CN115542329B - Depth Judgment Method of Shallow Water Low Frequency Sound Source Based on Modal Filtering - Google Patents

Abstract

A method for judging the depth of shallow water low-frequency sound sources based on modal filtering belongs to the technical field of shallow water low-frequency underwater target judgment. The present invention aims at judging the sound source depth when the existing hydrophone array aperture is limited, and the existing modal filtering technology-based water surface and underwater depth resolution method cannot simultaneously take into account the problems of no subspace overlap and the integrity of the modal space. Including establishing the expression of the sound field p(r,z _r ,z _s ) about the observation matrix V and the modal amplitude matrix a; dividing the observation matrix V into the notch subspace V ₀ and the free subspace V ₁ and performing singular value decomposition, Obtain the reduced-rank subspaces U ₀ and U ₁ to form matrix A, then obtain the orthogonal matrix β, determine the space H and space S, and compare the energy projected in the notch subspace with the energy projected in the entire orthogonal modal space to obtain The detection statistics are compared with the selected threshold to determine the depth of the sound source. The invention is used for sound source depth judgment.

Description

Depth Judgment Method of Shallow Water Low Frequency Sound Source Based on Modal Filtering

technical field

The invention relates to a shallow-water low-frequency sound source depth judgment method based on modal filtering, and belongs to the technical field of shallow-water low-frequency underwater target judgment.

Background technique

Shallow water depth judgment technology based on modal features is an important method in surface underwater target classification, especially when the aperture of hydrophone array is limited and high judgment estimation of sound source depth cannot be performed. Although the shallow water depth identification technology based on modal features cannot give the specific depth of the target, it can judge whether the target is on the water surface or underwater, and has applications in anti-submarine and marine biology.

According to the adiabatic approximation theory of normal waves, for shallow water low-frequency sound sources, the sound pressure at a receiving position can be expressed by a series of normal waves. The normal wave is affected by various environments, and the energy distribution of each order mode of the normal wave is mainly determined by the depth of the sound source and the receiving position. Vincent.E Premus simplifies the sound source localization part based on the Matched Field Processing (MFP) method, and then turns the sound source localization problem into a binary classification problem of target depth judgment. According to the characteristics that shallow sources are difficult to excite low-order modes, the mode space is divided into notched subspace and free subspace, and the energy ratio projected in the two subspaces is used as a statistic to judge the water surface and underwater targets.

However, in practice, whether it is a horizontal array or a vertical array composed of hydrophones, it often cannot achieve the aperture required for normal judgment, and there is overlap between the modal subspaces, which leads to a decrease in algorithm performance. In response to this problem, Vincent.EPremus applied the Scharf-Friedlander matched subspace detector to the modal filtering method to obtain a matched subspace discriminator (Matched Subspace Discriminator, MSD), and obtained orthogonal non-overlapping notches by the method of Scharf and Friedlander and free subspaces to improve the performance of the decision device. Ewen Conan multiplied the inverse of the observation matrix with the received sound pressure to obtain the modal magnitude vectors of the notch and free subspace, and calculated the energy ratio projected on the notch space through the modal magnitude vectors as the decision statistic. However, in the existing method, the normal data in the subspace will be removed at the same time when the subspace overlap is removed, thus affecting the judgment result.

In a shallow water waveguide, the modal distribution of the normal wave is closely related to the depth of the sound source. Although the modal strength of each order is difficult to obtain directly by the above methods, the energy of the projected subspace can be used to determine whether the target is underwater. Therefore, how to obtain a robust target decision algorithm when the aperture is limited remains to be paid attention to.

Contents of the invention

Aiming at the existing underwater depth resolution method based on modal filtering technology for judging the sound source depth when the aperture of the existing hydrophone array is limited, the problems of no subspace overlap and the integrity of the modal space cannot be taken into account at the same time. The present invention A method for judging the depth of low-frequency sound sources in shallow water based on modal filtering is provided.

