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In a shallow water waveguide, the low-frequency acoustic field can be viewed as a sum of normal modes. Warping transform provides an effective tool to filter the normal modes from the received signal of a single hydrophone, which can be used for source ranging and geoacoustic inversion. However, it should be noted that the conventional warping operator h(t) = t2+tr2 is only valid for a signal consisting of reflection dominated modes, where r represents the source range. In a waveguide with a strong thermocline or a surface channel where refracted modes dominate the received sound field, the dispersive characteristics of the waveguide become different and the performance of the warping operator h(t) = t2+tr2 will be significantly degraded. In this paper, the dispersive characteristics and warping transform of the refractive normal modes in a waveguide with a linearly decreased sound speed profile are discussed. The formulae for the horizontal wavenumber, the phase in frequency domain and the instantaneous phase in time domain of the refractive mode are deduced. Based on these formulae, the time warping and frequency warping operators verified by the simulated data are presented. Through time-axis stretching or compression, the time warping operator h(t) =tr-t2, where tr= r/c(h) and c(h) represents the bottom sound speed, can transform the refracted modes into single-tone components of frequencies determined by source range, sound speed gradient of water, bottom sound speed and mode number. The frequency warping operator h(f) = Df3, where D is a constant, can transform the refracted modes into separable impulsive sequences through frequency-axis stretching or compression and the time delay of the impulsive sequences changes linearly with the source range. As the warped modes are separated in time domain or frequency domain, these two operators can be used for filtering the refracted normal modes from the received signal. The theories in this paper are also applicable for refractive modes in the waveguide with a linearly increased sound speed profile or a linear variation of the square of the index of refraction (n2-linear sound speed profile).
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Keywords:
- warping transform /
- normal mode filtering /
- refractive normal mode
[1] Baraniuk R G, Jones D L 1995 IEEE T. Signal Proces. 43 2269
[2] Zeng J, Chapman N, Bonnel J 2013 J. Acoust. Soc. Am. 134 EL394
[3] Bonnel J, Nicolas B, Mars J I, Walker S C 2010 J. Acoust. Soc. Am. 128 719
[4] Bonnel J, Gervaise C, Nicolas B, Mars J I, Walker S C 2012 J. Acoust. Soc. Am. 131 119
[5] Bonnel J, Chapman N 2011 J. Acoust. Soc. Am. 130 EL101
[6] Bonnel J, Thode A M, Blackwell S B, Katherine K, Macrander A M 2014 J. Acoust. Soc. Am. 136 145
[7] Lu L C, Ma L 2015 Acta Phys. Sin. 64 024305 (in Chinese) [鹿力成, 马力 2015 物理学报 64 024305]
[8] Bonnel J, Touz G L, Nicolas B, Mars J I 2013 IEEE Signal Proc. Mag. 6 120
[9] Bonnel J, Gervaise C, Roux P, Nicolas B, Mars J I 2011 J. Acoust. Soc. Am. 130 61
[10] Touz G L, Nicolas B, Mars J I, Lacoume J 2009 IEEE Trans. Signal Proces 57 1783
[11] Niu H Q, Zhang R H, Li Z L 2014 J. Acoust. Soc. Am. 136 53
[12] Niu H Q, Zhang R H, Li Z L 2014 Sci. China: Ser. G 57 424
