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The polarization of the acoustic field in the ocean waveguide environment is a unique property that can be measured by using a particle velocity sensor in the water column. It can provide new ideas for locating and detecting the underwater target , so it is interesting to study the polarization. The polarization of a monochromatic signal has been described by the Stokes parameters, a set of four real-valued quantities in previous work. In this work, the Stokes parameters are extended to the broadband form, and the expression is simplified by using the nonstationary phase approximation, which reduces the complexity of the theoretical derivation and reveals the physical mechanism behind the significant variations in polarization with source depth and symmetrical depth. Theoretical analysis shows that the polarization characteristics in the ideal waveguide vary significantly in the sea surface, the sea bottom, the depth of the sound source and symmetrical depth. In this work the numerical simulation is used to verify the theoretical analysis and study the relationship between range and integral bandwidth when nonstationary phase approximation method is effective. The numerical results demonstrate that the simplified expression using the nonstationary phase approximation is effective and can better characterize the depth distribution characteristics of the polarization. Additionally, by normalizing the broadband Stokes parameters, the effect of range on the depth distribution characteristics of polarization can be removed. It means that the normalized broadband Stokes parameters are in theory free of the range and depend on the environment, the receiver depth and the source depth, which have the potential to be used for source depth estimation. Subsequently, focusing on normalized broadband Stokes parameters, we analyzes the effects of parameters such as source frequency, source depth, sound speed profile and water depth on the depth distribution characteristics of polarization. The analysis results show that environmental factors have great influence on the depth distribution characteristics of polarization. In the end, the validity of the nonstationary phase approximation and the range-independent property of the normalized broadband Stokes parameters are verified by the results of the RHUM-RUM experimental data processing. The findings provide a theoretical basis for passive target depth estimation based on polarization.
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Keywords:
- polarization /
- vector acoustic field /
- shallow sea /
- Stokes parameters
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图 1 横波的极化现象 (a) 线极化; (b) 圆极化; (c) 椭圆极化
Figure 1. Polarization of transverse wave: (a) Linear polarization; (b) circular polarization; (c) elliptical polarization.
图 2 质点椭圆运动轨迹
Figure 2. Elliptical path of acoustic particle motion.
图 3 归一化Stokes参数与极化状态的关系
Figure 3. Relation between normalized Stokes parameters and polarization states.
图 4 ${D_n}(z)$随$z$变化曲线
Figure 4. Curve of ${D_n}(z)$ with $z$.
图 5 ${E_n}(z)$随$z$变化曲线
Figure 5. Curve of ${E_n}(z)$ with z.
图 6 理想固体海底等声速波导环境下宽带Stokes参数深度分布曲线 (a) $S_0^{\left[ {{f_1}, {f_2}} \right]}$; (b) $S_1^{\left[ {{f_1}, {f_2}} \right]}$; (c) $S_2^{\left[ {{f_1}, {f_2}} \right]}$; (d) $S_3^{\left[ {{f_1}, {f_2}} \right]}$
Figure 6. Broadband Stokes parameters depth distribution curves for an ideal solid seabed isovelocity environment: (a) $S_0^{\left[ {{f_1}, {f_2}} \right]}$; (b) $S_1^{\left[ {{f_1}, {f_2}} \right]}$; (c) $S_2^{\left[ {{f_1}, {f_2}} \right]}$; (d) $S_3^{\left[ {{f_1}, {f_2}} \right]}$.
图 7 不同积分带宽条件下$S_0^{\left[ {{f_1}, {f_2}} \right]}$的误差值随距离的变化
Figure 7. Error value with range for different integration bandwidth.
