A method for estimating normal mode parameters based on tensor decomposition and its application

Abstract

This invention relates to a method and application for estimating normal mode parameters based on tensor decomposition, belonging to the field of underwater acoustic signal processing. It solves the problem of modal parameter estimation when the sampling vectors of the modal depth function are non-orthogonal. This invention utilizes the translation invariance of the sound field after distance compensation to transform the modal parameter estimation problem into a canonical polyadic decomposition (CPD) problem of a third-order tensor. The decomposed factor matrix is ​​then used to simultaneously estimate the modal depth function and the modal level wavenumber. Compared with existing technologies, this invention can achieve accurate estimation of the modal level wavenumber and modal depth function using fewer depth and distance dimension samples, provided that the modal depth function sampling vectors are linearly independent. It can be applied to underwater acoustic problems such as ground acoustic inversion and sound source depth classification.

Description

Technical Field

[0001] This invention belongs to the field of underwater acoustic signal processing technology, and relates to a method for estimating normal mode parameters based on tensor decomposition and its application. Background Technology

[0002] Normal mode theory describes the underwater acoustic field as a superposition of a series of discrete modes, each of which can be characterized by a modal depth function and its corresponding modal level wavenumber. Normal mode theory has been widely applied in underwater acoustic problems such as sound source localization and ground acoustic parameter inversion. Modal depth function estimation and modal level wavenumber estimation are collectively referred to as modal parameter estimation. Estimating modal parameters is often a crucial step in many underwater acoustic problems. However, modal parameter estimation has always faced two challenges: regarding modal level wavenumber estimation, sampling data from a single depth cannot recover the mode with a zero modal response at that depth; regarding modal depth function estimation, when the number of array elements is insufficient, the modal depth function sampling vectors are non-orthogonal, rendering traditional singular value decomposition (SVD) modal parameter estimation methods ineffective.

[0003] Modal wavenumber estimation requires sampling the sound field at a fixed depth and a certain distance aperture, and can be viewed as a spatial spectrum estimation problem. Classical modal wavenumber estimation methods are based on the Hankel function form of the sound field and employ Discrete Fourier Transform (DFT). The wavenumber resolution of this method is mainly limited by the distance aperture; since the horizontal wavenumber intervals of low-order modes are small, a large distance aperture is often required. High-resolution spectrum estimation methods have also been applied to modal wavenumber estimation problems, such as Prony spectrum estimation, Multiple signal classification (MUSIC), Estimation of signal parameters via rotational invariance techniques (ESPRIT), Autoregressive spectrum estimator, Compressive sensing (CS), and Sparse Bayesian learning. However, this single-depth spectral estimation method is significantly affected by the sampling depth, especially when the sampling depth happens to be near the zero point of a certain modal depth function. In such cases, the acoustic field measurement contains little or no information about that modal, making it difficult to estimate the mode. To avoid the influence of sampling depth, it is necessary to rely on sampling data from multiple depths. Song et al. proposed a high-resolution modal wavenumber estimation method based on a vertical array, and Akins and Kuperman also proposed the high-resolution wavenumber spectrum estimation method Modal-MUSIC. However, both of these methods require the number of array elements to be greater than the number of modes.

[0004] In modal depth function estimation, the classical method is to perform singular value decomposition (SVD) on the sound pressure data acquired by the vertical array. This requires that the modal depth function sampling vectors be strictly orthogonal, but in practice, such orthogonality is often difficult to guarantee. On the one hand, orthogonality requires dense sampling of modes along the depth direction, which means a large number of array elements; on the other hand, the seabed is not an absolutely rigid boundary, and there are modes below the seabed interface that cannot be sampled. To address the problem of modal depth function estimation when the sampling vectors are not orthogonal, Walker et al. used the frequency-wavenumber structure of broadband signals for mode separation and studied the influence of modal Doppler on modal depth function estimation. In addition, mode separation can also be achieved using time-frequency analysis, but this often requires a broadband sound source and the selection of an appropriate time starting point. Both of these methods require that the modes be separable in the frequency-wavenumber domain or the time-frequency domain, and separating low-order modes with short distances may be difficult. Niu et al., with known sound velocity profiles, constructed a dictionary matrix using the chasing method and the dispersion relation of broadband signals, applying Block Sparse Bayesian Learning to modal depth function estimation. Park et al. utilized the sparsity of sound pressure in the frequency-wavenumber domain to model the modal parameter estimation problem as a sparse signal reconstruction problem, also achieving modal parameter estimation when the modal depth function sampling vectors are non-orthogonal. Gazzah and Jesus used two identical vertical matrices with the same deployment depth for transmission and reception, respectively, modeling the modal depth function estimation problem as a matrix eigenvalue decomposition problem, thus avoiding the requirement for orthogonality of the sampled modes.

