## 2 Theory of GI with a TGSM Beam in Turbulent Ocean
Finally, the tensor form of the product of four response functions in the absence of turbulence is given as  
\[\exp\left(-\frac{i\pi}{\lambda z}\big{[}(\mathbf{u}_{1}-\mathbf{r}_{1})^{2}-( \mathbf{u}_{1}-\mathbf{r}_{2})^{2}+(\mathbf{u}_{2}-\mathbf{r}_{3})^{2}-( \mathbf{u}_{2}-\mathbf{r}_{4})^{2}\big{]}\right)=\exp\left[-\frac{ik}{2}\Big{(} \overline{\mathbf{r}}^{T}\overline{\mathbf{B}}\overline{\mathbf{r}}-\overline {\mathbf{r}}^{T}\overline{\mathbf{B}}\overline{\mathbf{u}}\Big{)}\right]\!. \tag{23}\]  
Here \(\overline{\mathbf{u}}=(u_{1},u_{1},u_{1},u_{1},u_{2},u_{2},u_{2},u_{2})\) is one column matrix, \(\overline{\mathbf{B}}=\left( \begin{array}{cc}\overline{\mathbf{B}}&0 \overline{\mathbf{I}}\\ 0\overline{\mathbf{I}}&\overline{\mathbf{B}}\end{array} \right)\,\,\overline{ \mathbf{B}}_{1}=\left( \begin{array}{cc}1/z&0\\ 0&-1/z\end{array} \right)\!\). On substituting Equations (17), (18), (21), and (23) into Equation (15) we obtain the expression  
\[\langle I_{1}(\mathbf{u}_{1})\rangle\langle I_{2}(\mathbf{u}_{2})\rangle=\frac {1}{\lambda^{4}[\det(\overline{\mathbf{B}}^{-1})]^{\frac{1}{2}}}\int\exp \left(-\frac{ik}{2}\overline{\mathbf{r}}^{T}\overline{\mathbf{N}}_{0}^{-1} \overline{\mathbf{r}}-\frac{ik}{2}\overline{\mathbf{r}}^{T}\overline{\mathbf{ S}}\overline{\mathbf{r}}\right)\exp\left[-\frac{ik}{2}\Big{(} \overline{\mathbf{r}}^{T}\overline{\mathbf{B}}\overline{\mathbf{r}}-2 \overline{\mathbf{r}}^{T}\overline{\mathbf{B}}\overline{\mathbf{u}}\Big{)} \right]\!d\overline{\mathbf{r}}, \tag{24}\]  
\[\Gamma(\mathbf{u}_{1},\mathbf{u}_{2})=\frac{1}{\lambda^{4}[\det(\overline{ \mathbf{B}}^{-1})]^{\frac{1}{2}}}\big{|}H(\mathbf{u}_{1})\big{|}^{2}\int\exp \left(-\frac{ik}{2}\overline{\mathbf{r}}^{T}\overline{\mathbf{M}}_{0}^{-1} \overline{\mathbf{r}}-\frac{ik}{2}\overline{\mathbf{r}}^{T}\overline{\mathbf{ S}}\overline{\mathbf{r}}\right)\exp\left[-\frac{ik}{2}\Big{(} \overline{\mathbf{r}}^{T}\overline{\mathbf{B}}\overline{\mathbf{r}}-2 \overline{\mathbf{r}}^{T}\overline{\mathbf{B}}\overline{\mathbf{u}}\Big{)} \right]\!d\overline{\mathbf{r}}. \tag{25}\]  
By applying the following integral formula  
\[\int d^{n}\mathbf{x}\exp(-\mathbf{x}^{T}\mathbf{A}\mathbf{x}+i\mathbf{k}^{T} \mathbf{x})=\pi^{n/2}(\det\mathbf{A})^{-1/2}\exp\left(-\frac{1}{4}\mathbf{k}^{T} \mathbf{A}^{-1}\mathbf{k}\right)\!, \tag{26}\]  
which is valid for \(n\)-dimensional vectors \(\mathbf{x}\), \(\mathbf{k}\) and a symmetric matrix \(\mathbf{A}\) with the positive definite real part, after integrating over \(\overline{\mathbf{r}}\), we have the form  
\[\langle I(\mathbf{u}_{1})\rangle\langle I(\mathbf{u}_{2})\rangle=\frac{\big{|}H( \mathbf{u}_{1})\big{|}^{2}}{[\det(\overline{\mathbf{B}}^{-1}\overline{ \mathbf{N}}_{0}^{-1}+\overline{\mathbf{B}}^{-1}\overline{\mathbf{S}}+\overline{ \mathbf{I}})]^{\frac{1}{2}}}\exp\left(-\frac{ik}{2}\overline{\mathbf{u}}^{T} \overline{\mathbf{N}}_{01}^{-1}\overline{\mathbf{u}}\right)\!, \tag{27}\]\[\Gamma(\mathbf{u}_{1},\mathbf{u}_{2})=\frac{\left|H(\mathbf{u}_{1})\right|^{2}}{ \left[\det(\mathbf{\overline{B}}^{-1}\mathbf{\widetilde{M}}_{0}^{-1}+\mathbf{ \overline{B}}^{-1}\mathbf{\overline{S}}+\mathbf{\overline{I}})\right]^{\frac{ 1}{2}}}\exp\!\left(-\frac{ik}{2}\mathbf{\overline{u}}^{T}\mathbf{\widetilde{M }}_{01}^{-1}\mathbf{\overline{u}}\right), \tag{28}\]  
where \(\mathbf{\widetilde{M}}_{01}^{-1}=\mathbf{\overline{B}}(\mathbf{\widetilde{M}}_ {0}^{-1}+\mathbf{\overline{S}}+\mathbf{\overline{B}})^{-1}\mathbf{\overline{B}}\) and \(\mathbf{\widetilde{N}}_{01}^{-1}=\mathbf{\overline{B}}(\mathbf{\widetilde{N }}_{0}^{-1}+\mathbf{\overline{S}}+\mathbf{\overline{B}})^{-1}\mathbf{ \overline{B}}\), \(\mathbf{\overline{I}}\) is a \(8\times 8\) unit matrix. Equations (27) and (28) provide a convenient way for studying the behavior of a ghost image formed with a TGSM beam in the presence of oceanic turbulence.