Transverse isotropic medium seismic wave group velocity distribution calculation method

Abstract

The invention relates to the technical field of seismic wave group velocity measurement and calculation, and discloses a calculation method for seismic wave group velocity distribution of a transverse isotropic medium. Obtaining a relation between a group velocity component and a group velocity direction through a polynomial fitting method; then, through the phase velocity directions of the seismic waves passing through a small number of times, based on a Christoffel equation, calculating group velocity components and group velocity directions of the seismic waves corresponding to the phase velocity directions, and obtaining coefficients of a formula of a polynomial fitting method; according to the method, the relationship between the group velocity component and the group velocity direction in the environment to be measured can be accurately obtained, the group velocity direction to be solved is determined, namely, the corresponding group velocity component can be obtained, the calculation amount is reduced, and the efficiency is improved.

Description

Technical Field

[0001] The present application relates to the technical field of seismic wave group velocity measurement, and in particular to a method for calculating the distribution of seismic wave group velocity in a transversely isotropic medium. Background Art

[0002] Transversely isotropic medium (TI) refers to a medium with a symmetry axis in the vertical direction. This medium has different physical properties in the vertical direction from the other two horizontal directions, showing anisotropy. In seismology, the seismic wave propagation characteristics of transversely isotropic medium are different from those of isotropic medium because its elastic properties are different in different directions. In transversely isotropic medium, the propagation characteristics of seismic waves vary with the propagation direction, which is often manifested as inconsistency in the phase angle and group angle, phase velocity and group velocity of seismic wave propagation.

[0003] When calculating the group velocity of seismic waves in transversely isotropic media, group velocities in various directions are usually used, that is, the distribution information of group velocity in different directions needs to be obtained. The usual practice is: first give the phase velocity direction n; then construct the Christoffel matrix, obtain the eigenvalues ​​and eigenvectors; then obtain the group velocity magnitude and ray direction corresponding to the phase velocity direction. Perform all-round sampling of the phase velocity direction and repeat the previous steps to obtain the distribution results of the group velocity in different ray directions.

[0004] However, the phase velocity direction is not one-to-one corresponding to the obtained ray direction. The final ray direction may not be within the required range. In this case, it is necessary to perform difference calculation and perform more precise sampling of the phase velocity direction to obtain more ray directions, so as to meet the distribution results of the group velocity in the required ray direction as much as possible. However, this requires a huge amount of calculation and has poor timeliness.

[0005] Therefore, there is an urgent need for a calculation method for the distribution of seismic wave group velocity in a transversely isotropic medium that can simplify the calculation and obtain accurate distribution results of group velocity in different ray directions. Summary of the invention

[0006] In order to solve the above-mentioned problem of cumbersome group velocity calculation, the present application provides a method for calculating the distribution of seismic wave group velocity in a transversely isotropic medium, comprising the following steps: S1: Obtain the elastic properties of the transversely isotropic medium in the test environment; S2: Select the phase velocity directions of m seismic waves, and calculate the group velocity component and group velocity direction of the seismic waves corresponding to each phase velocity direction through the Christoffel equation; S3: Substitute the group velocity component and group velocity direction of the seismic wave obtained in S2 into the polynomial fitting formula to obtain the group velocity component of the seismic wave corresponding to different group velocity directions; The formula of the polynomial fitting method is: ; Among them, v q is the component of the seismic wave group velocity in the qth direction, N q is the qth component of the seismic wave group velocity direction N, and 1≤q≤3, a0, a1, a p-1 and a p are the coefficients of the polynomial, p is the order, and p <m。

[0007] In some embodiments of the present application, the order p of the polynomial fitting formula is 4.

[0008] In some embodiments of the present application, the elastic property of the transversely isotropic medium is a fourth-order tensor , and 1≤i≤3, 1≤j≤3, 1≤k≤3, 1≤l≤3.

[0009] In some embodiments of the present application, the Christoffel equation is: ; In the formula, det represents the determinant calculation, c is the phase velocity, and n is the phase velocity direction. Represents a fourth-order tensor and the Christoffel matrix of the phase velocity direction n, where I is the identity matrix.

[0010] In some embodiments of the present application, the phase velocity direction , where θ is the angle between the phase velocity direction and the z-axis, and φ is the angle between the projection direction of the phase velocity in the horizontal direction and the x-axis.

