Signal-to-Noise Ratio Calculation

Under the assumption that an observation (or, equivalently, an exposure) of an object is made up of a certain number of frames (in order e.g. to reduce the total number of cosmic rays affecting the individual readouts and not to fill the potential wells of the detector' pixels, the conventions used in the following are as follows: Under these assumptions, $F$ can be written as

$\displaystyle F=S+n_p\,b~.$ (8)

The physical process of the emission of photons from an astronomical object can be statistically described in terms of a Poisson distribution. The standard error in the measurement of $F$ is then due to the intrinsic Poisson noise associated with $F$ and to the readnoise. These two contributions sum quadratically yielding for the variance of $F$

$\displaystyle \sigma_F^2=F+n_p\,r^2=S+n_p\,b+n_p\,r^2~.$ (9)

Since the signal $S$ from the object is estimated by subtraction from $F$ of the sky background

$\displaystyle S=F-n_p\,b~,$ (10)

the variance of $S$ is

$\displaystyle \sigma_S^2=\sigma_F^2+(n_p\,\sigma_b)^2=S+n_p\,b+n_p\,r^2+(n_p\,\sigma_b)^2~.$ (11)

$SNR$ is then

$\displaystyle SNR=\sqrt{n_{obs}}~\frac{S}{\sigma_S}=\frac{\sqrt{n_{obs}}\,S}
{\sqrt{S+n_p\,b+n_p\,r^2+(n_p\,\sigma_b)^2}}~,$ (12)

while $\sigma_{mag}$ is

$\displaystyle \sigma_{mag}=\frac{2.5\,\log\,e}{\sqrt{n_{obs}}~SNR}=
\frac{2.5\,\log\,e~\sqrt{S+n_p\,b+n_p\,r^2+(n_p\,\sigma_b)^2}}{\sqrt{n_{obs}}~S}~.$ (13)

The overall $SNR$ of a number $n_{fr}$ of repeated observations of a given field is finally given by

$\displaystyle SNR(T)=\sqrt{n_{fr}}\:SNR(t)~.$