Fast estimation method of sound field uncertainty based on adaptive dynamic orthogonal differential equation

Abstract

The present application relates to a kind of based on adaptive dynamic orthogonal differential equation sound field uncertainty fast estimation method, utilize singular value decomposition and dimension reduction and the method of finite difference realizes the tracking of random ray, thereby in the case of guaranteeing high precision, the fast estimation of the sound field uncertainty of dynamic ocean environment, reduce the operation amount of sound field uncertainty estimation, especially applicable to simulate the ray dynamic change track of deep sea 200Hz above medium-high frequency sound source.In the calculation multiple environmental parameters influence on sound field estimation, compared with the efficiency of traditional polynomial expansion method is higher.Compared with the complexity of traditional Monte Carlo algorithm is greatly reduced, while greatly saving the operation cost, with higher implementation value.Only need to calculate the sound velocity formula to obtain the sound velocity profile sample in a certain sea area, the sound field ray track and propagation loss characteristic uncertainty fast estimation can be realized, simple operation.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of ocean physics, ocean engineering and underwater acoustic engineering, and relates to a fast estimation method for sound field uncertainty based on adaptive dynamic orthogonal differential equation. The present application proposes a method based on singular value decomposition and dimension reduction to process the uncertainty transmission process of the marine environment-sound field, and solves the dynamic change range of the sound field ray trajectory in the dynamic marine field environment through an evolutionary random ray model, thereby realizing fast estimation of the sound field uncertainty. BACKGROUND

[0002] Sound waves are the main means for observing marine features, identifying foreign objects and communicating under the sea, but a significant feature is that underwater acoustic propagation is largely dependent on the environment or medium, so effective prediction of underwater acoustic propagation is not easy. Complex physical processes such as fronts, eddies and internal waves cause changes in the sound velocity profile in the deep sea, resulting in uncertainty in marine environment observation and prediction. This uncertainty is transmitted to the predicted sound field through the sound field model, resulting in nonlinear interaction and complex evolution of the model, leading to uncertainty in the estimation of the sound field. This uncertainty has a high degree of freedom to a certain extent, and the numerical method of wave equation is used to solve it, resulting in high degree of freedom changes in the sound field uncertainty, which makes the estimation of the sound field uncertainty require a large amount of computation.

[0003] At present, research on the observation and prediction of sound field uncertainty in dynamic marine environment is very active at home and abroad, such as traditional Monte Carlo method, polynomial chaos expansion proxy model method, etc., but these methods have certain problems.

[0004] (1) The traditional Monte Carlo method requires a large amount of calculation when there are multiple uncertain environmental parameters, and the convergence is slow;

[0005] (2) The polynomial chaos expansion method is an effective method for handling multi-dimensional and nonlinear problems, and has been widely used in the study of underwater acoustic uncertainty propagation in recent years. The existing research mostly applies this method to predict the sound field distribution under uncertain environment, but the number of uncertain environmental parameters selected by the research is small. With the increase in the number of environmental parameters, more polynomial expansion terms and training points are needed to ensure the convergence of the calculation results, which will greatly reduce the calculation efficiency.

[0006] The high degree of freedom disturbance of the sound velocity profile will lead to a very high dimension of the sound field uncertainty estimation, and the current related research is very few. The research on fast estimation of the sound field uncertainty of the marine environment still faces great challenges, and new principles and technical approaches must be sought. SUMMARY

[0007] Technical problems to be solved

[0008] In order to avoid the shortcomings of the prior art, the present application proposes a fast estimation method for sound field uncertainty based on adaptive dynamic orthogonal differential equation, which solves the bottleneck encountered in the current technology. The singular value decomposition, dimension reduction and finite difference method are used to realize the tracking of random rays, so as to realize the fast estimation of the sound field uncertainty of the dynamic marine environment under the condition of ensuring high precision, reduce the operation amount of the sound field uncertainty estimation, and is especially suitable for simulating the dynamic change trajectory of the ray of the deep sea 200Hz above the medium-high frequency sound source.

