## 4 Results
### Algorithm Accuracy Verification  
Three types of targets, including a simple sphere, cylinder and complex corner reflector are shown in Figure 6, which are tested to demonstrate the accuracy and efficiency of our approach. Firstly, unstructured meshing is performed using ANSYS software to divide the target surface into several triangular facets. The maximum grid size is set to be n\(\lambda\), where \(\lambda\) stands for the acoustic wavelength and n is associated with the definite value of \(\lambda\), as well as the size of the target. The detailed geometry size and the number of triangular patches of the targets are listed in Table 2.  
To calculate the TS value of the targets, the incident direction is rotated around the \(Z\) axis from 0 degree to 180 degrees with the interval of 1 degree at a frequency of 15 KHz, and \(\varphi=90^{{}^{\circ}}\). Moreover, when the frequency is fixed, the ray tube size of the grid on the virtual aperture is smaller and the TS value of the targets is more accurate. For the conventional SBR algorithm, the number of ray tubes at a specified incident direction are listed in Table 3.  
The target strength is defined as  
\[\text{TS}=10\lg\left|\frac{I_{r}}{I_{i}}\right|_{r=1} \tag{5}\]  
with \(I_{i}\) and \(I_{r}\) being the incident intensity and echo intensity of sound waves, and \(\left.I_{r}\right|_{r=1}\) being the echo intensity at 1 m from the equivalent acoustic center of the target. A rigid sphere of radius a that satisfies ka \(\gg 1\) and has a target strength of  
\[\text{TS}=10\lg\frac{a^{2}}{4} \tag{6}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Target** & **Size (m)** & **Maximum Mesh Size** & **Triangle Number** & **Node Number** \\ \hline Sphere & 2 m \(\times\) 2 m \(\times\) 2 m & 0.05 (0.5\(\lambda\)) & 11,000 & 5628 \\ Cylinder & 2 m \(\times\) 2 m \(\times\) 6 m & 0.05 (0.5\(\lambda\)) & 32,328 & 16,445 \\ Corner reflector & 62 m \(\times\) 7.5 m \(\times\) 11 m & 0.1 (1\(\lambda\)) & 294,850 & 147,427 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The geometry size and the number of triangular patches of three targets. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Sphere** & **Cylinder** & **Corner Reflector** \\ \hline Angle & 0.15 mm (0.0015\(\lambda\)) & 0.5 mm (0.005\(\lambda\)) & 3 mm (0.03\(\lambda\)) \\
0\({}^{\circ}\) & & 3999 \(\times\) 3999 & 3667 \(\times\) 2500 \\
60\({}^{\circ}\) & 13,327 \(\times\) 13,327 & 3999 \(\times\) 14,469 & 3667 \(\times\) 18,431 \\
90\({}^{\circ}\) & & 3999 \(\times\) 14,399 & 3667 \(\times\) 20,500 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The number of ray tubes at a specified incident direction in conventional SBR. Figure 6: Three types of targets: (**a**) sphere, (**b**) cylinder, (**c**) corner reflector. and the theoretical TS value of a rigid sphere with a radius of 1 m is \(-6\) dB. What's more, for a cylindrical object of length \(L\) and radius a, as shown in Figure 6, the target strength is expressed as  
\[\text{TS}=10\text{lg}\left[\frac{aL^{2}}{2\lambda}\left(\frac{\sin\beta}{\beta} \right)^{2}\cos^{2}\theta\right] \tag{7}\]  
When the radius, length of cylinder and wavelength are set to 1 m, 6 m and 0.1 m, respectively, the TS value of the ends and abeam direction of the cylinder is 32.69 dB and 22.56 dB. For the conventional SBR algorithm, we only compare the GPU-based SBR result with its theoretical value. Figure 7 illustrates that the TS value of the underwater target that is calculated by the conventional SBR algorithm shows good agreement with the theoretical value that is introduced above. Moreover, as shown in Figure 7a, the mean square error of the data is 0.05; this deviation may be partly due to the fact that invalid ray tubes do not participate in calculating the field integral field integral, and the ideal situation is that the entire target surface participates in the field integral. Figure 8 demonstrates that the TS value of the target has a good convergence effect at high frequencies.