[0005] In summary, to achieve modal parameter estimation in the case of non-orthogonality of modal depth function sampling vectors, existing methods fully utilize additional information, such as sparse priors, sound velocity profiles, multiple frequencies, and multiple sound sources. This can be explained from the perspective of matrix factorization: estimating modal parameters using the vertical matrix output is essentially a matrix factorization problem; however, matrix factorization does not have a unique solution. To obtain the desired unique solution, either the latent characteristics of the model must be further explored, or additional constraints must be imposed. This paper aims to propose a modal parameter estimation method that can jointly process multi-depth data and has a low dependence on the number of array elements and modal orthogonality. Summary of the Invention

[0006] Technical problems to be solved

[0007] To avoid the shortcomings of existing technologies, this invention proposes a method for estimating normal mode parameters based on tensor decomposition and its application.

[0008] Technical solution

[0009] A method for estimating normal mode parameters based on tensor decomposition, characterized by the following steps:

[0010] Step 1: Reconstruct the vertical matrix sampling data into a third-order tensor The sampling data are sound pressure data of a single-frequency sound source at different horizontal distances;

[0011] Step 2: Perform normalized multivariate decomposition on the third-order tensor to obtain three factor matrices:

[0012]

[0013] In the formula Let be any permutation matrix. It is a diagonal matrix with non-zero diagonal elements, and satisfies D1D2D3=I;

[0014] Step 3: Estimate the modal wavenumber and modal shape from the factor matrix obtained from tensor decomposition: [Examine the matrix...] The column vectors are normalized according to their maximum values ​​to obtain the modal depth function estimate, i.e., the modal shape; for or The wavenumber corresponding to the peak value of the wavenumber spectrum obtained by performing a Fourier transform on the column vector is the wavenumber of the mode to be determined.

[0015] The sound pressure data is represented by a vertical array located at r=0, containing N array elements with element depths z1, z2, ..., z. N The depth of the sound source is z s At distances r1, r2, ..., r L Transmit L single-frequency signals at intervals of Δ r The l-th signal received by the nth array element is:

[0016]

[0017] In the formula: ξ is a constant coefficient related to the sound source intensity, φ m (z) is the modal depth function (modal shape), k m Let M be the horizontal wavenumber of the m-th mode, and M be the number of modes.

[0018] The transmission of L single-frequency signals is without loss of generality, and L is assumed to be an even number.

[0019] The method for constructing the reconstructed third-order tensor in step 1 is as follows:

[0020]

[0021]

[0022]

[0023] In the formula (q=1,2,…Q;2≤Q≤L-1).

[0024] The third-order tensor is further expressed as:

[0025]

[0026] In the formula:

[0027]

[0028]

[0029] make Represents the space of real numbers. Representing complex space, tensor Recorded as

[0030] An application of the tensor decomposition-based normal mode parameter estimation method is characterized by its applicability to underwater acoustic problems, including but not limited to ground acoustic inversion and sound source depth classification.

[0031] Beneficial effects

[0032] This invention proposes a method and application for estimating normal mode parameters based on tensor decomposition, belonging to the field of underwater acoustic signal processing. It solves the problem of modal parameter estimation when the sampling vectors of the modal depth function are non-orthogonal. This invention utilizes the translation invariance of the sound field after distance compensation to transform the modal parameter estimation problem into a canonical polyadic decomposition (CPD) problem of a third-order tensor. The decomposed factor matrix is ​​then used to simultaneously estimate the modal depth function and the modal level wavenumber. Compared with existing technologies, this invention can achieve accurate estimation of the modal level wavenumber and modal depth function using fewer depth and distance dimension samples, under the condition that the modal depth function sampling vectors are linearly independent. It can be applied to underwater acoustic problems such as ground acoustic inversion and sound source depth classification.