[0009] Technical scheme

[0010] A fast estimation method for sound field uncertainty based on adaptive dynamic orthogonal differential equation, characterized by the following steps:

[0011] Step 1: introduce initial conditions: set the number of sound speed profile samples selected by the model as H, the number of rays selected as R, and give the initial conditions of the sound source:

[0012]

[0013]

[0014] Wherein: x d0 is the initial depth of the sound source, Ψ0 is the initial auxiliary variable, cosθ0 and sinθ0 are the initial angles of the sound field rays, and c(x0) represents the sound speed at the initial position x0;

[0015] Perform simulation operation of the ray model on each sound speed profile sample to obtain the related parameters of the sound field rays at a specific step "s" from the sound source, i.e. the ray parameters of the initial position, and express them in the form of a matrix:

[0016]

[0017]

[0018] Wherein: the dimension of the matrix is R*H;

[0019] Step 2, singular value decomposition and dimension reduction: find the mean of the row vectors of the initial matrix X r,d (s) and Ξ r,d (s) to obtain E η [X r,d (s)] and E n [Ξ r,d (s)], and then perform subtraction operation to obtain X r,d (s)-E η [X r,d (s)] and Ξ r, d(s)-E n [Ξ r,two new initial matrices of [X

[0020] X r,d (s)-E η [X r,d (s)] and Ξ r,d (s)-E n [Ξ r,d (s)] two new matrix singular value decomposition:

[0021]

[0022]

[0023] The diagonal matrix Σ1 and Σ2 obtained by singular value decomposition are arranged according to the importance of singular value, and dimension reduction processing is performed to obtain matrix X r,d (s) and Ξ r,d The mean vector, mode and mode coefficient matrix linear representation of [X

[0024]

[0025]

[0026] Wherein, E η [X r,d (s)] and E n [Ξ r,d (s)] are the mean variables of the ray parameter matrix and The column vectors of U1 and U2 are the modes of the dynamic orthogonal random ray model and , and The row vectors of U1 and U2 are the mode coefficients B of the dynamic orthogonal random ray model r,d and D r,d ;

[0027] Step 3: Finite difference derivation of ray parameters: based on the ray theory, the sound ray tracking ordinary differential equation set is derived, which expresses the differential relationship between adjacent positions of the sound field:

[0028]

[0029]

[0030] Wherein, C is the sound speed value matrix expression corresponding to all rays of the sound field under each sound speed profile, ▽ r,d C is the gradient matrix of the sound speed matrix in the distance r direction and the depth d direction, and the Cx variable is as follows:

[0031]

[0032] Substitute formula 7 and formula 8 into formula 9 and formula 10, and use the property of modal matrix eigenvector orthogonality to deduce and calculate, to obtain the mean variable corresponding to the ray position matrix and auxiliary variable matrix and modal and and the corresponding modal coefficient matrix B r,d and D r,d The differential relationship between two adjacent positions in the sound field:

[0033]

[0034]

[0035]

[0036]

[0037]

[0038]

[0039] Where H is the number of sound speed profile samples selected by the random ray model, cov is the covariance operation performed on the variable matrix therein;

[0040] Solve formula 12 to formula 17 using the finite difference algorithm to obtain the mean vector of the ray parameter matrix at the next position, the modal and the modal coefficient matrix, and reconstruct them using formula 7 and formula 8 to obtain the ray parameter matrix X of the next position in the sound field r,d (s+1) and Ξ r.d (s+1);

[0041] Step 4: Orthogonal correction of modal matrix: matrix A and matrix W are correction matrices, B r,d * and D r,d * are the corrected modal matrix and modal coefficient matrix; use the classic matrix orthogonal correction algorithm, gram-schmidt matrix orthogonal correction algorithm to calculate the correction matrices A and W;

[0042] Step 5: Repeat steps 3 and 4 until all rays enter the boundary range in the distance direction set by the model; allow the random ray to continue to enter the reflection domain by defining a "mirror symmetry" sound speed function at negative depth and greater than the depth of the seabed, and finally reflect back to the actual marine area as a post-processing step, so as to realize the boundary reflection problem.

