D. Benjamin Reeder: JASA Express Letters
[http://dx.doi.org/10.1121/1.4828979]
Published Online 10 December 2013
Clutter depth discrimination using the wavenumber spectrum D. Benjamin Reeder Department of Oceanography, Naval Postgraduate School, 833 Dyer Road, Monterey, California 93943 [email protected]
Abstract: Clutter depth is a key parameter in mid-frequency active sonar systems to discriminate between sources of clutter and targets of interest. A method is needed to remotely discriminate clutter depth by information contained in the backscattered signal—without a priori knowledge of that depth. Presented here is an efficient approach for clutter depth estimation using the structure in the wavenumber spectrum. Based on numerical simulations for a simple test case in a shallow water waveguide, this technique demonstrates the potential capability to discriminate between a clutter source in the water column vs one on the seabed. PACS numbers: 43.40.Vn, 43.60.Jn, 43.60.Pt [JL] Date Received: August 20, 2013 Date Accepted: October 22, 2013
1. Introduction Constructive and destructive interference processes produced by multipath propagation in the shallow water environment are exhibited in the complex structure of acoustic intensity patterns when plotted vs range and depth. This structure is a function of source/receiver geometry, acoustic signal characteristics, and physical parameters of the waveguide (e.g., sound speed profile, seabed topography, and acoustic properties). The acoustic signal observed by a receiver as a function of time and space therefore contains information about the source signal and the medium through which it propagates. For the case of an active sonar system, the backscattered signal contains information regarding the waveguide and acoustic scatterers, many of which are considered unwanted sources of clutter. For the case of a military sonar system, determining whether the source of a scattered signal is located on the seabed or occupies a position in the water column represents a significant capability to discriminate between sources of clutter and targets of interest. Presented here is an efficient method to remotely discriminate clutter source depth by information contained in the backscattered signal— without a priori knowledge of that depth. The intent of this paper is threefold: (a) Present the theoretical basis of the approach, (b) discuss the physics and rationale for using the energy which propagates along high-angle paths represented in the traditionally ignored continuous region of the wavenumber spectrum, and (c) extend the approach to clutter depth discrimination in the shallow water waveguide via numerical simulation in a simple test case to demonstrate the concept. 2. Theoretical development In a horizontally stratified (range-independent) waveguide with water column thickness h, sound speed c(z), and constant density q, the acoustic pressure due to a point source at (r, z) ¼ (0, zs) with harmonic time dependence (eixt) satisfies the inhomogeneous wave equation 1@ @ @2 dðrÞdðz zs Þ 2 r þ 2 þ k ðzÞ pðr; z; zs Þ ¼ 2 ; (1) @z r @r @r r
J. Acoust. Soc. Am. 135 (1), January 2014
EL1
D. Benjamin Reeder: JASA Express Letters
[http://dx.doi.org/10.1121/1.4828979]
Published Online 10 December 2013
where kðzÞ ¼ x=cðzÞ. Following the development of Frisk (1994), the application of the inverse zero-order Hankel transform operator to both sides of Eq. (1) gives the depth-dependent Helmholtz equation d2 2 2 þ ðk ðzÞ kr Þ gðkr ; z; zs Þ ¼ 2dðz zs Þ; (2) dz2 where the separation constant, kr, is the horizontal component of the acoustic wave vector and pðr; z; zs Þ is the solution to Eq. (2) for a given kr. The conjugate Hankel transform pair relationship between the pressure field and the depth-dependent Green’s function is ð1 gðkr ; z; zs ÞJ0 ðkr rÞkr dkr ; (3) pðr; z; zs Þ ¼ 0
gðkr ; z; zs Þ ¼
ð1
pðr; z; zs ÞJ0 ðkr rÞrdr;
(4)
0
where J0 ðkr rÞ is the zero-order Bessel function, and r is the range between the source at depth zs and receiver at depth z. The full solution for the pressure field due to a point source in a homogeneous (constant density and sound speed) fluid layer with arbitrary horizontally stratified boundaries [Frisk, 1994, p. 173, Eq. (6.19)] is expressed as # ð 1 " ikz jzzs j i e þ RS eikz ðzþzs Þ þ RB e2ikz h eikz ðzþzs Þ þ RS eikz jzzs j pðr; z; zs Þ ¼ J0 ðkr rÞkr dkr : kz ½1 RS RB e2ikz h 0 (5) The term inside the brackets is the depth-dependent Green’s function for this waveguide, which can be viewed as the waveguide’s spatial transfer function. The numerator of the integrand represents the finite-valued continuum of sinusoidal constituents of the pressure field which is a function of the source depth zs, receiver depth z, water depth h, sea surface and seabed reflection coefficients Rs and RB, and the vertical wavenumber kz which depends on angular frequency x and the depth-dependent sound speed c(z). The zero-order Bessel function modulates the pressure field as a function of horizontal wavenumber kr and range r. The denominator of the integrand introduces mode-type behavior into the solution at particular values of kz depending upon Rs, RB, and h which cause the denominator to tend to zero. These first order poles of the integrand represent a discrete set of resonances (singularities) which correspond to the eigenvalues of the propagating modes in the normal mode formalism. The eigenvalues represent resonances of the plane waves that constructively interfere at a finite set of distinct ray angles corresponding to discrete values of kr along the horizontal wavenumber axis; these resonating modes propagate in range in the waveguide. Discrete, trapped modes have horizontal wavenumbers that lie on the real axis in the complex kr-plane bounded by kmin ¼ x=cmax and kmax ¼ x=cmin , where cmax and cmin are the maximum and minimum sound speeds in the waveguide. In an ideal waveguide, the discrete region contains perfectly trapped modes representing energy propagating along non-bottom-reflected-surface-reflected paths (i.e., refracted-refracted, refracted-bottom-reflected, and refracted-surface-reflected) in the region defined by
wc x=cwc < k < x=c and modes representing energy propagating r max
along bottom min reflected-surface-reflected (BRSR) paths in the region defined by 0 < kr < x=cwc max . Importantly, the modes are perfectly trapped (the modal field is purely discrete) and the values of kr are real only in an ideal waveguide; in a realistic waveguide with attenuation, the values of kr are complex and the modes are imperfectly trapped, creating a modal continuum which represents modes with up- and down-going plane waves which
EL2 J. Acoust. Soc. Am. 135 (1), January 2014
D. Benjamin Reeder: Clutter depth discrimination
