Models for amplitude fluctuation of underwater acousticnarrow band signal based on modified modal scintillation index

An Liang Fang Shiliang Chen Lijun

(Key Laboratory of Underwater Acoustic Signal Processing of Ministry of Education， Southeast University， Nanjing 210096， China)

T he amplitude fluctuations of underwater narrow band signals have been studied for nearly 50 years.Researchers［1-4］summarized the most important causes of fluctuations in the power amplitudes of signals and noise propagating in the undersea acoustic environment.In recent years，it is still a focus in underwater signal processing.Katsnelson et al.［5-6］studied the frequency dependence and intensity fluctuations due to shallow water internal waves.Temporal variations of intensity fluctuations were found in experimental data.Comparisons of experimental results with theoretical estimates demonstrated good consistency.Colosi et al.［7-8］examined the second-and fourth-moment mode-amplitude statistics for lowfrequency ocean sound propagation through random soundspeed perturbations in deep and shallow-water environments.Nair et al.［9］evaluated the fluctuation observed in theReceivedsignal with relevance to underwater systems.They defined the fluctuation index k as the standard deviation of theReceivedsignal with noise normalized by the standard deviation of noise.Stojanovic et al.［10］studied the random signal variations in underwater communication systems introduced by surface waves， internal turbulence，fluctuations in the sound speed，and other small-scale phenomena.

Among the above causes leading to sound fluctuations，the source and receiver depth instability may be the most obvious one.According to the normal mode theory， the amplitude of the normal modes are related to the source and receiver depths as well as the sea depth，sound speed profiles， surface and bottom conditions， and the signal frequency.The scintillation of modal energies has often been used to characterize and understand acoustic wave propagation in a randomly fluctuating ocean waveguide.Premus［11］introduced the modal scintillation index(MSI)for the purpose of surface and submerged source discrimination in a shallow water waveguide.The MSI is defined as the variance of the modulus of modal excitation normalized by its expected value over some observation intervals.The MSI of different normal modes exhibits different distributions when theReceivedsignal contains sea noise and it directly casts the source classification problem as a binary hypothesis test.

The MSI definition has its evident deficiency.It depends on the source level and the source range intensively.In this paper，a modified modal scintillation index(MMSI)is defined as the variance of the modulus of modal excitation normalized by the square of its expected value over some observation intervals.It is proved in an analytical form that the MMSI is only depth dependent.The modal excitations obtained from simulations are used to calculate the MMSI-depth curve of sources with different sound levels and ranges under a noise-free condition.Then the pseudo-inverse mode filter is used to estimate the probability density functions(PDFs)of the MMSI with Gaussian noise in the sinusoidal signal.The simulation results of the normal mode propagation model show that sources at differ-ent depths have different distribution parameters with the same depth fluctuation variations.

1 Theory of Modified Modal Scintillation Index

1.1 Definition of modal scintillation index

According to the normal mode theory，the complex pressure field can be expressed in terms of a superposition of normal modes in the far field of an acoustic source［12］，

where z is the depth of the receiver;zsis the depth of the acoustic source;Φm(z)represents the eigenfunction of the m-th mode;A is the amplitude of the source signal;M is the number of the normal modes;and rsis the range between the source and the receiver.Suppose that the receiver is a vertical array with N hydrophones，and the depth of the i-th hydrophone is zi， i=1，2，…，N.The signal pReceivedby the hydrophone array can be written in a matrix form as

where Φ is the N × M matrix of normal mode functions;H is the M×1 vector of temporally varying modal excitations and depends on the depth of the acoustic source;and krmis the horizontal wavenumber.Premus［11］defined the modal scintillation index of the m-th mode as

For an acoustic source whose depth is fluctuating in response to wave interaction，the mode amplitude variance has a component which is a sensitive indicator of its mean depth.Mode amplitude fluctuations will exhibit high variance when the source is near the depth of a modal zero-crossing，where the derivative of the mode function is the maximum.Similarly，mode amplitudes fluctuate with a low variance for a source near a modal extremum，where the derivative of the mode function is zero.The critical property of the shallow waveguide which motivates the use of the modal scintillation index for binary depth classification is the fact that the normal modes are nearly sinusoidal and share a common zero-crossing at the surface as shown in Fig.1.Thus， a surface source with a given vertical motion variance will exhibit a high mode scintillation across all the modes.A submerged source with the same vertical motion variance will exhibit a low mode scintillation for at least one mode，due to its expected proximity to at least one modal extremum.

