## 2 Theoretical Analysis
### Transmission from the Target to the Receiver Plane  
As mentioned in the previous section, we assume that the surface of the target is isotropic and semi-rough. We describe the surface using a helpful model proposed by [PERSON] several years ago [PERSON] (1958). The model can be thought of so that, under some conditions, surface scattering is regarded as passing through a thin random phase screen. Therefore, we use Equations (12) and (13) to describe the action of the target scattering on laser beams. \[\langle T(\mathbf{t}_{1})T^{*}(\mathbf{t}_{2})\rangle=\frac{4\pi\beta^{2}}{k^ {2}}\exp\Biggl{[}-\frac{ik}{2}\overline{\mathbf{t}}^{T}\overline{\mathbf{T} \mathbf{t}}\Biggr{]}, \tag{12}\]  
\[\overline{\mathbf{T}}=\Bigg{(} \begin{array}{cc}\left(-\frac{2i}{kw_{R}^{2}} -\frac{2i}{lk_{c}^{2}}\right)\mathbf{I}&\frac{2i}{lk_{c}^{2}}\mathbf{I}\\ \frac{2i}{lk_{c}^{2}}\mathbf{I}&\left(-\frac{2i}{kw_{R}^{2}}-\frac{2i}{lk_{c}^ {2}}\right)\mathbf{I}\end{array} \Bigg{)}, \tag{13}\]  
where \"\(<>\)\" denotes an ensemble average, \"\(*\)\" denotes the complex conjugate, and \(\overline{\mathbf{t}}^{T}\mathbf{=}(\mathbf{t}_{1},\mathbf{t}_{2})^{T}\). \(w_{R}\) is the target size, \(l_{c}\) is the typical transverse correlation width, and \(\beta\) is the normalization parameter. Similar to Equation (7), after the combined action of atmospheric turbulence and target scattering on a laser beam, the atmospheric propagation obeys the following generalized Collins formula from the target plane to the receiver plane (S.A., 1991). \[\begin{array}{ll}W_{ij}(\overline{\mathbf{\rho}})=&\frac{k^{2}}{4\pi \det\left(\overline{\mathbf{B}}^{-1}\overline{\mathbf{t}}-\overline{ \mathbf{t}}^{T}\overline{\mathbf{B}}^{-1}\overline{\mathbf{\rho}}-\overline{ \mathbf{p}}^{T}\overline{\mathbf{B}}^{-1}\overline{\mathbf{t}}+\overline{ \mathbf{\rho}}^{T}\overline{\mathbf{B}}^{-1}\overline{\mathbf{\rho}}\right)} \times\exp\Biggl{[}-\frac{ik}{2}\left(\overline{\mathbf{t}}^{T}\overline{ \mathbf{P}}+\overline{\mathbf{t}}^{T}\overline{\mathbf{P}}\overline{\mathbf{ \rho}}+\overline{\mathbf{\rho}}^{T}\overline{\mathbf{P}}\overline{\mathbf{ \rho}}\right)\Biggr{]}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\where \(\mathbf{M}_{2\bar{\mathbf{ij}}}^{-1}\) denote the partially coherent complex curvature tensor in the receiver plane, respectively. According to the tensor method, we can express \(\mathbf{M}_{2\bar{\mathbf{ij}}}^{-1}\) as follows,  
\[ \begin{array}{ll}\mathbf{W}_{ij}(\boldsymbol{\rho}_{1},\boldsymbol{\rho}_{2}) =&\frac{4\pi\beta^{2}A_{i}A_{j}B_{ij}}{k^{2}\left[\det\left(\widetilde{ \mathbf{I}}+\widetilde{\mathbf{B}}\mathbf{M}_{1 ij}^{-1}+\widetilde{\mathbf{B} \mathbf{T}}+\widetilde{\mathbf{B}\mathbf{F}}\right)\right]^{\frac{1}{2}}\left[ \det\left(\widetilde{\mathbf{A}}+\widetilde{\mathbf{B}}\mathbf{M}_{0 ij}^{-1}+ \widetilde{\mathbf{B}\mathbf{F}}\right)\right]^{\frac{1}{2}}}\\ &\times\exp\left[-\frac{\boldsymbol{\rho}_{1}^{2}+\boldsymbol{\rho}_{2}^{2}}{4 \sigma_{ijat}^{2}}\right]\times\exp\left[-\frac{(\boldsymbol{\rho}_{2}- \boldsymbol{\rho}_{1})^{2}}{2\delta_{ijat}^{2}}\right]\times\exp\left[-\frac{ ik(\boldsymbol{\rho}_{1}^{2}-\boldsymbol{\rho}_{2}^{2})}{2R_{ijat}}\right]. \end{array}  \tag{17}\]  
Then, Equation (15) reduces to  
\[\mathbf{M}_{2 ij}^{-1}=\left( \begin{array}{cc}m_{21}\mathbf{I}&m_{22}\mathbf{ I}\\ m_{23}\mathbf{I}&m_{24}\mathbf{I}\end{array} \right)=\left( \begin{array}{cc}R_{ ijout}^{-1}+(-\frac{i}{2\delta_{ijout}^{2}}-\frac{i}{\delta_{ijout}^{2}}) \mathbf{I}&\frac{i}{\delta\delta_{ijout}^{2}}^{2}\mathbf{I}\\ \frac{i}{\delta\delta_{ijout}^{2}}\mathbf{I}&-R_{ijout}^{-1}+(-\frac{i}{2\delta _{ijout}^{2}}-\frac{i}{\delta\delta_{ijout}^{2}})\mathbf{I}\end{array} \right).