图 8 非平稳相位近似成立距离${r_1}$与积分带宽的关系
Figure 8. Relationship between ${r_1}$ and integration bandwidth
图 9 Pekeris波导环境下宽带Stokes参数深度分布曲线 (a) $S_0^{\left[ {{f_1}, {f_2}} \right]}$; (b) $S_1^{\left[ {{f_1}, {f_2}} \right]}$; (c) $S_2^{\left[ {{f_1}, {f_2}} \right]}$; (d) $S_3^{\left[ {{f_1}, {f_2}} \right]}$
Figure 9. Broadband Stokes parameters depth distribution curves for Pekeris waveguide: (a) $S_0^{\left[ {{f_1}, {f_2}} \right]}$; (b) $S_1^{\left[ {{f_1}, {f_2}} \right]}$; (c) $S_2^{\left[ {{f_1}, {f_2}} \right]}$; (d) $S_3^{\left[ {{f_1}, {f_2}} \right]}$.
图 10 Pekeris波导环境下归一化宽带Stokes参数深度分布曲线 (a) 归一化宽带Stokes参数$s_1^{\left[ {{f_1}, {f_2}} \right]}$; (b) 归一化宽带Stokes参数$s_3^{\left[ {{f_1}, {f_2}} \right]}$
Figure 10. Normalized broadband Stokes parameters depth distribution curves for Pekeris waveguide: (a) Normalized broadband Stokes parameter $s_1^{\left[ {{f_1}, {f_2}} \right]}$; (b) normalized broadband Stokes parameter $s_3^{\left[ {{f_1}, {f_2}} \right]}$.
图 11 Pekeris波导环境下归一化宽带Stokes参数的深度-距离分布 (a) 归一化宽带Stokes参数$s_1^{\left[ {{f_1}, {f_2}} \right]}$; (b) 归一化宽带Stokes参数$s_3^{\left[ {{f_1}, {f_2}} \right]}$
Figure 11. Depth-range distribution of normalized broadband Stokes parameters for Pekeris waveguide: (a) Normalized broadband Stokes parameter $s_1^{\left[ {{f_1}, {f_2}} \right]}$; (b) normalized broadband Stokes parameter $s_3^{\left[ {{f_1}, {f_2}} \right]}$.
图 12 不同频率范围的归一化宽带Stokes参数深度分布曲线 (a) $s_1^{\left[ {{f_1}, {f_2}} \right]}$(带宽$ {f_{{\text{c1}}}} $—125 Hz); (b) $s_1^{\left[ {{f_1}, {f_2}} \right]}$(带宽$ {f_{{\text{c1}}}} $—250 Hz); (c) $s_1^{\left[ {{f_1}, {f_2}} \right]}$(带宽$ {f_{{\text{c1}}}} $—500 Hz) ; (d) $s_3^{\left[ {{f_1}, {f_2}} \right]}$(带宽$ {f_{{\text{c1}}}} $—125 Hz) ; (e) $s_3^{\left[ {{f_1}, {f_2}} \right]}$(带宽$ {f_{{\text{c1}}}} $—250 Hz) ; (f) $s_3^{\left[ {{f_1}, {f_2}} \right]}$(带宽$ {f_{{\text{c1}}}} $—500 Hz)
Figure 12. Normalized broadband Stokes parameters depth distribution curves for different frequency ranges: (a) $s_1^{\left[ {{f_1}, {f_2}} \right]}$($ {f_{{\text{c1}}}} $—125 Hz); (b) $s_1^{\left[ {{f_1}, {f_2}} \right]}$($ {f_{{\text{c1}}}} $—250 Hz) ; (c) $s_1^{\left[ {{f_1}, {f_2}} \right]}$($ {f_{{\text{c1}}}} $—500 Hz) ; (d) $s_3^{\left[ {{f_1}, {f_2}} \right]}$($ {f_{{\text{c1}}}} $—125 Hz) ; (e) $s_3^{\left[ {{f_1}, {f_2}} \right]}$($ {f_{{\text{c1}}}} $—250 Hz) ; (f) $s_3^{\left[ {{f_1}, {f_2}} \right]}$($ {f_{{\text{c1}}}} $—500 Hz)
图 13 不同声源深度的归一化宽带Stokes参数深度分布 (a) $s_1^{\left[ {{f_1}, {f_2}} \right]}$; (b) $s_3^{\left[ {{f_1}, {f_2}} \right]}$
Figure 13. Normalized broadband Stokes parameters depth distributions for different sound source depths: (a) $s_1^{\left[ {{f_1}, {f_2}} \right]}$; (b) $s_3^{\left[ {{f_1}, {f_2}} \right]}$.
