Ocean acoustic field calculation method based on stable Chebyshev-Padé parabolic equation operator approximation

Abstract

The present invention discloses a method for calculating ocean sound field based on stable Chebyshev-Padé parabolic equation operator approximation. The present invention includes the following steps: First, obtain the original environmental information, and after rasterizing the environmental area, obtain the stepping position information; then, according to the current environmental information, set the operator approximation expansion interval at the initial stepping position; further, according to the current environmental information and the operator approximation expansion interval, solve the Chebyshev-Padé operator approximation coefficients under stability constraints to obtain the Chebyshev-Padé operator under stability constraints at the current stepping position, and then calculate the sound pressure value at the current stepping position and add it to the information required for the next stepping position calculation. Stepping recursion is performed until the end point of the stepping position is reached, thereby calculating the ocean sound field. The present invention improves the accuracy of operator approximation in the parabolic equation model and at the same time ensures the stability of the model, and is applicable to complex environments such as rapidly varying terrain.

Description

Technical Field

[0001] The present invention belongs to a method for calculating ocean acoustic fields in the technical field of underwater acoustic physics, and particularly relates to a method for calculating ocean acoustic fields based on the approximation of the Chebyshev-Padé parabolic equation operator with stability constraints. Background Art

[0002] Underwater acoustic modeling technology, as the basis of underwater acoustic science and engineering research, can provide direct guidance for solving specific underwater acoustic engineering problems such as communication, detection, positioning, and tracking. At the same time, it can also be used as a theoretical and model tool to explore underwater acoustic physical phenomena and laws. The underwater acoustic propagation model establishes a theoretical model of the relationship between ocean environmental parameters and acoustic propagation, and the research on its modeling and solving technology has received extensive attention. Early models often adopted relatively simple forms and made a large number of simplifications to the ocean environment. With the trend of ocean exploration and development gradually moving from coastal waters to the open sea and from shallow waters to deep waters, the demand of scientific research and engineering applications for the ability of the model to handle actual complex ocean environments is getting higher and higher.

[0003] The parabolic equation model uses the parabolic equation approximation for the original acoustic wave equation to obtain a parabolic partial differential equation containing a pseudo-differential operator, which is easier to solve than the original hyperbolic partial differential equation or elliptic partial differential equation, and has become an important type of model in the field of underwater acoustic modeling. Among them, the approximation of the Helmholtz radical operator or its functional form in the parabolic equation model is one of the most core technologies of the parabolic equation model. Padé rational approximation is the current mainstream operator approximation scheme. Compared with the commonly used Taylor approximation in the early stage, it also has good asymptotic performance in the region outside the Taylor series convergence domain |Z| < 1, and is called high-order operator approximation. However, the traditional Padé rational approximation only performs operator approximation at a single point at the origin. In actual problems, the eigenvalues of each mode are distributed in a certain interval. Therefore, this scheme still does not fully utilize environmental information in modeling, and there is still a large room for improvement in asymptotic performances such as operator accuracy. On the other hand, in some scenarios in the actual environment, it is necessary to consider the contribution of evanescent modes and medium attenuation to the acoustic field. Such situations often pose high requirements for the stability of the acoustic field calculation model. It is necessary to further develop high-precision and stable operator approximation methods to better calculate the ocean acoustic field in actual complex environments. Summary of the Invention

[0004] Aiming at the deficiencies of the existing underwater acoustic parabolic equation operator approximation, the present invention provides a method for calculating ocean acoustic fields based on the approximation of the Chebyshev-Padé parabolic equation operator with stability constraints.

