1.School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, China 2.Key Laboratory of Ocean Acoustics and Sensing, Northwestern Polytechnical University, Xi’an 710072, China 3.College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310058, China
Fund Project:Project supported by the Key Program of the National Natural Science Foundation of China (Grant No. 11534009)
Received Date:23 December 2019
Accepted Date:12 May 2020
Available Online:19 May 2020
Published Online:20 August 2020
Abstract:In this paper, the modal space detector (MSD) is investigated in shallow water environment when utilizing a vertical linear array. The processing gain of the MSD is derived, and the result demonstrates that the processing gain of the MSD degrades when the number of the propagated normal modes excited by the underwater acoustic source increases, and therefore the detection performance of the MSD decreases. By exploiting the orthogonality among the modal depth functions, the MSD can be decomposed into a group of modal subspace detectors (MSSDs). The processing gains of these MSSDs are derived as well and it is found out that the processing gain of a MSSD is in direct proportion to its corresponding modal attenuation coefficients. By designing a group of weighting coefficients based on the mode attenuation law, a weighted modal subspace detector (WMSSD) is proposed to alleviate the degradation of the processing gain processing of the MSD. We analyze the influences of acoustic source locations and sound velocity profiles (SVPs) on the detection performance of the WMSSD theoretically, and verify the theoretical analyses by comparing its processing gain with the MSD in simulation experiments. The results show that the WMSSD presents various processing gains versus different acoustic source locations. In the waveguide having a negatively-gradient SVP, there exists a ‘weak detection area’ for the WMSSD, that is, the processing gain of the WMSSD is smaller than that of the MSD when the acoustic source locations are close to sea surface. The reason is because there are inversion points on the lower-order modal depth functions and the depths of the inversion points are close to sea surface. In other most areas, the processing gain of the WMSSD is larger (even remarkably larger) than that of the MSD. In the waveguide having a positively-gradient SVP, due to the phenomenon that the modal inversion points of the lower-order modal depth functions are near sea floor, there is a contrary consequence, that is, the ‘weak detection area’ is close to sea floor. And meanwhile the WMSSD outperforms the MSD in other most areas as well. There are no modal inversion points in the waveguide having a constant SVP, and therefore the WMSSD always outperforms the MSD. Keywords:modal space detection/ modal attenuation coefficient/ processing gain/ modal inversion point
假设声源位于距离为18 km、深度为50 m处, 仿真中通过提高声源频率来增大波导中的传播模态数. 图2给出了MSD的检测概率随信噪比和虚警概率变化的曲线, 即检测性能曲线, 其中窄带声源中心频率f分别为50, 100, 300, 500和700 Hz时在波导中激发的传播模态数分别为3, 5, 16, 27和37. 从图2可以看出, 相同信噪比条件下, 随着传播模态数的增多, 对应MSD的检测概率随之下降, 下降幅度逐渐减缓. 图 2 不同传播模态数下MSD的检测性能曲线 (a) 检测概率随输入信噪比的变化, ${P_{{\rm{FA}}}} = 0.1$; (b) 检测概率随虚警概率的变化, $SNR = - 15\;{\rm{ dB}}$ Figure2. Detection performance curves of the MSD under various numbers of normal modes: (a) Probabilities of detection versus SNRs, ${P_{{\rm{FA}}}} = 0.1$; (b) probabilities of detection versus probabilities of false alarm, $SNR = - 15\;{\rm{dB}}$.
为进一步探究MSD检测性能变化的原因, 图3给出了不同模态数下MSD的处理增益, 对图中处理增益取对数表示, 即10lgG, 单位为分贝(dB). 由图3可知, 随着模态个数增多, MSD的处理增益逐渐减小. 这是由于随着传播模态数的增多, 引入了更多的噪声分量, 而信号能量保持不变, 从而导致MSD的处理增益下降. 同时也注意到随模态数的增加处理增益下降的速度在减缓, 这与检测性能曲线呈现的结果相一致. 图 3 MSD的处理增益随模态数的变化曲线 Figure3. The processing gains of the MSD versus the numbers of normal modes.
