Notes on noise calculation
The computation proposed here follows from R. Biswas et al. 2020. In the following the quantities
in bold refers to SNANA quantities.
The source count in a given band b in unit of ADU is given by
\[F_b = \kappa 10^{-0.4 m_b},\]
with \(m_b\) the magnitude of the source for the corresponding flux in ADU.
The sky count in a given band in unit of ADU is given by
\[C_b = \alpha 10^{-0.4 m_b^\mathrm{sky}},\]
where \(m_b^\mathrm{sky}\) is the sky magnitudes corresponding to a flux in \(\mathrm{ADU} \ \mathrm{arsec}^{-2}\).
Thus, the Signal-to-Noise Ratio (SNR) is
\[\mathrm{SNR} = \frac{F_b}{\left(F_b + C_b\right)^\frac{1}{2}}.\]
The magnitude of a source with a SNR equal to five, \(m_b^5\), allows to find the \(\kappa\) second degree equation
\[\kappa^2 - 25 \times 10^{0.8 m_b^5} \kappa - 25 \times 10^{0.4\left(2m_b^5 - m_b^\mathrm{sky}\right)} = 0.\]
This equation has a unique physical solution \(\kappa > 0\),
\[\kappa = \frac{25}{2}10^{-0.4m_b^5} \left(1 + \sqrt{1 + \frac{4\alpha}{25}10^{-0.4m_b^\mathrm{sky}}}\right).\]
Using this solution one can compute that:
\[\kappa = 25 \frac{\alpha}{\kappa} 10^{0.4\left(2m_b^5 - m_b^\mathrm{sky}\right)}\left[1 + \frac{\kappa}{\alpha}10^{-0.4\left(m_b^5 - m_b^\mathrm{sky}\right)}\right].\]
We will now describes the computation of the ratio \(\frac{\kappa}{\alpha}\):
If the source at the top of the atmosphere has a flux density \(F_\nu(\lambda)\), its ADU count is given by
\[F_b = \frac{\pi D^2 T}{4gh}\int_0^\infty F_\nu(\lambda) S^\mathrm{atm}(\lambda)S_b^\mathrm{syst}(\lambda)\lambda^{-1}d\lambda,\]
with \(D\) the mirror diameter, \(T\) the exposure time, \(g\) the CCD gain in photo-electron per ADU and \(h\) the Planck constant.
Thus, the magnitude of the source is
\[m_b = -2.5 \log_{10}\left(\frac{\int_0^\infty F_\nu(\lambda) S^\mathrm{atm}(\lambda)S_b^\mathrm{syst}(\lambda)\lambda^{-1}d\lambda}{\int_0^\infty F^\mathrm{ref}_\nu(\lambda) S^\mathrm{atm}(\lambda)S_b^\mathrm{syst}(\lambda)\lambda^{-1}d\lambda}\right),\]
with \(F^\mathrm{ref}_\nu(\lambda)\) the flux density of a reference source, in the AB magnitude system \(F^\mathrm{ref}_\nu(\lambda) = 3631 \ \mathrm{Jy}\). \(S^\mathrm{atm}(\lambda)\) is the atmosphere transmission
and \(S_b^\mathrm{syst}(\lambda)\) the system transmission including optics, filter, ccd efficiency.
We can write
\[F_b = C \times 10^{-0.4 m_b}\int_0^\infty F^\mathrm{ref}_\nu(\lambda) S^\mathrm{atm}(\lambda)S_b^\mathrm{syst}(\lambda)\lambda^{-1}d\lambda,\]
with \(C = \frac{\pi D^2 T}{4gh}\).
The \(\kappa\) value is thus given by
\[\kappa = C \times \int_0^\infty F^\mathrm{ref}_\nu(\lambda) S^\mathrm{atm}(\lambda)S_b^\mathrm{syst}(\lambda)\lambda^{-1}d\lambda.\]
For the sky, the same reasoning gives
\[C_b = C \times n_\mathrm{eff}p_\mathrm{size}^2 \times 10^{-0.4 m_b^\mathrm{sky}} \int_0^\infty F^\mathrm{ref}_\nu(\lambda)S_b^\mathrm{syst}(\lambda)\lambda^{-1}d\lambda,\]
where there is no more atmosphere transmission, \(p_\mathrm{size}\) is the length size of pixels with \(p_\mathrm{size}^2\) the corresponding pixel area in unit of \(\mathrm{arcsec}^2 \ \mathrm{pixel}^{-1}\) and, \(n_\mathrm{eff}\) is the effective number of pixel that is given by
\[\begin{split}n_\mathrm{eff} &= \left(\iint \mathrm{PSF}^2 dS\right)^{-1} \times p_\mathrm{size}^{-2}\\
&= 4 \pi \sigma_\mathrm{PSF}^2 p_\mathrm{size}^{-2}\\
&= \frac{\pi}{\ln2} \mathrm{FWHM}_\mathrm{PSF}^2 p_\mathrm{size}^{-2}\\
&= A \times p_\mathrm{size}^{-2},\end{split}\]
with the PSF width in unit of \(\mathrm{arcsec}\) and \(A\) is the noise equivalent area in \(\mathrm{arcsec}^{2}\). One can note that the
PSF sigma in units of \(\mathrm{pixel}\) is given by
\[\mathbf{PSF} = \frac{\mathrm{FWHM}_\mathrm{PSF}}{2\sqrt{2\ln2}}p_\mathrm{size}^{-1}.\]
Thus we have
\[\alpha = C \times A \times \int_0^\infty F^\mathrm{ref}_\nu(\lambda)S_b^\mathrm{syst}(\lambda)\lambda^{-1}d\lambda.\]
As previously stated, in the AB magnitude sytem the reference flux is constant, \(F^\mathrm{ref}_\nu(\lambda) = 3631 \ \mathrm{Jy}\)
and the ratio betweem \(\kappa\) and \(\alpha\) is
\[\frac{\kappa}{\alpha} = A^{-1} \frac{\int_0^\infty S^\mathrm{atm}(\lambda)S_b^\mathrm{syst}(\lambda)\lambda^{-1}d\lambda}{\int_0^\infty S_b^\mathrm{syst}(\lambda)\lambda^{-1}d\lambda}.\]
Using the approximation
\[\frac{\int_0^\infty S^\mathrm{atm}(\lambda)S_b^\mathrm{syst}(\lambda)\lambda^{-1}d\lambda}{\int_0^\infty S_b^\mathrm{syst}(\lambda)\lambda^{-1}d\lambda} \simeq 1,\]
we finally obtain
\[\kappa = 25 A 10^{0.4\left(2m_b^5 - m_b^\mathrm{sky}\right)}\left[1 + \frac{10^{-0.4\left(m_b^5 - m_b^\mathrm{sky}\right)}}{A}\right].\]
We define the zero-point ZPT as
\[\mathbf{ZPT} = 2.5\log_{10}\left(\kappa\right),\]
such that the source ADU count can be write
\[F_b = 10^{-0.4(m_b - \mathbf{ZPT})}.\]
The sky noise SKYSIG per pixel is given by
\[\sigma_\mathrm{sky}^2 = 10^{-0.4(m_b^\mathrm{sky} - \mathbf{ZPT})} \times p_\mathrm{size}^2.\]