[0005] The purpose of the present invention is achieved through the following technical solutions:

[0006] 1. A method for calculating ocean acoustic fields based on the stable Chebyshev-Padé parabolic equation operator approximation

[0007] Step 1: Get the original environment information, rasterize the environment area, get the step position information, and execute step 2 from the starting point of the step position;

[0008] Step 2: According to the current environment information, set the approximate expansion interval of the operator at the current step position;

[0009] Step 3: According to the current environmental information and the approximate expansion interval of the operator, solve the approximate coefficient of the Chebyshev-Padé operator under the stability constraint, obtain the Chebyshev-Padé operator under the stability constraint at the current step position, and then calculate the sound pressure value of the current step position and add it to the environmental information;

[0010] Step 4: Repeat steps 2 and 3 to solve and calculate and obtain the sound pressure value of the next step position until the end of the step position is reached, thereby calculating and obtaining the ocean sound field.

[0011] The environmental information includes sound velocity field, terrain and density field.

[0012] The step 2 is specifically as follows:

[0013] The wave number or opening angle range of the depth operator Z is determined according to the environmental information, and then the approximate expansion interval of the operator at the current step position is determined.

[0014] In the step 3, according to the current environment information, the approximate expansion interval of the operator and the stability constraint condition, the precision equation and the stability equation satisfied by the numerator rational coefficient and the denominator rational coefficient are constructed and solved to obtain the numerator rational coefficient and the denominator rational coefficient, thereby obtaining the Chebyshev-Padé operator under the stability constraint at the current step position, wherein the precision equation is the original operator function and the rational approximate operator function in the operator expansion interval [Z min , Z max ] on the first n+mn s The order Chebyshev basis corresponds to the equation with equal expansion coefficients, and the stability equation is n s The stability constraint correction equation for the rational approximate operator function is as follows:

[0015] f(Z)≈g n，m (Z)

[0016]

[0017] Expanded in the interval [Z min , Z max, i = 1, 2, ..., n + m - n s

[0018] g n，m (Z i ) = g i , g i ∈D, i = 1, 2, ..., n s

[0019] where f(Z) is the original operator function; g n，m (Z) is the rational approximation operator function; n and m are the orders of the numerator polynomial and the denominator polynomial of the rational approximation respectively; Z is the depth operator, Z j and Z l represent the j-th power and the l-th power of the operator Z; α j is the j-th molecular rational coefficient, with a total of n; β l is the l-th denominator rational coefficient, with a total of m; Z min and Z max are the minimum and maximum values of the operator expansion interval respectively; is the i-th Chebyshev expansion coefficient of the original operator function on the expansion interval, with a total of n + m - n s ones; is the i-th Chebyshev expansion coefficient of the rational approximation operator function on the expansion interval, with a total of n + m - n s ones; n s is the number of stability constraint conditions; Z i is the i-th stability constraint correction point set in the region where Re(Z) < -1, with a total of n s ones; Re(Z) represents taking the real part; D is the inner region of the unit circle that satisfies the stability condition; g i is the i-th stability correction mapping image point set on the region D corresponding to Z i , with a total of n s ones.

[0020] II. A computer device

[0021] The computer device includes a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, the steps of the method are implemented.

[0022] III. A computer-readable storage medium

[0023] A computer program is stored on the computer-readable storage medium, and when the computer program is executed by a processor, the steps of the method are implemented.

[0024] IV. A computer program product

[0025] The computer program product includes a computer program / instructions, and when the computer program / instructions are executed by a processor, the steps of the method are implemented.

[0026] The beneficial effects of the present invention are as follows:

[0027] 1) Compared with the traditional Padé approximation, the stable Chebyshev-Padé approximation in the present invention introduces a Chebyshev basis with good interval approximation, further improving the operator accuracy. It has a larger opening angle than the Padé approximation under the same error tolerance and can more accurately reflect the details of the sound field.

[0028] 2) When no stability constraint is imposed, there are stability problems in the model in some modal eigenvalue distribution scenarios such as a fast-varying seabed. However, after the stability constraint is imposed in the present invention, the stability is guaranteed. Description of the Drawings

[0029] Figure 1 is a flowchart of a method for calculating an ocean sound field based on a stable Chebyshev-Padé parabolic equation operator approximation proposed by the present invention.