D. Benjamin Reeder: JASA Express Letters
[http://dx.doi.org/10.1121/1.4828979]
Published Online 10 December 2013
only partly constructively interfere with each other due to loss of energy to the seabed. These imperfect (evanescent, leaky, virtual, improper) modes with complex eigenvalues dominate the continuous region, are associated with steep launch angles, and are more quickly attenuated in range compared to the discrete modes (Frisk and Lynch, 1984; Glattetre et al., 1989; Becker and Ballard, 2010; Cockrell and Schmidt, 2010). While the normal mode formalism requires the distinction in angular regimes separated by the critical angle, the distinction becomes less precise in realistic waveguides because attenuation displaces all of the modal eigenvalues into the first quadrant of the complex kr-plane. Each mode therefore possesses propagating and evanescent constituents (Jensen et al., 2000). This is consistent with Eq. (5). The finite-valued continuum of sinusoidal constituents of the pressure field in the numerator exists for all propagation angles, and modal behavior is superimposed onto the finite-valued continuum in the vicinity of the integrand’s poles. In other words, even though these poles dominate the magnitude of the Green’s function in the discrete region of the spectrum, the sinusoidal components of the field still exist throughout much of the wavenumber spectrum. While this has always been true, the modal continuum has been largely ignored, perhaps due to the historical focus on long-range propagation represented by high kr values. Recent technological developments and the modern operational environment facilitate the use of higher angle energy represented in the continuum portion of the wavenumber spectrum. The energy which propagates along BRSR paths creates sinusoidal structure in the wavenumber spectrum with periodicities corresponding to source depth, receiver depth, and water depth as predicted by the numerator of Eq. (5). In the kz domain, the absolute value of the sinusoidal modulations in the Green’s function has periods of p, such that jsinðkz zÞj ¼ 0 8 kz z ¼ p, which gives z ¼ p=Dkz ;
(6)
where Dkz is the width of the modulation between the nulls (Lauer, 1980). The interference pattern resulting from the multipath propagation in the waveguide generates the vertical wavenumber spectrum’s sinusoidal structure, the periodicity of which, defined by Dkz , provides an estimate of source depth, receiver depth, and water depth. The preceding paragraphs have presented the theoretical basis and rationale for the use of high-angle energy that contributes to the continuous portion of the wavenumber spectrum which has traditionally been overlooked in ocean acoustic wavenumber techniques; Sec. 3 extends the approach to clutter depth discrimination in the shallow water environment, including additional techniques to improve performance. 3. Numerical implementation and simulations Obtaining the Green’s function from the observed far-field pressure for a point source in an axisymmetric environment can be approximated by a Fourier transform of the complex pressure as a function of range times the square root of the range (Frisk and Lynch, 1984). The process involves the following steps, as illustrated and discussed below: (a) Demodulate the observed pressure to complex pressure as a function of time; (b) render complex pressure as a function of time to a function of range; (c) Fourier transform range-dependent complex pressure to horizontal wavenumber space [gðkr Þ]; (d) translate the Green’s function from horizontal wavenumber space to vertical wavenumber space [gðkz Þ], (e) Fourier transform gðkz Þ to the spatial (depth) domain: pobs ðtÞ ! pc ðtÞ ! pc ðrÞ ! gðkr Þ ! gðkz Þ ! z: fft
fft
(7)
Two examples illustrate the technique for a mid-frequency clutter point source in an axisymmetric, range-independent shallow water waveguide at two depths: (1) In the middle of the water column and (2) on the seabed. The sound speed profile used in the simulations represents a strong downward-refracting environment possessing a 10 m
J. Acoust. Soc. Am. 135 (1), January 2014
D. Benjamin Reeder: Clutter depth discrimination EL3
D. Benjamin Reeder: JASA Express Letters
[http://dx.doi.org/10.1121/1.4828979]
Published Online 10 December 2013
deep surface duct and a weak acoustic channel centered on 90 m water depth. The 200 m deep seabed is modeled as a homogeneous half-space having a sound speed of 1800 m/s. For simplicity, the two-way propagation problem is approximated as a one-way simulation from the clutter point source to the omnidirectional receiver. In each of these idealized simulations, the magnitude of the Green’s function is generated by the Scooter model (Porter, 2012) assuming that the spatial aperture fully populates the wavenumber spectrum. The Green’s function is plotted as a function of horizontal wavenumber the firstffi pin ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 panel and as a function of vertical wavenumber in the second panel, using kz ¼ kmax kr2 . Converting the x axis from the linear kr scale to the non-linear kz scale reveals consistent, periodic structure in jgðkz Þj, which facilitates use of the Fast Fourier Transform (FFT). Just as the FFT transforms a temporal signal in units of seconds to frequency space in units of inverse seconds, the FFT transforms the vertical wavenumber spectrum in units of inverse meters to the spatial (depth) domain in units of meters. Hence, the vertical wavenumber spectrum shown in the middle panels of Figs. 1 and 2 is Fourier-transformed into estimates of source depth, receiver depth, and water depth in the third panel. In the first example, the modeled source and receiver depths are 100 and 6 m, respectively. Acoustic frequency is 3 kHz. Figure 1 shows the magnitude of the Green’s function vs horizontal wavenumber [Fig. 1(a)] and vertical wavenumber [Fig. 1(b)], and estimated depths of the source, receiver, and seabed based on the FFT of the vertical wavenumber spectrum in Fig. 1(b). The vertical black lines in Figs. 1(a) and 1(b) indicate the angular regimes; the discrete spectrum lies to the right of kz2 and the continuous spectrum lies to the left of kz2. The widths (Dkz ) of the modulations in Fig. 1(b) correspond to the peaks in Fig. 1(c) produced by the FFT of the vertical wavenumber. More specifically, the results shown in the third panel are the FFT of the auto-correlation of the mean-removed vertical wavenumber spectrum. Removing
Fig. 1. Magnitude of the Green’s function vs horizontal wavenumber (a) and vertical wavenumber (b); FFT (c) of the vertical wavenumber spectrum in (b). Acoustic frequency, clutter source depth, and receiver depth are modeled at 3 kHz, 100 m, and 6 m, respectively.