Fig.1 Conceptual description of physics-based depth discrimination using MSI

Although the physical mechanism of utilizing the MSI to discriminate surface source and submerged source is reasonable， the definition in Eq.(5)is not a self-normalized statistic.The value of the MSI depends remarkably on the source level and source range.And this will be analytically proved in the following section.

1.2 Performance analysis of modal scintillation index

Consider the isovelocity waveguide as shown in Fig.2.The sound speed is a constant value c at all depths.D is the waveguide depth and ρ is the density of water.

and the corresponding eigenfunctions are given by

where kzmis the vertical wavenumber and it is given by

Fig.2 Schematic of the isovelocity waveguide

The horizontal wavenumber can be calculated as［12］

Suppose that the vertical motion of acoustic source Δz is random and that it follows the Gaussian distribution N(0，σ2)， and the mean acoustic source depth is zs0.The eigenfunctions in Eq.(7)can be modified as

where C=(2ρ/D) .Since the approximation condition

is satisfied in most low frequency narrow band signals in the far field situations(low order normal modes are dominant)， Eq.(9)can be simplified as

Evidently， the eigenfunctions Φm(zs)follows the Gaussian distribution N(μΦ，σ2Φ)， where

where K=A/(krmrs)1/2.Using the properties of the Gaussian random variable，hm(zs)follows the Gaussian distribution N(μKΦ，σ2KΦ)， where

And hm(zs) follows the folded Gaussian distribution［13］.Its mathematical expectation and variance are given by

Substituting Eqs.(12)， (15)and(16)into Eq.(5)，the modal scintillation index can be expressed as

where erf(·)is the error function and has the maximum value of 1.When KC?1，further simplification can be made as

Since K is the function of source level A and source range rs，SImis a source level and source range dependent statistic.

1.3 Definition of modified modal scintillation index

In section 1.2， it is analytically proved that the values of the MSI depend not only on the source depth，but also on the source level and source range.It means that a near submerged source may also exhibit high mode scintillation，and the binary discrimination method using the MSI is only valid when the surface source and submerged source have the same K value.Since this condition cannot always be satisfied，another statistic independent of K must be defined to work robustly.Define the MMSI as

Using a similar simplifying method，S'Imare given in an approximative form as

Eq.(20)shows that S'Imis a source depth dependent statistic only and independent of the source level and the source range.

In practice，the MMSI must be estimated from theReceivedsignal with noise.The model with additive noise in signalsReceivedby the array is

where n is the additive ambient noise vector{n1，n2，…nN}Tand ni(i=1，2，…N)is the noise at the i-th hydrophone.The modal excitation vector can be computed from samples of theReceivedpressure field via the pseudo-inverse calculation given by

where Φ+=(ΦHΦ)-1ΦHrepresents the pseudo-inverse of Φ and the superscript H denotes conjugate transpose.Suppose that the additive noise n is the zero mean Gaussian random process，and^H is the unbiased estimation of H.Substituting Eq.(22)into Eq.(19)， the MMSI can be estimated from theReceivedsignal.

2 Simulation results and discussion

To illustrate the utility of the modified modal scintillation index，a simulation experiment is performed for the Pekeris waveguide［12］.The simulation geometry is depicted in Fig.3.The waveguide depth D is 100 m;the sound speed in the water c is 1 500 m/s;the density of the water ρ is 1 000 kg/m3;the sound speed in the bottom cbis 2 000 m/s;and the density of the bottom ρbis 1 000 kg/m3.The source is a narrow band source and its frequency is 70 Hz.The receiver is a fully spanning ver-tical array consisting of 41 hydrophones equally spaced at 2.5 m.There are six modes that can propagate in the Pekeris waveguide at the frequency of 70 Hz.The mode shapes are depicted in Fig.4.The vertical motion of source follows the Gaussian distribution N(0，1).

Fig.3 Schematic of the Pekeris waveguide

Fig.4 Normal modes in Pekeris waveguide

The simulation experiment consists of three parts.First，the MSI and the MMSI are calculated by using the analytical form for sources of different sound levels with the source depth varying from 5 to 90 m.Secondly， the MSI and the MMSI are calculated by using the analytical form for sources of different ranges with the source depth varying from 5 to 90 m.The first two parts are under the condition free of noise.In the third part， the MMSI distribution is simulated with noise in theReceivedsignal for surface and submerged sources.

2.1 MSI and MMSI of sources with different sound levels

The Kraken normal mode model［14］is used to calculate the eigenfunctions Φm， the horizontal wavenumber krmand the vertical wavenumber kzm.Then Φm， krmand kzmare inserted into Eq.(4)， Eq.(5)and Eq.(19)to calculate the modal excitations H analytically for acoustic source with the source levels of 160 and 130 dB.The source level is in units of dB re:1 μPa ·m.The source range is 5 km.The MSI-depth and MMSI-depth curves are depicted in Fig.5.

