A method and system for generalized Hamiltonian operator symmetry and amplitude preservation.

Abstract

This invention discloses a method and system for generalized Hamiltonian operator symmetry and amplitude preservation. The method includes: determining the dimensions of the decomposed equations based on different one-way wave equations; establishing amplitude-preserving forms and boundary conditions for up- and down-going wave equations with different dimensions; adjusting the traditional one-way wave processing flow based on the modified source function and initial / boundary conditions until a one-way wavefield recursive formula satisfying the amplitude-preserving condition is obtained; running the modified one-way wave program to obtain the amplitude-preserving wavefield and verifying it with the results of a finite difference wavefield or analytical solution wavefield. By utilizing the symmetry of the one-way wave up- and down-going wave propagation operators, the amplitude-preserving characteristics of the propagation operator are achieved, and the reciprocity principle is satisfied under conservative Hamiltonian dynamics.

Description

Technical Field

[0001] This invention relates to the technical field of generalized Hamiltonian dynamical systems, and more particularly to a method and system for operator symmetry and amplitude preservation in generalized Hamiltonian systems. Background Technology

[0002] In recent years, the study of inverse problems based on wave theory has been a hot topic and a challenge in many fields of physics, including exploration seismology, mid- and far-infrared electromagnetic waves, and microwave imaging and inversion based on wave dynamics. The wave propagation equations in different media are wave forms of the generalized Hamiltonian canonical equations.

[0003] Given the inherent limitations of the full-waveform inversion method—namely, its theoretical completeness but numerous shortcomings in practical applications, such as the initial model deviating significantly from the actual model leading to iterative non-convergence, the lack of low-frequency data, the absence of large-offset observation data, unknown seismic wavelet waveforms, unknown subsurface medium density, and noise interference—currently, the main approaches to studying the dynamic characteristics of one-way waves include the directional wave field decomposition method. This method decomposes the full-wave equation of a sound wave into two coupled up-going and down-going one-way wave equations. In a homogeneous medium, the up-going and down-going wave fields are decoupled, allowing this method to accurately characterize the kinematic and dynamic features of the wave. When the medium velocity gradient is small, the traditional one-way wave approximation can yield highly accurate results. The iterative solution of this method can simultaneously simulate wave transmission and reflection phenomena, but due to its high computational cost, this method does not specifically address how to preserve the amplitude of the one-way wave equation. Another approach is the amplitude correction method using the transport equation. This method demonstrates that the travel time accuracy obtained by the traditional one-way function equation and the two-way function equation is consistent, but the transport equation results differ. This error correction can improve the amplitude information of the one-way wave equation. The third method involves reciprocity conservation principle correction. Wapenaar proposed a normalized energy flow decomposition method for layered media, which can correct the traditional one-way wave equation, making it obey the reciprocity principle and achieving amplitude calculation accuracy that matches the WKBJ solution. The limitation of this method is that the discussed one-way wave true amplitude formula is only applicable to local layered media. The fourth method is multi-order one-way wave modeling. Kiyashchenko et al. proposed an iterative calculation method for coupled uplink and downlink waves based on perturbation theory in 2005: the full wave equation is decomposed into three equations: uplink, downlink, and error terms. At the zeroth iteration, independent uplink and downlink waves are calculated. At the first iteration, the wave field obtained from the previous iteration is used to calculate the error term, which is then used as a source function term to compensate for the uplink and downlink wave equations. This process is repeated until the required calculation accuracy is met. The computational cost of this method is five times that of the traditional method. Similar to the first method, this method does not address how to guarantee the true amplitude output of the one-way wave. Summary of the Invention

[0004] The purpose of this section is to outline some aspects of embodiments of the present invention and to briefly describe some preferred embodiments. Simplifications or omissions may be made in this section, as well as in the abstract and title of this application, to avoid obscuring the purpose of these documents; however, such simplifications or omissions should not be construed as limiting the scope of the invention.

[0005] In view of the problem that the one-way wave operator of the acoustic medium wave equation does not preserve amplitude in the existing technology, the present invention is proposed.