声源位置同上, 图4给出了不同传播模态数下归一化的各阶MSSD处理增益(这里为便于展示${G_m}$的变化趋势未对其取对数)的分布, 其中阶数大于8的处理增益都近似为零, 图中未予展示. 由图4可知, 尽管模态数增多, 处理增益仍主要集中在少数几阶MSSD, 且各阶MSSD的分布呈现随阶数的增大逐渐减小的趋势; 此外, 还观察到当传播模态数较多时, 各阶MSSD的处理增益分布存在起伏. 由文中(24)式可知, 第m阶MSSD的处理增益不仅与模态衰减系数有关, 它使得MSSD的处理增益随阶数呈下降趋势; 此外, 第m阶MSSD的处理增益还与声源深度上的模态函数幅值$\phi _m^{}\left( {{z_{\rm{s}}}} \right)$有关, 由于$\phi _m^{}\left( {{z_{\rm{s}}}} \right)$随阶数振荡变化, 尤其是当传播模态数较多时, $\phi _m^{}\left( {{z_{\rm{s}}}} \right)$随阶数振荡变化的更剧烈, 进而使得各阶MSSD的处理增益存在起伏; 例如当M = 37时, 此时$\phi _1^{}\left( {{z_{\rm{s}}}} \right) < \phi _2^{}\left( {{z_{\rm{s}}}} \right)$使得第一阶MSSD的处理增益小于第二阶MSSD. 图 4 各阶MSSD处理增益随阶数的变化 Figure4. The processing gains of MSSD versus the orders of normal modes.
声源位置同上. 利用(17)式和(26)式分别计算了该声源位置处MSD, WMSSD和OWMSSD的处理增益, f = 100 Hz时对应的处理增益分别为17.41 dB, 20.10 dB和20.11 dB, f = 300 Hz时对应的处理增益分别14.90 dB, 19.18 dB和19.25 dB. 可以看出, WMSSD的处理增益相较于MSD有很大幅度的提升, 并且接近于OWMSSD. 图5和图6分别给出了这两种频率(100和300 Hz)下相应的MSD, WMSSD和OWMSSD的检测性能曲线. 由图5和图6可知, 对于该位置的声源, WMSSD与OWMSSD的检测性能曲线几乎重合, 即WMSSD实现了理论上所能达到的最优检测性能, 相较于MSD有了显著的提升; 取检测概率为0.5时所需的信噪比门限为比较对象(下同), 对于100和300 Hz的声源, WMSSD的信噪比门限分别下降了1.5和3 dB. 对比图5和图6也可看出, 虽然WMSSD对较高频率声源的性能提升幅度较大, 但当频率增大时其检测性能仍有所下降, 这由传播模态数增多导致. 图 5 MSD, WMSSD, OWMSSD的检测性能曲线, f = 100 Hz (a) 检测概率随输入信噪比的变化, ${P_{{\rm{FA}}}} = 0.1$; (b) 检测概率随虚警概率的变化, $SNR = - 15\;{\rm{dB}}$ Figure5. Detection performance curves of the MSD, WMSSD and OWMSSD with f = 100 Hz: (a) Probabilities of detection versus SNR, ${P_{{\rm{FA}}}} = 0.1$; (b) probabilities of detection versus probabilities of false alarm, $SNR = - 15\;{\rm{dB}}$.
图 6 MSD, WMSSD, OWMSSD的检测性能曲线, f = 300 Hz (a) 检测概率随输入信噪比的变化, ${P_{{\rm{FA}}}} = 0.1$; (b) 检测概率随虚警概率的变化, $SNR = - 15\;{\rm{dB}}$ Figure6. Detection performance curves of the MSD, WMSSD and OWMSSD with f = 300 Hz: (a) Probabilities of detection versus SNR, ${P_{{\rm{FA}}}} = 0.1$; (b) probabilities of detection versus probabilities of false alarm, $SNR = - 15\;{\rm{dB}}$.
式中, ${z_{{\rm{rev}}}}$为反转点处深度. 由(28)式可知, 由于各阶水平波数不同, 各阶模态深度函数具有迥异的反转点深度. 由于各阶水平波数随阶数增大而逐渐减小, 高阶模态函数的反转点处声速更大. 因此对于负梯度声速剖面, 各阶模态函数的反转点深度将随阶数增大而变小. 以图1中的负梯度声速波导为例, 各阶模态函数及其反转点深度如图8所示. 由图8(b)可知, 反转点深度随阶数增大而逐渐变小. 结合图8(a)可知, 在反转点以浅, 模态函数呈指数迅速衰减, 相应的模态函数值很小; 在反转点以深, 模态函数呈周期性的振荡变化. 对于较高阶模态函数, 若$c\left( z \right) < 2{\text{π}}f/{k_{rm}}$对所有深度都成立, 此时不存在反转点或者说反转深度为零, 如图8(b)中第8阶及以后的模态函数不再存在反转点. 图 8 各阶模态函数及其反转点深度, f = 300 Hz (a) 波导环境中的各阶模态函数分布; (b) 各阶模态函数的反转点深度 Figure8. The modal depth functions and their turning-depths with f = 300 Hz: (a) Each modal depth function in the waveguide; (b) the turning-depth of each modal depth function.