[0030] Figure 2 is a schematic diagram of the environmental scenario corresponding to the underwater acoustic propagation problem in the embodiment.

[0031] Figure 3 is a comparison diagram of the operator approximation performance in the embodiment scenario.

[0032] Figure 4 is a comparison diagram of the calculation results of the sound propagation loss field in the embodiment scenario. Detailed Embodiments

[0033] To make the objectives, technical solutions, and advantages of the implementation of the present invention clearer, the technical solutions of the present invention will be described in more detail below in conjunction with the embodiments of the present invention and the corresponding drawings.

[0034] The present invention proposes a method for calculating an ocean sound field based on a stable Chebyshev-Padé parabolic equation operator approximation, which includes a Chebyshev-Padé rational approximation operator with a stability constraint in an underwater acoustic parabolic equation model. The present invention improves the accuracy of the operator approximation in the parabolic equation model and simultaneously guarantees the stability of the model, and is applicable to the sound field calculation in complex environments such as fast-varying terrains.

[0035] As Figure 1 shown, the present invention includes the following steps:

[0036] Step 1: Obtain the original environmental information. After rasterizing the environmental area (i.e., the computational area), obtain the stepping position information, and start executing Step 2 from the starting point of the stepping position; among them, the environmental information includes the sound speed field, terrain, and density field.

[0037] In this embodiment, as Figure 2 shown, the embodiment scenario is a shallow - sea wedge - shaped slope benchmark, where the water depth linearly decreases from 200 m to 0 within the range of 0 - 4 km, the sound speed c w in the water layer = 1500 m / s, the density ρ w = 1.0 g / cm 3 ; the sound speed c b at the seabed = 1700 m / s, the density ρ b = 1.5 g / cm 3 , the attenuation coefficient attn = 0; the sound source is located at a distance of 0 and a depth z s = 100 m, and the frequency f = 25 Hz.

[0038] Step 2: According to the current environmental information, set the operator approximation expansion interval at the current stepping position.

[0039] Specifically, Step 2 is as follows:

[0040] Determine the wavenumber or the opening - angle range of the depth operator Z according to the environmental information, and then determine the operator approximation expansion interval at the current stepping position. In this embodiment, take [Z min， Z max = [-0.5, 0].

[0041] Specifically as follows:

[0042] According to the modal Fourier analysis theory, the physical meaning of the depth operator Z in two - dimensional cases can be represented by the wavenumber or the opening - angle:

[0043]

[0044] p(Z)ζ m = p(-sin 2 φ m )ζ m (2)

[0045] where p(Z) is an operator function of any polynomial order, k z，m is the vertical wavenumber corresponding to the m - th mode, ζ m is the depth eigen - function corresponding to the m - th mode, φ m is the horizontal propagation opening - angle corresponding to the m - th mode, and k 0 is the reference wavenumber.

[0046] In the three-dimensional case, similarly, the operator can be expressed as a function of the opening angles in two spatial dimensions, and can also be represented by the phase velocity anomaly (or refractive index anomaly) and the horizontal opening angle.

[0047] According to the relationship between the above operator function and the function of the wave number or opening angle, the wave number or opening angle range involved in the calculation can be obtained from the environmental information, and then the reasonable operator distribution interval [Z min ,Z max can be accurately or approximately determined as the subsequent operator approximation expansion interval.

[0048] Step 3: According to the current environmental information and the operator approximation expansion interval, solve the Chebyshev-Padé operator approximation coefficients under the stability constraint to obtain the Chebyshev-Padé rational operator under the stability constraint at the current stepping position, and then calculate the sound pressure value at the current stepping position and add it to the environmental information;