[0011] In some embodiments of the present application, the expression of the Christoffel matrix is: ; in, represents the stress generated in the e direction and the strain in the f direction, and n1, n2 and n3 represent the three components in the phase velocity direction respectively.

[0012] In some embodiments of the present application, in S3, the group velocity component and group velocity direction of the seismic wave obtained in S2 are substituted into a polynomial fitting formula to obtain the group velocity component of the seismic wave corresponding to different group velocity directions, specifically: S31: using the group velocity component and group velocity direction of the seismic wave obtained in S2 to solve the coefficients of the polynomial fitting equation by the least square method; S32: Substitute the coefficients of the polynomial to be solved into the polynomial fitting formula to obtain the polynomial fitting formula with the determined coefficients; S33: Determine the direction of the group velocity to be measured, and obtain the group velocity component corresponding to the direction of the group velocity to be measured according to the direction of the group velocity to be measured and the polynomial fitting formula of the determined coefficients.

[0013] The embodiments of the present application have the following technical effects: In this application, the relationship between the group velocity component and the group velocity direction is obtained by the polynomial fitting method; then, by passing the phase velocity direction of the seismic wave a small number of times, based on the Christoffel equation, the group velocity component and group velocity direction of the seismic wave corresponding to each phase velocity direction are calculated to obtain the coefficients of the polynomial fitting method formula. This method can accurately obtain the relationship between the group velocity component and the group velocity direction in the test environment, determine the group velocity direction to be obtained, and obtain the corresponding group velocity component, reduce the amount of calculation, and improve efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] In order to more clearly illustrate the specific implementation methods of the present application or the technical solutions in the prior art, the drawings required for use in the specific implementation methods or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are some implementation methods of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0015] Figure 1 It is a flow chart of a method for calculating the seismic wave group velocity distribution in a transversely isotropic medium provided in an embodiment of the present application; Figure 2 is a comparison diagram of the calculation results of the seismic wave group velocity distribution in a transversely isotropic medium provided by the embodiment of the present application and the conventional seismic wave group velocity distribution calculation method; Figure 3 It is a structural schematic diagram of an electronic device provided in an embodiment of the present application. DETAILED DESCRIPTION

[0016] In order to make the purpose, technical solution and advantages of the present application clearer, the technical solution of the present application will be described clearly and completely below. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in the field without creative work belong to the scope of protection of the present application.

[0017] Based on the problem of large amount of calculation when obtaining the group velocity of seismic waves, this application uses the polynomial fitting method to obtain the relationship between the group velocity component and the group velocity direction; then through a small number of times of passing the phase velocity direction of the seismic wave, based on the Christoffel equation, the group velocity component and group velocity direction of the seismic wave corresponding to each phase velocity direction are calculated to obtain the coefficients of the polynomial fitting method formula. This greatly reduces the number of calculations and the amount of calculation.

[0018] The present application provides a method for calculating the distribution of seismic wave group velocity in a transversely isotropic medium, comprising the following steps: S1: Obtain the elastic properties of the transversely isotropic medium in the test environment; In some embodiments of the present application, the elastic property of the transversely isotropic medium is a fourth-order tensor , and 1≤i≤3, 1≤j≤3, 1≤k≤3, 1≤l≤3.

[0019] When describing the elastic properties of anisotropic media, the fourth-order tensor is called the elasticity tensor or stiffness tensor. This tensor contains all the information describing how a medium responds to an external force. In seismology and solid mechanics, the fourth-order tensor The subscripts i, j, k, and l represent: The two indices i and j represent the direction of the force. In three-dimensional space, they can take the values ​​1 (x direction), 2 (y direction) or 3 (z direction), indicating the direction in which the force is applied.

[0020] The two indices k and l represent the direction of the strain or displacement. Similarly, they can also take the values ​​of 1, 2 or 3, indicating the direction in which the medium is strained or displaced.

[0021] Each component of the elastic tensor describes the relationship between the stress in the j direction, the strain in the k direction, and the stress in the l direction when a unit force is applied in the i direction. In short, It describes how applying a force in the i direction affects the stress in the j direction and how this effect depends on the strains in the k and l directions.

[0022] For example, It represents the stress in the y direction when a unit force is applied in the x direction, and this effect is related to the strains in the x and y directions.