当声源位于若干阶模态函数的反转深度以浅时, 相应的声源模态幅值${\phi _m}\left( {{z_{\rm{s}}}} \right)$很小, 这种情况下由于声源模态幅值的作用, 各阶MSSD处理增益的变化趋势为低阶和高阶较小, 中阶较大, 这与加权系数的变化趋势不一致, 如图9(a)所示, 导致WMSSD的处理增益较差并小于MSD. 当声源位于各阶反转深度以深时, 加权系数较大程度地刻画了各阶MSSD处理增益的变化趋势, 如图9(b)所示, 进而WMSSD的处理增益较大并大于MSD. 当声源位于某一深度上时, WMSSD的处理增益与MSD相等, 该深度即为图7(b)中的临界深度. 图 9 各阶MSSD的加权系数与处理增益, f = 300 Hz, 声源距离15 km (a)声源深度10 m; (b) 声源深度80 m Figure9. The weighting coefficients and the processing gains of the MSSD with f = 300 Hz and source range of 15 km: (a) Source depth of 10 m; (b) source depth of 80 m.
对于不同频率的声源, 临界深度随声源距离的变化如图10所示. 由图10可知, 临界深度随声源距离的增大而减小. 这是由于当声源距离增大时, 衰减指数的作用更加剧烈, 信号能量更多的集中在低阶MSSD, 相较于图9(a), 图11所示的各阶MSSD归一化的加权系数与处理增益的变化趋势更加一致, 进而WMSSD的处理增益提高, 临界深度变浅. 同时由图10可以看出, 随声源距离增大低频声源临界深度更快减小为零. 这是由于当声源频率增大时, 模态个数增加, 存在反转点的模态数也增多, 因而声源模态幅值的影响增大, 最终导致临界深度随声源距离的变化速率减缓. 图 10 不同频率时临界深度随距离的变化图 Figure10. The critical depths versus ranges under various frequencies.
图 11 各阶MSSD归一化的加权系数与处理增益, f = 300 Hz, 声源深度10 m, 声源距离25 km Figure11. The weighting coefficients and the processing gains of the MSSD with f = 300 Hz, source depth of 10 m and source range of 25 km.
图 13 各阶模态函数的反转深度, f = 300 Hz Figure13. The turning-depth of each modal depth function with f = 300 Hz.
图 14 两种声速剖面波导中的各阶模态函数, f = 300 Hz (a) 等声速剖面; (b) 正梯度声速剖面 Figure14. Each modal depth function in the two kinds of waveguides with f = 300 Hz: (a) Constant SVP; (b) positive gradient SVP.
图15给出了两种声速剖面波导下WMSSD的处理增益(单位为dB), 图中黑线标出了不同距离上的临界深度. 在等声速波导中, 各阶模态函数不存在反转点且呈周期振荡变化, 声源深度上模态函数幅值对各阶MSSD处理增益的影响较小, 因此当声源位于不同深度上时, 各阶MSSD处理增益的变化趋势总是与各阶衰减指数相一致, 使得WMSSD的处理增益在所有深度上都大于MSD, 因此不存在临界深度, 如图15(a)所示. 在正梯度声速波导中, 各阶模态函数的反转深度随阶数增大而增大(与负梯度声速的情况相反), 在反转点以深, 模态函数呈指数衰减, 并很快接近于零. 当声源位于若干阶模态的反转深度以深时, 声源模态幅值对各阶MSSD的非中心参量产生显著影响, 且随着深度增加, 产生的影响也增大. 因此, WMSSD的处理增益下降, 进而出现临界深度, 在临界深度以深WMSSD的处理增益小于MSD, 如图15(b)所示. 而在临界深度以浅的大部分区域, WMSSD的处理增益大于MSD. 临界深度随距离变化的原因与负梯度声速波导相同. 对比图15(a)和图15(b)也可知, 等声速波导中WMSSD的处理增益大于正梯度声速波导. 图 15 两种声速剖面下, 不同声源位置处的WMSSD处理增益, f = 300 Hz (a)等声速剖面; (b) 正梯度声速剖面 Figure15. The processing gains of the WMSSD versus acoustic source locations with f = 300 Hz: (a) Constant SVP; (b) positive gradient SVP.