[0006] Therefore, the purpose of this invention is to provide a method for generalized Hamiltonian operator symmetry and amplitude preservation, which aims to achieve amplitude preservation characteristics of the propagation operator by utilizing the symmetry of the one-way wave up and down propagation operator, and to satisfy the reciprocity principle under conservative Hamiltonian dynamics.

[0007] To solve the above-mentioned technical problems, the present invention provides the following technical solution:

[0008] In a first aspect, this invention provides a method for operator symmetry and amplitude preservation of generalized Hamiltonian, which includes the following specific steps: determining the dimensions of the decomposed equations based on different one-way wave equations; establishing amplitude-preserving forms and boundary conditions for up- and down-going wave equations with different dimensions; adjusting the one-way wave field recursive formula based on the modified source function and initial boundary conditions according to the traditional one-way wave processing flow until the amplitude-preserving conditions are met; running the modified one-way wave program to obtain the amplitude-preserving wave field and verifying it with the results of the finite difference wave field or analytical solution wave field.

[0009] As a preferred embodiment of the generalized Hamiltonian operator symmetry and amplitude preservation method described in this invention, the dimensions include the dimensions of the output constant density sound pressure equation and the dimensions of each physical quantity in the equation. The expression of the constant density sound pressure equation is:

[0010]

[0011] The dimensions of each physical quantity in formula ① are: sound pressure [p] = N / m 2 The speed of sound propagation [v] = m / s, Laplace operator Second-order partial derivatives of time The three-dimensional spatial coordinate dimension is [x=(x,y,z)]=m, and the time coordinate dimension is [t]=s, [x s If m is the location of the source function, then the dimension of the source function is source function [f] = N / m. 4 .

[0012] As a preferred embodiment of the operator symmetry and amplitude-preserving method of the generalized Hamiltonian described in this invention, the decomposition includes two strategies:

[0013] Strategy 1 involves decomposing the two-way wave acoustic equation ① into coupled one-way wave equations:

[0014] make Then equation ① can be decomposed into:

[0015]

[0016] Where D and U are the downward and upward traveling wave fields, respectively, with dimensions of N / m. 3 ;

[0017] Quasi-differential operators Its dimension is 1 / m, and its frequency-wavenumber domain expression is: Quasi-differential operators Its dimension is 1 / m 2 .

[0018] As a preferred embodiment of the generalized Hamiltonian operator symmetry and amplitude preservation method described in this invention, strategy two includes decomposing the two-way wave acoustic equation ① into coupled one-way wave equations:

[0019] make Then equation ① can be decomposed into:

[0020]

[0021] Among them, D ＊ U ＊ These are the decomposed wave fields of the down-traveling and up-traveling waves, respectively, which have dimensions different from D and U.

[0022] As a preferred embodiment of the operator symmetry and amplitude preservation method of the generalized Hamiltonian described in this invention, the amplitude preservation form and boundary conditions include establishing the true amplitude equation and boundary conditions of the downwave through equations ② and ③, the expressions of which are as follows:

[0023] The true amplitude equation and boundary conditions of the downwave established using equation ② are as follows:

[0024]

[0025] The true amplitude equation and boundary conditions of the downwave established using equation ③ are as follows:

[0026]

[0027] As a preferred embodiment of the operator symmetry and amplitude preservation method of the generalized Hamiltonian described in this invention, the traditional single-pass wave processing flow includes selecting the generalized screen GSP operator as the method for single-pass wave amplitude preservation, and determining the approximate solution of the single-pass wave operator L as follows:

[0028] Based on the approximate solution of the generalized screen operator for the one-way wave operator Λ in the frequency wavenumber domain of equation ⑤ for:

[0029]

[0030] Wherein, the plane coordinate x T = (x, y), where ω is angular momentum and v0 is the reference velocity at depth z; For lateral velocity change; when At that time, let Λ and The first to third derivatives are equal, and the optimal optimization parameters are obtained by solving for a1 = 0.5 and b1 = 0.25. The three terms in formula ⑥ are the phase shift term, the phase correction term, and the wide-angle finite difference correction term, respectively. The operators corresponding to these three terms are called the one-way wave generalized screen propagation operator, the phase correction operator, and the wide-angle amplitude correction operator.