A method for judging the depth of a low-frequency sound source in shallow water based on modal filtering in the present invention includes:

Set the underwater depth of the sound source as z _s , and obtain _the sound field p(r,z _{r ,} z _s ), the sound field p(r, z _r , z _s ) is an expression about the observation matrix V and the modal amplitude matrix a;

Divide the observation matrix V into notch subspace V ₀ and free subspace V ₁ ; perform singular value decomposition on the notch subspace V ₀ to obtain a linearly independent rank-reduced subspace U ₀ , and perform singular value decomposition on the free subspace V ₁ , A linearly independent reduced-rank subspace U ₁ is obtained; the reduced-rank subspace U ₀ and the reduced-rank subspace U ₁ form a matrix A;

Process the matrix A to obtain the orthogonal matrix β; according to the size of the reduced-rank subspace U ₀ and the reduced-rank subspace U ₁ , the orthogonal matrix β is divided into space H and space S, and space H corresponds to the notch subspace after orthogonalization, The space S corresponds to the free subspace after orthogonalization;

Calculate the projection matrix P _H of the space H and the projection matrix P _S of the space S; and then calculate the signal energy E _H projected on the space H by the hydrophone array received signal W and the signal energy E _S projected on the space S; the signal The energy E _H is the energy projected in the notch subspace, and the sum of E _H + E _S is the energy projected in the entire orthogonal modal space; the energy projected in the notch subspace and the energy projected in the entire orthogonal modal space The detection statistic is obtained by doing the ratio, and compared with the selected threshold, if the detection statistic is greater than the selected threshold, it is determined that the sound source is a submerged source, otherwise it is a surface source.

According to the modal filter-based shallow water low-frequency sound source depth judgment method of the present invention, the underwater depth z _r is greater than the underwater depth z _s .

According to the shallow water low-frequency sound source depth judgment method based on modal filtering of the present invention, the expression of the sound field p(r, z _r , z _s ) is:

为模态m关于深度的模态函数，k_rm为模态m的水平波数。In the formula, X(f) is the amplitude of the sound field at frequency f, j is an imaginary number, ρ(z _s ) is the water density at the sound source, M is the mode number of the sound field propagating in the environment, and m is the ordinal number of the mode;

is the mode function of mode m with respect to depth, and k _rm is the horizontal wavenumber of mode m.

According to the shallow water low-frequency sound source depth judgment method based on modal filtering of the present invention, the horizontal wave number k _rm of the mode m is:

ω为声源信号的角频率，c为声速梯度中的最大声速，ω＝2πf，/>

为模态m的垂直波数。where k is a constant,

ω is the angular frequency of the sound source signal, c is the maximum sound velocity in the sound velocity gradient, ω=2πf, />

is the vertical wavenumber of mode m.

According to the shallow-water low-frequency sound source depth judgment method based on modal filtering of the present invention, the hydrophone array is set as an N-element horizontal array, and the sound field is sampled at the distance r ₁ , r ₂ ,..., r _N positions of the sound source, and the water The signal W received by the earphone array is the vector of the sound field p:

W = Va;

r _i =r ₁ +(i-1)d cosθ,i=1,2,...,N,

In the formula, d is the array element spacing, θ is the angle between the sound source and the hydrophone array in the horizontal direction;

Divide the observation matrix V into the notched subspace V ₀ and the free subspace V ₁ , and divide the modal amplitude matrix a into the modal amplitude a ₀ corresponding to the notched subspace V ₀ and the mode amplitude a 0 corresponding to the free subspace V ₁ state amplitude a ₁ , get:

The column vector v _m of the observation matrix V is:

In the formula, R is a vector composed of r _i ;

The modal amplitude matrix a is:

a=[a ₁ ,a ₂ ,…,a _M ] ^T ,

where κ is a constant:

According to the shallow-water low-frequency sound source depth judgment method based on modal filtering of the present invention, the hydrophone array receiving signal W is regarded as the weighted sum of a _m corresponding to each mode, and the weight vector of each mode is the observation matrix V Column vector v _m ; assuming that the mode number of the notch subspace V ₀ is M ₀ , then:

为对应于陷波子空间V₀的M₀个列向量，/>

为对应于模态幅度a₀的M₀个元素；/>

为对应于自由子空间V₁的M-M₀个列向量，/>

为对应于模态幅度a₁的M-M₀个元素；In the formula

are M ₀ column vectors corresponding to the notch subspace V ₀ , />

be the M ₀ elements corresponding to the modal amplitude a ₀ ;/>

be MM ₀ column vectors corresponding to the free subspace V ₁ , />

be ₀ elements of MM corresponding to the modal amplitude a ₁ ;

Singular value decomposition is performed on the notch subspace V ₀ to obtain a linearly independent reduced-rank subspace U ₀ :

U ₀ ∈ ^{C N×q} ,

In the formula, C is a set of complex matrices, and q is the column number of the reduced-rank subspace U ₀ ;

Singular value decomposition is performed on the free subspace V ₁ to obtain a linearly independent reduced-rank subspace U ₁ :

U ₁ ∈ C ^N×t ,

In the formula, t is the column number of the reduced-rank subspace U ₁ ;

And then get the matrix A:

A=[U ₀ U ₁ ].

According to the shallow-water low-frequency sound source depth judgment method based on modal filtering of the present invention, in order to eliminate the overlapping part of the reduced-rank subspace U ₀ and the reduced-rank subspace U ₁ , the two reduced-rank subspaces are subjected to orthogonal processing, so that the reduced-rank subspace U ₀ and the column vectors of the reduced-rank subspace U ₁ form a set of orthogonal basis of the observation space:

According to the matrix A=[U ₀ U ₁ ],

The column vector [u ₁ ,…,u _q+t ] of the matrix A∈C ^N×(q+ t) is a set of linearly independent vectors in the observation space, then there is an orthogonal matrix β:

β=[β ₁ ,...,β _q+t ],

Let A=[β ₁ ,…,β _q+t ]B _A ,

where B _A is the unit upper triangular matrix:

B _A ∈ ^{C (q+t)×(q+t)} ;

The relationship between the column vector [u ₁ ,…,u _q+t ] and the orthogonal matrix β is as follows:

According to the shallow-water low-frequency sound source depth judgment method based on modal filtering of the present invention, according to the size of the reduced-rank subspace U ₀ and the reduced-rank subspace U ₁ , the orthogonal matrix β is divided into space H and space S:

From this we get:

In the formula, span( ) represents the space spanned by the column vectors of .

According to the shallow water low-frequency sound source depth judgment method based on modal filtering of the present invention, the projection y _H of the hydrophone array received signal W on the space H and the projection y _S on the space S are:

The signal energy E H projected on the space H by the hydrophone array receiving signal _W and the signal energy E _S projected on the space S are respectively:

E _H ＝tr(W ^H P _H W)

E _S ＝tr(W ^H P _S W)

Where tr(·) represents the trace of the matrix, and W ^H is the conjugate transpose of W.

According to the shallow water low-frequency sound source depth judgment method based on modal filtering of the present invention, the detection statistic is L(z _s ):

The selected threshold is η:

If L(z _s )≤η, the sound source is a surface source;

If L(z _s )>η, the sound source is submerged source.

Beneficial effects of the present invention: the method of the present invention divides the mode space into notched waves and free subspaces according to the fact that shallow sources are difficult to excite low-order modes in a sound velocity profile environment with negative clines. The non-overlapping notch modal subspace and free modal subspace are obtained by the Schmidt orthogonal method, which eliminates the overlap of subspaces caused by insufficient array aperture and maintains the integrity of the modal space. Through the energy ratio projected on the notched subspace and the free subspace, a judgment is made on whether the sound source is on the water surface or underwater.

