is adopted.
- The combining method is an average. The images are actually combined using a median or the artificial skepticism method. These methods increase the output noise slightly.
- The number of input images contributing to a given output pixel is constant True over much of the mosaic. At the "gutters" between images and near bad columns it varies strongly. This is ignored.
- The scaling factors are the same for all images. This is typically true to within 5%, if the images were taken under photometric conditions. The average scaling factor is adopted.
- We know the sky background levels and photometric scaling factors perfectly. Not quite true, but very close. The sky levels are known to about 0.1% and the relative error on the photometric zero-points is 0.006 mags.
The noise in electrons, σe, of a single CCD
image is given by
where Re is the read noise in electrons
and Fe is the flux in electrons of a given
pixel. Working in DN we have:
where σDN is the noise in DN,
FDN is the flux in DN
and g is the gain in DN/electron.
The goal now is to find an effective read noise
and an effective gain for the stacked images.
These are values that would fit into the places
occupied by Re and g in equation 3 and
give a reaonable approximation of the noise properties
of the MegaPipe stacks.
For the input images, we can neglect read noise (4
electrons vs. 1000 electrons for a typical image) so equation 3
becomes:
This approximation will be used later.
Let us examine the stacking process:
The each input input has a sky level
subtracted and is then scaled to
reflect the photometric calibration.
The scaled, sky-subtracted images are
then averaged together.
If
the flux of the output image is FO,
the flux in the input image is FI
the flux of the sky is FS
and the scaling factor is f
and there are N images
then:
If the scaling factors, the sky levels and the input
fluxes all roughly the same then we can simplify the equation:
Or, rearanging:
Now, let's examine the noise in the output image, σO
Assuming the scaling factor and the sky level have
no associated errors we have:
inserting the simplified version of the noise for the
input images from (4) above we get:
If we make the assumption that the scaling factors and
input fluxes are similar for all the images, then:
We now have the noise in the output image in terms
of the flux in the input images. Using (8) above,
we can change this get noise as a function
of the flux in the output images:
or rearranging things slightly:
The first term is the fixed noise, which due
the sky noise in the input images. The second
term is the Poisson noise due to the sources.
We compare equation 14 to equation 3, we
can identify the element corresponding to
the effective gain, namely:
The new "read noise", which is actually due the
sky noise is given by:
again by analogy with equation 3.
Inserting the value for effective gain and simplifying
we get:
Note that the effective readnoise identified here
has nothing to do with the original read noise
of the image. It is simply the value,
which plugged blindy into equation 3, will
give the more or less the correct noise values.
The following table gives the effective gain and effective read noise for the image stacks produced by MegaPipe based on the Elixir LSB images. It also lists the average sky level and the average scaling factor. The original indivdual images had the sky level subtracted and were then multiplied by the scaling factor before being stacked. If you want to convert the stacks back to a simliar scale and background level as the original images, divide them by the scaling factor and add the sky level. Remember that this will change the zero-point by 2.5 log10 (scaling factor).