## 2 Theory of GI with a TGSM Beam in Turbulent Ocean
By applying Equations (1)-(4), the average intensity received by detectors \(D_{1}\) and \(D_{2}\) can be expressed as  
\[ \begin{array}{ll}I_{1}(\mathbf{u}_{1})&=\int_{\omega}^{\infty}W(\mathbf{r}_ {1},\mathbf{r}_{2})h_{1}(\mathbf{r}_{1},\mathbf{u}_{1})h_{2}^{\star}(\mathbf{ r}_{2},\mathbf{u}_{1})d^{2}\mathbf{r}_{2},\\ &=\frac{1}{\lambda^{2}z^{2}}\int_{-\infty}^{\infty}d^{2}\mathbf{r}_{1}d^{2} \mathbf{r}_{2}[H(\mathbf{u}_{1})]^{2}W(\mathbf{r}_{1},\mathbf{r}_{2})\exp \biggl{[}-\frac{i\pi}{\lambda z}(\mathbf{u}_{1}-\mathbf{r}_{1})^{2}\biggr{]} \\ &\times\exp\biggl{[}\frac{i\pi}{\lambda z}(\mathbf{u}_{1}-\mathbf{r}_{2})^{2} \biggr{]}\times\langle\exp\bigl{[}\psi(\mathbf{r}_{1},\mathbf{u}_{1},z)+\psi^ {\star}(\mathbf{r}_{2},\mathbf{u}_{1},z)\bigr{]}\rangle_{m^{\prime}},\\ I_{2}(\mathbf{u}_{2})&=\int_{-\infty}^{\infty}W(\mathbf{r}_{1},\mathbf{r}_{2} )h_{1}(\mathbf{r}_{1},\mathbf{u}_{2})h_{1}^{\star}(\mathbf{r}_{2},\mathbf{u}_ {2})d^{2}\mathbf{r}_{1}d^{2}\mathbf{r}_{2},\\ &=\frac{1}{\lambda^{2}z^{2}}\int_{-\infty}^{\infty}d^{2}\mathbf{r}_{1}d^{2} \mathbf{r}_{2}W(\mathbf{r}_{1},\mathbf{r}_{2})\exp\bigl{[}-\frac{i\pi}{ \lambda z}(\mathbf{u}_{2}-\mathbf{r}_{1})^{2}\bigr{]}\\ &\times\exp\Bigl{[}\frac{i\pi}{\lambda z}(\mathbf{u}_{2}-\mathbf{r}_{2})^{2} \Bigr{]}\times\langle\exp\bigl{[}\psi(\mathbf{r}_{1},\mathbf{u}_{2},z)+\psi^ {\star}(\mathbf{r}_{2},\mathbf{u}_{2},z)\bigr{]}\rangle_{m^{\prime}}\end{array}  \tag{5}\]  
respectively. The angular brackets with subscript \"\(M\)\" denote the ensemble average over the fluctuations of turbulence. Under the condition that the oceanic turbulence is isotropic and homogeneous, the second-order statistics of phase perturbations can be represented as [43]  
\[\langle\exp[\psi^{\star}(\mathbf{r}_{1},\mathbf{u}_{1},z)+\psi(\mathbf{r}_{2 },\mathbf{u}_{2},z)]\rangle_{m}=\exp\Bigl{\{}-M[\bigl{(}\mathbf{u}_{1}- \mathbf{u}_{2}\bigr{)}^{2}+(\mathbf{u}_{1}-\mathbf{u}_{2})(\mathbf{r}_{1}- \mathbf{r}_{2})+(\mathbf{r}_{1}-\mathbf{r}_{2})^{2}]\Bigr{\}}. \tag{6}\]  
The parameter \(M\) is expressed as  
\[M=\frac{\pi^{2}k^{2}z}{3}\int_{0}^{\infty}x^{3}\Phi_{n}(\kappa,\eta,\varepsilon,\chi_{T},\omega)dx, \tag{7}\]  
where \(\Phi_{n}\) is the one-dimensional power spectrum of refractive index fluctuations given by  
\[\Phi_{n}(\kappa,\eta,\varepsilon,\chi_{T},\omega)=0.388\times 10^{-8}\varepsilon ^{-1/3}\kappa^{-11/3}[1+2.35(\kappa\eta)^{2/3}]f(\kappa,\omega,\chi_{T}). \tag{8}\]  
Here \(\kappa\) is the magnitude of three-dimensional spatial frequency vector. \(\varepsilon\) is the rate of dissipation of turbulent kinetic energy per unit mass of fluid, usually varying from \(10^{-2}\) to \(10^{-8}\) m\({}^{2}\)/s\({}^{3}\), and \(\eta=10^{-3}\) m is the Kolmogorov microscale, the function \(f\) takes the form  
\[f(\kappa,\omega,\chi_{T})=\frac{\chi_{T}}{\omega^{2}}(\omega^{2}e^{-A_{T} \delta}+e^{-A_{S}\delta}-2\omega e^{-A_{TS}\delta}). \tag{9}\]  
\(\chi_{T}\) in Equation (9) is the dissipation rate of mean-square temperature, which may vary from \(10^{-10}\) to \(10^{-4}\) K\({}^{2}\)/s; \(\omega\) (\(-5<\omega<0\)) represents the ratio of temperature and salinity contributions to the refractive index spectrum. In the low bound value, it means that no fluctuations in salinity concentration, while the salinity fluctuations dominate the major role when \(\omega\) approaches zero. The other parameters in Equation (9) are set as \(A_{T}=1.863\times 10^{-2}\), \(A_{S}=1.9\times 10^{-4}\), \(A_{TS}=9.41\times 10^{-3}\), \(\delta=8.284(\kappa\eta)^{4/3}+12.978(\kappa\eta)^{2}\). The reason for the chosen of these parameter values can be found in [43] and these values are widely used in the calculation of laser beams propagation in oceanic turbulence. It is known that one cannot acquire any information of object through measuring the average intensity in path one or two. The reason is that in path one, no object is located, and the bucket detector \(D_{2}\) only measures the total intensity from the object with no resolution.