## 2 Theory and Analysis
### Interference Structure of Ocean Bottom Reflection Coefficient in Angle-Frequency Domain  
In order to describe the interference structure caused by the layered ocean bottom in Figure 2b, the theoretical expression of the interference structure of the layered ocean bottom reflection coefficient in the angle-frequency domain was deduced in this paper.  
The reflection coefficient of the two-layer ocean bottom \(V\) can be expressed as follows [23]:  
\[V=\frac{V_{\text{ws}}+V_{\text{sb}}\text{e}^{2\phi}}{1+V_{\text{ws}}V_{\text{ sb}}\text{e}^{2\phi}} \tag{2}\]  
where, \(\phi=k_{\text{s}}dsin\varphi_{\text{s}}\) is the vertical phase shift of sound waves from the seawater penetrating the sedimentary layer to the basement, \(k_{\text{s}}\) is the number of acoustic waves in sedimentary layer, and \(V_{\text{ws}}=\frac{c_{\text{s}}\rho_{\text{s}}/sin\varphi_{\text{s}}-c_{ \text{w}}\rho_{\text{w}}/sin\varphi_{\text{w}}}{c_{\text{s}}\rho_{\text{s}}/ sin\varphi_{\text{s}}+c_{\text{w}}\rho_{\text{w}}/sin\varphi_{\text{w}}}\) and \(V_{\text{sb}}=\frac{c_{\text{s}}\rho_{\text{b}}/sin\varphi_{\text{b}}-c_{ \text{s}}\rho_{\text{s}}/sin\varphi_{\text{s}}}{c_{\text{s}}\rho_{\text{b}}/ sin\varphi_{\text{b}}+c_{\text{w}}\rho_{\text{w}}/sin\varphi_{\text{s}}}\) are the Rayleigh reflection coefficients of sound waves at the seawater-sediment interface and the sediment-basement interface, respectively. \(c_{\text{w}}\), \(c_{\text{s}}\), and \(c_{\text{b}}\) are the sound speed, \(\rho_{\text{w}}\), \(\rho_{\text{s}}\), and \(\rho_{\text{b}}\) are the density, and \(\varphi_{\text{w}}\), \(\varphi_{\text{s}}\), and \(\varphi_{\text{b}}\) are the grazing angle of sea water, sediment, and basement, respectively. \(d\) is the thickness of the sedimentary layer. The high-speed sediment refers to \(c_{\text{s}}>c_{\text{w}}\).  
The interference condition is shown as Equation (3), when the ocean bottom grazing angle is greater than the critical angle of the seawater-basement:  
\[\frac{4f/d\sin\varphi_{\text{s}}}{c_{\text{s}}}=n(n=1,2,3\cdots)\quad\text{ while}\quad\varphi_{\text{w}}\geq\arccos(\frac{c_{\text{w}}}{c_{\text{b}}}) \tag{3}\]  
where, \(f\) is the frequency, and \(n\) is the order of the interference fringe. When \(n\) is an odd number, it corresponds to the quarter-wave layer, and when \(n\) is an even number, it corresponds to the half-wave layer. The two cases respectively correspond to the dark and bright fringes of the interference structure in the range-frequency domain of the deep sea.  
When the ocean bottom grazing angle is greater than the seawater-sediment critical angle and is less than the seawater-basement critical angle, \(\varphi_{\text{s}}\) satisfies the small grazing angle condition, and the sea bottom reflection coefficient \(V\) is:  
\[V=\frac{V_{\text{ws}}+\text{e}^{i(Q\varphi_{\text{s}}-\pi)}\text{e}^{2\phi}}{ 1+V_{\text{ws}}\text{e}^{i(Q\varphi_{\text{s}}-\pi)}\text{e}^{2\phi}}=\frac{V_ {\text{ws}}+\text{e}^{i(Q\varphi_{\text{s}}-\pi+2k_{\text{s}}d\sin\varphi_{ \text{s}})}}{1+V_{\text{ws}}\text{e}^{i(Q\varphi_{\text{s}}-\pi+2k_{\text{s}}d \sin\varphi_{\text{s}})}} \tag{4}\]  
where \(Q=\frac{2\rho_{\text{b}}/\rho_{\text{s}}}{\sqrt{1-(c_{\text{s}}/c_{\text{b}} )^{2}}}\). The interference conditions are:  
\[\frac{4 fd\sin\varphi_{\text{s}}}{c_{\text{s}}}+\frac{Q\varphi_{\text{s}}}{ \pi}-1=n(n=1,2,3\cdots)\ \ \text{while}\ \arccos(\frac{c_{\text{w}}}{c_{\text{s}}})\leq\varphi_{\text{w}}\leq\arccos( \frac{c_{\text{w}}}{c_{\text{b}}}) \tag{5}\]  
Figure 3 shows the first 12-order interference fringes of the bottom reflection coefficient in the angle-frequency domain. It can be seen from Figure 3 that Equations (3) and (5) could accurately describe the interference fringes of the bottom reflection coefficient in the angle-frequency domain.