somewhat straightforward. The higher-order field moments and the fourth-order target moment can be reduced to a sum of the products of second-order moments by application of the Gaussian moment-factoring theorem. The [ac]{}-coupling removes several of these terms, and we are left with integrals and Fourier transforms of the product of Gaussian terms, the turbulence correlation functions, and the target’s intensity-reflection coefficient, ${\mathcal{T}}({\bm\rho})$. To simplify our final SNR expressions, we define two new terms, $$\begin{aligned}
\label{eq:ATp}A'_T & = \int\! d{\bm\rho}\, {\mathcal{T}}^2({\bm\rho}), \\
\label{eq:Gamma}\Gamma &= \frac{1}{(4\pi\beta)^2}\int \! d\bm\nu\, \ e^{-|\bm\nu|^2/2}O(\bm\nu,4\sqrt{\beta}),\end{aligned}$$ where $O(\bm\zeta, D)$ is the two-circle overlap function for circles of diameter $D$, $$\begin{aligned}
\lefteqn{\hspace*{-.1in}O(\bm\zeta,D) = } \nonumber \\[.05in]
&&\hspace*{-.2in}\left\{\begin{array}{ll}
\frac{D^2}{2}\left[ \cos^{-1} \left( \frac{|\bm\zeta|}{D}\right) -\frac{|\bm\zeta|}{D}\sqrt{1-\frac{|\bm\zeta|^2}{D^2}} \right], & \mbox{for $|\bm\zeta|\le D$} \\[.05in]
0, & \text{elsewhere,}
\end{array}\right.\end{aligned}$$ $\beta = A_b/\pi a_0^2$, and we have utilized the dimensionless difference coordinates $\bm{\nu} = \rho_Lk_0(\bm{\rho}'-\bm{\rho}'')/L$, where ${\bm\rho}'$ and ${\bm\rho}''$ are coordinates at the bucket detector.

Both $A'_T$ and $\Gamma$ have significant physical interpretations. Similar to $A_T$, $A'_T$ is another measure of the reflective area of the target, and thus directly related to the number of on-target resolution cells, and the subsequent amount of time it takes to form an image. $\Gamma$ is a metric for the spatial 