[0043] The number of sound speed profile samples H = 500.

[0044] The number of rays R = 102.

[0045] Advantages

[0046] The application provides a fast uncertainty estimation method for a sound field based on adaptive dynamic orthogonal differential equations, singular value decomposition and dimension reduction, and finite difference method are used to track random rays, so that the fast uncertainty estimation of the sound field in a dynamic marine environment is realized under the condition of ensuring high precision, the operation amount of the sound field uncertainty estimation is reduced, and the method is particularly suitable for simulating the dynamic change trajectory of a middle-high frequency sound source above 200Hz in a deep sea.

[0047] Advantages are embodied in:

[0048] (1) The application establishes a connection between sound speed profile disturbance and sound field uncertainty estimation, realizes sound field simulation under high degree of freedom environmental disturbance, and has more practical significance than traditional methods for solving the relationship between a single environmental factor and a sound field.

[0049] (2) The application uses the idea of dynamic orthogonal differential equation evolution random field, and through the method of singular value decomposition and dimension reduction, the sound field initial position ray parameter matrix obtained after executing the ray model is decomposed into a mean vector, a mode and a mode coefficient matrix.

[0050] (3) The application does not need to simulate complex marine dynamic processes and marine meteorological processes such as internal waves, vortices and fronts, has high precision, simple implementation, and only needs to calculate the sound speed profile sample in a certain sea area through a sound speed formula, so that the fast uncertainty estimation of the sound field ray trajectory and propagation loss characteristics can be realized, and the operation is simple.

[0051] (4) In the fields of target detection and positioning, submarine communication, etc., the traditional simulation is based on a certain deterministic marine environment, that is, a certain sound speed profile, without considering the influence of the dynamic change of the marine environment on the sound field model. BRIEF DESCRIPTION OF DRAWINGS

[0052] Figure 1(a) Mapping of the data sampling area; (b) Sound velocity profile samples obtained from sampling in the Northwest Pacific Ocean, with 1000 sound velocity profiles selected for plotting (longitude range: 124°~142°; latitude range: 0°~20°N); (c) Deterministic ray trajectory simulation (four sound velocity profile samples were randomly selected and executed using the Bellhop deterministic ray model, with the number of rays preset to 102, the sound source frequency to 500Hz, and the sound source depth to 200m); (d) Images of the four randomly selected sound velocity profiles, corresponding to the simulation images of the four deterministic ray models in (c); (e) Comparison of the depths of the endpoints of each ray calculated using the Bellhop deterministic ray model under the four randomly selected sound velocity profile samples.

[0053] Figure 2 (a) Flowchart of the acoustic field uncertainty estimation method based on adaptive dynamic orthogonal differential equations; (b) Schematic diagram of the seabed and sea surface reflection processing.

[0054] Figure 3 (a) Study on the singular value convergence characteristics of X and Ξ variables under 102 rays and 500 sound velocity profiles; (b) Study on the singular value convergence characteristics of X and Ξ variables under 102 rays and 1000 sound velocity profiles; (c) Study on the singular value convergence characteristics of X and Ξ variables under 1002 rays and 500 sound velocity profiles; (e) Study on the singular value convergence characteristics of X and Ξ variables under 1002 rays and 1000 sound velocity profiles.

[0055] Figure 4 (a) Mean squared error images of the positions of rays at different angles under 102 rays and 500 sound velocity profiles with different numbers of singularities; (b) Mean squared error images of rays at different angles under 102 rays and 1000 sound velocity profiles with different numbers of singularities; (c) Mean squared error images of rays at different angles under 1002 rays and 500 sound velocity profiles with different numbers of singularities; (d) Mean squared error images of rays at different angles under 1002 rays and 1000 sound velocity profiles with different numbers of singularities.