Description of drawings

Fig. 1 is the schematic diagram of the shallow water low-frequency sound source depth judgment method based on modal filtering in the present invention; SVD represents singular value decomposition among the figure; Schmidt represents Schmidt orthogonal method;

Fig. 2 is the sound velocity profile in the simulated environment water depth of 160m;

Fig. 3 is a graph of detection statistics as a function of sound source depth.

Detailed ways

The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only some, not all, embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by persons of ordinary skill in the art without making creative efforts belong to the protection scope of the present invention.

It should be noted that, in the case of no conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other.

The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, but not as a limitation of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 1. As shown in FIG. 1 , the present invention provides a method for judging the depth of low-frequency sound sources in shallow water based on modal filtering, including:

Set the underwater depth of the sound source as z _s , and _{obtain the sound field p(r,z r , z s} ), the sound field p(r, z _r , z _s ) is an expression about the observation matrix V and the modal amplitude matrix a;

Divide the observation matrix V into notch subspace V ₀ and free subspace V ₁ ; perform singular value decomposition on the notch subspace V ₀ to obtain a linearly independent rank-reduced subspace U ₀ , and perform singular value decomposition on the free subspace V ₁ , A linearly independent reduced-rank subspace U ₁ is obtained; the reduced-rank subspace U ₀ and the reduced-rank subspace U ₁ form a matrix A;

Process the matrix A to obtain the orthogonal matrix β; according to the size of the reduced-rank subspace U ₀ and the reduced-rank subspace U ₁ , the orthogonal matrix β is divided into space H and space S, and space H corresponds to the notch subspace after orthogonalization, The space S corresponds to the free subspace after orthogonalization;

Calculate the projection matrix P _H of the space H and the projection matrix P _S of the space S; and then calculate the signal energy E _H projected on the space H by the hydrophone array received signal W and the signal energy E _S projected on the space S; the signal The energy E _H is the energy projected in the notch subspace, and the sum of E _H + E _S is the energy projected in the entire orthogonal modal space; the energy projected in the notch subspace and the energy projected in the entire orthogonal modal space The detection statistic is obtained by doing the ratio, and compared with the selected threshold, if the detection statistic is greater than the selected threshold, it is determined that the sound source is a submerged source, otherwise it is a surface source.

As an example, the submerged depth z _r is greater than the submerged depth z _s .

Furthermore, according to the adiabatic approximation theory of normal waves, the expression of the sound field p(r,z _r ,z _s ) excited by a sound source at depth z _s at depth z _r in a shallow water waveguide is:

为模态m关于深度的模态函数，k_rm为模态m的水平波数。In the formula, X(f) is the amplitude of the sound field at frequency f, j is an imaginary number, ρ(z _s ) is the water density at the sound source, M is the mode number of the sound field propagating in the environment, and m is the ordinal number of the mode;

is the mode function of mode m with respect to depth, and k _rm is the horizontal wavenumber of mode m.

The horizontal wavenumber k _rm of mode m is:

ω为声源信号的角频率，c为声速梯度中的最大声速，ω＝2πf，/>

为模态m的垂直波数。where k is a constant,

ω is the angular frequency of the sound source signal, c is the maximum sound velocity in the sound velocity gradient, ω=2πf, />

is the vertical wavenumber of mode m.

k_rm，M这些与频率相关的参数可视为常数。Considering the simplified model, a single-frequency signal is used in the derivation, so X(f),

k _rm , M these frequency-related parameters can be regarded as constants.

