Gayana 68(2): 502-507, 2004

MEAN AND VARIANCE OF THE FORWARD FIELDS PROPAGATED THROUGH 3D RANDOM INTERNAL WAVES WITH RAYLEIGHT-BORN SCATTERING

Purnima Ratilal, Tianrun Chen & Nicholas C. Makris

Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. (purnima@mit.edu)

ABSTRACT

The mean and variance of the acoustic field forward propagated through an ocean waveguide containing random internal waves is modelled using a normal mode formulation [1] that takes into account the 3-D interaction of the acoustic field with medium inhomogeneities. The effects of dispersion, attenuation and redistribution of modal energy due to random multiple scattering in the forward azimuth are treated analytically. The inhomogeneous medium's scatter function density is modelled using the Rayleigh-Born approximation to Green's theorem to account for random fluctuations in density and compressibility caused by the internal waves. The generalized waveguide extinction theorem [2] is applied to determine the attenuation due to scattering from the internal wave inhomogeneities. Simulations for typical continental shelf environments quantitatively show how coherent and incoherent acoustic field intensities vary with range, depth and internal wave field properties.

INTRODUCTION

Here, we model the mean field, variance, and mean total intensity of the forward field propagated through an ocean waveguide containing temporally and spatially random 3-D internal waves. We apply the general normal mode formulation developed in Ref. [1] based on Green's theorem to analytically determine the statistical moments of the forward field. This formulation is convenient because it takes into account attenuation and dispersion of the mean forward field due to multiple scattering from random inhomogeneities. It also accounts for the steady state redistribution of modal energies after multiple scattering through the random waveguide.

Inhomogeneities arising from internal wave disturbances typically have relatively small differences in density and compressibility from the surrounding medium. A convenient approach for modelling their scatter function is to apply the first-order Rayleigh-Born approximation to Green's theorem [3]. This first-order approximation leads to a purely real scatter function since the imaginary part of the scatter function is too small to be obtained by first order theory. The real scatter function accounts for dispersion in the mean forward field due to scattering, but not attenuation which depends on the imaginary part of the scatter function. To determine the imaginary part using the Rayleigh-Born approach requires higher order approximations that may be cumbersome to obtain. Here, we efficiently determine attenuation in the forward field due to scattering by applying the waveguide extinction theorem [2]. The waveguide extinction theorem relates power loss in the forward azimuth for any given mode to the total scattered power in all directions from that mode. The latter can be approximated fairly accurately using the first order Rayleigh-Born theory.

The present formulation is advantageous in that field moments for the full 3D forward scattering problem can be obtained by directly evaluating analytic expressions for field moments given knowledge of the moments of the scatter function density of the random medium. This formulation can then be readily applied to evaluate performance constraints expected in a variety of underwater acoustic applications in continental shelf waters including communication, passive source localization, active target localization and geo-acoustic inversion.

FORMULATION

Internal waves in shallow water often occur in the region between warm water at the sea surface and cool water deeper in the water column.[4,5] Here, we model the internal wave field as a set of disturbances propagating along the boundary between strata in a two layer wa+ter column, as illustrated in Fig. 1, although many other internal wave models and parametrizations could have been chosen given the general formulation of Ref. [1]. In the absence of internal waves, the boundary separating the warm upper medium with density r₁ and sound speed c₁ and the cool lower medium with density r₂ and sound speed c₂ is level. In the presence of internal waves, a part of the lower medium protrudes into the upper medium and a part of the upper medium protrudes into the lower medium. We model the protrusion of the lower medium into the upper medium as a volumetric inhomogeneity that scatters the sound field, and vice versa for the lower medium.

To formulate the problem, we place the origin of the coordinate system at the sea surface. The z-axis points downward and normal to the interface between horizontal strata. The level boundary separating the upper and lower medium when there are no internal waves in the waveguide is at z=D. The source coordinates are defined by r₀=(-x₀,0,z₀) and the receiver coordinates by r=(x,0,z). Spatial cylindrical (r,f,z) and spherical systems (r,q,f) are defined by x=rsinqcosf, y=rsinqsinf, z=rcosq, and r=x²+y². The horizontal and vertical wavenumber components for the nth mode are respectively x_n=ksina_n and g_n=kcosa_n, where a_n is the elevation angle of the mode measured from the z-axis. Moreover, k²= x_n²+g_n ², and the wavenumber magnitude k equals the angular frequency w divided by the sound speed c in the object layer. The azimuthal angle of the modal plane wave is denoted by b, where 0 < b< 2p. The geometry of spatial and wavenumber coordinates is shown in Ref. 6.

Statistical Model of Random Internal Wave Inhomogeneities

The displacement h(x,y) of the internal wave boundary, as illustrated in Fig. 1, is modelled as a zero-mean Gaussian random process in space with < h > = 0, and variance <h²> = s_h². Since the displacement of the internal wave cannot exceed the water surface or the sea bottom, we limit the probability distribution for h to a maximum displacement h_m above and below the mean. The probability density function of the internal wave displacement is,

(1)

where,

(2)

is a normlization constant, and P(b) is the cummulative distribution function for a normalized Gaussian random variable.