[0049] In Step 3, according to the current environmental information, the operator approximation expansion interval and the stability constraint conditions, construct and solve the accuracy equation and stability equation satisfied by the rational coefficients of the numerator and denominator to obtain the rational coefficients of the numerator and denominator, so as to obtain the Chebyshev-Padé operator under the stability constraint at the current stepping position; where the accuracy equation is that the original operator function and the rational approximation operator function are in the operator expansion interval [Z min， Z max for the first n + m - n s order Chebyshev basis corresponding expansion coefficients are equal equations, and the stability equation is n s stability constraint correction equations for the rational approximation operator function, and the corresponding formulas are as follows:

[0050] ψ[(n + 1)Δr, z] = f(Z)ψ(nΔr, z) (3)

[0051]

[0052]

[0053] Expanded in the interval [Z min ,Z max , i = 1, 2,..., n + m - n s (6)

[0054] g n，m (Z i ) = g i ,g i ∈D, i = 1, 2,..., n s (7)

[0055] Among them, formula (3) is the step-by-step form solution of the underwater acoustic parabolic equation model, formula (6) and formula (7) are the accuracy and stability equations respectively, ψ[(n + 1)Δr, z] is the sound pressure value at the (n + 1)-th step position; ψ(nΔr, z) is the sound pressure value at the n-th step position; k 0 is the reference wave number; Δr is the distance step; Z is the depth operator; f(Z) is the original operator function; g n，m (Z) is the rational approximation operator function; n and m are the orders of the numerator polynomial and the denominator polynomial of the rational approximation respectively; Z is the depth operator, Z j and Z l represent the j-th power and the l-th power of the operator Z; α j is the j-th molecular rational coefficient, with a total of n; β l is the l-th denominator rational coefficient, with a total of m; Z min and Z max are the minimum value and the maximum value of the operator expansion interval respectively; is the i-th Chebyshev expansion coefficient of the original operator function on the expansion interval, with a total of n + m - n s ones; is the i-th Chebyshev expansion coefficient of the rational approximation operator function on the expansion interval, with a total of n + m - n s ones; n s is the stability constraint condition number; Z i is the i-th stability constraint correction point set in the region where Re(Z) < -1, with a total of n s ones; Re(Z) represents taking the real part; D is the interior region of the unit circle that satisfies the stability condition, and satisfies D = |g n，m (Z)| ≤ 1, |g n，m (Z)| represents taking the modulus of the rational approximation operator function; g i is the i-th stability correction mapping image point set on the region D corresponding to Z i , with a total of n s ones.

[0056] Step 4: Repeat Step 2 and Step 3, solve and calculate to obtain the sound pressure value at the next step position until reaching the end point of the step position, so as to calculate and obtain the ocean sound field.

[0057] Figure 3 is the comparison diagram of the operator approximation performance in the embodiment scenario, and 4 types of operator approximations including Padé approximation of the same order, Padé approximation with stability constraints, Chebyshev-Padé approximation, and Chebyshev-Padé approximation with stability constraints are compared. Figure 3 (a) is the comparison of the operator accuracy performance, and the error tolerance is set to 10 -5, the maximum opening angles corresponding to the four types of approximations are approximately 40°, 36°, 52°, and 50° respectively; Figure 3 In (b) of Figure 3 , it is a comparison of the operator stability. Among the four types of approximations, only the Padé approximation with stability constraints and the Chebyshev-Padé approximation with stability constraints can map the operator Z to the interior of the stability region and have stability; while for the two types of approximations without stability constraints, there are image points mapped outside the stability region, resulting in stability problems. Therefore, the Chebyshev-Padé approximation can improve the operator accuracy and has a larger opening angle than the Padé approximation under the same error tolerance, and the Chebyshev-Padé approximation with stability constraints can ensure stability.

[0058] The comparison of the calculation results of the sound propagation loss field in the embodiment scenario is as Figure 4 shown. Figure 4 In (a) of Figure 4 , it is the calculation result of the Chebyshev-Padé approximation without stability constraints. There are obvious numerical oscillations in the sound propagation loss field. Especially, the propagation loss in the seabed sediment layer is greatly affected by the oscillations and cannot correctly represent the details of the sound field; Figure 4 In (b) of Figure 4 , it is the calculation result of the Chebyshev-Padé approximation with stability constraints, showing the expected stability and obtaining accurate propagation loss results.

