assumption is satisfied for $C_n^2 \le 10^{-14}\,{\rm m}^{-2/3}$. Moreover, this situation has no spatial-averaging of the turbulence at the target, and thus represents a worst-case scenario for the SNR.

For ${\bm\rho}'_\ell$ and ${\bm\rho}''_\ell$ being coordinates at the CCD or bucket detectors, and ${\bm\rho}_\ell$ being coordinates at the target, the preceding coherence-length assumptions imply that $$\begin{aligned}
{\langle e^{\psi^*_m({\bm\rho}'_1, {\bm\rho}_1)}&e^{\psi_m({\bm\rho}''_1, {\bm\rho}_1)}e^{\psi^*_m({\bm\rho}'_2, {\bm\rho}_2)}e^{\psi_m({\bm\rho}''_2, {\bm\rho}_2)} \rangle}\nonumber \\
 &\approx {\langle e^{\psi^*_m(\bm 0, {\bm\rho}_1)}e^{\psi_m(\bm 0, {\bm\rho}_1)}e^{\psi^*_m(\bm 0, {\bm\rho}_2)}e^{\psi_m(\bm 0, {\bm\rho}_2)} \rangle}\nonumber \\
  &= {\langle e^{2{\rm Re}[\psi_m(\bm 0, {\bm\rho}_1)]+2{\rm Re}[\psi_m(\bm 0, {\bm\rho}_2)]} \rangle}.\end{aligned}$$ Decomposing the turbulence function into its log-amplitude and phase components as $\psi_m({\bm\rho}_1,{\bm\rho}_2)=\chi_m({\bm\rho}_1,{\bm\rho}_2)+i\phi_m({\bm\rho}_1,{\bm\rho}_2)$, and taking the logamplitude to be Gaussian distributed with mean equal to minus its variance [@Strohbehn1978], then gives $$\begin{aligned}
{\langle e^{\psi^*_m({\bm\rho}'_1, {\bm\rho}_1)}e^{\psi_m({\bm\rho}''_1, {\bm\rho}_1)}e^{\psi^*_m({\bm\rho}'_2, {\bm\rho}_2)}e^{\psi_m({\bm\rho}''_2, {\bm\rho}_2)} \rangle} \nonumber \\[.05in]
\approx e^{4 K_{\chi_m} ({\bm\rho}_1-{\bm\rho}_2)},\end{aligned}$$ where $$K_{\chi_m} ({\bm\rho}_1-{\bm\rho}_2) = {\langle \Delta\chi_m (\bm 0, {\bm\rho}_1)\Delta\chi_m (\bm 0, {\bm\rho}_2) \rangle},$$ for $\Delta\chi_m \equiv \chi_m - {\langle \chi_m \rangle}$, is the log-amplitude covariance function. The logamplitude variance will be taken to be [@Osche2002] $$\begin{aligned}
\sigma_m^2=K_{\chi_m} (\bm 0)=0.124 \,C_{n,m}^2 k_0^{7/6}L^{11/6},\end{aligned}$$ where we have used the Rytov-approximation expression, whose validity is ensured by our assumption of weak-to-medium strength turbulence.

Even with these simplifications, the SNR evaluation is quite tedious, if 