图 14 声速剖面示意图
Figure 14. Sound speed profile.
图 15 不同声速剖面的归一化宽带Stokes参数深度分布曲线 (a) $s_1^{\left[ {{f_1}, {f_2}} \right]}$(正梯度); (b) $s_1^{\left[ {{f_1}, {f_2}} \right]}$(负梯度); (c) $s_1^{\left[ {{f_1}, {f_2}} \right]}$(负跃层); (d) $s_3^{\left[ {{f_1}, {f_2}} \right]}$(正梯度); (e) $s_3^{\left[ {{f_1}, {f_2}} \right]}$(负梯度); (f) $s_3^{\left[ {{f_1}, {f_2}} \right]}$(负跃层)
Figure 15. Normalized broadband Stokes parameters depth distribution curves for different sound speed profiles: (a) $s_1^{\left[ {{f_1}, {f_2}} \right]}$(positive gradient); (b) $s_1^{\left[ {{f_1}, {f_2}} \right]}$(negative gradient); (c) $s_1^{\left[ {{f_1}, {f_2}} \right]}$(negative thermocline); (d) $s_3^{\left[ {{f_1}, {f_2}} \right]}$(positive gradient); (e) $s_3^{\left[ {{f_1}, {f_2}} \right]}$(negative gradient); (f) $s_3^{\left[ {{f_1}, {f_2}} \right]}$(negative thermocline).
图 16 不同海深环境的归一化宽带Stokes参数深度分布曲线 (a) $s_1^{\left[ {{f_1}, {f_2}} \right]}$(海深50 m); (b) $s_1^{\left[ {{f_1}, {f_2}} \right]}$(海深100 m); (c) $s_1^{\left[ {{f_1}, {f_2}} \right]}$(海深200 m); (d) $s_3^{\left[ {{f_1}, {f_2}} \right]}$(海深50 m); (e) $s_3^{\left[ {{f_1}, {f_2}} \right]}$(海深100 m); (f) $s_3^{\left[ {{f_1}, {f_2}} \right]}$(海深200 m)
Figure 16. Normalized broadband Stokes parameters depth distribution curves for different depths of water column: (a) $s_1^{\left[ {{f_1}, {f_2}} \right]}$(depth 50 m); (b) $s_1^{\left[ {{f_1}, {f_2}} \right]}$(depth 100 m); (c) $s_1^{\left[ {{f_1}, {f_2}} \right]}$(depth 200 m); (d) $s_3^{\left[ {{f_1}, {f_2}} \right]}$(depth 50 m); (e) $s_3^{\left[ {{f_1}, {f_2}} \right]}$(depth 100 m); (f) $s_3^{\left[ {{f_1}, {f_2}} \right]}$(depth 200 m).
图 17 声压通道倒谱图
Figure 17. Cepstrogram of pressure channel.
图 18 目标方位估计结果
Figure 18. Results of ship azimuth estimation.
图 19 质点振速信号的时频图 (a) 水平质点振速${v_r}$; (b) 垂直质点振速${v_z}$
Figure 19. Spectrogram of the particle velocity: (a) horizontal particle velocity ${v_r}$; (b) vertical particle velocity ${v_z}$.
图 20 归一化宽带Stokes参数随时间的变化 (a) $s_1^{\left[ {{f_1}, {f_2}} \right]}$; (b) $s_2^{\left[ {{f_1}, {f_2}} \right]}$; (c) $s_3^{\left[ {{f_1}, {f_2}} \right]}$
Figure 20. Time-varying curves of the normalized broadband Stokes parameters: (a) $s_1^{\left[ {{f_1}, {f_2}} \right]}$; (b) $s_2^{\left[ {{f_1}, {f_2}} \right]}$; (c) $s_3^{\left[ {{f_1}, {f_2}} \right]}$.
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