EL4 J. Acoust. Soc. Am. 135 (1), January 2014
D. Benjamin Reeder: Clutter depth discrimination
D. Benjamin Reeder: JASA Express Letters
[http://dx.doi.org/10.1121/1.4828979]
Published Online 10 December 2013
Fig. 2. Plots similar to Fig. 1; environment, acoustic frequency, and receiver depth are identical to the first example, but in this case, the modeled depth of the clutter source is 200 m.
the mean and using the auto-correlation acts as an in-band noise filter that increases the signal-to-noise ratio (SNR) of the signal and produces a cleaner result. The FFT in Fig. 1(c) of the vertical wavenumber spectrum in Fig. 1(b) provides estimates of the depths of the receiver, source, and seabed of 7, 103, and 206 m, which are close to the modeled receiver, source, and seabed depths of 6, 100, and 200 m. In a realistic scenario, the receiver and seabed depths will be known a priori, leading to the conclusion that the peak at 103 m corresponds to the depth of the clutter source. In the second example, the modeled source depth is set to 200 m (on the seabed) while the environment, acoustic frequency, and receiver depth are identical to the first example. The FFT in Fig. 2(c) shows peaks at approximately 7 and 206 m. Given that the receiver depth is known to be 6 m, it can be reasonably concluded that the peak at 206 m corresponds to a clutter source on the seabed. 4. Discussion and conclusion Source localization methods have been investigated extensively over the years in ocean acoustics, many of which seek to exploit the interference pattern resulting from multipath propagation in the ocean waveguide. Some of these methods include matched field processing which attempts to maximize an objective function which correlates the modeled and measured acoustic fields (Bucker, 1976; Baggeroer et al., 1988; Fawcett et al., 1996); modal decomposition methods which seek to match the modeled and measured modal amplitudes as a function of depth (Shang, 1985; Yang, 1987; Glattetre et al., 1989); matched mode methods which endeavor to match modeled and observed mode amplitudes as measured on a horizontal line array by use of the frequency-wavenumber (f-k) transform (Nicolas et al., 2006); and waveguide invariant approaches which are based on Chuprov’s (1982) parameterization relating range and frequency to the slope of the striations in acoustic pressure in a frequency-range plot (Brekhovskikh and Lysanov, 1991; D’Spain and Kuperman, 1999). The waveguide invariant is commonly interpreted in
J. Acoust. Soc. Am. 135 (1), January 2014
D. Benjamin Reeder: Clutter depth discrimination EL5
D. Benjamin Reeder: JASA Express Letters
[http://dx.doi.org/10.1121/1.4828979]
Published Online 10 December 2013
terms of constructive and destructive interference of propagating normal modes (Turgut and Orr, 2010), but it has also been described in terms of ray theory (Gerstoft et al., 2001) and variations in eigenray arrival times (Harrison, 2011), and related to the wavenumber integration technique (Cockrell and Schmidt, 2010). Upon close inspection, the current technique can be loosely interpreted to be a dimensionally-reduced form of both the waveguide invariant process (WIP) and the mode matching method (MMM) using the f-k representation mentioned above. In the former method (WIP), the broadband acoustic intensity is plotted as a function of frequency and range in order to visualize the slope of the striations which is then used to infer the parameter of interest. The current approach essentially utilizes a signal from a single frequency bin in the WIP frequency-range plot as the acoustic pressure portion of the integrand in the onedimensional Hankel transform that is integrated over range. In the latter method (MMM), the broadband acoustic pressure is plotted as a function of frequency and wavenumber by use of a two-dimensional Fourier transform. The current approach essentially uses the amplitude of the wavenumber spectrum found in a single frequency bin of the MMM f-k plot. While analogous in terms of exploiting the physics of multipath spatial interference patterns found in f-r and f-k plots of acoustic pressure, the current method offers several advantages over waveguide invariant and mode matching methods; specifically, it does not rely on uncertain assumptions such as the value of the waveguide invariant parameter b, and it is computationally efficient (i.e., a narrowband vs broadband process, does not require propagation modeling, objective functions, or image processing). Further, the current approach requires only two FFT’s—the first to transform the acoustic pressure into the horizontal wavenumber spectrum, and the second to transform the vertical wavenumber spectrum to depth. Finally, the method produces a simple result of depth, defined by amplitude above a chosen threshold. It is important to note that the other source localization methods mentioned above rely on discrete propagating modes as represented by modal eigenvalues, while this current technique relies specifically on the modal continuum. As such, the modes contributing to the continuum portion of the wavenumber spectrum are associated with launch angles above the critical angle and will therefore decay more quickly in range due to a greater number of waveguide boundary interactions per unit range than those waterborne modes associated with the discrete portion of the spectrum which dominate the field downrange. It is clear that this technique will perform best at shorter ranges (order of kilometers) in order to exploit the high-angle energy before it is completely attenuated. New developments in equipment and signal processing techniques designed specifically to enhance and capture this high-angle energy will improve its performance. It is anticipated that this depth discrimination method could be successfully implemented in the shallow water environment within the context of a horizontal acoustic fisheries survey application, a coherent multi-static active sonar application, or a high duty cycle sonar application, complemented by a system of distributed netted sensors. While