[0031] As a preferred embodiment of the generalized Hamiltonian operator symmetry and amplitude preservation method described in this invention, the recursive formula includes a recursive formula for obtaining the one-way wave field p with sound pressure dimension based on down-sweep wave fields of different dimensions:

[0032] According to equation ④, the dimension is N / m 3 The recursive formula for the one-way down-slip wave field p with sound pressure dimensions, obtained after dimension normalization, for the down-slip wave field of D is:

[0033]

[0034] Where Δz is the depth step size, z s It is the depth of the source function.

[0035] According to equation ⑤, the dimension is... D ＊ The recursive formula for the one-way downslope wave field p, which has sound pressure dimensions after dimension normalization, is as follows:

[0036]

[0037] Where Δz is the depth step size, z s It is the depth of the source function.

[0038] Secondly, embodiments of the present invention provide a generalized Hamiltonian operator symmetry and amplitude-preserving system, which implements the steps of the above method using any computer programming language, and compiles and runs the program on any operating system and any hardware architecture.

[0039] Thirdly, embodiments of the present invention provide a computer device, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement any step of the above-described method.

[0040] Fourthly, embodiments of the present invention provide a computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements any step of the above-described method.

[0041] The beneficial effects of this invention are as follows: Based on the acoustic medium wave equation under the generalized Hamiltonian dynamic system, the amplitude preservation characteristic of the propagation operator is achieved through the symmetry of the one-way wave up and down propagation operator, and the reciprocity principle under the conservative Hamiltonian dynamic system is satisfied. Attached Figure Description

[0042] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. Wherein:

[0043] Figure 1 This is a flowchart of the operator symmetry and amplitude preservation method of the generalized Hamiltonian of this invention.

[0044] Figure 2 This is a schematic diagram comparing the results of the operator symmetry of the generalized Hamiltonian of this invention with the amplitude-preserving method. Detailed Implementation

[0045] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0046] Many specific details are set forth in the following description in order to provide a full understanding of the invention. However, the invention may also be practiced in other ways different from those described herein, and those skilled in the art can make similar extensions without departing from the spirit of the invention. Therefore, the invention is not limited to the specific embodiments disclosed below.

[0047] Secondly, the term "one embodiment" or "embodiment" as used herein refers to a specific feature, structure, or characteristic that may be included in at least one implementation of the present invention. The phrase "in one embodiment" appearing in different places in this specification does not necessarily refer to the same embodiment, nor is it a single or selective embodiment that is mutually exclusive with other embodiments.

[0048] Secondly, the present invention is described in detail with reference to the schematic diagrams. When detailing the embodiments of the present invention, for ease of explanation, the cross-sectional views illustrating the device structure may be partially enlarged, not according to the usual scale. Furthermore, the schematic diagrams are merely examples and should not limit the scope of protection of the present invention. In addition, actual fabrication should include three-dimensional spatial dimensions of length, width, and depth.

[0049] Example 1

[0050] Reference Figure 1 This is the first embodiment of the present invention, which provides a method for generalized Hamiltonian operator symmetry and amplitude preservation. The method includes the following specific steps:

[0051] S1. Determine the dimensions of the decomposition equation based on different one-way wave equations.

[0052] The dimensions include the dimensions of the constant density sound pressure equation and the dimensions of each physical quantity in the equation. The expression for the constant density sound pressure equation is:

[0053]

[0054] The dimensions of each physical quantity in formula ① are: sound pressure [p] = N / m 2 The speed of sound propagation [v] = m / s, Laplace operator Second-order partial derivatives of time The three-dimensional spatial coordinate dimension is [x=(x,y,z)]=m, and the time coordinate dimension is [t]=s, [x s If m is the location of the source function, then the dimension of the source function is source function [f] = N / m. 4 .

