## 2 Theory and Analysis
### Range-Frequency Interference Structure of Shadow Zone  
The distribution of the interference structure in Figure 2b is not quite identical to that in Figure 3, as the interference structure in Figure 2b is composed of the arrival energy at multiple angles, and the angle component is not singular. It can be seen from Equation (1) that the interference structure of the shadow zone is mainly composed of B, SB, BS, and SBS, the energies of these four multi-paths are similar, and the shadow zone interference structure caused by the ocean bottom reflection is complicated when simultaneously taking into account the grazing angle components of the four multi-paths. In order to describe the interference structure with a single grazing angle, we could set a reasonable sound source and receiver depth to make the receiving grazing angles of SB and BS the same, and then the sound pressure received in the shadow zone could be approximated as [20]:  
\[p(x,z;f)\approx\frac{S(f)\sqrt{W}}{4\pi}\bigg{(}\frac{\sqrt{F_{\rm B}}}{R_{ \rm B}}V_{\rm B}e^{i2\pi ft_{\rm B}}-\frac{\sqrt{F_{\rm SB}}}{R_{\rm SB}}V_{ \rm SB}e^{i2\pi ft_{\rm SB}}-\frac{\sqrt{F_{\rm BS}}}{R_{\rm BS}}V_{\rm BS}e^{ i2\pi ft_{\rm BS}}+\frac{\sqrt{F_{\rm SBS}}}{R_{\rm SBS}}V_{\rm SBS}e^{i2\pi ft_{\rm SBS}} \bigg{)} \tag{6}\]  
To simplify the derivation, the approximate form of Equation (2) can be written as follows:  
\[V\approx V_{\rm ws}+W_{\rm ws}V_{\rm sb}W_{\rm sw}e^{i2\Phi} \tag{7}\]  
\(W_{\rm ws}\) is the transmission coefficient of the seawater-sediment interface. The approximate expression only slightly affects the amplitude of the ocean bottom reflection coefficient, but has no effect on the interference period caused by the two-layer seabed [11]. It was considered that the focus factor \(F\) and slant range \(R\) are almost the same. Since the transceiver depth is at the sea surface, the difference of the grazing angles of the four multi-path arrivals (B, SB, BS, and SBS) is small. It was approximately considered that \(V_{\rm ws}\), \(W_{\rm ws}\), and \(V_{\rm sb}\) corresponding to the four are almost the same. Equation (6) was divided into two parts:  
\[p(x,z;f)=p_{1}(x,z;f)+p_{2}(x,z;f) \tag{8}\]  
where, \(p_{1}\) is the part which is not related to the interference of the ocean bottom reflection coefficient:  
\[p_{1}(x,z;f)\approx\frac{S(f)\sqrt{WF}}{4\pi R}V_{\rm ws}\Big{(}e^{i2\pi ft_{ \rm B}}-e^{i2\pi ft_{\rm SB}}-e^{i2\pi ft_{\rm BS}}+e^{i2\pi ft_{\rm SBS}} \Big{)} \tag{9}\]  
Defining \(\Delta t_{1}=t_{\rm SB}-t_{\rm B}\), \(\Delta t_{2}=t_{\rm BS}-t_{\rm SBS}\), then the coherent superposition part can be approximately expressed as:  
\[ \begin{array}{l}e^{i2\pi ft_{\rm B}}-e^{i2\pi ft_{\rm SB}}=e^{i2\pi ft_{\rm SB }}(e^{-i2\pi f/\Delta t_{1}}-1)\\ e^{i2\pi ft_{\rm BS}}-e^{i2\pi ft_{\rm SBS}}=e^{i2\pi ft_{\rm BS}}(1-e^{i2 \pi/\Delta t_{2}})\end{array}  \tag{10}\]  
Figure 3: The interference structure of seabed reflection coefficients in the angle-frequency domain (the red line is the quarter-wave layer (dark fringe), and the blue line is the half-wave layer (bright fringe)). Considering that the receiving grazing angles of SB and BS are the same, namely, \(\varphi_{\text{rSB}}=\varphi_{\text{rBS}}\), and the time delay of the two sound rays is equal, namely, \(t_{\text{SB}}=t_{\text{BS}}\), then Equation (9) can be approximately expressed as:  
\[p_{1}(x,z;f)\approx\frac{S(f)\sqrt{WF}}{4\pi R}V_{\text{ws}}e^{2\pi ft_{\text{SB }}}(e^{-i2\pi f\Delta t_{1}}+e^{i2\pi f\Delta t_{2}}-2) \tag{11}\]  
\(p_{2}\) is the other part which is related to interference of the ocean bottom reflection coefficient:  
\[ \begin{array}{cc}p_{2}(x,z;f)\approx\frac{S(f)\sqrt{WF}}{4\pi R}W_{\text{ws} }V_{\text{sb}}W_{\text{sw}}&\left(e^{i2\Phi_{\text{bs}}}e^{2\pi ft_{\text{Bs}} }-e^{i2\Phi_{\text{SB}}}e^{i2\pi ft_{\text{SB}}}\right.\\ &\left.-e^{i2\Phi_{\text{RS}}}e^{i2\pi ft_{\text{BS}}}+e^{i2\Phi_{\text{RS}}}e^ {i2\pi ft_{\text{SB}}}\right)\end{array}