the approach is deceptively simple, it is important to note that the above results are numerical simulations, unconstrained by practical issues such as range aperture and spatial sampling of the pressure field, which a real-world implementation would necessarily address. Uncertainty will be directly proportional to the effect of any physical mechanism which diffuses the coherence of the received acoustic pressure (e.g., Doppler shift, seabed scatter, internal waves, sea surface roughness) during the observation window and inversely proportional to SNR and range aperture. For example, there exists a tradeoff between aperture and Doppler—the greater the aperture, the greater the likelihood for Doppler shift. For a specific system operating in a certain frequency band within a particular source/receiver geometry, the minimum aperture and spatial sampling requirements would need to be determined to avoid aliasing and minimize error. These questions are the focus of an ongoing investigation. In summary, a method has been presented to remotely discriminate clutter source depth using information contained in the backscattered signal—without a priori knowledge of clutter source depth. The theoretical basis and rationale for the use of
EL6 J. Acoust. Soc. Am. 135 (1), January 2014
D. Benjamin Reeder: Clutter depth discrimination
D. Benjamin Reeder: JASA Express Letters
[http://dx.doi.org/10.1121/1.4828979]
Published Online 10 December 2013
high-angle energy that contributes to the continuous portion of the wavenumber spectrum has been presented, followed by application of the method to clutter depth discrimination in the shallow water environment. Specifically, the interference pattern resulting from multipath propagation in the waveguide generates sinusoidal structure in the vertical wavenumber spectrum, the periodicities of which correspond to source depth, receiver depth, and water depth. It is shown that once the Green’s function is obtained from the Hankel transform of the acoustic pressure observed over a spatial aperture, the clutter depth can be estimated by Fourier-transforming the vertical wavenumber spectrum to the spatial (depth) domain. Numerical simulations for a simple test case in a shallow water waveguide demonstrate this approach provides the potential capability to discriminate between a clutter source in the water column vs one on the seabed. Acknowledgments The author thanks the editor for the very insightful comments which greatly improved this paper. This work was supported by the Office of Naval Research. References and links Baggeroer, A. B., Kuperman, W. A., and Schmidt, H. (1988). “Matched field processing: Source localization in correlated noise as an optimum parameter estimation problem,” J. Acoust. Soc. Am. 83(2), 571–587. Becker, K. M., and Ballard, M. S. (2010). “Accounting for water-column variability in shallow-water waveguide characterizations based on modal eigenvalues,” in Shallow-Water Acoustics, Second International ShallowWater Acoustics Conference, edited by J. Simmen, E. S. Livingston, J. X. Zhou, and F. H. Li (AIP, Melville, NY), pp. 46–52. Brekhovskikh, L. M., and Lysanov, Y. P. (1991). Fundamentals of Ocean Acoustics, 2nd ed. (Springer, New York), pp. 140–145. Bucker, H. P. (1976). “Use of calculated sound fields and matched field detection to locate sound sources in shallow water,” J. Acoust. Soc. Am. 59(5), 368–373. Chuprov, S. D. (1982). “Interference structure of a sound field in a layered ocean,” in Ocean Acoustics, Current Status, edited by L. M. Brekhovskikh and I. B. Andreyeva (Nauka, Moscow), pp. 71–91. Cockrell, K. L., and Schmidt, H. (2011). “A modal Wentzel-Kramers-Brillouin approach to calculating the waveguide invariant for non-ideal waveguides,” J. Acoust. Soc. Am. 130, 72–83. D’Spain, G. L., and Kuperman, W. A. (1999). “Application of waveguide invariants to analysis of spectrograms from shallow water environments that vary in range and azimuth,” J. Acoust. Soc. Am. 106, 2454–2468. Fawcett, J. A., Yeremy, M. L., and Chapman, N. R. (1996). “Matched-field source localization in a rangedependent environment,” J. Acoust. Soc. Am. 99(1), 272–282. Frisk, G. V. (1994). Ocean and Seabed Acoustics: A Theory of Wave Propagation (PTR Prentice-Hall, Englewood Cliffs, NJ), Chap. 6. Frisk, G. V., and Lynch, J. F. (1984). “Shallow water waveguide characterization using the Hankel transform,” J. Acoust. Soc. Am. 76(1), 205–216. Gerstoft, P., D’Spain, G. L., Kuperman, W. A., and Hodgkiss, W. S. (2001). “Calculating the waveguide invariant (beta) by ray-theoretic approaches,” MPL Technical Memo No. TM-468, Marine Physical Laboratory, University of California, San Diego. Glattetre, J., Knudsen, T., and Sostrand, K. (1989). “Mode interference and mode filtering in shallow water: A comparison of acoustic measurements and modeling,” J. Acoust. Soc. Am. 86(2), 680–690. Harrison, C. H. (2011). “The relation between the waveguide invariant, multipath impulse response, and ray cycles,” J. Acoust. Soc. Am. 129(5), 2863–2877. Jensen, F. B., Kuperman, W. A., Porter, M. B., and Schmidt, H. (2000). Computational Ocean Acoustics (Springer, New York), Chaps. 2, 3, and 5. Lauer, R. B. (1980). “Wavenumber acoustics: Passive localization and multipath decomposition,” Acoustics, Speech and Signal Processing, IEEE International Conference on ICASSP’80, Vol. 5, pp. 1026–1029. Nicolas, B., Mars, J. I., and Lacoume, J.-L. (2006). “Source depth estimation using a horizontal array by matched-mode processing in the frequency-wavenumber domain” EURASIP J. Appl. Signal Process. 1, 1–16. Porter, M. B. (2012). “The Scooter FFP model,” http://oalib.hlsresearch.com/Modes/AcousticsToolbox/ manual_html/node63.html (Last viewed November 9, 2013). Shang, E. C. (1985). “Source depth estimation in waveguides,” J. Acoust. Soc. Am. 77(4), 1413–1418. Turgut, A., and Orr, M. (2010). “Broadband source localization using horizontal-beam acoustic intensity striations,” J. Acoust. Soc. Am. 127(1), 73–83. Yang, T. C. (1987). “A method of range and depth estimation my modal decomposition,” J. Acoust. Soc. Am. 82, 1736–1745.
