## 2 Spatial Distribution Characteristics of the Signal and Noise Fields  
### Investigation of the Spatial Distribution of the Signal Field  
In a uniformly shallow sea, the characteristics of the medium do not change significantly with the depth; therefore, the sound field can be studied using ray theory. The accuracy of ray theory for calculating the sound field is correlated with the number of sound rays examined in the study. The larger the number of sound rays considered, the less the sound rays reflected from the bottom of the water contribute to the sound field. Thus, the direct sound and the first-order reflection rays of the sea surface are mainly considered. The sound pressure field is  
\[\mathrm{p}(\mathrm{r},\mathrm{z},\mathrm{t})=\frac{1}{R_{00}}exp[i(kR_{00}- \omega t)]-G_{10}\frac{1}{R_{10}}exp[i(kR_{10}-\omega t)] \tag{1}\]  
The direct sound path \(R_{00}=\sqrt{r^{2}+\left(z-z_{s}\right)^{2}}\) and the first surface reflection sound path \(R_{10}=\sqrt{r^{2}+\left(z+z_{s}\right)^{2}}\), where \(\mathrm{r}\) is the horizontal distance between the sound source and the receiving hydrophone and \(z_{s}\) is the depth of sound source placement; \(G_{10}\) is the absolute value of the surface reflection coefficient. Separating out the time variable gives  
\[\mathrm{p}(\mathrm{r},\mathrm{z})=\frac{1}{R_{00}R_{10}}exp(i\mathrm{kR})\{R_ {10}exp[i\mathrm{k}(R_{00}-R)]-G_{10}R_{00}exp[ik(R_{10}-R)]\} \tag{2}\]  
where  
\[R_{00}-R=\sqrt{r^{2}+z^{2}}\Bigg{(}\sqrt{1+\frac{z_{s}^{2}-2z_{s}z}{r^{2}+z^{ 2}}}-1\Bigg{)} \tag{3}\]  
When the value of \(\mathrm{r}\) is greater than 3 times the depth of the sea, i.e., \((z_{s}^{2}-2z_{s}z)/(r^{2}+z^{2})\ll 1\), Formula (3) can be approximated:  
\[R_{00}-R\approx\sqrt{r^{2}+z^{2}}\Bigg{(}\sqrt{1+2.\frac{z_{s}^{2}-2z_{s}z}{2 (r^{2}+z^{2})}+\left[\frac{z_{s}^{2}-2z_{s}z}{2(r^{2}+z^{2})}\right]^{2}}-1 \Bigg{)}=\frac{z_{s}^{2}-2z_{s}z}{2(r^{2}+z^{2})} \tag{4}\]  
Similarly:  
\[R_{00}-R\approx\frac{z_{s}^{2}+2z_{s}z}{2(r^{2}+z^{2})} \tag{5}\]  
Substituting (4) and (5) into Equation (2), we obtain  
\[ \begin{array}{rl}\mathrm{p}=\frac{1}{R_{00}R_{10}}exp\Big{[}&ik\Big{(}R+ \frac{z_{s}^{2}}{2\sqrt{r^{2}+z^{2}}}\Big{)}\Big{]}\\ &\times\Big{\{}(R_{10}-G_{10}R_{00})cos\Big{[}k\Big{(}\frac{z_{s}z}{\sqrt{r^{2} +z^{2}}}\Big{)}\Big{]}\\ &-i(R_{10}-G_{10}R_{00})sin\Big{[}k\Big{(}\frac{z_{s}z}{\sqrt{r^{2}+z^{2}}} \Big{)}\Big{]}\Big{\}}\end{array}  \tag{6}\]  
Then, the mean square sound pressure in the sound field is  
\[ \begin{array}{rl}\overline{\mathrm{p}^{2}}&=\frac{1}{2}\frac{1}{\left(r^{2} +z^{2}+z^{2}\right)^{2}-4z_{s}^{2}z^{2}}\Big{[}\left(1+G_{10}^{2}\right) \left(r^{2}+z^{2}+z_{s}^{2}\right)+2\left(1-G_{10}^{2}\right)z_{s}z\\ &\qquad-2G_{10}\sqrt{\left(r^{2}+z^{2}+z_{s}^{2}\right)^{2}-4z_{s}^{2}z^{2}} cos\left(\frac{2k_{s}z}{\sqrt{r^{2}+z^{2}}}\right)\Big{]}\end{array}  \tag{7}\]When focusing on the sound field distribution in the central area of the water body, there is an approximate value of \(\mathbf{z}\approx z_{s}/2\); after substitution into Equation (7), this gives  
\[ \begin{array}{rl}\overline{\mathbf{p}^{2}}&=\frac{1}{2}\frac{1}{\left(r^{2}+ \frac{5}{4}z_{s}^{2}\right)^{2}-z_{s}^{4}}\left[\left(1+G_{10}^{2}\right)\left( r^{2}+\frac{5}{4}z_{s}^{2}\right)+\left(1-G_{10}^{2}\right)z_{s}^{2}\right.\\ &\qquad\left.-2G_{10}\sqrt{\left(r^{2}+\frac{5}{4}z_{s}^{2}\right)^{2}-z_{s}^{ 4}}cos\left(\frac{2 bz_{s}z}{\sqrt{r^{2}+z^{2}}}\right)\right]\end{array}  \tag{8}\]  
If the signal is a broadband signal, the following formula can be obtained:  
\[\overline{\overline{\mathbf{p}^{2}}_{\Delta f}}=\frac{1}{\Delta f}\int_{f_{0} -0.5\Delta f}^{f_{0}+0.5\Delta f}\overline{\mathbf{p}(f)^{2}}df \tag{9}\]  
Further derivation can be written as follows:  
\[