Set the hydrophone array as an N-element horizontal array, the depth is z _r , and the sound field is sampled at the distance r ₁ , r ₂ ,…, r _N from the sound source. Without considering the noise, the hydrophone array receives the signal W is the vector of the sound field p, expressed as:

W = Va;

r _i =r ₁ +(i-1)d cosθ,i=1,2,...,N,

In the formula, d is the array element spacing, θ is the angle between the sound source and the hydrophone array in the horizontal direction;

In a shallow water environment with negative clines, low-order modes that are difficult to be excited by shallow sources are called trapped modes, and modes whose phase velocity is less than the maximum sound velocity in water are defined as trapped modes. Divide the observation matrix V into the notched subspace V ₀ and the free subspace V ₁ , and divide the modal amplitude matrix a into the modal amplitude a ₀ corresponding to the notched subspace V ₀ and the mode amplitude a 0 corresponding to the free subspace V ₁ state amplitude a ₁ , get:

The column vector v _m of the observation matrix V is:

In the formula, R is a vector composed of r _i ;

The modal amplitude matrix a is:

a=[a ₁ ,a ₂ ,…,a _M ] ^T ,

where κ is a constant:

Further, the received signal W of the hydrophone array is regarded as the weighted sum of a _m corresponding to each mode, and the weight vector of each mode is the column vector v _m of the observation matrix V; by the expression of the modal magnitude matrix a It can be seen that when the depth and aperture of the horizontal receiving array are known for single-frequency signals, the size of a _m depends on the sound source depth z _s .

Assuming that the modal number of the notch subspace V ₀ is M ₀ , then:

为对应于陷波子空间V₀的M₀个列向量，/>

为对应于模态幅度a₀的M₀个元素；/>

为对应于自由子空间V₁的M-M₀个列向量，/>

为对应于模态幅度a₁的M-M₀个元素；In the formula

are M ₀ column vectors corresponding to the notch subspace V ₀ , />

be the M ₀ elements corresponding to the modal amplitude a ₀ ;/>

be MM ₀ column vectors corresponding to the free subspace V ₁ , />

be ₀ elements of MM corresponding to the modal amplitude a ₁ ;

Singular value decomposition is performed on the notch subspace V ₀ , and the main eigenvalue vector is taken to obtain a linearly independent reduced-rank subspace U ₀ :

U ₀ ∈ ^{C N×q} ,

In the formula, C is a set of complex matrices, and q is the column number of the reduced-rank subspace U ₀ ;

Singular value decomposition is performed on the free subspace V ₁ , and the main eigenvalue vector is taken to obtain a linearly independent reduced-rank subspace U ₁ :

U ₁ ∈ C ^N×t ,

In the formula, t is the column number of the reduced-rank subspace U ₁ ;

In the direction of the column vectors of U ₀ and U ₁ , the subspaces V ₀ and V ₁ have the maximum energy.

And then get the matrix A:

A=[U ₀ U ₁ ].

In order to eliminate the overlapping part of the reduced-rank subspace U ₀ and the reduced-rank subspace U ₁ , the two reduced-rank subspaces are processed orthogonally, so that the column vectors of the reduced-rank subspace U ₀ and the reduced-rank subspace U ₁ form a group of observation spaces Orthogonal basis:

According to the matrix A=[U ₀ U ₁ ],

The column vector [u ₁ ,…,u _q+t ] of the matrix A∈C ^N×(q+ t) is a set of linearly independent vectors in the observation space, then there is an orthogonal matrix β:

β=[β ₁ ,...,β _q+t ],

Let A=[β ₁ ,…,β _q+t ]B _A ,

In the formula, B _A is the unit upper triangular matrix, and the diagonal elements of the unit upper triangular matrix are all 1:

B _A ∈ ^{C (q+t)×(q+t)} ;

The relationship between the column vector [u ₁ ,…,u _q+t ] and the orthogonal matrix β is as follows:

Based on the above formula, a set of orthogonal basis [β ₁ ,…,β _q+t ] spanning space A can be obtained.

According to the size of the reduced-rank subspace U ₀ and the reduced-rank subspace U ₁ , the orthogonal matrix β is divided into space H and space S, corresponding to the notch and free subspace after orthogonalization:

This results in two orthogonal subspaces that do not overlap and form the entire modal space:

In the formula, span( ) represents the space spanned by the column vectors of .

