Modal wavenumber estimation by combining physical informed neural network.

The horizontal wavenumbers and modal depth functions are estimated by block sparse Bayesian learning (BSBL) for broadband signals received by a vertical line array in shallow-water waveguides. The dictionary matrix consists of multi-frequency modal depth functions derived from shooting methods given a large set of hypothetical horizontal wavenumbers. The dispersion relation for multi-frequency horizontal wavenumbers is also taken into account to generate the dictionary. In this dictionary, only a few of the entries are used to describe the pressure field. These entries represent the modal depth functions and associated wavenumbers. With the constraint of block sparsity, the BSBL approach is shown to retrieve the horizontal wavenumbers and corresponding modal depth functions with high precision, while a priori knowledge of sea bottom, moving source, and source locations is not needed. The performance is demonstrated by simulations and experimental data.

In shallow water, modal wavenumbers are useful for many applications. This paper presents an approach for wavenumber estimation of range-independent waveguide based on pressure and its finite difference (FD) data matrices built on the vertical linear array (VLA) data. The core of this method is that wavenumber information is expressed in the singular value decomposition (SVD) of the FD data matrix. Therefore, wavenumbers can be estimated directly by combining the SVDs of the pressure data matrix and FD data matrix. The method is validated on simulated data. The dependence of the proposed method on relevant parameters is discussed.

Source depth estimation is always a problem in underwater acoustic area, because depth estimation is a nonlinear problem. Traditional depth estimation methods use a vertical line array, which has disadvantage of poor mobility due to the size of sensor array. In order to estimate source depth with a horizontal line array, we propose a matched-mode depth estimation method based on autoregressive (AR) wavenumber estimation for a moving source in shallow water waveguides. First, we estimate the mode wavenumbers using the improved AR modal wavenumber spectrum. Second, according to the mode wavenumber estimation, we estimate the mode amplitudes by the wavenumber spectrum, which is obtained by generalized Hankel transform. Finally, we estimate source depth estimation by the peak of source depth function wherein the data mode best matches the replica mode that is calculated using a propagation model. Compared with synthetic aperture beamforming, the proposed method exhibits a better performance in source depth estimation under low signal-to-noise ratio or the small range span. The robustness of the proposed method is illustrated by simulating the performance in mismatched environment.

This paper describes that in shallow water the scattered wave from the target is detected by converging the sound wave to the target using the phase conjugate wave. The phase conjugate wave radiated from the vertical linear transducer array (VLA) is converged to the sound source position. It was confirmed that the phase conjugate wave converged at the sound source position by the experiment. The sound source is replaced with a pseudo sound source. It is assumed that the sound wave is transmitted from a pseudo sound source, and obtains the wave form in the element of VLA by the calculation. The calculated waveform is excuted time reversal processing and it reradiated from VLA. And it was confirmed that the phase conjugate wave converged at the position of the pseudo sound source by the experiment. Consequently, it was confirmed to move the focus of the phase conjugate wave without a real sound source, and to be able to detect the target.

Acoustic source bearing modeling in shallow water waveguides

The difference in intensity fluctuation between a surface signal and a submerged signal can be used to discriminate the source depth in the shallow water waveguide. However, the properties of fluctuation distinction in intensity between surface source and submerge source are rarely studied, resulting in the poor performance of fluctuation-based methods sometimes. In this work, the intensity fluctuations caused by source depth fluctuations is analyzed by differential method and variance statistics method to figure out the physics of intensity fluctuations, and to present the suitable quantity for depth discrimination. Firstly, the derivative expression of real part, power and amplitude of the pressure field are respectively derived from the normal mode theory, hence their fluctuation quantities are specified. Then, the numerical examples, including the factor of receiver depth, source frequency and source range, are treated to verify the derivation and summarize the characteristic of fluctuations. The property of the intensity fluctuations is also compared with the distribution of normalized quantity on the depth dimension. Finally, the SWellEx-96 experimental data processing results validate the conclusions. The processing results of derivation, simulation data and experimental data show that the interactions between modes excited by the source give rise to the fluctuations. Meanwhile, the property of the interactions is affected by the received depth, source range and source frequency. Amplitude is capable of building a quantity based on fluctuations for source depth discrimination. For surface and submerged sources with lower frequency, their distinction in their fluctuations is stably represented by amplitude. In the SWellEx-96 experiment, the normalized values calculated from the amplitude of surface noise and submerged sound sources show the lowest difference of 0.5 and the highest difference of 2.5, indicating the effectiveness of using amplitude fluctuations to discriminate the sound source depth.