[0033] Compared with the prior art, the beneficial effects of the present invention are:

[0034] 1. It solves the dependence on modal orthogonality and only relies on the linear independence of the modal depth function sampling vector, thus greatly reducing the sampling requirements for the distance and depth dimensions. Even if the number of array elements is less than the number of modes, accurate modal parameter estimation can be achieved.

[0035] 2. The modal horizontal wavenumber estimation has high accuracy and relatively low requirements for distance aperture, making it a promising application in environments with varying horizontality. Attached Figure Description

[0036] Figure 1 This is a flowchart illustrating an embodiment of the present invention.

[0037] Figure 2 This is a schematic diagram of the transmit / receive configuration used for modal parameter estimation in an embodiment of the present invention.

[0038] Figure 3 This is a flowchart of the tensor decomposition algorithm in an embodiment of the present invention.

[0039] Figure 4 This is a comparison chart of modal level wavenumber estimation results obtained by different methods in an embodiment of the present invention:

[0040] (a) 4.5 km from the sampling aperture

[0041] (b) 3km from the sampling aperture

[0042] (c) 1.5km from the sampling aperture

[0043] Figure 5 This is a comparison chart of modal depth function estimation results obtained by different methods in an embodiment of the present invention.

[0044] Figure 6 As shown in the embodiments of the present invention, the curves of modal parameter estimation error versus signal-to-noise ratio under different parameters are as follows:

[0045] (a) Modal parameter estimation performance curves for different numbers of array elements

[0046] (b) Modal parameter estimation performance curves at different aperture distances

[0047] (c) Modal parameter estimation performance curves under different tensor construction parameters

[0048] Figure 7 This is a graph showing the variation of modal parameter estimation error with distance aperture for different methods in an embodiment of the present invention. Detailed Implementation

[0049] The present invention will now be further described in conjunction with the embodiments and accompanying drawings:

[0050] Please see Figure 1 This invention discloses a mode extraction method based on tensor decomposition, which mainly includes data acquisition, tensor representation, tensor decomposition, mode wavenumber estimation, and mode shape estimation. Specifically, it includes the following steps:

[0051] (1) Use a vertical array to collect sound pressure data of a single-frequency sound source at different horizontal distances.

[0052] Please see Figure 2The receiving array used for mode extraction is an N-element vertical array with element depths z1, z2, ..., z. N The sound source is located at r = 0. The depth of the sound source used is z. s In sequence at distances r1, r2, ..., r L The system transmits L single-frequency signals at a distance interval of Δ (without loss of generality, we can assume L is an even number). r According to the normal mode theory, the depth is z n The distance at which the array element receives the signal is r. l The sound pressure signal emitted by the sound source at that location is

[0053]

[0054] In the formula, ξ is a constant coefficient related to the sound source intensity, and φ m (z) is the modal depth function (modal shape), k m Let be the horizontal wavenumber of the m-th mode.

[0055] (2) Reconstruct the vertical matrix sampling data into a third-order tensor

[0056] The third-order tensor is constructed as follows: First, the received signal of the entire vertical array is represented in matrix form.

[0057]

[0058] Then, compensation is performed based on the distance to obtain...

[0059]

[0060] The distance-compensated signal is constructed as a third-order tensor of N×(L–Q+1)×Q.

[0061]

[0062] In the formula (q = 1, 2, ..., Q; 2 ≤ Q ≤ L-1), this third-order tensor can be further expressed as:

[0063]

[0064] In the formula

[0065]

[0066]

[0067]

[0068] Further Represents the space of real numbers. Representing complex space, tensor Can be recorded as

[0069] (3) Perform CPD decomposition on the third-order tensor

[0070] For third-order tensors Decomposition yields estimates of the factor matrices A, B, and C. Due to the inherent uniqueness of tensor decomposition, the resulting estimates have the following form.

[0071]

[0072] In the formula Represents an arbitrary permutation matrix. All are diagonal matrices with non-zero diagonal elements, and satisfy D1D2D3=I.