The array received signal is projected onto the processed notched subspace H and the free subspace S. The projection matrix P _Y of space Y can be calculated by the following formula:

P _Y ＝Y(Y ^H Y) ^-1 Y ^H , and P _Y ＝P _Y *P _Y .

Thus, the projection y _H of the hydrophone array received signal W on the space H and the projection y _S on the space S are:

The signal energy E H projected on the space H by the hydrophone array receiving signal _W and the signal energy E _S projected on the space S are respectively:

E _H ＝tr(W ^H P _H W)

E _S ＝tr(W ^H P _S W)

Where tr(·) represents the trace of the matrix, and W ^H is the conjugate transpose of W.

In the following, the depth judgment problem is simplified as a binary hypothesis problem. Assume that H ₀ indicates that the sound source is near the water surface and is a surface source, and assume that H ₁ indicates that the sound source is underwater and is a submerged source. as shown below

In the formula, z _lim is the judgment depth, generally 5-10m. From the expression of the modal amplitude matrix a, it can be seen that when the target is in the far field, the energy distribution in the modal space is mainly affected by the depth of the sound source. Since the surface source is difficult to couple to the low-order modes, if more energy is detected in the high-order mode subspace, the sound source is a surface source, otherwise the sound source is a submerged source. According to this, the energy ratio projected to the notched subspace and the free subspace can be established as the detection statistic

The detection statistic is L(z _s ):

The selected threshold is η:

If L(z _s )≤η, the sound source is a surface source;

If L(z _s )>η, the sound source is submerged source.

Since the two orthogonal subspaces in the method of the present invention form a complete modal space, there is almost no energy loss; while in the existing VE method, the use of subspace projection will eventually discard the overlapping part of the two subspaces, so the theoretical The received signal energy projected onto the overlapping portion is discarded. Therefore, the method of the present invention obtains more accurate detection results.

When the energy of the received signal is constant, z _s ≤ z _lim , it is difficult for the sound source to couple to the low-order mode, so the energy is more concentrated in the free subspace S of the high-order mode, and L(z _s ) is in the depth range of the sound source Inside should be smaller. When the water depth of the simulated environment is 160m, the sound velocity profile is shown in Figure 2. The frequency of the sound source is 353 Hz, and the 20-element uniform linear array is fixed on the bottom of the water. The distance between the array elements is 15 m. For the convenience of analysis, the sound source is located in the end-fire direction of the array.

In Fig. 2, p1, p6, and p16 are the phase velocities corresponding to modes 1, 6, and 16, respectively. Only at depths where the modal phase velocity is greater than the speed of sound can the mode propagate normally. In this environment, the variation of L(z _s ) with the sound source depth is shown in Figure 3. In the range where the sound source depth is smaller than the decision depth, the energy ratio L(z _s ) is the smallest; when the sound source depth is greater than the decision depth In the depth range of , the energy ratio L(z _s ) changes constantly, and decreases gradually with the depth near the bottom, but all of them are greater than the energy ratio of the surface source. Similar features exist at other frequencies and in environments with a negative gradient of sound velocity. Therefore, in the method of the present invention, it is feasible to use L(z _s ) as the detection statistic. When selecting the decision depth, you can refer to the curve of L(z _s ) changing with the depth of the sound source.

The performance of the algorithm is evaluated by using the receiver operating characteristic curve ROC, and the detection here refers to the detection of the submerged source. Then, the detection probability P _d is the probability of determining the submerged source as the submerged source, and the false alarm probability P _f is the probability of determining the surface source as the submerged source. In other words, the detection probability P _d can be used as a measure of the algorithm's ability to judge submerged sources, while the ability to judge surface sources can be evaluated using 1-P _f . When selecting the threshold, you can refer to the ROC curve and select the corresponding threshold according to the preset false alarm probability. Judgment of the sound source depth is carried out through the decision index and the threshold.