In this model, a positive value for h indicates that the internal wave inhomogeneity is in the lower medium, while a negative value for h indicates an internal wave element in the upper medium. We assume that the internal wave height has characteristic length scales l_xand l_y in the x and y directions respectively, which correspond to the range and cross-range coherence lengths of the stationary random process h(x,y) obtained from its 2-D autocorrelation or spectrum. The internal waves are then coherent over an area A_c=l_xl_y in the horizontal plane.

Scatter Function of an Internal Wave Inhomogeneity

Given an inhomogeneity with characteristic lengths l_xand l_y in the x and y directions respectively and height h, its scatter function can be determined from first-order Rayleigh-Born approximation to Green's theorem. For an incoming plane wave in the direction k_i=(k, a_i, b_i) and scattered plane wave in the direction k=(k, a, b), the scatter function is,

(3)

where,

(4)

is the cosine of the angle between the incident and scattered plane wave directions, and G_k is the fractional change in compressibilty and G_d is the fractional change in density of the inhomogeneity from the surrounding medium. Since each internal

wave inhomogeneity occupies an area of A_c=l_xl_y in the horizontal plane, there are 1/A_c inhomogeneities per unit area in the waveguide. The plane wave scatter function per unit area of the inhomogeneities is then given by,

(5)

The mean field, variance, and expected total intensity of the forward field propagated through the ocean waveguide containing random internal waves is expressed analytically using the formulation of Ref. [1]. For a source at r₀=(-x₀,0,z₀) and a receiver at r=(x,0,z), the mean forward field is given by,

(6)

(7)

is the incident field from mode m when there are no inhomogeneities in the medium, u_m(z) is the amplitude of the mode shape at depth z, and,

	(8)
000	(7)

is the modal horizontal wavenumber change arising from scattering with the inhomogeneities, where,

(10)

and z_h=D+h/2 is the depth of the centroid of the scatter function per unit area. Equation 9 is an expected value over the possible heights h of the internal wave disturbance. We observe that for each mode m, the horizontal wavenumber change depends on the scatter function amplitude in the forward azimuth where b=b_i=0. The modal horizontal wavenumber change is complex and it leads to both dispersion and attenuation in the mean forward field.

As discussed in Sec. IIB, the scatter function S_h of the internal wave inhomogeneity in Eq. 3 is purely real. As a result, using Eq. 3 in Eq. 9 will lead to a modal horizontal wavenumber change that is purely real. This provides us with the dispersion, but not the attenuation in the mean forward field as a result of scattering. Here we derive the modal attenuation coefficients I{n_m} by applying the waveguide extinction theorem[2]. For mode m, this is found to be,

(11)

The variance of the forward field at the receiver can be expressed as,

(12)

(13)

is defined in Ref. [1] to be the exponential coefficient of modal field variance. The variance of the forward field depends on the first and second order moments of the scatter function density of the random medium. The exponential coefficient of modal field variance describes how energy in incident mode m is coupled to scattered modes p after random multiple scattering through the waveguide.

The mean forward field of Eq. 6 is also called the coherent field, the magnitude square of which is proportional to the coherent intensity. The variance of the forward field in Eq. 12 provides a measure of the incoherent intensity. The total intensity at the receiver of the forward field is the sum of the coherent and incoherent intensities. The coherent field dominates at short ranges from the source and in slightly random media, while the incoherent field dominates in highly random media. It should be noted that in a non-random waveguide m_m=0 so that the variance of the forward field is zero, from Eq. 12. This is expected since the field is fully coherent in this case.

ILLUSTRATIVES EXAMPLES

Here we provide examples illustrating the dependencies of the forward field moments on parameters of a random internal wave field in a typical shallow water waveguide. We investigate the effects of internal wave height on the forward field moments. The coherent, incoherent, and total intensities of the forward field are compared for these different internal wave conditions.

In all examples, a water column of H_w=100 m depth is used to simulate a typical continental shelf environment, as shown in Fig. 1. The water column is comprised of a warm upper layer overlying a cool lower layer. The bottom is comprised of a sediment half-space made of sand. The source is located at z₀=50 m depth in the water column with frequency 415 Hz and a source strength of 0 dB re 1 mPa @ 1m. The internal wave field propagates randomly along the boundary between the warm and cool water layers at D=30 m depth. We consider two scenarios for the internal wave height standard deviation. In the first scenario, the internal wave disturbances have a height standard deviation of s_h=1 m which is smaller than the acoustic wavelength of l=3.6 m. We consider this waveguide to be slightly random. In the second scenario, the internal wave disturbances have a height standard deviation of s_h =4 m which is on the order of but a little larger than the acoustic wavelength. We consider this waveguide to be highly random. In these scenarios, we assume that the internal wave disturbances have characteristic length scales of l_x=l_y=250 m in the x and y directions.

