## 2 Theory of GI with a TGSM Beam in Turbulent Ocean
\begin{array}{ll}\Gamma(\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}_{1})\Big{|}^{2}W(\mathbf{ r}_{1},\mathbf{r}_{4})W(\mathbf{r}_{3},\mathbf{r}_{2})\\ &\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)]\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{16}\]  
It follows from Equation (15) that the information of the object is contained in \(\Gamma(\mathbf{u}_{1},\mathbf{u}_{2})\), but it involves eight-folded integrals. Note that the integral functions in Equation (15) or Equation (16) are all exponential functions with quadratic terms. Therefore, it can be greatly simplified with the help of the tensor method [44]. The terms in the right-hand of Equation (13) have the following alternative tensor forms  
\[W(\mathbf{r}_{1},\mathbf{r}_{2})W(\mathbf{r}_{3},\mathbf{r}_{4})=\exp\Biggl{[} -\frac{ik}{2}\Bigl{(}\overline{r}^{\intercal}\widetilde{\mathbf{N}}_{0}^{-1} \overline{r}\Bigr{)}\Biggr{]}, \tag{17}\]  
\[W(\mathbf{r}_{1},\mathbf{r}_{4})W(\mathbf{r}_{3},\mathbf{r}_{2})=\exp\Biggl{[} -\frac{ik}{2}\Bigl{(}\overline{r}^{\intercal}\widetilde{\mathbf{M}}_{0}^{-1} \overline{r}\Bigr{)}\Biggr{]}, \tag{18}\]where \(\overline{r}=(x_{1},y_{1},x_{2},y_{2},x_{3},y_{3},x_{4},y_{4})\) is one column matrix, the superscript \"\(T\)\" denotes the transpose of the matrix, and \(\overline{\mathbf{N}}_{0}^{-1}\) and \(\overline{\mathbf{M}}_{0}^{-1}\) are \(8\times 8\) matrices, given by  
\[\overline{\mathbf{N}}_{0}^{-1}=\left( \begin{array}{cc}\overline{\mathbf{N}}^ {-1}&0\overline{\mathbf{I}}\\ 0\overline{\mathbf{I}}&\overline{\mathbf{N}}^{-1}\end{array} \right)\,\, \overline{\mathbf{M}}_{0}^{-1}=\left( \begin{array}{cc}\overline{\mathbf{M}}_ {1}^{-1}&\overline{\mathbf{M}}_{2}^{-1}\\ \left(\overline{\mathbf{M}}_{2}^{-1}\right)^{T}&\overline{\mathbf{M}}_{1}^{-1} \end{array} \right) \tag{19}\]  
with  
\[\overline{\mathbf{N}}^{-1}=\left( \begin{array}{cc}c_{1}\mathbf{I}&c_{2} \mathbf{I}+\mu\mathbf{J}\\ c_{2}\mathbf{I}+\mu\mathbf{J}^{T}&c_{1}\mathbf{I}\end{array} \right)\,\, \overline{\mathbf{M}}_{1}^{-1}=\left( \begin{array}{cc}c_{1}\mathbf{I}&0 \mathbf{I}\\ 0\mathbf{I}&c_{1}\mathbf{I}\end{array} \right)\,\,\overline{\mathbf{M}}_{2}^{-1 }=\left( \begin{array}{cc}0\mathbf{I}&c_{2}\mathbf{I}+\mu\mathbf{J}\\ c_{2}\mathbf{I}+\mu\mathbf{J}&0\mathbf{I}\end{array} \right). \tag{20}\]  
Here \(c_{1}=-\frac{i}{2\hbar x_{0}^{2}}-\frac{i}{\hbar x_{0}^{2}}\), \(c_{2}=\frac{i}{\hbar x_{0}^{2}}\), \(\mathbf{I}=\left( \begin{array}{cc}1&0\\ 0&1\end{array} \right)\) and \(\mathbf{J}=\left( \begin{array}{cc}0&1\\ -1&0\end{array} \right)\) are \(2\times 2\) identity and anti-symmetric matrices, respectively. The fourth-order statistics induced by the oceanic turbulence can be rewritten as  
\[F_{4}=\exp\left(-\frac{ik}{2}\overline{r}^{T}\overline{\mathbf{S}}\overline{r }\right)\!, \tag{21}\]  
where  
\[\overline{\mathbf{S}}=\frac{2M}{k}\left( \begin{array}{cc}\overline{\mathbf{ S}}&0\overline{\mathbf{I}}\\ 0\overline{\mathbf{I}}&\overline{\mathbf{S}}\end{array} \right)\!, \tag{22}\]  
\(\overline{I}\) in Equation (22) is an \(4\times 4\) identity matrix and \(\overline{\mathbf{S}}=\left( \begin{array}{cc}\mathbf{I}&-\mathbf{I}\\ -\mathbf{I}&\mathbf{I}\end{array} \right)\).