[0055] Decomposition includes the following two strategies:

[0056] Strategy 1 involves decomposing the two-way wave acoustic equation ① into coupled one-way wave equations:

[0057] make Then equation ① can be decomposed into:

[0058]

[0059] Where D and U are the downward and upward traveling wave fields, respectively, with dimensions of N / m. 3 ;

[0060] Quasi-differential operators Its dimension is 1 / m, and its frequency-wavenumber domain expression is: Quasi-differential operators Its dimension is 1 / m 2 .

[0061] Strategy 2 involves decomposing the two-way wave acoustic equation ① into coupled one-way wave equations:

[0062] make Then equation ① can be decomposed into:

[0063]

[0064] Among them, D ＊ U ＊ These are the decomposed wave fields of the down-traveling and up-traveling waves, respectively, which have dimensions different from D and U.

[0065] S2. Establish amplitude-preserving forms and boundary conditions for up- and down-wave equations with different dimensions.

[0066] The amplitude-preserving form and boundary conditions include establishing the true amplitude equation and boundary conditions of the downwave through equations ② and ③, and their expressions are as follows:

[0067] The true amplitude equation of the downward wave, established using equation ②, and the boundary conditions are written as follows:

[0068]

[0069] The true amplitude equation of the downward wave, established using equation ③, and the boundary conditions are written as follows:

[0070]

[0071] S3, based on the traditional single-wave processing flow, adjusts according to the modified source function and initial boundary value conditions until the single-pass wavefield recursive formula that satisfies the amplitude preservation condition is obtained.

[0072] Traditional single-pass wave processing involves selecting the generalized screen GSP operator as the method for amplitude preservation in single-pass wave processing. The approximate solution for the single-pass wave operator L is determined as follows:

[0073] Based on the approximate solution of the generalized screen operator for the one-way wave operator Λ in the frequency wavenumber domain of equation ⑤ for:

[0074]

[0075] Wherein, the plane coordinate x T = (x, y), where ω is angular momentum and v0 is the reference velocity at depth z; For lateral velocity change; when At that time, let Λ and The first to third derivatives are equal, and the optimal optimization parameters are solved as a1 = 0.5 and b1 = 0.25. The three terms in formula ⑥ are the phase shift term, the phase correction term, and the wide-angle finite difference correction term, respectively. The operators corresponding to these three terms are called the one-way wave generalized screen propagation operator, the phase correction operator, and the wide-angle amplitude correction operator.

[0076] The recursive formula includes a recursive formula for obtaining the one-way wave field p with sound pressure dimension based on the down-sweep wave field with different dimensions.

[0077] According to equation ④, the dimension is N / m 3 The recursive formula for the one-way down-slip wave field p with sound pressure dimensions, obtained after dimension normalization, for the down-slip wave field of D is:

[0078]

[0079] Where Δz represents the depth step size, z s Indicates the depth of the source function.

[0080] According to equation ⑤, the dimension is... D ＊ The recursive formula for the one-way downslope wave field p, which has sound pressure dimensions after dimension normalization, is as follows:

[0081]

[0082] Where Δz represents the depth step size, z s Indicates the depth of the source function.

[0083] S4. Run the modified one-way wave program to obtain the amplitude-preserving wave field and verify it with the results of the finite difference wave field or the analytical solution wave field.

[0084] Furthermore, this embodiment also provides a generalized Hamiltonian operator symmetry and amplitude-preserving system, which implements the steps of the method using any computer programming language, and compiles and runs the program on any operating system and any hardware architecture.

[0085] This embodiment also provides a computer device applicable to the generalized Hamiltonian operator symmetry and amplitude-preserving method, including a memory and a processor; the memory is used to store computer-executable instructions, and the processor is used to execute the computer-executable instructions to realize the power distribution area household-transformer relationship identification method proposed in the above embodiment.

[0086] The computer device can be a terminal, comprising a processor, memory, communication interface, display screen, and input devices connected via a system bus. The processor provides computing and control capabilities. The memory includes non-volatile storage media and internal memory. The non-volatile storage media stores the operating system and computer programs. The internal memory provides an environment for the operation of the operating system and computer programs stored in the non-volatile storage media. The communication interface is used for wired or wireless communication with external terminals; wireless communication can be achieved through Wi-Fi, carrier networks, NFC (Near Field Communication), or other technologies. The display screen can be an LCD screen or an e-ink screen. The input devices can be a touch layer covering the display screen, buttons, a trackball, or a touchpad on the computer device's casing, or an external keyboard, touchpad, or mouse.