Most acoustic sources found in the ocean environment are spatially complex and broadband. In the case of shallow water propagation, these source characteristics complicate the analysis of received acoustic data considerably. A common approach to the broadband problem is to decompose the received signal into a set of narrow-band lines. This then allows the problem to be treated as a multiplicity of narrow-band problems. Here a model-based approach is developed for the processing of data received on a vertical array from a broadband source where it is assumed that the propagation is governed by the normal-mode model. The goal of the processor is to provide an enhanced (filtered) version of the pressure at the array and the modal functions. Thus a pre-processor is actually developed, since one could think of several applications for these enhanced quantities such as localization, modal estimation, etc. It is well-known that in normal-mode theory a different modal structure evolves for each temporal frequency; thus it is not surprising that the model-based solution to this problem results in a scheme that requires a “bank” of narrow-band model-based processors—each with its own underlying modal structure for the narrow frequency band it operates over. The “optimal” Bayesian solution to the broadband pressure field enhancement and modal function extraction problem is developed. It is shown how this broadband processor can be implemented (using a suboptimal scheme) in pseudo real time due to its inherent parallel structure. A set of noisy broadband data is synthesized to demonstrate how to construct the processor and achieve a minimum variance (optimal Bayesian) design. It is shown that both broadband pressure-field and modal function estimates can be extracted illustrating the feasibility of this approach.

The output of a vertical linear array is used to infer about the parameters of the normal mode model that describes acoustic propagation in a shallow water. Existing subspace algorithms perform singular vector decomposition of the array data matrix to estimate the sampled model functions. Estimates are exact only if the sensing array is totally covering the water column. We design a new subspace algorithm free from this very restrictive requirement. We use two short hydrophone arrays and activate a monochromatic source at different depths. Estimates of both the modal functions and the wave numbers are obtained in a fully automatic and search-free manner. The algorithm can be qualified as truly high resolution in the sense that, while using short sensing arrays, estimation error becomes arbitrarily low if observation noise is arbitrarily low. This method compares advantageously to existing subspace techniques, as well as transform-domain techniques that require impulsive sources, among other constraints. With two (eigen and singular) vector decompositions, the proposed technique has the complexity of a regular subspace algorithm.

The shallow ocean is an uncertain, varying, dispersive environment dominated by ambient and shipping noise as well as temperature fluctuations that alter sound propagation throughout creating a large number of environmental variations. The need to develop processors that are capable of tracking these changes while simultaneously providing enhanced signals implies a stochastic as well as an adaptive design is required. The stochastic requirement follows directly from the multitude of variations created by uncertain parameters and noise. An adaptive processor providing enhanced signal estimates for acoustic hydrophone measurements on a vertical array as well as enhanced modal function and wavenumber estimates is developed. A normal-mode model is transformed to state-space form and incorporated directly into the processor enabling the signal enhancement capabilities. Data synthesized from the well-known Hudson Canyon experiment is used to demonstrate the viability of this approach.

Estimating the depth of a moving source with unknown source range is always a challenging problem in shallow water waveguides. The method of estimating the current motion source depth is sensitive to the unknown initial range and requires the horizontal synthetic aperture length formed by the motion of the source to be much longer than the modal interference period. Presented in this work is a method to estimate the depth of moving source based on the incoherently matched beam-wavenumber. In the beam-wavenumber domain, each peak amplitude only contains the modal excitation related to source depth, and each peak position corresponds to the mode propagation angle and the horizontal wavenumber. In this method, the received data are first used to perform beam-formed transformation in the vertical depth and horizontal synthetic aperture direction, and transformed into the beam-wavenumber domain. Then beam-wavenumber peak amplitudes are extracted and incoherently matched with the modal depth function to estimate the source depth. The proposed method is used to eliminate the unknown distance dependent term and improves the mode resolution by performing mode separation in the beam-wavenumber two-dimensional domain. The prominent feature of this method lies in realizing the source depth estimation at the unknown initial range and the horizontal synthetic aperture length which is smaller than the mode interference period. The simulation and SWellEx-96 experimental data processing results validate the superior performance of the proposed method.

The Shallow-water environment has significant influences on the acoustic scattering from objects, but the characteristics are quite different from those of free-space problems. A high-performance numerical model of conventional and fast multipole BEM based on Burton-Miller formulation for acoustic scattering from objects in shallow-water waveguides is established for the first time. The numerical model is also combined with the well-developed sound propagation theory for ocean acoustics; the mirror image method is adopted for the near-field problem; and, the normal mode method (i.e. the famous open-source code kraken) is chosen for the far-field problem. The time and space complexities of conventional and fast multipole BEM are compared and analyzed by the computation of different-scale problems. The algorithms are validated by comparison with the scattering results of a rigid sphere situated in a shallow-water waveguide, computed by each method. The horizontal directivity and the frequency responses of scattering from different objects in shallow-water waveguides are computed, and the shallow-water target echoes are obtained by inverse FFT. The results show that, the complexity of conventional BEM is at least O(N2), and the fast multipole BEM is O(N log N). Combined with these two methods, various-scale problems can be computed efficiently. The multipath effects of shallow-water waveguides bring in obvious interferences in the space domain; a ‘comb filter’ effect in the frequency domain; and, also large extension of the target echoes in the time domain. Such characteristics are rather like that of sound propagation problems in the ocean waveguide. Further, it can be found that, at least for simple objects, the frequency characteristics of the scattering are controlled by the shallow-water waveguide, and different objects only take on modulation effects on the amplitude of the frequency response, which are completely different from those of free-space problems.

