# A Method for Estimating Source Depth Based on the Adjacent Mode Group Acoustic Pressure Field
## 2 Theory
### Mode Extraction  
Singular value decomposition (SVD) can decompose the measurement matrix of any row full rank into the following:  
\[A_{N\times M}=USV^{\dagger} \tag{6}\]  
where the complex matrix \(A\) of \(N\times M\) can be decomposed into three matrices, \(S\) is a semi-positive diagonal matrix, \(U,V\) are orthogonal matrices and \(\dagger\) represents the conjugate transpose. The acoustic pressure data \(P\) are collected through a vertical array, and the matrix \(P\) is decomposed by SVD:  
\[P=e^{i\pi/4}\overline{\Phi}\Sigma R \tag{7}\]  
where the column vector (\(\overline{\Phi}_{n}\)) of \(\overline{\Phi}\) corresponds to different order mode functions and the diagonal elements of \(\Sigma\) consist of the singular values of the matrix \(P\). Usually, in practical applications, the obtained sound pressure matrix \(P\) is not a square matrix, so we use the cross spectral density matrix \(C\) for singular value decomposition:  
\[C=PP^{\dagger}\approx\Phi\Lambda^{2}\Phi^{\dagger} \tag{8}\]  
Similarly, \(\Lambda\) is a diagonal matrix, where the singular values gradually decrease on the diagonal, and the column vector of \(\Phi\) is the mode corresponding to each singular value. The experimental data used in the study presented in this paper were from a sea trial conducted at the SACLANT center in the Mediterranean Sea near the west coast of Italy in October 1993 [19]. This experiment collected mainly waveguide acoustic pressure data through a shallow water vertical array and was conducted in a flat waveguide area between 120 m and 140 m of the isobath. In the test, the acoustic pressure data were obtained through a vertical array of 48 hydrophones spanning a 94 m aperture; three groups of test data were obtained, as shown in Table 2.  
The three sets of data correspond to static source test data with a source frequency of 170 Hz (A2601-1), moving source test data with a source frequency of 170 Hz (P2701) and static source test data with a source frequency of 350 Hz (A2601-2). This study used mainly experimental data from a moving source with a source frequency of 170 Hz, as this set of data was consistent with the data requirements of singular value decomposition mode extraction. The data composition was \(48\times N\), where 48 corresponds to the number of vertical array hydrophones, and \(N\) is related to the time of data acquisition. The longer the elapsed time, the greater the value of \(N\). In order to verify the accuracy of the mode extraction results, a one-to-one simulation was conducted in the Acoustic Toolbox v2.2 (AT, an open-source software for sound field modeling calculations developed by [PERSON]) based on the waveguide environment \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Value** \\ \hline Receiver Depths & 0-150 (m) \\ Sound Speed: Waveguide & 1500 (m/s) \\ Sound Speed: Bottom & 2000 (m/s) \\ Rho: Water & 1 (g/cm\({}^{3}\)) \\ Rho: Bottom & 2 (g/cm\({}^{3}\)) \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._ \begin{table}
\begin{tabular}{c c c} \hline
**Data Name** & **Meaning** & **Data Quantity** \\ \hline A2601-1 & 170 Hz static source on 26 October & 59.6 MB \\ P2701 & 170 Hz movable source on 27 October & 59.6 MB \\ A2601-2 & 350 Hz static source on 26 October & 29.8 MB \\ \hline \end{tabular}
\end{table} 
Table 2: SACLANT simulation data. corresponding to the SACLANT data, the sound speed profile [20] of which is shown in Figure 3. The source was placed at a depth of 80 m in the waveguide, with the corresponding simulation mode distribution shown in Figure 4.  
Due to the correlation between the depth of the source and the order of the waveguide mode, the greater the value of the corresponding mode at the depth of the source, the greater the effect of that mode. In order to obtain better mode extraction results, it was necessary to limit the data used.