[0059] The above has introduced the present invention in detail. The content described in the embodiments of this specification is only an example of the implementation form of the inventive concept. The protection scope of the present invention should not be regarded as limited to the specific forms stated in the embodiments. The protection scope of the present invention also extends to equivalent technical means that those skilled in the art can think of according to the inventive concept.

Claims

1. A method for calculating ocean acoustic fields based on the stable Chebyshev-Padé parabolic equation operator approximation, characterized in that: The steps include: Step 1: Get the original environment information, rasterize the environment area, get the step position information, and execute step 2 from the starting point of the step position; Step 2: According to the current environment information, set the approximate expansion interval of the operator at the current step position; Step 3: According to the current environmental information and the approximate expansion interval of the operator, solve the approximate coefficient of the Chebyshev-Padé operator under the stability constraint, obtain the Chebyshev-Padé operator under the stability constraint at the current step position, and then calculate the sound pressure value of the current step position and add it to the environmental information; In the step 3, according to the current environment information, the approximate expansion interval of the operator and the stability constraint condition, the precision equation and the stability equation satisfied by the numerator rational coefficient and the denominator rational coefficient are constructed and solved to obtain the numerator rational coefficient and the denominator rational coefficient, thereby obtaining the Chebyshev-Padé operator under the stability constraint at the current step position, wherein the precision equation is the original operator function and the rational approximate operator function in the operator expansion interval [Z min ,Z max ] on the first n+mn s The order Chebyshev basis corresponds to the equation with equal expansion coefficients, and the stability equation is n s The stability constraint correction equation for the rational approximate operator function is as follows: f(Z)≈g n,m (WITH) Expanded in the interval [Z min ,Z max ],i=1,2,…,n+mn s g n,m (Z i )=g i ,g i ∈D,i=1,2,…,n s Among them, f(Z) is the original operator function; g n,m (Z) is the rational approximation operator function; n and m are the numerator polynomial order and denominator polynomial order of the rational approximation, respectively; Z is the depth operator, Z j With Z l represents the jth and lth powers of the operator Z; α j is the jth numerator rational coefficient, a total of n; β l is the lth denominator rational coefficient, m in total; Z min and Z max are the minimum and maximum values ​​of the operator expansion interval respectively; f c i is the i-th order Chebyshev expansion coefficient of the original operator function on the expansion interval, a total of n+mn s indivual; is the i-th order Chebyshev expansion coefficient of the rational approximation operator function on the expansion interval, a total of n+mn s pcs; n s is the number of stability constraints; Z i is the ith stability constraint correction point set in the region Re(Z)<-1, with a total of n s Re(Z) represents the real part; D is the area inside the unit circle that satisfies the stability condition; g i is the i-th set on region D and Z i The corresponding stability correction map image points, a total of n s indivual; Step 4: Repeat steps 2 and 3 to solve and calculate and obtain the sound pressure value of the next step position until the end of the step position is reached, thereby calculating and obtaining the ocean sound field.

2. The method for calculating ocean acoustic field based on the stable Chebyshev-Padé parabolic equation operator approximation according to claim 1, characterized in that: The environmental information includes sound velocity field, terrain and density field.

3. The method for calculating ocean acoustic field based on the stable Chebyshev-Padé parabolic equation operator approximation according to claim 1, characterized in that: The step 2 is specifically as follows: The wave number or opening angle range of the depth operator Z is determined according to the environmental information, and then the approximate expansion interval of the operator at the current step position is determined.

4. A computer device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that: When the processor executes the computer program, the steps of the method according to any one of claims 1 to 3 are implemented.

5. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 3 are implemented.

6. A computer program product comprising a computer program / instructions, characterized in that When the computer program / instructions are executed by a processor, the steps of the method according to any one of claims 1 to 3 are implemented.

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