[0056] Figure 5 :against Figure 1 A comparison of the results of the sound field estimation model of the adaptive dynamic orthogonal differential equation for four sound velocity profiles (using 500 sound velocity profile samples) and the results of the ray model.

[0057] Figure 6 against Figure 1 The deviation between the simulated location at 6km and the ray model calculation results is shown in the acoustic field estimation model of the adaptive dynamic orthogonal differential equation of the four acoustic velocity profiles (using 500 acoustic velocity profile samples). Detailed Implementation

[0058] The present invention will now be further described in conjunction with the embodiments and accompanying drawings:

[0059] A fast estimation method for acoustic field uncertainty based on adaptive dynamic orthogonal differential equations is characterized by the following steps:

[0060] Step 1: Introduce initial conditions: Set the number of sound velocity profile samples selected for the model to H = 500, and the number of selected rays to R = 102. The initial conditions of the sound source are given as shown in equations (1) and (2):

[0061]

[0062]

[0063] Where x d0 Let Ψ0 be the initial depth of the sound source, cosθ0 and sinθ0 be the initial auxiliary variables, and c(x0) represent the sound speed at the initial position x0.

[0064] For each sound velocity profile sample, a ray model simulation is performed to obtain the relevant parameters of the sound field ray at a specific step length "s" from the sound source, i.e., the ray parameters at the initial position, and these parameters are represented as a matrix X. r,d (s) and Ξ r,d (s), as shown in equations (3) and (4), where the dimension of the matrix is ​​R*H.

[0065]

[0066]

[0067] Step 2: Singular Value Decomposition and Dimensionality Reduction: First, the initial matrix X constructed from equations (3) and (4) is... r,d (s) and Ξ r,d (s) Calculate the mean of the row vectors to obtain E. η [X r,d (s)] and E n [Ξ r,d (s)], then perform subtraction to get X r,d (s)-E η [X r,d (s)] and Ξ r,d (s)-E n [Ξ r,d [s] Two new initial matrices.

[0068] For the constructed X r,d (s)-E η [X r,d (s)] and Ξ r,d(s)-E n [Ξ r,d The process of the two new matrix singular value decompositions can be represented as equations (5) and (6), and the diagonal matrices Σ1 and Σ2 obtained by singular value decomposition are arranged according to the importance of singular values, and are processed by dimension reduction, so that the matrix X r,d (s) and Ξ r,d The mean vector, mode and mode coefficient matrix linear representation of (s) and Ξ

[0069]

[0070]

[0071]

[0072]

[0073] Step 3: Finite difference derivation of ray parameters: based on ray theory, the sound ray tracking ordinary differential equation set can be derived, as shown in equations (9) and (10), which expresses the differential relationship between adjacent positions of sound field rays.

[0074]

[0075]

[0076] Where C is the sound speed value matrix corresponding to all rays of the sound field under each sound speed profile, ∇ r,d C is the gradient matrix of the sound speed matrix in the distance r direction and the depth d direction, and the Cx variable is expressed as shown in equation (11):

[0077]

[0078] By substituting equations (7) and (8) into equations (9) and (10) respectively, and using the property of the orthogonal eigenvectors of the mode matrix, the mean variables corresponding to the ray position matrix and the auxiliary variable matrix and mode and and the corresponding mode coefficient matrix B r,d and D r,d The differential relationship between two adjacent positions in the sound field is shown in equations (17) to (22).

[0079]

[0080]

[0081]

[0082]

[0083]

[0084]

[0085] Where H is the number of the sample of the sound velocity profile selected by the random ray model, cov is the covariance operation on the variable matrix in it.