[0023] S2: Select the phase velocity directions of m seismic waves, and calculate the group velocity component and group velocity direction of the seismic waves corresponding to each phase velocity direction through the Christoffel equation; In some embodiments of the present application, the Christoffel equation is: ; (1) In the formula, det represents the determinant calculation, c is the phase velocity, and n is the phase velocity direction. Represents a fourth-order tensor and the Christoffel matrix of the phase velocity direction n, where I is the identity matrix.

[0024] In some embodiments of the present application, the phase velocity direction , where θ is the angle between the phase velocity direction and the z-axis, and φ is the angle between the horizontal projection of the phase velocity and the x-axis.

[0025] Transversely isotropic (TI) media are widely used anisotropic media, in which the phase velocity and group velocity are related to only five independent elastic parameters. ), then the expression of the Christoffel matrix is: ; (2) in, represents the stress generated in the e direction and the strain in the f direction, and n1, n2 and n3 represent the three components in the phase velocity direction respectively. It is used to refer to the , , ,……, , The e-th direction is the same as the fourth-order tensor The jth direction and the fth direction in the fourth-order tensor are the same The kth direction in .

[0026] S3: Substitute the group velocity component and group velocity direction of the seismic wave obtained in S2 into the polynomial fitting formula to obtain the group velocity component of the seismic wave corresponding to different group velocity directions; For each phase velocity direction, the formula of the polynomial fitting method is: ; (8) Among them, v q is the component of the seismic wave group velocity in the qth direction, N q is the qth component of the seismic wave group velocity direction N, and 1≤q≤3, representing the three directions of X, Y and Z respectively. a0, a1, a p-1 and a p are the coefficients of the polynomial, p is the order, and p <m。

[0027] Substituting formula (2) into formula (1) yields: (3) The above formula can be rewritten as (4) From this, we can get the phase velocities (qP, qSV, qSH). qP refers to quasi-longitudinal waves (or P waves), qSV refers to quasi-vertical shear waves (or SV waves), and qSH refers to quasi-horizontal shear waves (or SH waves).

[0028] According to the relationship between phase velocity and group velocity, the group velocity component can be obtained: (5) The group velocity component can be used to obtain the magnitude and direction of the group velocity: (6) (7) And because the phase velocity c is a first-order homogeneous function about the phase velocity direction n: , which means that there is a relatively simple functional relationship between the two, and α represents a constant. The relationship between the group velocity component and the group velocity direction of the seismic wave is similar to the relationship between the phase velocity magnitude and the phase velocity direction. It is inferred that there may also be a functional relationship between the group velocity component and the group velocity direction of the seismic wave. Based on this, the multi-order polynomial fitting method is used to obtain the relationship between the group velocity component and the group velocity direction, as shown in formula (8).

[0029] According to formula (4), the phase velocity c is a fourth-order function of the phase velocity direction n.

[0030] From formula (5), we can know that the group velocity component With the phase velocity direction n q It is a function of no more than fourth order. From formula (6), we can see that the group velocity v and the phase velocity direction are also functions of no more than fourth order, and the two have the same order.

[0031] In some embodiments of the present application, the order p of the polynomial fitting formula is 4.

[0032] In some embodiments of the present application, in S3, the group velocity component and group velocity direction of the seismic wave obtained in S2 are substituted into a polynomial fitting formula to obtain the group velocity component of the seismic wave corresponding to different group velocity directions, specifically: S31: The group velocity component and group velocity direction of the seismic wave obtained in S2 are used to solve the coefficients of the polynomial fitting formula by the least square method. The formula is as follows: ; Where a0, a1, a2, a3, and a4 are the coefficients of the polynomial, N (1) 、N (2) ,…,N (m) are the direction of the seismic wave group velocity obtained in S2, v (1) 、v(2) ,…,v (m) are the seismic wave group velocity components obtained in S2 respectively; S32: Substitute the coefficients of the polynomial to be solved into the polynomial fitting formula to obtain the polynomial fitting formula with the determined coefficients; S33: Determine the direction of the group velocity to be measured, and obtain the group velocity component corresponding to the direction of the group velocity to be measured according to the direction of the group velocity to be measured and a polynomial fitting formula with determined coefficients.

[0033] Figure 2 : This is a comparison diagram of the calculation results of the seismic wave group velocity distribution of the transverse isotropic medium provided by the embodiment of the present application and the conventional seismic wave group velocity distribution calculation method. The black fitted line represents the calculation results of the seismic wave group velocity distribution of the transverse isotropic medium, and the red line represents the results of the conventional seismic wave group velocity distribution calculation method. In the N1 and N3 directions, the results of the two calculation methods are basically completely overlapped. Therefore, the calculation method of the seismic wave group velocity distribution of the transverse isotropic medium provided by the present application is not only simple to calculate, but also accurate.

