planar 2-D object with random, microscopic height variations—having depths on the order of several wavelengths, and transverse variations on the order a wavelength—and an average intensity-reflection coefficient ${\mathcal{T}}({\bm\rho})$. Thus, following laser-radar theory [@Shapiro1981], we shall assume that this target has a random field-reflection coefficient, $T({\bm\rho})$, that is a zero-mean, complex-valued Gaussian random process characterized by the autocorrelation function $$\begin{aligned}
\label{eq:Scatter}
{\langle T^*({\bm\rho}_1)T({\bm\rho}_2) \rangle} = \lambda_0^2 {\mathcal{T}}({\bm\rho}_1) \delta({\bm\rho}_1-{\bm\rho}_2),\end{aligned}$$ where the deterministic pattern ${\mathcal{T}}({\bm\rho})$ is what we are tying to image.

Target reflection is a passive process, so we require $|T({\bm\rho})| \le 1$, which is in conflict with the Gaussian statistics and the delta-function term in Eq. (\[eq:Scatter\]). That delta function, however, leads to quasi-Lambertian reflection, implying that at standoff distances a realistic bucket detector will only capture a very small fraction of the reflected light. Hence using the reflection statistics from the preceding paragraph is a reasonable approximation for ghost imaging of rough-surfaced targets. However, because some light may be absorbed by or transmitted through the target, the quantum model for target reflection needs to be $$\begin{aligned}
\label{eq:TargetLoss}
{\hat{E}}_T({\bm\rho},t) = T({\bm\rho}){\hat{E}}'_S({\bm\rho},t) +\sqrt{1-|T({\bm\rho})|^2}\,{\hat{E}}_{vac,S}({\bm\rho},t)\end{aligned}$$ Here ${\hat{E}}_{vac,S}({\bm\rho},t)$ is a vacuum-state field operator, needed to preserve the free-field commutator relations given, for the source fields, 