[0073] There are many algorithms for calculating tensor CPD. The following section uses the commonly used Alternating Least Squares (ALS) method as an example to further illustrate the process of solving the CPD of a third-order tensor. It can be expanded into matrix form in three different ways:

[0074]

[0075]

[0076]

[0077] In the formula, vec(·) represents the matrix straightening operator, which expands the matrix column-wise into a long vector, ⊙ represents the Khatri-Rao product, and the superscript "T" indicates transpose. The three matrices mentioned above are called tensors. The CPD decomposition problem of tensors is transformed into finding factor matrices A, B, and C that satisfy the following conditions: (mode-1, mode-2, and mode-3 expansions are given.)

[0078]

[0079]

[0080]

[0081] This problem can be solved iteratively. The iterative formula is as follows:

[0082]

[0083]

[0084]

[0085] Where: A q , B q , C q represent the factor matrices in the q-th iteration, represents the generalized inverse. Define the fitting error

[0086]

[0087] (4) Estimate the modal wavenumber and modal depth function from the factor matrices obtained by tensor decomposition

[0088] Modal depth function estimation: Normalize the column vectors of the matrix by their maximum values.

[0089] Modal horizontal wavenumber estimation: When L - Q + 1 > Q, let represent the m-th column element of, and calculate the wavenumber spectrum of using the discrete Fourier transform

[0090]

[0091] Where the k corresponding to the peak of the beam spectrum S(k) is the estimated value of the modal wavenumber. When L - Q + 1 < Q, can be taken as the m-th column of, and the subsequent steps are the same as

[0092] The present invention will be further described below in conjunction with specific simulation examples.

[0093] Simulation example:

[0094] To illustrate the effectiveness of the present invention, consider a typical scenario where the sampling vectors of the modal depth function are non-orthogonal. In a shallow sea environment, the sea depth is 100 m, the sound speed in water is constant at 1500 m / s, and the seabed condition is an absolutely hard seabed. Use the acoustic field simulation program Kraken based on modal theory to generate simulation data. In the above ideal waveguide environment, the modal horizontal wavenumber and modal depth function can be expressed as

[0095]

[0096] where f represents the frequency and m is a positive integer not exceeding 2fH / c + 1 / 2. When the frequency is 100 Hz, there are 13 modes in this environment, and the values of the modal horizontal wavenumber are shown in Table 1.

[0097] Table 1 Modal horizontal wavenumber values (accuracy 10 -4 )

[0098]

[0099]

[0100] Without loss of generality, we use the synthesized sound field of the first four modes for simulation analysis. In the simulation, a 25-element vertical array is uniformly distributed at 2m intervals in the depth range of [52,100]m, the sound source depth is 100m, and the distance to the sampling interval is 15m.

[0101] Figure 4 The modal level wavenumber spectra of three methods—CPD, CS, and SVD—are presented at different distance apertures (4.5 km, 3 km, and 1.5 km). In the figure, solid lines represent the wavenumber spectrum, red circles represent wavenumber peaks, and gray solid dots represent the true values ​​of the level wavenumbers. It can be seen that at a distance aperture of 4.5 km, all three methods can achieve accurate modal level wavenumber estimation. As the distance aperture decreases, the main lobes of the wavenumber spectra of all three methods tend to broaden. The peak value of the CPD method consistently corresponds accurately to the true modal level wavenumber; the CS and SVD methods show deviations in their level wavenumber estimates at a distance aperture of 1.5 km, and the CS method can no longer correctly estimate the fourth-order mode. In contrast, the CPD method requires only a shorter distance aperture (fewer distance sampling points) to achieve accurate modal parameter estimation, thus showing better application prospects in environments with varying distances.

[0102] Figure 5 These are the modal depth function estimation results for three methods at a distance of 4.5 km from the aperture. It can be seen that the CPD and CS methods achieve accurate modal depth function estimation, while the SVD method's estimated modal depth function differs significantly from the true value and exhibits higher wavenumber spectral sidelobes than the CPD method. This is mainly due to the fact that the modal depth function sampling vector and the range sampling vector do not strictly satisfy orthogonality. It should be noted that the estimation results of the CS method are obtained with carefully selected hyperparameters, while the CPD method does not require setting hyperparameters.

