## 2 Theoretical Model  
A complete active sonar system consists of a source, a receiver, and a target. For the active sonar system in shallow water waveguides, the source transmits an acoustic signal, and the acoustic signal propagates forward. When it encounters the target, it scatters and the scattered field is received by the receiver. A front view of the relative positional relationship between the source, the target, and the receiver is shown in Figure 1, which includes two cases: Figure 1a is the monostatic configuration, and Figure 1b is the bistatic configuration. In Figure 1, \(r_{1}\) is the distance from the source to the target, \(r_{2}\) is the distance from the target to the receiver, the ocean depth is denoted as H, \(z_{s}\) is the depth of the source, \(z_{t}\) is the depth of the target, and \(z_{t}\) is the depth of the receiver. Figure 1: Front view of the relative positional relationship: (**a**) monostatic configuration; (**b**) bistatic configuration. The acoustic path in the sonar configuration has two parts: the source to the target path (r\({}_{1}\)) and the target to the receiver path (r\({}_{2}\)). For a target in the far field, the field point acoustic pressure for small grazing angles can be expressed as:  
\[ \begin{array}{ll}\mathrm{p}(\mathrm{r}_{1},\mathrm{z}_{\mathrm{t}};\mathrm{r} _{2},\mathrm{z}_{\mathrm{r}};\mathrm{z}_{\mathrm{s}};\omega)&=\mathrm{C}^{2} \sum\limits_{\mathrm{m}}\sum\limits_{\mathrm{n}}\Psi_{\mathrm{m}}(\mathrm{z}_ {\mathrm{s}})\Psi_{\mathrm{m}}(\mathrm{z}_{\mathrm{t}})\frac{e^{\mathrm{i} \mathrm{k}\mathrm{m}\mathrm{r}_{1}}}{\sqrt{\mathrm{k}\mathrm{m}\mathrm{r}_{1}}} \\ &\times\mathrm{S}(\mathrm{\alpha}_{\mathrm{m}},\Phi_{0};\mathrm{\alpha}_{ \mathrm{n}},\Phi)\Psi_{\mathrm{n}}(\mathrm{z}_{\mathrm{t}})\Psi_{\mathrm{n}}( \mathrm{z}_{\mathrm{r}})\frac{e^{\mathrm{i}\mathrm{k}\mathrm{m}\mathrm{r}_{2}} }{\sqrt{\mathrm{k}\mathrm{m}\mathrm{r}_{2}}}\end{array} . \tag{1}\]  
The interpretation of Equation (1) is straightforward. The acoustic pressure amplitude of the scattered field is the product of the far-field scattering function and the mode function of the incident and scattered fields. The azimuth of the scattering function is obtained jointly from the source and the field point. Where \(\mathrm{C}=\mathrm{i}\mathrm{e}^{\mathrm{i}\pi/4}/\mathrm{\rho}\sqrt{8\pi}\) is the normalization constant, \(\mathrm{k}_{\mathrm{m}}\) and \(\mathrm{k}_{\mathrm{n}}\) are the horizontal wavenumbers of the mth and nth normal modes, respectively. Supposing \(\mathrm{B}_{\mathrm{m}}=\Psi_{\mathrm{m}}(\mathrm{z}_{\mathrm{s}})\Psi_{ \mathrm{m}}(\mathrm{z}_{\mathrm{t}})\frac{e^{\mathrm{i}\mathrm{k}\mathrm{m} \mathrm{r}_{1}}}{\sqrt{\mathrm{k}\mathrm{m}\mathrm{r}_{1}}}\), it is the effect of the mth propagating mode on the sound field, when the acoustic signal propagates between the source and the target. \(\mathrm{B}_{\mathrm{n}}=\Psi_{\mathrm{n}}(\mathrm{z}_{\mathrm{t}})\Psi_{ \mathrm{n}}(\mathrm{z}_{\mathrm{r}})\frac{e^{\mathrm{i}\mathrm{k}\mathrm{m} \mathrm{r}_{2}}}{\sqrt{\mathrm{k}\mathrm{m}\mathrm{r}_{2}}}\) is the effect of the nth scattering mode, when the acoustic signal is coupled with the target and propagates to the receiver. The scattering function [21]\(\mathrm{S}(\mathrm{\alpha}_{\mathrm{m}},\Phi_{0};\mathrm{\alpha}_{\mathrm{n}},\Phi)\), where \(\mathrm{\alpha}_{\mathrm{m}}\) and \(\Phi_{0}\) are the grazing and azimuth angles of the incident mode m, respectively, and \(\mathrm{\alpha}_{\mathrm{n}}\) and \(\Phi\) are the grazing and azimuth angles, respectively, corresponding to the scattered mode, is an important property of the target that can reflect the shape, structure, and material of the target, as well as the scattering angle information. The acoustic intensity of the field scattered from a far-field target can be written as \(\mathrm{I}=\left|\mathrm{p}\right|^{2}\).