J. Acoust. Soc. Am. 135 (1), January 2014
D. Benjamin Reeder: Clutter depth discrimination EL7
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[http://dx.doi.org/10.1121/1.4828979]
Published Online 10 December 2013
Clutter depth discrimination using the wavenumber spectrum D. Benjamin Reeder Department of Oceanography, Naval Postgraduate School, 833 Dyer Road, Monterey, California 93943 [email protected]
Abstract: Clutter depth is a key parameter in mid-frequency active sonar systems to discriminate between sources of clutter and targets of interest. A method is needed to remotely discriminate clutter depth by information contained in the backscattered signal—without a priori knowledge of that depth. Presented here is an efficient approach for clutter depth estimation using the structure in the wavenumber spectrum. Based on numerical simulations for a simple test case in a shallow water waveguide, this technique demonstrates the potential capability to discriminate between a clutter source in the water column vs one on the seabed. PACS numbers: 43.40.Vn, 43.60.Jn, 43.60.Pt [JL] Date Received: August 20, 2013 Date Accepted: October 22, 2013
1. Introduction Constructive and destructive interference processes produced by multipath propagation in the shallow water environment are exhibited in the complex structure of acoustic intensity patterns when plotted vs range and depth. This structure is a function of source/receiver geometry, acoustic signal characteristics, and physical parameters of the waveguide (e.g., sound speed profile, seabed topography, and acoustic properties). The acoustic signal observed by a receiver as a function of time and space therefore contains information about the source signal and the medium through which it propagates. For the case of an active sonar system, the backscattered signal contains information regarding the waveguide and acoustic scatterers, many of which are considered unwanted sources of clutter. For the case of a military sonar system, determining whether the source of a scattered signal is located on the seabed or occupies a position in the water column represents a significant capability to discriminate between sources of clutter and targets of interest. Presented here is an efficient method to remotely discriminate clutter source depth by information contained in the backscattered signal— without a priori knowledge of that depth. The intent of this paper is threefold: (a) Present the theoretical basis of the approach, (b) discuss the physics and rationale for using the energy which propagates along high-angle paths represented in the traditionally ignored continuous region of the wavenumber spectrum, and (c) extend the approach to clutter depth discrimination in the shallow water waveguide via numerical simulation in a simple test case to demonstrate the concept. 2. Theoretical development In a horizontally stratified (range-independent) waveguide with water column thickness h, sound speed c(z), and constant density q, the acoustic pressure due to a point source at (r, z) ¼ (0, zs) with harmonic time dependence (eixt) satisfies the inhomogeneous wave equation 1@ @ @2 dðrÞdðz zs Þ 2 r þ 2 þ k ðzÞ pðr; z; zs Þ ¼ 2 ; (1) @z r @r @r r
J. Acoust. Soc. Am. 135 (1), January 2014
EL1
D. Benjamin Reeder: JASA Express Letters
[http://dx.doi.org/10.1121/1.4828979]
Published Online 10 December 2013
where kðzÞ ¼ x=cðzÞ. Following the development of Frisk (1994), the application of the inverse zero-order Hankel transform operator to both sides of Eq. (1) gives the depth-dependent Helmholtz equation d2 2 2 þ ðk ðzÞ kr Þ gðkr ; z; zs Þ ¼ 2dðz zs Þ; (2) dz2 where the separation constant, kr, is the horizontal component of the acoustic wave vector and pðr; z; zs Þ is the solution to Eq. (2) for a given kr. The conjugate Hankel transform pair relationship between the pressure field and the depth-dependent Green’s function is ð1 gðkr ; z; zs ÞJ0 ðkr rÞkr dkr ; (3) pðr; z; zs Þ ¼ 0
gðkr ; z; zs Þ ¼
ð1
pðr; z; zs ÞJ0 ðkr rÞrdr;
(4)
0
where J0 ðkr rÞ is the zero-order Bessel function, and r is the range between the source at depth zs and receiver at depth z. The full solution for the pressure field due to a point source in a homogeneous (constant density and sound speed) fluid layer with arbitrary horizontally stratified boundaries [Frisk, 1994, p. 173, Eq. (6.19)] is expressed as # ð 1 " ikz jzzs j i e þ RS eikz ðzþzs Þ þ RB e2ikz h eikz ðzþzs Þ þ RS eikz jzzs j pðr; z; zs Þ ¼ J0 ðkr rÞkr dkr : kz ½1 RS RB e2ikz h 0 (5) The term inside the brackets is the depth-dependent Green’s function for this waveguide, which can be viewed as the waveguide’s spatial transfer function. The numerator of the integrand represents the finite-valued continuum of sinusoidal constituents of the pressure field which is a function of the source depth zs, receiver depth z, water depth h, sea surface and seabed reflection coefficients Rs and RB, and the vertical wavenumber kz which depends on angular frequency x and the depth-dependent sound speed c(z). The zero-order Bessel function modulates the pressure field as a function of horizontal wavenumber kr and range r. The denominator of the integrand introduces mode-type behavior into the solution at particular values of kz depending upon Rs, RB, and h which cause the denominator to tend to zero. These first order poles of the integrand represent a discrete set of resonances (singularities) which correspond to the eigenvalues of the propagating modes in the normal mode formalism. The eigenvalues represent resonances of the plane waves that constructively interfere at a finite set of distinct ray angles corresponding to discrete values of kr along the horizontal wavenumber axis; these resonating modes propagate in range in the waveguide. Discrete, trapped modes have horizontal wavenumbers that lie on the real axis in the complex kr-plane bounded by kmin ¼ x=cmax and kmax ¼ x=cmin , where cmax and cmin are the maximum and minimum sound speeds in the waveguide. In an ideal waveguide, the discrete region contains perfectly trapped modes representing energy propagating along non-bottom-reflected-surface-reflected paths (i.e., refracted-refracted, refracted-bottom-reflected, and refracted-surface-reflected) in the region defined by
wc x=cwc < k < x=c and modes representing energy propagating r max
along bottom min reflected-surface-reflected (BRSR) paths in the region defined by 0 < kr < x=cwc max . Importantly, the modes are perfectly trapped (the modal field is purely discrete) and the values of kr are real only in an ideal waveguide; in a realistic waveguide with attenuation, the values of kr are complex and the modes are imperfectly trapped, creating a modal continuum which represents modes with up- and down-going plane waves which
EL2 J. Acoust. Soc. Am. 135 (1), January 2014
