# A Method for Estimating Source Depth Based on the Adjacent Mode Group Acoustic Pressure Field
## 2 Theory
### AMG Acoustic Pressure Field Acquisition  
The distribution of an acoustic pressure field is a combination of numerous AMG acoustic pressure fields, which leads to a more complex distribution of singular points in the entire waveguide. The AMG acoustic pressure field can be obtained on the basis of mode extraction and the decomposition of the acoustic pressure field in normal mode theory, and can be decomposed into vector product forms in the horizontal and vertical directions:  
\[P(r,z)=Z(z)R(r) \tag{1}\]  
It is assumed that in the far field of the ideal shallow waveguide, the upper layer is a uniform water layer, and the lower layer is the hard uniform seafloor. \(\rho_{1}\) is the density of waveguide, \(\rho_{2}\) is the density of the bottom, the speed of sound in the waveguide is \(c_{1}\) and the speed of sound at the bottom is \(c_{2}\). Here, the reference system consists of horizontal and vertical axes \(r\) and \(z\), and the origin is at the sea level. If the acoustic pressure field \(P\) is known and the accurate mode distribution \(Z\) is obtained through mode extraction, the  
Figure 1: Flow diagram of the proposed method.  
horizontal vector \(R\) can be obtained through Equation (1) to obtain the simulated AMG acoustic pressure field. The known simulated acoustic pressure is [18]:  
\[P(r,z)=2\pi\omega\rho\sum_{n=I,q}\sin(\zeta_{l}z)F(z_{0},b_{n})\mathrm{H}_{0}^{( 1)}(b_{n}r) \tag{2}\]  
\[F(z_{0},b_{n})=\frac{\xi_{n}\mathrm{sin}(\xi_{n}z_{0})}{\xi_{n}H-\sin{(\xi_{n} H)}\mathrm{cos}(\xi_{n}H)-(\rho_{1}/\rho_{2})^{2}\mathrm{tan}(\xi_{n}H) \mathrm{sin}^{2}(\xi_{n}H)} \tag{3}\]  
\[\xi_{n}=\sqrt{k_{1}^{2}-b_{n}^{2}} \tag{4}\]  
\[k_{i}=\omega/c_{i},i=1,2 \tag{5}\]  
Here, \(H\) is the depth of the waveguide (\(0\leq z\leq H\)), \(\omega\) is the cyclic frequency of the signal, \(z_{0}\) is the depth of the source, \(n\) is the serial number of the normal wave, \(b_{n}\) is the \(n\)th eigenvalue and \(\mathrm{H}_{0}^{(1)}(x)\) is the zero-order Hankel function of the first kind.  
By replacing \(\sin(\zeta_{l}z)\) in Equation (2) with the mode distribution \(Z\) obtained through mode extraction, the acoustic pressure can be obtained through mode extraction. Through the above steps, we can obtain the acoustic pressure field, as shown in Figure 2b. Figure 2 shows the simulated acoustic pressure field and calculated acoustic pressure field obtained by (6,7)-AMG when the source is at the bottom of the sea; the darker the color, the lower the acoustic pressure value; the simulated shallow water environmental parameters [13] are shown in Table 1. Here, the Euclidean distance is used to calculate the similarity of two images to achieve image matching. The Euclidean distances in Figure 2a,b are only 0.0072 and the difference is small, which verifies that obtaining the acoustic pressure field distribution of AMGs through mode extraction and calculation is feasible.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Source Frequency & 50 (Hz) \\ Number of Media & 1 \\ Top Option & Vacuum \\ Waveguide Depth & 150 (m) \\ Bottom Option & Acoustic: Elastic \\ Lower Limit of Phase Velocity & 0 (m/s) \\ Upper Limit of Phase Velocity & 2000 (m/s) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation of shallow ocean environment parameters.  
Figure 2: Distribution of the simulated acoustic pressure field and calculated acoustic pressure field through mode extraction: **(a)** (6,7)-AMG simulated acoustic pressure field; **(b)** (6,7)-AMG calculated acoustic pressure field through mode extraction.