In this article, a method of passively localizing a narrow-band source in range and depth in a waveguide is presented based on "matching" predicted normal mode amplitudes to measured mode amplitudes. The modes are measured by using a vertical array of hydrophones and performing mode filtering. Previous studies of mode filtering have considered only the overdetermined case, i.e., where there are more hydrophones than discrete modes present in • the waveguide. In this study, mode filtering is considered for the underdetermined case, i.e., where there are fewer hydrophones than the total number of discrete modes in the waveguide, but only a subset of the total number of modes is to be estimated. Previous studies of matched field localization have been based on matching the entire pressure field. In this study, the pressure field is expressed in terms of normal modes, and only a subset of the total number of modes is used for localization. Using a subset of modes allows trade-offs to be made between localization accuracy, computational complexity, and sensitivity to environmental mismatch. In this article, the matched mode localization method is presented, and the dependence of its localization accuracy on the number of modes used and environmental conditions is demonstrated. The effects of array length and hydrophone spacing on mode estimation error, and hence on localization accuracy, are also demonstrated for the particular method of mode estimation used here. Other methods of mode estimation may produce different results. Finally, the effects of mismatch between the assumed and actual environment due to water depth variation are explored. It is shown that localization accuracy in range is proportional to the mode interference distance between the lowest and highest modes used to localize, and that as few as six modes can be used for ranging. It is also shown that the array length need not be any longer than the depth extent of the highest mode to be estimated, and that the hydrophone spacing must be no greater than half the vertical wavelength of the highest mode that contributes significantly to the sound field (not just the highest mode to be estimated). Localization is most sensitive to environmental mismatch effects that contribute to changes in the phase of the horizontal component of the mode amplitudes. Because a subset of modes is used for localization instead of the entire pressure field, this method of localization can be fairly insensitive to certain kinds of environmental mismatch.
A ray theory approach for simulating the propagation of broadband signals interacting with a layered ocean bottom is presented. The range-invariant environment consists of the ocean and an arbitrary number of bottom layers. Each bottom layer has profiles of compressional and shear wave velocity, attenuation, and density. Eigenrays, including those with multiple interactions and refractions in the ocean and bottom layering, are found using a thorough and efficient algorithm based on an analysis of the sound velocity profile and the dependence of ray path geometry on the Snell invariant. The time series at a receiver due to an arbitrary source waveform is obtained by constructing a frequency domain transfer function from the eigenray characteristics. Several example applications demonstrate the potential use of this approach and show that reflection from relatively simple layering can severely distort a received time series.
A method is presented for calculating the complex plane-wave reflection coefficient of an acoustic wave impinging on an ocean bottom consisting of a continuously stratified solid sediment overlying a semi-infinite homogeneous solid substrate. The depth separated wave equations for the potentials are represented by the “propagator” matrix [F. Gilbert and G. E. Backus, Geophys. 31, 326 (1966).] The method is unique in that the propagator is calculated by direct numerical integration. The reflection coefficient is then accurately computed without recourse to approximation by homogeneous layers or by special functions. The choice of differential equations determines the frequency range for which the method is valid. For gradients typical of marine sediments the Helmholtz equations with depth dependent wave velocities are applicable for frequencies above 10 Hz. Numerical results for a typical turbidite layer for frequencies from 20 to 200 Hz are presented. They show that sediment shear waves can significantly increase the bottom reflection loss for thin (35-m) sediment layers. [Work supported by the Naval Ocean Research and Development Activity and the Naval Electronic Systems Command.]
The group velocity for a normal mode can be calculated without invoking a finite difference approximation requiring a second eigenmode calculation. The reciprocity relation is employed in a derivation of the normal mode group velocity and attenuation coefficient. The group velocity thus calculated is more accurate than a comparable finite difference approximation. Arbitrarily arranged layers of solid and fluid media are considered. PACS numbers: 43.30. Bp, 43.30.Jx, 43.20.Hq, 43.20. Bi INTRODUCTIONFor low-frequency long range underwater acoustic propagation problems, a normal mode description has the advantage of being a direct wave theory capable of dealing with complicated depth dependent media. Various physical quantities may be computed once the modal eigenvalues and eigenfunctions are obtained. Two quantities typically of interest are the modal attenuation coefficients and the group velocities. For fluid media the attenuation of a given mode can be calculated using a perturbation approach through integrations involving the depth dependence of the absorption coefficients of the medium and of the modal eigenfunction at the frequency of interest. In calculating the group velocity, the usual approach I involves a second eigenvalue calculation at a nearby frequency followed by dividing the difference between the two eigenvalues into the corresponding frequency difference. The additional eigenvalue calculation required doubles the computational burden. This paper describes an exact group velocity calculation, similar to that used for calculating the mode attenuation coefficient, not requiring an additional eigenvalue calculation. The derivation of this approach is based on the reciprocity principle 2 and is valid for mixed fluid and solid layers. A general discussion 3 shows the relationship between group velocity and energy flux. The calculation of normal mode group velocity as described in our paper has also been proposed and applied to the layered fluid. 4 The present paper considers the problem from another viewpoint for a geometry frequently assumed in underwater acoustic propagation. We begin by describing in Sec. I the horizontally stratified model and our notation. In Sec. II the attenuation and group velocity formulas, for an arbitrary arrangement of horizontal fluid or solid layers, are derived from the reciprocity relation. Section III gives an example of numerical results. Section IV summarizes our results. The Appendix contains a derivation of the reciprocity relation in the form used here. //•. is the depth ofthejth layer from the surface. The last layer N is a homogeneous fluid or solid half-space. The formulation developed below can also be applied to an adiabatic description 5 of problems having range variable parameters. Under these circumstances the shear potential •p and the compressional potential 4 of an acoustic wave are well defined and satisfy 6 the simple wave equations C} 0t 2' V2•/'= C• 0t 2' (l) More complicated equations would be required if additional terms related to density, shear ...
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