## 2 Theory and Analysis  
In the ray theory, the approximate theoretical expression of the range-frequency interference structure in the shadow zone of the deep sea is [20]:  
\[\left|p(r,z;f)\right|_{2}\approx\frac{|S(f)|^{2}}{(4\pi)^{2}}\frac{4W_{\rm FB} }{R_{\rm B}^{2}}|V_{\rm B}|^{2}[1-\cos(2\pi f\Delta t_{1})][1-\cos(2\pi f\Delta t _{2})] \tag{1}\]  
where, \(|S(f)|\) is the sound source spectrum, \(F_{\rm B}\) is the focus factor, \(R_{\rm B}\) is the length of the sea bottom reflection path, \(W\) is the radiated sound power in unit solid angle, \(|V_{\rm B}|\) is the ocean bottom reflection coefficient, \(\Delta t_{1}\) is the time delay difference of arrival between the sea bottom primary reflection (B) and the sea surface-bottom reflection (SB), and \(\Delta t_{2}\) is the time delay difference of arrival between the sea bottom-surface reflection (BS) and the sea surface-bottom-surface reflection (SBS). Equation (1) describes the two types of interference structures caused by the time delay difference of multi-path arrival in the shadow zone of the deep sea, when the depth of the sound source and the hydrophone is located at the sea surface. The change in \(|V_{\rm B}|\) with frequency has no effect on the sound field interference structure if the ocean bottom is regarded as a semi-infinite seafloor [20]. However, the actual ocean bottom has a complex stratification, especially in deep water, where a certain thickness of sediment layer is formed after long-term and stable deposition. Accordingly, it is more representative to model the ocean bottom as a two-layer seafloor. Equation (1) was applied to the situations where it was not necessary to separate water-sediment reflected ray and sediment-basement reflected ray, i.e., the thickness of the sediment layer was not too large, or the seafloor-reflected energy of the tens of meters was taken into account. The simulation sound speed profile used the measured profile of the East Indian Ocean as shown in Figure 1, which is an incomplete sound channel. The simulation parameters were as follows: sound source frequency \(f\) = 50-1000 Hz, sound source depth \(s_{d}\) = 200 m,receiving depth \(r_{d}\) = 200 m, and receiving range \(r_{r}\) = 0-25 km. Table 1 shows the simulation values of the geoacoustic parameters for the above two types of ocean bottom models. A third interference structure appeared in the shadow zone due to the layered structure of the ocean bottom, when the ocean bottom was modeled as two layers. Figure 2 shows the range-frequency interference structure of the deep sea under the semi-infinite ocean bottom and the two-layer ocean bottom. The results of Figure 2 were obtained by the Bellhop algorithm. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Ocean Bottom Modal**} & \multirow{2}{*}{\begin{tabular}{c} **Sound** \\ **Speed (m/s)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Density** \\ **(g/cm3)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Attenuation** \\ **Coefficient** \\ **(dB/\(\lambda\))** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Thickness** \\ **(m)** \\ \end{tabular} } \\ \hline Semi-infinite ocean bottom & 1700.00 & 1.80 & 0.50 & - \\ \hline Layered ocean bottom & sediment & 1600.00 & 1.50 & 0.20 & 20.0 \\ \cline{2-6}  & basement & 1800.00 & 2.00 & 1.00 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation values of geoacoustic parameters. Figure 1: The simulation sound speed profile. Figure 2: Range–frequency interference structures in the deep sea, (**a**) semi–infinite ocean bottom, (**b**) layered ocean bottom. Figure 2a shows the interference structure of two regions, including the direct wave zone in the distance of 0-5 km and the shadow zone in the distance of 5-25 km. The time delay difference of arrival between the direct wave and the primary reflected wave from the sea surface formed the interference structure in the direct wave region [20].