The results in Fig.5 show that both the MSI and the MMSI exhibit high mode amplitude fluctuation variances at the depth of modal zero-crossing.In Figs.5(a)， (c)and(e)，there is a great disparity between the two curves of sources with different source levels.In Figs.5(b)，(d)and(f)，the two curves of sources with different source levels match well.This means that the MSI is a source level sensitive statistic while the MMSI is not.

Fig.5 MSI-depth and MMSI-depth curves for different source levels.(a)MSI-depth curve of Mode 1;(b)MMSI-depth curve of Mode 1;(c)MSI-depth curve of Mode 3;(d)MMSI-depth curve of Mode 3;(e)MSI-depth curve of Mode 5;(f)MMSI-depth curve of Mode 5

2.2 MSI and MMSI of sources with different ranges

The same algorithm as in Section 2.1 is used to calculate the modal excitations H for acoustic source with the source ranges of 5 and 10 km.The source level is 160 dB.The results are depicted in Fig.6.

Fig.6 MSI-depth and MMSI-depth curves for different source ranges.(a)MSI-depth curve of Mode 1;(b)MMSI-depth curve of Mode 1;(c)MSI-depth curve of Mode 3;(d)MMSI-depth curve of Mode 3;(e)MSI-depth curve of Mode 5;(f)MMSI-depth curve of Mode 5

The results in Fig.6 show that both the MSI and the MMSI exhibit high mode amplitude fluctuation variances at the depth of modal zero-crossing.In Figs.6(a)， (c)and(e)，there is a small disparity between the two curves of sources with different source levels.In Figs.6(b)， (d)and(f)，the two curves of sources with different source levels match well.This means that the range variation does not affect the MSI as greatly as the source level variation does.The MMSI is a self-normalized statistic and independent of source range.

2.3 MMSI probability density functions estimation

In Section 2.1 and Section 2.2， both the MSI and the MMSI are analytically calculated without noise.But in real applications， the ambient noise cannot be ignored.The hydrophone array receives the signal from the acoustic source as well as the ambient noise.The noise field in this simulation experiment is modeled as the spatial white Gaussian noise with a noise spectrum level of 65 dB re:1 μPa·m.The Kraken normal mode model is also used to calculate Φm， krmand kzm.Considering the effect of the ambient noise，only the estimation of the modal excitations H can be obtained from Eq.(22)by using the pseudo-inverse mode filter.In this case， the statistic MMSI is modeled as a distribution depending on the vertical motion and ambient noise.

A submerged source and a surface source are considered in the simulation experiment.Fig.7 shows the estimated PDFs of six modified modal scintillation indices obtained from Monte Carlo simulations using 1 000 trials under each hypothesis.The submerged source PDF is denoted as p (MMSI| Hsub)and the surface source PDF is denoted as p (MMSI| Hsurf).The source range is 5 km and the source level is 160 dB in each case.

Fig.7 shows that the PDFs of the modified modal scintillation indices of the submerged source separates to the left of that of the surface source in the direction of small values of the MMSI in most cases except for mode 4.The positions of the PDFs of surface sources are relatively fixed on the axis of log(MMSI)，while the positions of the PDFs of submerged sources shifted along the axis of log(MMSI).This phenomenon shows that the surface source is near the depth of modal zero-crossing for all the modes and exhibits a high variance of mode amplitudes.The submerged source is near the depth of mode zerocrossing for some modes(mode 3 and mode 4)and near the depth of mode extremums for other modes(mode 1，mode 2， mode 5 and mode 6).This attribute can be used to discriminate the submerged and surface sources.

Fig.7 Estimated PDF for MMSI.(a)Mode 1;(b)Mode 2;(c)Mode 3;(d)Mode 4;(e)Mode 5;(f)Mode 6

3 Conclusion

A modified modal scin tillation index is proposed in this paper.It is analytically proved that the MMSI is a depth dependent signature and independent of the source level and range under the condition of the ideal waveguide，while the MSI is both source level and range dependent.A simulation experiment which consists of three parts is performed for the Pekeris waveguide to illustrate the utility of the modified modal scintillation index.The simulation results show that the MMSI is a self-normalized statistic while the MSI is not.The MMSI probability density functions of submerged and surface sources separate from each other in most modes with the same vertical motion variance.And this attribute can be regarded as the signature to discriminate underwater acoustic sources.

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