Figure 1: Geometry of the shallow water waveguide with Hw=100 m depth water column comprised of a warm upper layer with density r1=1.0242 g/cm3 and sound speed c1=1520 m/s overlying a cool lower layer with density r2=1.0249 g/cm3 and sound speed c2=1500 m/s. The bottom sediment half space is composed of sand with density rb =1.9 g/cm3 and sound speed cb==1700 m/s. The attenuation in the water column is a=6e-5 dB/l and in the bottom is ab=0.8 dB/l . The 3D internal wave field propagates along the boundary between the warm and cool water at D=30 m.

Figure 2 shows intensity of the coherent, incoherent and expected total intensity for the slightly random waveguide in (a) and the highly random waveguide in (b). In the slightly random waveguide, the incoherent intensity is small compared to the coherent intensity even at long ranges up to 50 km. The total field maintains the range and depth dependent structure due to modal interference. The situation changes in the highly random waveguide as shown in Fig. 2(b). The coherent intensity decays rapidly as a function of range from the source due to severe attenuation arising from internal wave scattering. The incoherent component dominates the expected total intensity and the field becomes predominanty incoherent beyond a couple of kilometers in range. The expected total intensity decays monotonically with range in this waveguide at sufficiently long ranges and no longer exhibits a coherent modal interference structure in range and depth.

Figure 2: Intensities of the coherent field, incoherent field, and total field at 415 Hz as a function of range and depth in the shallow water waveguide of Fig. 1. The internal wave disturbances have a height standard deviation of sh=1 m in Fig. (a) and sh=4 m in Fig. (b). The characteristic lengths are lx=ly=250 m. The source is at 50 m depth with strength 0 dB re 1 mPa @ 1m. The acoustic intensity in the static waveguide with no internal waves is also plotted as the unperturbed intensity for comparison.

CONCLUSION

The statistical moments of the forward field propagated through an ocean waveguide with

random internal waves are modelled using a normal mode formulation that takes into account the 3-D interaction of the acoustic field with medium inhomogeneities. The formulation analytically expresses the effect of dispersion, attenuation and redistribution of modal energies due to random scattering with the internal wave inhomogeneities. Calculations for a typical continental shelf environment show that the acoustic field becomes incoherent at short ranges in a waveguide where rms internal wave amplitudes at the mixed layer are on the order of the acoustic wavelength. Calculations of the moments of the acoustic field resulting from 3D interactions with random internal wave inhomogeneities may likely be inaccurately modeled by Monte-Carlo simulations using 2-D propagation and scattering models for ranges where the Fresnel length exceeds the characteristic cross-range length scale of the inhomogeneities.

References

P. Ratilal, N. C. Makris, «Mean and covariance of the forward field propagated through a stratified ocean waveguide with three-dimensional random inhomogeneities,» submitted to J. Acoust. Soc. Am.         [ Links ] [1]

P. Ratilal & N. C. Makris, «Extinction theorem for object scattering in a stratified medium», J. Acoust. Soc. Am., Vol. 110, 2924-2945, (2001).         [ Links ] [2]

P. M. Morse & K. U. Ingard, Theoretical Acoustics, (Princeton University Press, New Jersey, 1986).         [ Links ] [3]

P. Hursky, M. B. Porter, B. D. Cornuelle, W. S. Hodgkiss & W. A. Kupeman, «Adjoint modeling for acoustic inversion,» J. Acoust. Soc. Am., Vol. 115, 607-619, (2004).         [ Links ] [4]

J. R. Apel, M. Badiey, C.-S. Chiu, S. Finette, R. Headrick, J. Kemp, J. F. Lynch, A. Newhall, M. H. Orr, B. H. Pasewark, D. Tielbuerger, A. Turgut, K. Heydt & S. Wolf, «An overview of the 1995 SWARM shallow-water internal wave acoustic scattering experiment,» IEEE J. Ocean. Eng., Vol. 22, 465-500, (1997).         [ Links ] [5]

N. C. Makris, F. Ingenito, & W. A. Kuperman, «Detection of a submerged object insonified by surface noise in an ocean waveguide,» J. Acoust. Soc. Am., Vol. 96, 1703-1724, (1994).         [ Links ] [6]

T. Duda & J. Preisig, «A modeling study of acoustic propagation through moving shallow water solitary wave packets,» IEEE J. Oceanic Eng., Vol. 24, 16-32, (1999).         [ Links ] [7]

J. Colosi, R. Beardsley, J. Lynch, G. Gawarkiewicz, C. Chiu & A. Scotti, «Observations of nonlinear internal waves on the outer New England continental shelf during the summer Shelfbreak Primer study,» J. Geophys. Res., Vol. 106, 9587-9601, (2001).         [ Links ] [8]

D. B. Creamer, «Scintillating shallow water waveguides,» J. Acoust. Soc. Am., Vol. 99, 2825-2838 (1996).         [ Links ] [9]