[13] Zhang R H, Li F H 1999 Sci. China: Ser. A 29 241 (in Chinese) [张仁和, 李风华 1999 中国科学A辑 29 241]
[14] Qi Y B, Zhou S H, Zhang R H, Ren Y 2015 Acta Phys. Sin. 64 74301 (in Chinese) [戚聿波, 周士弘, 张仁和, 任云 2015 物理学报 64 74301]
[15] Zhou S H, Qi Y B, Ren Y 2014 Sci. China: Ser. G 57 225
[16] Qi Y B, Zhou S H, Zhang R H, Ren Y 2015 J. Comput. Acoust. 23 1550003
[17] Qi Y B, Zhou S H, Ren Y, Liu J J, Wang D J, Feng X Q 2015 Acta Acoust. 40 144 (in Chinese) [戚聿波, 周士弘, 任云, 刘建军, 王德俊, 冯希强 2015 声学学报 40 144]
[18] Qi Y B, Zhou S H, Zhang R H, Zhang B, Ren Y 2014 Acta Phys. Sin. 63 044303 (in Chinese) [戚聿波, 周士弘, 张仁和, 张波, 任云 2014 物理学报 63 044303]
[19] Jensen F B, Kuperman W A, Porter M B, Schmidt H 2011 Computational Ocean Acoustics (2nd Ed.) (New York: Springer) p408
[20] Cockrell K L, Schmidt H 2011 J. Acoust. Soc. Am. 130 72
[21] Brekhovskih L M 1980 Waves in Layered Media (2nd Ed.) (New York: Academy Press) p6
[22] Bender C M, Orszag SA 1978 Advanced Mathematical Methods for Scientists and Engineers (New York: McGraw-Hill) p276
[23] Porter M B 1991 The KRAKEN Normal Mode Program (La Spezia: SACLANT Undersea Research Centre) p1
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[1] Baraniuk R G, Jones D L 1995 IEEE T. Signal Proces. 43 2269
[2] Zeng J, Chapman N, Bonnel J 2013 J. Acoust. Soc. Am. 134 EL394
[3] Bonnel J, Nicolas B, Mars J I, Walker S C 2010 J. Acoust. Soc. Am. 128 719
[4] Bonnel J, Gervaise C, Nicolas B, Mars J I, Walker S C 2012 J. Acoust. Soc. Am. 131 119
[5] Bonnel J, Chapman N 2011 J. Acoust. Soc. Am. 130 EL101
[6] Bonnel J, Thode A M, Blackwell S B, Katherine K, Macrander A M 2014 J. Acoust. Soc. Am. 136 145
[7] Lu L C, Ma L 2015 Acta Phys. Sin. 64 024305 (in Chinese) [鹿力成, 马力 2015 物理学报 64 024305]
[8] Bonnel J, Touz G L, Nicolas B, Mars J I 2013 IEEE Signal Proc. Mag. 6 120
[9] Bonnel J, Gervaise C, Roux P, Nicolas B, Mars J I 2011 J. Acoust. Soc. Am. 130 61
[10] Touz G L, Nicolas B, Mars J I, Lacoume J 2009 IEEE Trans. Signal Proces 57 1783
[11] Niu H Q, Zhang R H, Li Z L 2014 J. Acoust. Soc. Am. 136 53
[12] Niu H Q, Zhang R H, Li Z L 2014 Sci. China: Ser. G 57 424
[13] Zhang R H, Li F H 1999 Sci. China: Ser. A 29 241 (in Chinese) [张仁和, 李风华 1999 中国科学A辑 29 241]
[14] Qi Y B, Zhou S H, Zhang R H, Ren Y 2015 Acta Phys. Sin. 64 74301 (in Chinese) [戚聿波, 周士弘, 张仁和, 任云 2015 物理学报 64 74301]
[15] Zhou S H, Qi Y B, Ren Y 2014 Sci. China: Ser. G 57 225
[16] Qi Y B, Zhou S H, Zhang R H, Ren Y 2015 J. Comput. Acoust. 23 1550003
[17] Qi Y B, Zhou S H, Ren Y, Liu J J, Wang D J, Feng X Q 2015 Acta Acoust. 40 144 (in Chinese) [戚聿波, 周士弘, 任云, 刘建军, 王德俊, 冯希强 2015 声学学报 40 144]
[18] Qi Y B, Zhou S H, Zhang R H, Zhang B, Ren Y 2014 Acta Phys. Sin. 63 044303 (in Chinese) [戚聿波, 周士弘, 张仁和, 张波, 任云 2014 物理学报 63 044303]
[19] Jensen F B, Kuperman W A, Porter M B, Schmidt H 2011 Computational Ocean Acoustics (2nd Ed.) (New York: Springer) p408
[20] Cockrell K L, Schmidt H 2011 J. Acoust. Soc. Am. 130 72
[21] Brekhovskih L M 1980 Waves in Layered Media (2nd Ed.) (New York: Academy Press) p6
[22] Bender C M, Orszag SA 1978 Advanced Mathematical Methods for Scientists and Engineers (New York: McGraw-Hill) p276
[23] Porter M B 1991 The KRAKEN Normal Mode Program (La Spezia: SACLANT Undersea Research Centre) p1
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