D. Benjamin Reeder: Clutter depth discrimination
D. Benjamin Reeder: JASA Express Letters
[http://dx.doi.org/10.1121/1.4828979]
Published Online 10 December 2013
only partly constructively interfere with each other due to loss of energy to the seabed. These imperfect (evanescent, leaky, virtual, improper) modes with complex eigenvalues dominate the continuous region, are associated with steep launch angles, and are more quickly attenuated in range compared to the discrete modes (Frisk and Lynch, 1984; Glattetre et al., 1989; Becker and Ballard, 2010; Cockrell and Schmidt, 2010). While the normal mode formalism requires the distinction in angular regimes separated by the critical angle, the distinction becomes less precise in realistic waveguides because attenuation displaces all of the modal eigenvalues into the first quadrant of the complex kr-plane. Each mode therefore possesses propagating and evanescent constituents (Jensen et al., 2000). This is consistent with Eq. (5). The finite-valued continuum of sinusoidal constituents of the pressure field in the numerator exists for all propagation angles, and modal behavior is superimposed onto the finite-valued continuum in the vicinity of the integrand’s poles. In other words, even though these poles dominate the magnitude of the Green’s function in the discrete region of the spectrum, the sinusoidal components of the field still exist throughout much of the wavenumber spectrum. While this has always been true, the modal continuum has been largely ignored, perhaps due to the historical focus on long-range propagation represented by high kr values. Recent technological developments and the modern operational environment facilitate the use of higher angle energy represented in the continuum portion of the wavenumber spectrum. The energy which propagates along BRSR paths creates sinusoidal structure in the wavenumber spectrum with periodicities corresponding to source depth, receiver depth, and water depth as predicted by the numerator of Eq. (5). In the kz domain, the absolute value of the sinusoidal modulations in the Green’s function has periods of p, such that jsinðkz zÞj ¼ 0 8 kz z ¼ p, which gives z ¼ p=Dkz ;
(6)
where Dkz is the width of the modulation between the nulls (Lauer, 1980). The interference pattern resulting from the multipath propagation in the waveguide generates the vertical wavenumber spectrum’s sinusoidal structure, the periodicity of which, defined by Dkz , provides an estimate of source depth, receiver depth, and water depth. The preceding paragraphs have presented the theoretical basis and rationale for the use of high-angle energy that contributes to the continuous portion of the wavenumber spectrum which has traditionally been overlooked in ocean acoustic wavenumber techniques; Sec. 3 extends the approach to clutter depth discrimination in the shallow water environment, including additional techniques to improve performance. 3. Numerical implementation and simulations Obtaining the Green’s function from the observed far-field pressure for a point source in an axisymmetric environment can be approximated by a Fourier transform of the complex pressure as a function of range times the square root of the range (Frisk and Lynch, 1984). The process involves the following steps, as illustrated and discussed below: (a) Demodulate the observed pressure to complex pressure as a function of time; (b) render complex pressure as a function of time to a function of range; (c) Fourier transform range-dependent complex pressure to horizontal wavenumber space [gðkr Þ]; (d) translate the Green’s function from horizontal wavenumber space to vertical wavenumber space [gðkz Þ], (e) Fourier transform gðkz Þ to the spatial (depth) domain: pobs ðtÞ ! pc ðtÞ ! pc ðrÞ ! gðkr Þ ! gðkz Þ ! z: fft
fft
(7)
Two examples illustrate the technique for a mid-frequency clutter point source in an axisymmetric, range-independent shallow water waveguide at two depths: (1) In the middle of the water column and (2) on the seabed. The sound speed profile used in the simulations represents a strong downward-refracting environment possessing a 10 m
J. Acoust. Soc. Am. 135 (1), January 2014
D. Benjamin Reeder: Clutter depth discrimination EL3
D. Benjamin Reeder: JASA Express Letters
[http://dx.doi.org/10.1121/1.4828979]
Published Online 10 December 2013
deep surface duct and a weak acoustic channel centered on 90 m water depth. The 200 m deep seabed is modeled as a homogeneous half-space having a sound speed of 1800 m/s. For simplicity, the two-way propagation problem is approximated as a one-way simulation from the clutter point source to the omnidirectional receiver. In each of these idealized simulations, the magnitude of the Green’s function is generated by the Scooter model (Porter, 2012) assuming that the spatial aperture fully populates the wavenumber spectrum. The Green’s function is plotted as a function of horizontal wavenumber the firstffi pin ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 panel and as a function of vertical wavenumber in the second panel, using kz ¼ kmax kr2 . Converting the x axis from the linear kr scale to the non-linear kz scale reveals consistent, periodic structure in jgðkz Þj, which facilitates use of the Fast Fourier Transform (FFT). Just as the FFT transforms a temporal signal in units of seconds to frequency space in units of inverse seconds, the FFT transforms the vertical wavenumber spectrum in units of inverse meters to the spatial (depth) domain in units of meters. Hence, the vertical wavenumber spectrum shown in the middle panels of Figs. 1 and 2 is Fourier-transformed into estimates of source depth, receiver depth, and water depth in the third panel. In the first example, the modeled source and receiver depths are 100 and 6 m, respectively. Acoustic frequency is 3 kHz. Figure 1 shows the magnitude of the Green’s function vs horizontal wavenumber [Fig. 1(a)] and vertical wavenumber [Fig. 1(b)], and estimated depths of the source, receiver, and seabed based on the FFT of the vertical wavenumber spectrum in Fig. 1(b). The vertical black lines in Figs. 1(a) and 1(b) indicate the angular regimes; the discrete spectrum lies to the right of kz2 and the continuous spectrum lies to the left of kz2. The widths (Dkz ) of the modulations in Fig. 1(b) correspond to the peaks in Fig. 1(c) produced by the FFT of the vertical wavenumber. More specifically, the results shown in the third panel are the FFT of the auto-correlation of the mean-removed vertical wavenumber spectrum. Removing
Fig. 1. Magnitude of the Green’s function vs horizontal wavenumber (a) and vertical wavenumber (b); FFT (c) of the vertical wavenumber spectrum in (b). Acoustic frequency, clutter source depth, and receiver depth are modeled at 3 kHz, 100 m, and 6 m, respectively.
