## 2 Theory of GI with a TGSM Beam in Turbulent Ocean
However, the information of the object can be retrieved by measuring the fourth-order correlation function (FOCF) between \(D_{1}\) and \(D_{2}\). According to the optical coherence theory, the FOCF \(G^{(2)}\) between \(D_{1}\) and \(D_{2}\) is defined as  
\[ \begin{array}{ll}G^{2}(\mathbf{u}_{1},\mathbf{u}_{2})&=\langle I_{1}(\mathbf{u }_{1})I_{2}(\mathbf{u}_{2})\rangle=\langle E(\mathbf{u}_{1})E(\mathbf{u}_{1}) E(\mathbf{u}_{2})E^{*}(\mathbf{u}_{2})\rangle\\ &=\int_{-\infty}^{+\infty}d^{2}\mathbf{r}_{1}d^{2}\mathbf{r}_{2}d^{2}\mathbf{r }_{3}d^{2}\mathbf{r}_{4}\Big{\{}E(\mathbf{r}_{1})E^{*}(\mathbf{r}_{2})E( \mathbf{r}_{3})E^{*}(\mathbf{r}_{4})\rangle_{s}\Big{\{}h_{1}(\mathbf{r}_{1}, \mathbf{u}_{1})h_{1}{}^{*}(\mathbf{r}_{2},\mathbf{u}_{1})h_{2}(\mathbf{r}_{3}, \mathbf{u}_{2})h_{2}{}^{*}(\mathbf{r}_{4},\mathbf{u}_{2})\rangle_{m}.\end{array}  \tag{11}\]  
In the derivation of Equation (11), it is assumed that the fluctuations induced by the source and the turbulence are statistical independent. On substituting Equations (3) and (4) into Equation (11), it turns out to be  
\[ \begin{array}{ll}G^{2}(\mathbf{u}_{1},\mathbf{u}_{2})&=\frac{1}{\lambda^{4 }z^{4}}\int_{-\infty}^{+\infty}d^{2}\mathbf{r}_{1}d^{2}\mathbf{r}_{2}d^{2} \mathbf{r}_{3}d^{2}\mathbf{r}_{4}\Big{|}H(\mathbf{u}_{2})\Big{|}^{2}\Big{\langle} E(\mathbf{r}_{1})E^{*}(\mathbf{r}_{2})E(\mathbf{r}_{3})E^{*}(\mathbf{r}_{4}) \rangle_{s}\\ &\times\exp\Bigl{(}-\frac{i\pi}{\lambda z}\Bigl{[}\left(\mathbf{u}_{1}- \mathbf{r}_{1}\right)^{2}-\left(\mathbf{u}_{1}-\mathbf{r}_{2}\right)^{2}+ \left(\mathbf{u}_{2}-\mathbf{r}_{3}\right)^{2}-\left(\mathbf{u}_{2}-\mathbf{r }_{4}\right)^{2}\Bigr{]}\Bigr{\}}\\ &\times\langle\exp[\psi(\mathbf{r}_{1},\mathbf{u}_{1},z)+\psi^{*}(\mathbf{r}_ {2},\mathbf{u}_{1},z)+\psi(\mathbf{r}_{3},\mathbf{u}_{2},z)+\psi^{*}(\mathbf{ r}_{4},\mathbf{u}_{2},z)]\rangle_{m},\end{array}  \tag{12}\]  
To further simplify Equation (12), it is assumed that the statistics of the source fluctuations obey the Gaussian statistics. Applying the Gaussian moment theorem [42], the fourth-order statistics can be expressed in terms of summation of second-order statistics, i.e.,  
\[ \begin{array}{ll}\Big{\langle}E(\mathbf{r}_{1})E^{*}(\mathbf{r}_{2})E( \mathbf{r}_{3})E^{*}(\mathbf{r}_{4})\rangle_{s}=W(\mathbf{r}_{1},\mathbf{r}_{ 2})W(\mathbf{r}_{3},\mathbf{r}_{4})+W(\mathbf{r}_{1},\mathbf{r}_{4})W(\mathbf{ r}_{3},\mathbf{r}_{2}),\end{array}  \tag{13}\]  
where \(W(\mathbf{r}_{i},\mathbf{r}_{j})=\Big{\langle}E(\mathbf{r}_{i})E*\Big{(} \mathbf{r}_{j}\Big{)}\Big{\rangle}_{s}\), \((i,j=1,2,3,4)\) stands for the second-order statistics of the source, the expression for \(W(\mathbf{r}_{i},\mathbf{r}_{j})\) is shown in Equation (1). While it is difficult to find the analytical expression for the fourth-order statistics of oceanic turbulence shown in the last term in Equation (12), here we assume that the statistics of turbulence in path one and in path two are independent. The fourth-order statistics of turbulence are reasonably simplified as  
\[ \begin{array}{ll}F_{4}&=\langle\exp[\psi(\mathbf{r}_{1},\mathbf{u}_{1},z)+ \psi^{*}(\mathbf{r}_{2},\mathbf{u}_{1},z)+\psi(\mathbf{r}_{3},\mathbf{u}_{2}, z)+\psi^{*}(\mathbf{r}_{4},\mathbf{u}_{2},z)]\rangle_{m}\\ &=\langle\exp[\psi(\mathbf{r}_{1},\mathbf{u}_{1},z)+\psi^{*}(\mathbf{r}_{2}, \mathbf{u}_{1},z)]\rangle_{m}\langle\exp[\psi(\mathbf{r}_{3},\mathbf{u}_{2}, z)+\psi^{*}(\mathbf{r}_{4},\mathbf{u}_{2},z)]\rangle_{m}.\end{array}  \tag{14}\]  
On substituting Equations (13) and (14) into Equation (12), we derive the following expression  
\[G^{2}(\mathbf{u}_{1},\mathbf{u}_{2})=\langle I_{1}(\mathbf{u}_{1})\rangle \langle I_{2}(\mathbf{u}_{2})\rangle+\Gamma(\mathbf{u}_{1},\mathbf{u}_{2}), \tag{15}\]  
with  
\[