[0034] Some embodiments of the present application further provide a system for calculating the distribution of seismic wave group velocity in a transversely isotropic medium, which executes any of the above-mentioned methods for calculating the distribution of seismic wave group velocity in a transversely isotropic medium, such as Figure 3 As shown, the system includes an elastic medium acquisition module, a coefficient determination parameter acquisition module, and a calculation module; An elastic medium acquisition module is used to acquire the elastic properties of the transversely isotropic medium in the environment to be tested; The coefficient determination parameter acquisition module is used to select the phase velocity directions of m seismic waves, calculate the group velocity component and group velocity direction of the seismic waves corresponding to each phase velocity direction through the Christoffel equation, and obtain the parameters required for coefficient determination; A calculation module is used to obtain the group velocity components and group velocity directions of the seismic waves by a polynomial fitting method to obtain the group velocity components corresponding to the seismic waves in different group velocity directions; Some embodiments of the present application further provide an electronic device, the electronic device comprising: Processor and memory; The processor is used to execute the steps of any of the above-mentioned methods for calculating the seismic wave group velocity distribution in a transversely isotropic medium by calling the program or instruction stored in the memory.

[0035] Figure 3 Schematic diagram of the structure of an electronic device provided in an embodiment of the present application. Figure 3 , the electronic device 500 includes one or more processors 501 and a memory 502.

[0036] The processor 501 may be a central processing unit (CPU) or other forms of processing units having data processing capabilities and / or instruction execution capabilities, and may control other components in the electronic device 500 to perform desired functions.

[0037] The memory 502 may include one or more computer program products, and the computer program products may include various forms of computer-readable storage media, such as volatile memory and / or non-volatile memory. Volatile memory may include, for example, random access memory (RAM) and / or cache memory (cache), etc. Non-volatile memory may include, for example, read-only memory (ROM), hard disk, flash memory, etc. One or more computer program instructions may be stored on a computer-readable storage medium, and the processor 501 may run the program instructions to implement the calculation method of the seismic wave group velocity distribution of the transversely isotropic medium in any embodiment of the present application described above and / or other desired functions. Various contents such as initial external parameters, thresholds, etc. may also be stored in the computer-readable storage medium.

[0038] In one example, the electronic device 500 may further include: an input device 503 and an output device 504, which are interconnected via a bus system and / or other forms of connection mechanisms (not shown). The input device 503 may include, for example, a keyboard, a mouse, etc. The output device 504 may output various information to the outside, including prompt information, etc. The output device 504 may include, for example, a display, a speaker, a printer, a communication network and a remote output device connected thereto, etc.

[0039] Of course, to simplify, Figure 3 Only some of the components related to the present application in the electronic device 500 are shown, and components such as a bus, an input / output interface, etc. are omitted. In addition, according to specific application conditions, the electronic device 500 may also include any other appropriate components.

[0040] In addition to the above-mentioned methods and devices, an embodiment of the present application may also be a computer program product, which includes computer program instructions, which, when executed by a processor, enable the processor to execute the method for calculating the seismic wave group velocity distribution in a transversely isotropic medium provided in any embodiment of the present application.

[0041] The computer program product may be written in any combination of one or more programming languages ​​to write program codes for performing the operations of the embodiments of the present application, including object-oriented programming languages, such as Java, C++, etc., and conventional procedural programming languages, such as "C" language or similar programming languages. The program code may be executed entirely on the user computing device, partially on the user device, as an independent software package, partially on the user computing device and partially on a remote computing device, or entirely on a remote computing device or server.

[0042] In addition, another aspect of the present application provides a storage medium, which stores a computer executable program, and the computer executable program is called by a processor to execute the steps of the method for calculating the seismic wave group velocity distribution in a transversely isotropic medium as described in any of the above items.

[0043] The computer readable storage medium can adopt any combination of one or more readable media. The readable medium can be a readable signal medium or a readable storage medium. The readable storage medium can include, for example, but is not limited to, a system, device or device of electricity, magnetism, light, electromagnetic, infrared, or semiconductor, or any combination of the above. More specific examples (non-exhaustive list) of readable storage media include: an electrical connection with one or more wires, a portable disk, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), an optical fiber, a portable compact disk read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the above.