[0087] This embodiment also provides a storage medium on which a computer program is stored. When the program is executed by a processor, it implements the method for realizing the operator symmetry and amplitude preservation of generalized Hamilton as proposed in the above embodiments.

[0088] The storage medium proposed in this embodiment and the data storage method proposed in the above embodiments belong to the same inventive concept. Technical details not described in detail in this embodiment can be found in the above embodiments, and this embodiment has the same beneficial effects as the above embodiments.

[0089] In summary, based on the acoustic medium wave equation under the generalized Hamiltonian dynamic system, the amplitude preservation characteristic of the propagation operator is achieved by utilizing the symmetry of the up and down propagation operators of the one-way wave, thus satisfying the reciprocity principle requirements under the Hamiltonian dynamic system.

[0090] Example 2

[0091] Reference Figure 1 and Figure 2 This is the second embodiment of the present invention, which differs from the first embodiment in that it provides a specific application of the method of the present invention.

[0092] The system of first-order differential equations of the generalized Hamiltonian canonical equations is in the form of:

[0093]

[0094] in, For momentum, Let M be the force, M be the mass, and q = (q1, ..., q2) n ) is a position vector. For velocity vector, Here, T is the acceleration vector; T is the kinetic energy function; V is the potential energy function; H = T + V is the Hamiltonian, representing the total energy.

[0095] Based on the decomposition forms of the uplink and downlink waves of the one-way wave equation with different dimensions, amplitude-preserving forms of the one-way wave equation expressed in different dimensions and their initial-boundary value conditions are proposed. Furthermore, within the existing framework for realizing one-way waves, amplitude preservation processing of one-way waves is achieved by modifying the source function and initial-boundary value conditions. The specific technical methods include the following steps:

[0096] S1: Determine the dimensions of different one-way wave decomposition equations, that is, what kind of physical quantity the one-way wave equation under discussion is.

[0097] The constant density sound pressure equation and the dimensions of each physical quantity in the equation are shown below:

[0098] The expression for the constant density sound pressure equation is:

[0099]

[0100] The dimensions of each physical quantity in formula ① are: sound pressure [p] = N / m 2 The speed of sound propagation [v] = m / s, Laplace operator Second-order partial derivatives of time The three-dimensional spatial coordinate dimension is [x=(x,y,z)]=m, and the time coordinate dimension is [t]=s, [x s If m is the location of the source function, then the dimension of the source function is source function [f] = N / m. 4 .

[0101] Output the coupled one-way wave equations expressed in different dimensions, as well as the dimensions of each physical quantity in the equations.

[0102] Depending on the expected observations of the researchers, the constant density acoustic wave equation can be decomposed into coupled one-way wave forms expressed in different dimensions.

[0103] This invention provides only two exemplary decomposition strategies, as shown in the following steps:

[0104] Strategy 1 for decomposing the two-way acoustic equation ① into coupled one-way wave equations:

[0105] make Then equation ① can be decomposed into:

[0106]

[0107] Where D and U are the downward and upward traveling wave fields, respectively, with dimensions of N / m. 3 ;

[0108] Quasi-differential operators Its dimension is 1 / m, and its frequency-wavenumber domain expression is: Quasi-differential operators Its dimension is 1 / m 2 .

[0109] Strategy 2 involves decomposing the two-way wave acoustic equation ① into coupled one-way wave equations:

[0110] make Then equation ① can be decomposed into:

[0111]

[0112] Among them, D ＊ U ＊ These are the decomposed wave fields of the down-traveling and up-traveling waves, respectively, which have dimensions different from D and U.

[0113] S2: Give the amplitude-preserving form of the up and down wave equations with different dimensions and the initial and boundary conditions.