[0086] Solving the equations (12) to (17) using the finite difference algorithm, the mean vector of the ray parameter matrix at the next position, the mode and the mode coefficient matrix can be obtained, and using the equations (7) and (8) to reconstruct them, the ray parameter matrix X at the next position can be obtained. r,d (s+1) and Ξ r.d (s+1).

[0087] Step 4: Orthogonal correction of the mode matrix: in the calculation process using the finite difference algorithm, the mode matrix and the mode coefficient matrix at each step need to be corrected to ensure that the eigenvectors of the mode matrix and the mode coefficient matrix are orthogonal to each other after each calculation (the correction matrices are matrix A and matrix W, B r,d * and D r,d * the corrected mode matrix and the mode coefficient matrix).

[0088]

[0089]

[0090] B r,d * =A -1 B r,d (20)

[0091] D r,d * =W -1 D r,d (21)

[0092] The present application adopts the classical matrix orthogonal correction algorithm, gram-schmidt matrix orthogonal correction algorithm to calculate the correction matrix A and matrix W. Taking the calculation of matrix A as an example, it can be divided into three steps:

[0093] (1) using equation (22) to calculate the gram matrix K of the mode matrix of the random ray model, F ilwhere αi is the value of the ith row, 1th column of the modal matrix, and the expression of K is given by equation (23) where αi k where αi is the value of the ith row, 1th column of the modal matrix, and the expression of K is given by equation (23) where αi

[0094]

[0095]

[0096] (2)Gram matrix eigenvalue decomposition to get VΣV T ;

[0097] K=VΣV T (24)

[0098] (3) by equation (25) to calculate the orthogonal correction matrix A;

[0099] A=VΣ -1 / 2 V T =K -1 / 2 (25)

[0100] Step 5: repeat step 3 and step 4 until all the rays enter the model set distance direction of the boundary range. In the process of repeated iteration, the problem of reflection of the seabed and sea surface boundary also needs to be handled, and the present application allows the random rays to continue to enter the reflection domain by defining the "mirror symmetry" sound speed function at negative depth and greater than the seabed depth, and finally the reflection back to the actual marine area as a post-processing step, so as to realize the boundary reflection problem. (Later combined Figure 2 (b) make a further explanation)

[0101] Thus, the sound field uncertainty estimation model based on adaptive dynamic orthogonal differential equation is finally built, and the sound field uncertainty estimation under dynamic marine environment is realized, that is, the fast solution of the sound field ray trajectory under each sound speed profile.

[0102] The method uses the idea of singular value decomposition and dimension reduction, and performs singular value decomposition on the ray parameter matrix calculated under each sound speed profile sample, and represents it with the mean vector, mode and modal coefficient matrix. Then, by reducing the unimportant singular values, the dimension of the matrix is reduced, and by means of finite difference algorithm, the mean vector, mode and modal coefficient matrix corresponding to the ray parameter matrix of each position of the sound field are derived, and then the dynamic change range of all ray trajectories of the sound field is solved, and the uncertainty of the sound field is quickly estimated.

[0103] The application establishes an end-to-end mapping model based on a large number of sound speed profile samples and sound field uncertainty estimation, does not need to simulate ocean dynamic processes such as internal waves and vortices, can realize nonlinear mapping from dynamic ocean environment to sound field prediction, and can solve the sound field ray trajectory change range in the dynamic ocean environment by using the model and part of the sound speed profile samples, realize the estimation of the sound field uncertainty, has small calculation amount and simple realization, and is suitable for directly acquiring the parameters of the ocean environment, i.e., the sound speed profile, by using a temperature sound speed measuring instrument CTD to realize the quasi-real-time acquisition of the sound speed profile, so that the change range of the sound field ray trajectory is directly simulated.