[0044] It should be noted that the terms used in this application are only for describing specific embodiments, rather than limiting the scope of this application. As shown in the specification of this application, unless the context clearly indicates an exception, the words "one", "a", "a kind of" and / or "the" do not specifically refer to the singular, but may also include the plural. The terms "include", "comprise" or any other variant thereof are intended to cover non-exclusive inclusion, so that the process, method or device including a series of elements includes not only those elements, but also includes other elements not explicitly listed, or also includes elements inherent to such process, method or device. In the absence of more restrictions, the elements defined by the sentence "include one..." do not exclude the presence of other identical elements in the process, method or device including the elements.

[0045] It should also be noted that the terms "center", "up", "down", "left", "right", "vertical", "horizontal", "inside", "outside", etc., indicating the orientation or position relationship, are based on the orientation or position relationship shown in the drawings, and are only for the convenience of describing the present application and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation on the present application. Unless otherwise clearly specified and limited, the terms "installed", "connected", "connected", etc. should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection, or it can be an indirect connection through an intermediate medium, or it can be a connection between the two elements. For those of ordinary skill in the art, the specific meanings of the above terms in this application can be understood according to specific circumstances.

[0046] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present application, rather than to limit it. Although the present application has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the technical solutions of the embodiments of the present application.

Claims

1. A method for calculating the distribution of seismic wave group velocity in a transversely isotropic medium, characterized in that: The steps include: S1: Obtain the elastic properties of the transversely isotropic medium in the test environment; S2: Select the phase velocity directions of m seismic waves, and calculate the group velocity component and group velocity direction of the seismic waves corresponding to each phase velocity direction through the Christoffel equation; S3: using the group velocity component and group velocity direction of the seismic wave obtained in S2, the group velocity component corresponding to the seismic wave in different group velocity directions is obtained by a polynomial fitting method; The formula of the polynomial fitting method is: ； Among them, v q is the component of the seismic wave group velocity in the qth direction, N q is the qth component of the seismic wave group velocity direction N, and 1≤q≤3, a0, a1, a p-1 and a p are the coefficients of the polynomial, p is the order, and p <m。 2. The method for calculating the seismic wave group velocity distribution in a transversely isotropic medium according to claim 1, characterized in that: The order p of the polynomial fitting formula is 4.

3. The method for calculating the seismic wave group velocity distribution in a transversely isotropic medium according to claim 1, characterized in that: The elastic properties of a transversely isotropic medium are fourth-order tensors , and 1≤i≤3, 1≤j≤3, 1≤k≤3, 1≤l≤3.

4. The method for calculating the seismic wave group velocity distribution in a transversely isotropic medium according to claim 3, characterized in that: The Christoffel equation is: ； In the formula, det represents the determinant calculation, c is the phase velocity, and n is the phase velocity direction. Represents a fourth-order tensor and the Christoffel matrix of the phase velocity direction n, where I is the identity matrix.

5. The method for calculating the seismic wave group velocity distribution in a transversely isotropic medium according to claim 4, characterized in that: Phase velocity direction , where θ is the angle between the phase velocity direction and the z-axis, and φ is the angle between the projection of the phase velocity in the horizontal direction and the x-axis.

6. The method for calculating the seismic wave group velocity distribution in a transversely isotropic medium according to claim 4, characterized in that: The expression of the Christoffel matrix is: ； in, represents the stress generated in the e direction and the strain in the f direction, and n1, n2 and n3 represent the three components in the phase velocity direction respectively.

7. The method for calculating the seismic wave group velocity distribution in a transversely isotropic medium according to claim 2, characterized in that: In S3, the group velocity component and group velocity direction of the seismic wave obtained in S2 are substituted into the polynomial fitting formula to obtain the group velocity component of the seismic wave corresponding to different group velocity directions, specifically: S31: using the group velocity component and group velocity direction of the seismic wave obtained in S2 to solve the coefficients of the polynomial fitting equation by the least square method; S32: Substitute the coefficients of the polynomial to be solved into the polynomial fitting formula to obtain the polynomial fitting formula with the determined coefficients; S33: Determine the direction of the group velocity to be measured, and obtain the group velocity component corresponding to the direction of the group velocity to be measured according to the direction of the group velocity to be measured and the polynomial fitting formula of the determined coefficients.

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