EL4 J. Acoust. Soc. Am. 135 (1), January 2014
D. Benjamin Reeder: Clutter depth discrimination
D. Benjamin Reeder: JASA Express Letters
[http://dx.doi.org/10.1121/1.4828979]
Published Online 10 December 2013
Fig. 2. Plots similar to Fig. 1; environment, acoustic frequency, and receiver depth are identical to the first example, but in this case, the modeled depth of the clutter source is 200 m.
the mean and using the auto-correlation acts as an in-band noise filter that increases the signal-to-noise ratio (SNR) of the signal and produces a cleaner result. The FFT in Fig. 1(c) of the vertical wavenumber spectrum in Fig. 1(b) provides estimates of the depths of the receiver, source, and seabed of 7, 103, and 206 m, which are close to the modeled receiver, source, and seabed depths of 6, 100, and 200 m. In a realistic scenario, the receiver and seabed depths will be known a priori, leading to the conclusion that the peak at 103 m corresponds to the depth of the clutter source. In the second example, the modeled source depth is set to 200 m (on the seabed) while the environment, acoustic frequency, and receiver depth are identical to the first example. The FFT in Fig. 2(c) shows peaks at approximately 7 and 206 m. Given that the receiver depth is known to be 6 m, it can be reasonably concluded that the peak at 206 m corresponds to a clutter source on the seabed. 4. Discussion and conclusion Source localization methods have been investigated extensively over the years in ocean acoustics, many of which seek to exploit the interference pattern resulting from multipath propagation in the ocean waveguide. Some of these methods include matched field processing which attempts to maximize an objective function which correlates the modeled and measured acoustic fields (Bucker, 1976; Baggeroer et al., 1988; Fawcett et al., 1996); modal decomposition methods which seek to match the modeled and measured modal amplitudes as a function of depth (Shang, 1985; Yang, 1987; Glattetre et al., 1989); matched mode methods which endeavor to match modeled and observed mode amplitudes as measured on a horizontal line array by use of the frequency-wavenumber (f-k) transform (Nicolas et al., 2006); and waveguide invariant approaches which are based on Chuprov’s (1982) parameterization relating range and frequency to the slope of the striations in acoustic pressure in a frequency-range plot (Brekhovskikh and Lysanov, 1991; D’Spain and Kuperman, 1999). The waveguide invariant is commonly interpreted in
J. Acoust. Soc. Am. 135 (1), January 2014
D. Benjamin Reeder: Clutter depth discrimination EL5
D. Benjamin Reeder: JASA Express Letters
[http://dx.doi.org/10.1121/1.4828979]
Published Online 10 December 2013
terms of constructive and destructive interference of propagating normal modes (Turgut and Orr, 2010), but it has also been described in terms of ray theory (Gerstoft et al., 2001) and variations in eigenray arrival times (Harrison, 2011), and related to the wavenumber integration technique (Cockrell and Schmidt, 2010). Upon close inspection, the current technique can be loosely interpreted to be a dimensionally-reduced form of both the waveguide invariant process (WIP) and the mode matching method (MMM) using the f-k representation mentioned above. In the former method (WIP), the broadband acoustic intensity is plotted as a function of frequency and range in order to visualize the slope of the striations which is then used to infer the parameter of interest. The current approach essentially utilizes a signal from a single frequency bin in the WIP frequency-range plot as the acoustic pressure portion of the integrand in the onedimensional Hankel transform that is integrated over range. In the latter method (MMM), the broadband acoustic pressure is plotted as a function of frequency and wavenumber by use of a two-dimensional Fourier transform. The current approach essentially uses the amplitude of the wavenumber spectrum found in a single frequency bin of the MMM f-k plot. While analogous in terms of exploiting the physics of multipath spatial interference patterns found in f-r and f-k plots of acoustic pressure, the current method offers several advantages over waveguide invariant and mode matching methods; specifically, it does not rely on uncertain assumptions such as the value of the waveguide invariant parameter b, and it is computationally efficient (i.e., a narrowband vs broadband process, does not require propagation modeling, objective functions, or image processing). Further, the current approach requires only two FFT’s—the first to transform the acoustic pressure into the horizontal wavenumber spectrum, and the second to transform the vertical wavenumber spectrum to depth. Finally, the method produces a simple result of depth, defined by amplitude above a chosen threshold. It is important to note that the other source localization methods mentioned above rely on discrete propagating modes as represented by modal eigenvalues, while this current technique relies specifically on the modal continuum. As such, the modes contributing to the continuum portion of the wavenumber spectrum are associated with launch angles above the critical angle and will therefore decay more quickly in range due to a greater number of waveguide boundary interactions per unit range than those waterborne modes associated with the discrete portion of the spectrum which dominate the field downrange. It is clear that this technique will perform best at shorter ranges (order of kilometers) in order to exploit the high-angle energy before it is completely attenuated. New developments in equipment and signal processing techniques designed specifically to enhance and capture this high-angle energy will improve its performance. It is anticipated that this depth discrimination method could be successfully implemented in the shallow water environment within the context of a horizontal acoustic fisheries survey application, a coherent multi-static active sonar application, or a high duty cycle sonar application, complemented by a system of distributed netted sensors. While the approach is deceptively simple, it is important to note that the above