averaging of the target-induced speckle over the bucket detector. For very small detectors $\Gamma \approx 1$, while for large detectors it is inversely proportional to the area of the bucket detector, viz. $\Gamma \approx 1/2\beta$. To further simplify our SNR expressions, we make two final assumptions: first that $A'_T/\rho_L^2\gg 30$, which is equivalent to saying that the ghost image consists of at least $10 \times 10$ resolution cells; and second, that the bucket detector is at least as large as the area of the source beam, or $\beta \ge 1$. Applying these conditions we arrive at the following SNR expressions for pseudothermal and SPDC reflective ghost imaging through atmospheric turbulence,

$$\begin{gathered}
\label{eq:CSNR}
\text{SNR}_C = \frac{\displaystyle {\mathcal{T}}^2({\bm\rho}_p)}{\displaystyle\frac{T_0}{T_I}\Delta^2\text{Source} + \Delta^2\text{Path} + \frac{\Omega_BT_0^2}{T_I}\Delta^2\text{Detect}+\frac{T_0}{T_I}\Delta^2\text{Mix}} \\
\label{eq:QSNR}
\text{SNR}_Q = \frac{\displaystyle {\mathcal{T}}^2({\bm\rho}_p) \left[1+\frac{1}{4\sqrt{\pi}\mathcal{I}}\right]^2}{\displaystyle \frac{4}{\Omega_BT_I}\Delta^2\text{Source} + \Delta^2\text{Path}\left[1+\frac{1}{4\sqrt{\pi}\mathcal{I}}\right]^2 + \frac{T_0 4\sqrt{2}}{T_I}\Delta^2\text{Detect}\left[1+\frac{1}{4\sqrt{\pi}\mathcal{I}}\right]+ \frac{T_0}{T_I}\Delta^2\text{Mix}\left[\frac{2}{\sqrt{3}}+\frac{1}{4\sqrt{\pi}\mathcal{I}}\right]}.\end{gathered}$$

where, for ease of comparison with the pseudothermal imager, we have inverted the image coordinates in the SPDC case.

The terms that appear in the noise denominators of Eqs. (\[eq:CSNR\]) and (\[eq:QSNR\]) are as follows: $$\begin{aligned}
&\Delta^2\text{Source} = \frac{A'_T (1+\beta^{-1})e^{4(\sigma_R^2+\sigma_S^2+\sigma_T^2)}}{\sqrt{2\pi}\rho_L^2} , \\
&\Delta^2\text{Path} = {\mathcal{T}}^2({\bm\rho}_p)\left[e^{4(\sigma_R^2+\sigma_S^2+\sigma_T^2)}\left(\Gamma+1\right)-1\right], \\
&\Delta^2\text{Detect} =\frac{{\mathcal{T}}({\bm\rho}_p) \rho_L^2 \sqrt{\pi}}{16 \sqrt{2} A_p \eta^2 \mathcal{I}^2}\frac{L^2}{A_b}, \\
&\Delta^2\text{Mix} = \frac{{\mathcal{T}}({\bm\rho}_p)}{\eta\mathcal{I}}\frac{ L^2}{A_b}e^{4\sigma_R^2}\nonumber\\
&\quad\quad+\frac{\pi\rho_L^2{\mathcal{T}}^2({\bm\rho}_p)}{A_p\eta\mathcal{I}}\left[\frac{4}{3}+\beta^{-1}\right]e^{4(\sigma_S^2+\sigma_T^2)},\end{aligned}$$ where $\sigma^2_\ell$ for $\ell = R, 