[0114] Depending on the dimensions, the uplink and downlink wave fields can have many different expressions. In actual numerical implementation, the uplink and downlink wave fields D in equations ② and ③ are... ＊ with U ＊ The principle of spatial reciprocity must be satisfied to achieve amplitude preservation. Therefore, the true amplitude equations and boundary conditions for downwaves with different dimensions are established as follows:

[0115] The true amplitude equation of the downward wave, established using equation ②, and the boundary conditions are written as follows:

[0116]

[0117] The true amplitude equation and boundary conditions of the downwave established using equation ③ are as follows:

[0118]

[0119] S3: Select the traditional one-way wave processing flow. Based on this flow, and according to the modified source function and initial boundary value conditions, formulate a one-way wave field recursive formula that satisfies the amplitude preservation condition.

[0120] By selecting the traditional one-way wave equation processing procedure, an approximate solution for the one-way wave operator Λ is determined:

[0121] Traditional methods for solving the one-way wave operator Λ include phase-shifted PS, split-step Fourier SSF, frequency spatial domain finite-difference WXFD, Fourier finite-difference FFD, generalized screen GSP, and local cosine basis LCB, all of which can be applied in this method; here, the GSP operator is used as an example of the amplitude-preserving implementation method for one-way waves. The generalized screen operator approximate solution of the frequency wavenumber domain one-way wave operator Λ in equation ⑤ is shown. for:

[0122]

[0123] Wherein, the plane coordinate x T = (x, y), where ω is angular momentum and v0 is the reference velocity at depth z; For lateral velocity change; when At that time, let Λ and The first to third derivatives are equal, and the optimal optimization parameters are obtained by solving for a1 = 0.5 and b1 = 0.25. The three terms in formula ⑥ are the phase shift term, the phase correction term, and the wide-angle finite difference correction term, respectively. The operators corresponding to these three terms are called the one-way wave generalized screen propagation operator, the phase correction operator, and the wide-angle amplitude correction operator.

[0124] Based on the down-slip wave fields of different dimensions, a recursive formula for the down-slip wave field p of a one-way wave with sound pressure dimension is derived.

[0125] According to equation ④, the dimension is N / m 3 The recursive formula for the one-way down-slip wave field p with sound pressure dimensions, obtained after dimension normalization, for the down-slip wave field of D is:

[0126]

[0127] Where Δz represents the depth step size, z s Indicates the depth of the source function.

[0128] According to equation ⑤, the dimension is... D ＊ The recursive formula for the one-way downslope wave field p, which has sound pressure dimensions after dimension normalization, is as follows:

[0129]

[0130] Where Δz represents the depth step size, z s Indicates the depth of the source function.

[0131] S4: Run the modified single-pass wave program to obtain the amplitude-preserving wave field and verify it with the finite difference wave field or analytical solution wave field results.

[0132] like Figure 2 As shown, Figure (a) shows the wave field values ​​generated by the one-way wave operator of the acoustic medium without amplitude preservation; Figure (b) shows the amplitude-preserving propagation wave field values ​​of the sound wave calculated using the operator's symmetry (the field value dimensions are sound pressure N / m). 2 After amplitude preservation processing, propagation amplitude compensation was achieved at a propagation angle of nearly 90°, so that the one-way wave propagation effect and the two-way wave propagation effect remained consistent.

[0133] In summary, this invention primarily addresses the lack of amplitude preservation in the one-way wave operator of the acoustic medium wave equation under the generalized Hamiltonian dynamical system. By employing the symmetry of the one-way wave propagation operators (upward and downward propagation), it achieves the goal of preserving the amplitude characteristics of the propagation operator while satisfying the reciprocity principle under the conservative Hamiltonian dynamical system. Based on the traditional one-way wave technique, without altering the original computational framework, the method of ensuring the amplitude characteristics of the one-way wave propagation process in the generalized Hamiltonian wave form through dimensional normalization is necessary and has broad theoretical significance and application value.

[0134] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the present invention.