results are numerical simulations, unconstrained by practical issues such as range aperture and spatial sampling of the pressure field, which a real-world implementation would necessarily address. Uncertainty will be directly proportional to the effect of any physical mechanism which diffuses the coherence of the received acoustic pressure (e.g., Doppler shift, seabed scatter, internal waves, sea surface roughness) during the observation window and inversely proportional to SNR and range aperture. For example, there exists a tradeoff between aperture and Doppler—the greater the aperture, the greater the likelihood for Doppler shift. For a specific system operating in a certain frequency band within a particular source/receiver geometry, the minimum aperture and spatial sampling requirements would need to be determined to avoid aliasing and minimize error. These questions are the focus of an ongoing investigation. In summary, a method has been presented to remotely discriminate clutter source depth using information contained in the backscattered signal—without a priori knowledge of clutter source depth. The theoretical basis and rationale for the use of
EL6 J. Acoust. Soc. Am. 135 (1), January 2014
D. Benjamin Reeder: Clutter depth discrimination
D. Benjamin Reeder: JASA Express Letters
[http://dx.doi.org/10.1121/1.4828979]
Published Online 10 December 2013
high-angle energy that contributes to the continuous portion of the wavenumber spectrum has been presented, followed by application of the method to clutter depth discrimination in the shallow water environment. Specifically, the interference pattern resulting from multipath propagation in the waveguide generates sinusoidal structure in the vertical wavenumber spectrum, the periodicities of which correspond to source depth, receiver depth, and water depth. It is shown that once the Green’s function is obtained from the Hankel transform of the acoustic pressure observed over a spatial aperture, the clutter depth can be estimated by Fourier-transforming the vertical wavenumber spectrum to the spatial (depth) domain. Numerical simulations for a simple test case in a shallow water waveguide demonstrate this approach provides the potential capability to discriminate between a clutter source in the water column vs one on the seabed. Acknowledgments The author thanks the editor for the very insightful comments which greatly improved this paper. This work was supported by the Office of Naval Research. References and links Baggeroer, A. B., Kuperman, W. A., and Schmidt, H. (1988). “Matched field processing: Source localization in correlated noise as an optimum parameter estimation problem,” J. Acoust. Soc. Am. 83(2), 571–587. Becker, K. M., and Ballard, M. S. (2010). “Accounting for water-column variability in shallow-water waveguide characterizations based on modal eigenvalues,” in Shallow-Water Acoustics, Second International ShallowWater Acoustics Conference, edited by J. Simmen, E. S. Livingston, J. X. Zhou, and F. H. Li (AIP, Melville, NY), pp. 46–52. Brekhovskikh, L. M., and Lysanov, Y. P. (1991). Fundamentals of Ocean Acoustics, 2nd ed. (Springer, New York), pp. 140–145. Bucker, H. P. (1976). “Use of calculated sound fields and matched field detection to locate sound sources in shallow water,” J. Acoust. Soc. Am. 59(5), 368–373. Chuprov, S. D. (1982). “Interference structure of a sound field in a layered ocean,” in Ocean Acoustics, Current Status, edited by L. M. Brekhovskikh and I. B. Andreyeva (Nauka, Moscow), pp. 71–91. Cockrell, K. L., and Schmidt, H. (2011). “A modal Wentzel-Kramers-Brillouin approach to calculating the waveguide invariant for non-ideal waveguides,” J. Acoust. Soc. Am. 130, 72–83. D’Spain, G. L., and Kuperman, W. A. (1999). “Application of waveguide invariants to analysis of spectrograms from shallow water environments that vary in range and azimuth,” J. Acoust. Soc. Am. 106, 2454–2468. Fawcett, J. A., Yeremy, M. L., and Chapman, N. R. (1996). “Matched-field source localization in a rangedependent environment,” J. Acoust. Soc. Am. 99(1), 272–282. Frisk, G. V. (1994). Ocean and Seabed Acoustics: A Theory of Wave Propagation (PTR Prentice-Hall, Englewood Cliffs, NJ), Chap. 6. Frisk, G. V., and Lynch, J. F. (1984). “Shallow water waveguide characterization using the Hankel transform,” J. Acoust. Soc. Am. 76(1), 205–216. Gerstoft, P., D’Spain, G. L., Kuperman, W. A., and Hodgkiss, W. S. (2001). “Calculating the waveguide invariant (beta) by ray-theoretic approaches,” MPL Technical Memo No. TM-468, Marine Physical Laboratory, University of California, San Diego. Glattetre, J., Knudsen, T., and Sostrand, K. (1989). “Mode interference and mode filtering in shallow water: A comparison of acoustic measurements and modeling,” J. Acoust. Soc. Am. 86(2), 680–690. Harrison, C. H. (2011). “The relation between the waveguide invariant, multipath impulse response, and ray cycles,” J. Acoust. Soc. Am. 129(5), 2863–2877. Jensen, F. B., Kuperman, W. A., Porter, M. B., and Schmidt, H. (2000). Computational Ocean Acoustics (Springer, New York), Chaps. 2, 3, and 5. Lauer, R. B. (1980). “Wavenumber acoustics: Passive localization and multipath decomposition,” Acoustics, Speech and Signal Processing, IEEE International Conference on ICASSP’80, Vol. 5, pp. 1026–1029. Nicolas, B., Mars, J. I., and Lacoume, J.-L. (2006). “Source depth estimation using a horizontal array by matched-mode processing in the frequency-wavenumber domain” EURASIP J. Appl. Signal Process. 1, 1–16. Porter, M. B. (2012). “The Scooter FFP model,” http://oalib.hlsresearch.com/Modes/AcousticsToolbox/ manual_html/node63.html (Last viewed November 9, 2013). Shang, E. C. (1985). “Source depth estimation in waveguides,” J. Acoust. Soc. Am. 77(4), 1413–1418. Turgut, A., and Orr, M. (2010). “Broadband source localization using horizontal-beam acoustic intensity striations,” J. Acoust. Soc. Am. 127(1), 73–83. Yang, T. C. (1987). “A method of range and depth estimation my modal decomposition,” J. Acoust. Soc. Am. 82, 1736–1745.
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D. Benjamin Reeder: Clutter depth discrimination EL7
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