[0104] Figure 1 (a) The data sampling region is plotted, (b) the sound speed profile samples obtained by sampling in the northwest Pacific Ocean are given, (c) the sound field simulation calculation under four randomly selected sound speed profile samples is given. It can be seen that with the change of time, the ocean environment field changes constantly, leading to the disturbance of the sound speed profile, and further affecting the trajectory change of the ray, (d) the images of the four selected sound speed profile samples are given, (e) the contrast chart of the depths of the endpoints of the rays calculated by using the bellhop deterministic ray model under the four randomly selected sound speed profile samples, which intuitively reflects the influence of the disturbance of the ocean environment field on the sound line.

[0105] Figure 2 The implementation framework flowchart of the sound field uncertainty estimation method of the dynamic orthogonal differential equation and the schematic diagram of the sea surface and seabed boundary reflection processing are given. The method mainly includes four processes: (1) initial condition acquisition: firstly, H sound speed profile samples required by the application are selected, the bellhop ray model is executed for each sound speed profile, the ray parameter values of the initial positions under each sound speed profile sample are obtained, the ray parameter values are constructed into a matrix, the singular value decomposition and dimension reduction processing are performed on the constructed matrix, the mean vector of the initial position, the mode and the mode coefficient matrix are obtained, (2) finite difference implementation: the differential numerical relationship between the adjacent positions of the sound field rays is obtained by deducing the dynamic orthogonal differential equation and the ray theory model, the mean vector, the mode and the mode coefficient matrix of all positions of the sound field are obtained by using the forward difference algorithm, wherein the mode and the mode coefficient matrix need to be orthogonally corrected at each step to ensure the orthogonalization of the characteristic vectors, (3) boundary reflection problem: the sound speed function at the negative depth and greater than the seabed depth is defined, the random rays are allowed to continue to enter the reflection domain, and finally the reflection back to the actual ocean region is taken as a post-processing step, so that the boundary reflection problem is realized, i.e. Figure 2 (b) the schematic process is given, and (4) the finally calculated result is compared with the result calculated by using the deterministic ray model to verify.

[0106] Figure 3The singular value convergence characteristic images under different ray numbers and different sound speed profile sample numbers are given, and it is clearly shown that different ray numbers and different sound speed profile sample numbers both affect the dimension of singular value convergence, and it can be seen from the images that the dimension of variable X convergence is about 5%-10% of the minimum value between the ray number and the sound speed profile sample number, and the dimension of variable Ξ convergence is about 5%-10% of the minimum value between the ray number and the sound speed profile sample number.

[0107] Figure 4 The mean square errors of the sound field angle rays corresponding to different singular value numbers (different dimensions) under different ray numbers and different sound speed profile sample numbers are given, and it can be more intuitively seen that the dimension of convergence increases, and the accuracy of the calculation is higher, and the dimension corresponding to the position of the image convergence is the dimension used in the application.

[0108] Figure 5 First, the ray number is set to 102, the sound speed profile sample number is selected to be 500, the farthest distance is set to 10km, and the sound field uncertainty estimation method calculated and solved by the adaptive dynamic orthogonal differential equation is given Figure 1 (d) The dynamic trajectory of the random ray under the sound speed sample is shown, and the ray trajectory calculated by the ray model simulation is compared.

[0109] Figure 6 From the comparison and analysis between the position of the ray endpoint calculated by using the method proposed in the application and the position calculated by the deterministic model, it can be further seen that the sound field uncertainty method of the adaptive dynamic orthogonal differential equation proposed in the application maintains high accuracy after reducing the reasonable dimension, and compared with the traditional Monte Carlo method, the accuracy requirement is ensured, and the calculation efficiency is greatly improved.

[0110] The application achieves obvious implementation effects in typical examples, the sound field uncertainty fast estimation method based on the adaptive dynamic orthogonal differential equation has superior performance and good robustness, the sound speed profile and the sound field model are connected, and singular value decomposition and dimension reduction processing are used, so that the sound field uncertainty fast estimation in the dynamic marine environment is realized. In the case of ensuring high accuracy, the calculation amount and operation cost are greatly reduced, and the implementation is simple.