Although the invention is described herein with reference to specific embodiments, it should be understood that these embodiments are merely illustrative of the principles and applications of the invention. It is therefore to be understood that numerous modifications may be made to the exemplary embodiments and that other arrangements may be devised without departing from the spirit and scope of the invention as defined by the appended claims. It shall be understood that different dependent claims and features described herein may be combined in a different way than that described in the original claims. It will also be appreciated that features described in connection with individual embodiments can be used in other described embodiments.

Claims (9)

1. A shallow water low frequency sound source depth judging method based on modal filtering is characterized by comprising the following steps,

setting the underwater depth of the sound source to be z _s At underwater depth z _r Acquisition of the sound source z through hydrophone array on the horizontal plane of the position _s Sound field p (r, z) at horizontal distance r _r ,z _s ) The sound field p (r, z _r ,z _s ) Is an expression for the observation matrix V and the modal amplitude matrix a;

dividing the observation matrix V into notch subspaces V ₀ And free subspace V ₁ The method comprises the steps of carrying out a first treatment on the surface of the For notch subspace V ₀ Singular value decomposition is carried out to obtain a linear independent rank reduction subspace U ₀ For free subspace V ₁ Singular value decomposition is carried out to obtain a linear independent rank reduction subspace U ₁ The method comprises the steps of carrying out a first treatment on the surface of the Rank reduction subspace U ₀ And rank reduction subspace U ₁ Forming a matrix A;

processing the matrix A to obtain an orthogonal matrix beta; according to the rank reduction subspace U ₀ And rank reduction subspace U ₁ Dividing an orthogonal matrix beta into a space H and a space S, wherein the space H corresponds to an orthogonalized notch subspace, and the space S corresponds to an orthogonalized free subspace;

calculating a projection matrix P of the space H _H Projection matrix P of sum space S _S The method comprises the steps of carrying out a first treatment on the surface of the Then the hydrophone array receiving signal W is obtained through calculation and projected in the space HSignal energy E of (2) _H And signal energy E projected in space S _S The method comprises the steps of carrying out a first treatment on the surface of the The signal energy E _H For projection of energy in the notch subspace, E _H +E _S The sum of (2) is the energy projected in the whole orthogonal mode space; the energy projected in the notch subspace is compared with the energy projected in the whole orthogonal mode space to obtain detection statistics, the detection statistics are compared with a selected threshold, if the detection statistics are larger than the selected threshold, the sound source is judged to be a submerged source, and otherwise, the sound source is judged to be a surface source;

depth z under water _r Greater than the underwater depth z _s 。

2. The method for determining the depth of a shallow water low frequency sound source based on modal filtering as claimed in claim 1, wherein,

sound field p (r, z) _r ,z _s ) The expression of (2) is:

where X (f) is the amplitude of the sound field at frequency f, j is the imaginary number, ρ (z _s ) The water density of the sound source is M, the number of modes of the sound field propagating in the environment is M, and M is the ordinal number of the modes;

as a mode function of mode m with respect to depth, k _rm The horizontal wavenumber for mode m.

3. The method for determining the depth of a shallow water low frequency sound source based on modal filtering as claimed in claim 2, wherein,

horizontal wavenumber k of mode m _rm The method comprises the following steps:

wherein k is a constant value, and the formula is shown in the specification,

ω is the angular frequency of the sound source signal, c is the maximum sound velocity in the sound velocity gradient, ω=2pi,/and->

Is the vertical wavenumber of mode m.

4. A shallow water low frequency sound source depth decision method based on modal filtering as claimed in claim 3, wherein,

setting hydrophone array as N-element horizontal array, and setting the hydrophone array to be distant from sound source r ₁ ,r ₂ ,…,r _N The position samples the sound field, and the hydrophone array receives a signal W as a vector of the sound field p:

W＝Va；

r _i ＝r ₁ +(i-1)dcosθ,i＝1,2,…,N，

wherein d is the array element spacing, and θ is the included angle between the sound source and the hydrophone array in the horizontal direction;

dividing the observation matrix V into notch subspaces V ₀ And free subspace V ₁ And the modal amplitude matrix a is divided into a notch subspace V ₀ Corresponding modal amplitude a ₀ And with free subspace V ₁ Corresponding modal amplitude a ₁ The method comprises the following steps of:

wherein the column vector V of the matrix V is observed _m The method comprises the following steps:

wherein R is R _i A vector of components;

the modal amplitude matrix a is:

a＝[a ₁ ,a ₂ ,…,a _M ] ^T ,

wherein κ is a constant:

5. the method for determining the depth of a shallow water low frequency sound source based on modal filtering as claimed in claim 4, wherein,

the hydrophone array received signal W is regarded as a corresponding to each mode _m The weight vector of each mode is the column vector V of the observation matrix V _m The method comprises the steps of carrying out a first treatment on the surface of the Let us assume a notch subspace V ₀ The number of modes is M ₀ Then:

in the middle of

To correspond to notch subspace V ₀ M of (2) ₀ Individual column vectors>

To correspond to the modal amplitude a ₀ M of (2) ₀ An element; />

To correspond to the free subspace V ₁ M-M of (2) ₀ Individual column vectors>

To correspond to the modal amplitude a ₁ M-M of (2) ₀ An element;

for notch subspace V ₀ Singular value decompositionSolving to obtain a linearly independent rank reduction subspace U ₀ ：

U ₀ ∈C ^N×q ，

Wherein C is complex matrix set, q is reduced rank subspace U ₀ The number of columns of (a);

for free subspace V ₁ Singular value decomposition is carried out to obtain a linear independent rank reduction subspace U ₁ ：

U ₁ ∈C ^N×t ，

Wherein t is a rank reduction subspace U ₁ The number of columns of (a);

and then obtaining a matrix A:

A＝[U ₀ U ₁ ]。

6. the method for determining the depth of a shallow water low frequency sound source based on modal filtering as claimed in claim 5, wherein,

to eliminate rank reduction subspace U ₀ And rank reduction subspace U ₁ Is orthogonal processed to two rank reduction subspaces, so that rank reduction subspace U ₀ And rank reduction subspace U ₁ Is spread out as a set of orthogonal basis of observation space:

according to matrix a= [ U ] ₀ U ₁ ]，

Matrix A εC ^N×(q+t) Column vector [ u ] ₁ ,…,u _q+t ]For a set of linearly independent vectors of the observation space, then there is an orthogonal matrix β:

β＝[β ₁ ,…,β _q+t ]，

let A= [ beta ] ₁ ,…,β _q+t ]B _A ，

In B of _A The unit is a triangular array:

B _A ∈C ^(q+t)×(q+t) ；

column vector [ u ] ₁ ,…,u _q+t ]The relationship with the orthogonal matrix β is as follows:

7. the method for determining the depth of a shallow water low frequency sound source based on modal filtering as claimed in claim 6, wherein,

according to the rank reduction subspace U ₀ And rank reduction subspace U ₁ Dividing the orthogonal matrix β into space H and space S:

this gives:

the space formed by column vectors of span (·) expression in the formula.

8. The method for determining the depth of a shallow water low frequency sound source based on modal filtering as claimed in claim 7,

projection y of hydrophone array received signal W onto space H _H And projection y onto space S _S The method comprises the following steps:

the hydrophone array receives signal energy E projected in space H from signal W _H And signal energy E projected in space S _S The method comprises the following steps of:

E _H ＝tr(W ^H P _H W)

E _S ＝tr(W ^H P _S W)

wherein tr (. Cndot.) represents the trace of the matrix, W ^H Is the conjugate transpose of W.

9. The method for determining the depth of a shallow water low frequency sound source based on modal filtering as claimed in claim 8,

the detection statistic is L (z) _s )：

The selected threshold is η:

if L (z) _s ) If eta is less than or equal to eta, the sound source is a surface source;

if L (z) _s ) > η, the sound source is a inundation source.

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