[0103] Figure 6 The curves showing the variation of modal parameter estimation error with signal-to-noise ratio (SNR) under different parameters are presented. Here, the SNR is defined as...

[0104]

[0105] Where p n ε represents the frequency domain data of the nth element. n This represents the corresponding additive Gaussian noise. Monte Carlo simulations are used to analyze the parameter estimation performance of the proposed method. The root mean square error of modal level wavenumber estimation is defined as:

[0106]

[0107] in Let be the estimated level wavenumber of the m-th mode in the j-th Monte Carlo experiment, and J be the total number of Monte Carlo experiments. Similarly, for the modal depth function, we have:

[0108]

[0109] in Let be the estimated sample vector value of the m-th modal depth function in the j-th Monte Carlo experiment. Since there are N modal depth function estimates at various depths, the average value was used when calculating the total error level. It can be seen that, overall, the modal parameter estimation error decreases with increasing signal-to-noise ratio, and also decreases with increasing array element number, range aperture, and tensor construction parameter Q. However, when Q increases to a certain extent (e.g., from 50 to 100), the estimation error essentially stops decreasing. Clearly, a larger Q value corresponds to a larger computational cost. Therefore, in practical applications, a suitable Q value should be chosen by balancing the error level with the computational cost.

[0110] Figure 7 The curves showing the estimation errors of the CPD and SVD methods as a function of distance aperture under different signal-to-noise ratios (SNRs) are presented. At the same SNR, the error level of the CPD method is significantly lower than that of the SVD method. Even with a 10 dB difference in SNR (10 dB for CPD and 20 dB for SVD), the estimation error of the CPD method remains lower than that of the SVD method. It should be noted that Monte Carlo simulations were not performed on the CS method because the selection of hyperparameters significantly affects the results at different SNRs and the number of peaks in the wavenumber spectrum is difficult to determine.

[0111] In summary, this invention discloses a modal parameter estimation method based on tensor decomposition. This method can achieve accurate estimation of modal level wavenumbers and modal depth functions using fewer depth and distance dimension samples, provided that the modal depth function sampling vectors are linearly independent. It can be applied to underwater acoustic problems such as ground acoustic inversion and sound source depth classification.

[0112] It should be noted that the above are merely typical embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for estimating normal mode parameters based on tensor decomposition, characterized in that... The steps are as follows: Step 1: Reconstruct the vertical matrix sampling data into a third-order tensor The sampling data consists of sound pressure data from a single-frequency sound source at different horizontal distances. The sound pressure data is from a vertical array located at... place, including There are array elements, and the depths of the array elements are respectively... The depth of the sound source is At different distances emission Sub-frequency signal, distance interval is ;No. The element receives the first The signal is: In the formula: It is a constant coefficient related to the sound source intensity. For modal depth function, for The horizontal wavenumber of the first mode, The number of modes; Step 2: Perform normalized multivariate decomposition on the third-order tensor to obtain three factor matrices: In the formula Let be any permutation matrix. Let be a diagonal matrix whose diagonal elements are not zero, and satisfy the following conditions: ; Step 3: Estimate the modal wavenumber and modal shape from the factor matrix obtained from tensor decomposition: For the matrix The column vectors are normalized according to their maximum values ​​to obtain the modal depth function estimate, i.e., the modal shape; for or The wavenumber corresponding to the peak value of the wavenumber spectrum obtained by performing a Fourier transform on the column vector is the wavenumber of the mode to be determined.

2. The method for estimating normal mode parameters based on tensor decomposition according to claim 1, characterized in that: The launch Without loss of generality, let the sub-single-frequency signal be assumed It is an even number.

3. The method for estimating normal mode parameters based on tensor decomposition according to claim 1, characterized in that: The method for constructing the reconstructed third-order tensor in step 1 is as follows: In the formula ; The third-order tensor is further expressed as: In the formula: , ， ， ， make , , , Represents the space of real numbers. Representing complex space, tensor Recorded as .

4. An application of the tensor decomposition-based normal mode parameter estimation method according to any one of claims 1 to 3, characterized in that: It is applied to ground acoustic inversion and sound source depth classification.

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