Claims

1. A method for fast estimation of sound field uncertainty based on adaptive dynamic orthogonal differential equations, characterized by The steps are as follows: Step 1: Introducing initial conditions: setting the number of sound speed profile samples selected by the model as H, the number of rays selected as R, and giving the initial conditions of the sound source: where: x d0 is the initial depth of the source, Ψ0is the initial auxiliary variable, cosθ0and sinθ0are the initial angles of the sound field rays, and c(x0) represents the sound speed at the initial position x0; Perform simulation operation of the ray model for each sound speed profile sample to obtain the relevant parameters of the sound field rays at a specific step length "s" from the sound source, that is, the ray parameters of the initial position, and express them in the form of a matrix: Wherein: the dimension of the matrix is R*H; Step 2, singular value decomposition and dimension reduction: on the initial matrix X r,d (s) and Ξ r,d (s) to get E η [X r,d (s)] and E n [Ξr, d (s)] and subtract to get X r,d (s)-E η [X r,d (s)] and Ξ r,d (s)-E n [Ξ r,d (s)] two new initial matrices; X r,d (s)-E η [X r,d (s)] and Ξ r,d (s)-E n [Ξ r,d (s)] two new matrix singular value decompositions: The diagonal matrices Σ1 and Σ2 obtained by singular value decomposition are arranged according to the importance of singular values, and dimension reduction processing is performed to obtain matrices X r,d (s) and Ξ r,d The mean vector, mode and mode coefficient matrix linear representation of (s) Among them, E η [X r,d (s)] and E n [Ξ r,d [s] represents the mean variable of the ray parameter matrix. and The column vectors of U1 and U2 represent the modes of the dynamic orthogonal random ray model. and The expression, and The row vector is the modal coefficient B of the dynamic orthogonal random ray model. r,d and D r,d The representation of; Step 3: Finite difference derivation of ray parameters: deriving the ordinary differential equation set of sound line tracking based on the ray theory, and the equation set expresses the differential relationship between adjacent positions of the rays in the sound field: where C is a matrix of sound speed values corresponding to all the rays of the sound field under each sound speed profile, is the gradient matrix of the sound speed matrix in the distance r direction and the depth d direction, and the Cx variable is as follows: The formula 7 and formula 8 are brought into formula 9 and formula 10 respectively, and the mean variables corresponding to the ray position matrix and auxiliary variable matrix are obtained by deducing and calculating using the orthogonal property of modal matrix eigenvectors and modal and and the corresponding modal coefficient matrix B r,d and D r,d The differential relationship between two adjacent positions in the sound field: Where H is the number of sound speed profile samples selected by the random ray model, and cov is the covariance operation performed on the variable matrix; The finite difference algorithm is used to solve the equations 12 to 17 to obtain the mean vector of the ray parameter matrix of the next position, the mode and the mode coefficient matrix, which are reconstructed by using the equations 7 and 8 to obtain the ray parameter matrix X of the next position of the sound field r,d (s+1) and Ξ r.d (s+1); Step 4: Orthogonal correction of modal matrix: matrix A and matrix W are correction matrices, B r,d * and D r,d * are the corrected modal matrix and modal coefficient matrix; the classic matrix orthogonal correction algorithm, gram-schmidt matrix orthogonal correction algorithm is used to calculate the correction matrix A and matrix W; Step 5: Repeating steps 3 and 4 until all rays enter the boundary range in the distance direction set by the model; allowing the random rays to continue to enter the reflection domain by defining the "mirror symmetry" sound speed function at the negative depth and greater than the depth of the seabed, and finally taking the reflection back to the actual marine area as a post-processing step, so as to realize the boundary reflection problem.

2. The method of claim 1, wherein: The number of sound speed profile samples H = 500.

3. The method of claim 1, wherein: The number of rays R = 102.

Images