Claims

1. A method for operator symmetry and amplitude preservation in generalized Hamiltonian, characterized in that: Includes the following steps, Determine the dimensions of the decomposition equations based on different one-way wave equations; Establish amplitude-preserving forms and boundary conditions for up- and down-wave equations with different dimensions; Based on the traditional single-wave processing flow, adjustments are made according to the modified source function and initial boundary value conditions until the single-pass wave field recursive formula that meets the amplitude preservation condition is obtained. Run the modified one-way wave program to obtain the amplitude-preserving wave field and verify it with the results of the finite difference wave field or the analytical solution wave field; The dimensions include the dimensions of the constant density sound pressure equation and the dimensions of each physical quantity in the equation. The expression for the constant density sound pressure equation is: The dimensions of each physical quantity in formula ① are: sound pressure [p] = N / m 2 The speed of sound propagation [v] = m / s, Laplace operator Second-order partial derivatives of time The three-dimensional spatial coordinate dimension is [x=(x,y,z)]=m, and the time coordinate dimension is [t]=s, [x s If m is the location of the source function, then the dimension of the source function is source function [f] = N / m. 4 ; The decomposition includes strategy one and strategy two; Strategy 1 involves decomposing the two-way wave acoustic equation ① into coupled one-way wave equations: make Then equation ① can be decomposed into: Where D and U are the downward and upward traveling wave fields, respectively, with dimensions of N / m. 3 ; Quasi-differential operators Its dimension is 1 / m, and its frequency-wavenumber domain expression is: Quasi-differential operators Its dimension is 1 / m 2 .

2. The method for operator symmetry and amplitude preservation of generalized Hamiltonian according to claim 1, characterized in that: Strategy 2 includes decomposing the two-way wave acoustic equation ① into coupled one-way wave equations: make Then equation ① can be decomposed into: Among them, D * D * These are the decomposed wave fields of the down-traveling and up-traveling waves, respectively, which have dimensions different from D and U.

3. The method for operator symmetry and amplitude preservation of generalized Hamiltonian according to claim 2, characterized in that: The amplitude preservation form and boundary conditions include establishing the true amplitude equation and boundary conditions of the downwave through equations ② and ③, the expressions of which are as follows: The true amplitude equation and boundary conditions of the downwave established using equation ② are as follows: The true amplitude equation and boundary conditions for the downward wave established using equation ③ are as follows:

4. The method for operator symmetry and amplitude preservation of generalized Hamiltonian according to claim 3, characterized in that: The traditional single-pass wave processing flow includes selecting the generalized screen GSP operator as the method for single-pass wave amplitude preservation, and determining the approximate solution of the single-pass wave operator L as follows: Based on the approximate solution of the generalized screen operator for the one-way wave operator Λ in the frequency wavenumber domain of equation ⑤ for: Wherein, the plane coordinate x T = (x, y), where ω is angular momentum and v0 is the reference velocity at depth z; For lateral velocity change; when At that time, let Λ and The first to third derivatives are equal, and the optimal optimization parameters are solved as a1 = 0.5 and b1 = 0.

25. The three terms in formula ⑥ are the phase shift term, the phase correction term, and the wide-angle finite difference correction term, respectively. The operators corresponding to these three terms are called the one-way wave generalized screen propagation operator, the phase correction operator, and the wide-angle amplitude correction operator.

5. The method for operator symmetry and amplitude preservation of generalized Hamiltonian according to claim 4, characterized in that: The recursive formula includes a recursive formula for obtaining the one-way wave field p with sound pressure dimension based on the down-sweep wave fields of different dimensions. According to equation ④, the dimension is N / m 3 The recursive formula for the one-way down-slip wave field p with sound pressure dimensions, obtained after dimension normalization, for the down-slip wave field of D is: Where Δz is the depth step size, z s It is the depth of the source function; According to equation ⑤, the dimension is... D * The recursive formula for the one-way downslope wave field p, which has sound pressure dimensions after dimension normalization, is as follows: Where Δz is the depth step size, z s It is the depth of the source function.

6. A generalized Hamiltonian operator symmetry and amplitude-preserving system, characterized in that: Implement the steps of any of the methods described in claims 1 to 5 using any computer programming language, and compile and run the program on any operating system and any hardware architecture.

7. 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, it implements the steps of the method according to any one of claims 1 to 5.

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

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