Shallow water waveguides in the presence of internal waves can make a time-varying, horizontally stratified medium that can significantly affect broadband pulse propagation. In this paper, we analyze a unique data set obtained during the passage of an internal wave event in a region where a broadband sound source (270 to 330 Hz) was received by an L-shaped hydrophone array about 20 km away in 80 meters on the New Jersey continental shelf. During the time 20:30 and 22:07 GMT, on 17 August 2006, an approaching IW affected the dispersion characteristic of the broadband LFM chirp signal while passing an acoustic track in shallow water waveguide. Modal behavior is examined before and after the internal wave front crossed the source-receiver track. While dispersion characteristics of the signals changed, modal arrival time reversal occurred. The corresponding group and phase velocities that signals experienced during this phenomenon are analyzed using the theory of horizontal rays and vertical modes. These results have motivated theoretical and modeling studies of the waveguide behavior since it was first reported in 2010. [Work supported by ONR322OA.]

A survey is presented of recent related work (comprising numerical modeling and field experiments) on the determination of seafloor geoacoustic properties in shallow water from monochromatic pressure field measurements obtained using synthetic aperture and towed arrays. Two related techniques are discussed, their common foundation lying in the Hankel transform inversion of the pressure versus range data to obtain a depth-dependent Green's function versus horizontal wave number. Characteristic features of the Green's function, such as the peaks associated with normal mode propagation in the shallow water waveguide, are used to extract the geoacoustic properties (compressional and shear) as a function of depth in the bottom. INTRODUCTION Two related methods for determining geoacoustic properties in shallow water waveguides have recently been under development (Frisk and Lynch, 1984; Frisk et al., 1985; Kuperman et al., 1985; Werby and Tango, 1985). In both cases, the required input data are the magnitude and phase versus range of the pressure field due to a monochromatic (CW) point source. These data are numerically Hankel transformed to obtain the depth-dependent Green's function versus horizontal wave number. In the context of normal mode theory, the Green's function contains information about the nature of the discrete and continuous modal spectra as well as the plane-wave reflection coefficients of the waveguide boundaries. Specifically, the Green's function contains prominent peaks at horizontal wave numbers corresponding to the eigen-values for any trapped and virtual modes excited in the waveguide. The positions and magnitudes of these modal peaks are sensitive to the acoustic properties of the bottom, and this sensitivity is the basis of the two inverse methods for determining geoacoustic models in shallow water. The WHOI method utilizes an exact Hankel transform inversion of synthetic aperture array data obtained by towing a CW source away from two moored receivers. A geoacoustic model is obtained by computing the theoretical Green's function for various values of the bottom parameters and determining the parameter set which provides the best agreement with the experimental Green's function, particularly in the positions and relative magnitudes of the modal peaks. The NORDA technique involves an approximate Hankel transform inversion in which the beam formed output of a towed array due to the self-noise of the tow ship is computed. Layer sound velocity estimates are obtained from the identification of modal critical angles in the variable-offset beam former response. THEORY Consider a water column of thickness h, characterized by a constant density and a sound speed c(z), and bounded at the surface and bottom by horizontally stratified media as shown in Fig. 1. Then the spatial part of the acoustic pressure p, due to a point source at r =.0 and z = Zo with harmonic time satisfies the inhomogeneous Helmholtz equation(Mathematical equation available in full paper) The Green's function satisfies along with impedance boundary conditions at the surface and bottom. These boundary conditions can be ex pressed in terms of the plane-wave reflection coefficients RS(kr) and RB(kr) of the surface and bottom respectively (Frisk et al., 1980).

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).

Based on the physical acoustic method (the Kirchhoff's approximation) and planar element method, the problem of the echo intensity of the target in a shallow water waveguide is studied. The physical acoustic method is a kind of high frequency approximate calculation method, which ignores the contribution to the scattering field of the shadow of the target. The planar elements method is a method of surface integral numerical discretization based on the physical acoustic method, which is to approximate the target surface with a large number of small planar polygons (or plates and surfaces). In this paper, the target echo is decomposed into the direct wave, sea surface reflection wave and sea bottom reflection wave. The direct wave is obtained by using the planar elements method directly, which is mono-static scattering. The sea surface and sea bottom reflection wave is analyzed by the mirror image method firstly, and the reflection waves are obtained by using the planar elements method, which include mono-static and bi-static scattering. The total scattering field of the target in an homogeneous wave guide is the summation of all the waves mentioned above. The target strength of the BeTSSi-Sub (Benchmark Target Strength Simulation Submarine) simple submarine in free field, and in a shallow water homogeneous wave guide, are calculated at different directions. The BeTSSi-Sub model is a benchmark model and is created and meshed using Ansys soft. In the computational examples, we suppose that the water depth is 100m, the sea surface is the absolute soft interface, and the bottom is the absolute hard interface. The target strength of a submarine under water 20m to 80m was calculated at 1 kHz. The results show that the echo intensity of the target in a shallow water waveguide is different from that of the free field due to the influence of upper and lower interface, and the echo intensity of the target is continuously changing with the depth of target.