A calculation method for seismic wave group velocity distribution in transversely isotropic media

Abstract

The present application relates to the technical field of seismic wave group velocity measurement and calculation, and discloses a method for calculating the group velocity distribution of seismic waves in a transversely isotropic medium. The relationship between group velocity components and group velocity directions is obtained through a polynomial fitting method. Then, by passing the phase velocity directions of the seismic waves a small number of times, the group velocity components and group velocity directions of the seismic waves corresponding to each phase velocity direction are calculated based on the Christoffel equation, and the coefficients of the polynomial fitting method formula are obtained. This method can accurately obtain the relationship between the group velocity components and group velocity directions in the environment to be measured, determine the group velocity direction to be obtained, and thus obtain the corresponding group velocity components, reducing the amount of calculation and improving efficiency.

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] A transversely isotropic medium (TI) is a medium with an axis of symmetry in the vertical direction. Such a medium exhibits different physical properties in the vertical direction than in the other two horizontal directions, thus exhibiting anisotropy. In seismology, the seismic wave propagation characteristics of a transversely isotropic medium differ from those of an isotropic medium because its elastic properties vary in different directions. In a transversely isotropic medium, the propagation characteristics of seismic waves vary with the propagation direction, often manifesting as inconsistencies in the phase angle and group angle, as well as the phase velocity and group velocity.

[0003] When calculating seismic group velocity in transversely isotropic media, group velocities in various directions are often required. This means that the distribution of group velocities in different directions is required. A common approach is to first determine the phase velocity direction n; then construct a Christoffel matrix and calculate the eigenvalues ​​and eigenvectors; and finally, obtain the group velocity magnitude and ray direction corresponding to that phase velocity direction. The phase velocity direction is sampled omnidirectionally, and the previous steps are repeated to obtain the distribution of group velocities in different ray directions.

[0004] However, the phase velocity direction and the obtained ray direction do not correspond one-to-one. The final ray direction may not be within the required range. In this case, it is necessary to perform interpolation calculations 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 directions as much as possible. However, this method requires a large amount of calculations and has poor timeliness.

[0005] Therefore, there is an urgent need for a calculation method for the distribution of seismic wave group velocity in transversely isotropic media 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 group velocity distribution of seismic waves in a transversely isotropic medium, comprising the following steps:

[0007] S1: Obtain the elastic properties of the transversely isotropic medium in the test environment;

[0008] S2: Select m phase velocity directions of seismic waves and calculate the group velocity component and group velocity direction of the seismic wave corresponding to each phase velocity direction using the Christoffel equation;

[0009] S3: Substitute the group velocity components and group velocity directions of the seismic waves obtained in S2 into the polynomial fitting formula to obtain the group velocity components of the seismic waves corresponding to different group velocity directions;

[0010] The formula of the polynomial fitting method is:

[0011] ;

[0012] 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, p <m。

[0013] In some embodiments of the present application, the order p of the polynomial fitting formula is 4.

[0014] In some embodiments of the present application, the elastic properties of the transversely isotropic medium are fourth-order tensors , and 1≤i≤3, 1≤j≤3, 1≤k≤3, 1≤l≤3.

[0015] In some embodiments of the present application, the Christoffel equation is:

[0016] ;

[0017] In the formula, det represents the determinant calculation, c is the phase velocity, 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.

[0018] 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.

[0019] In some embodiments of the present application, the expression of the Christoffel matrix is:

[0020] ;

[0021] in, represents the stress generated in the e-th direction and the strain in the f-th direction, and n1, n2 and n3 represent the three components of the phase velocity direction respectively.

[0022] 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:

[0023] 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;

[0024] 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;

[0025] S33: Determine the group velocity direction to be measured, and obtain the group velocity component corresponding to the group velocity direction to be measured according to the group velocity direction to be measured and the polynomial fitting formula of the determined coefficients.

[0026] The embodiments of the present application have the following technical effects:

[0027] In this application, a polynomial fitting method is used to determine the relationship between group velocity components and group velocity directions. Then, based on a small number of phase velocity directions of the seismic waves, the Christoffel equation is used to calculate the group velocity components and group velocity directions corresponding to each phase velocity direction, thereby obtaining the coefficients of the polynomial fitting method formula. This method can accurately determine the relationship between group velocity components and group velocity directions in the test environment, determine the group velocity direction to be determined, and thus obtain the corresponding group velocity components, reducing the amount of calculation and improving efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] In order to more clearly illustrate the specific implementation methods of the present application or the technical solutions in the prior art, the following is a brief introduction to the drawings required for use in the specific implementation methods or the description of the prior art. 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 any creative work.

[0029] Figure 1 This is a flow chart of a method for calculating seismic wave group velocity distribution in a transversely isotropic medium provided in an embodiment of the present application;

[0030] Figure 2 1 is a comparison diagram of the calculation results of the seismic wave group velocity distribution in a transversely isotropic medium provided by an embodiment of the present application and a conventional seismic wave group velocity distribution calculation method;

[0031] Figure 3 This is a structural diagram of an electronic device provided in an embodiment of the present application. DETAILED DESCRIPTION

[0032] To make the objectives, technical solutions, and advantages of this application more clear, the technical solutions of this application are described clearly and completely below. Obviously, the embodiments described are only some of the embodiments of this application, not all of them. Based on the embodiments in this application, all other embodiments obtained by persons of ordinary skill in the art without inventive effort are within the scope of protection of this application.

[0033] To address the computational complexity of determining seismic wave group velocity, this application uses a polynomial fitting method to determine the relationship between group velocity components and group velocity directions. By performing a small number of passes through the phase velocity directions of the seismic waves, the Christoffel equation is used to calculate the group velocity components and group velocity directions corresponding to each phase velocity direction, thereby obtaining the coefficients for the polynomial fitting method formula. This significantly reduces the number of calculations and the computational complexity.

[0034] The present application provides a method for calculating the distribution of seismic wave group velocity in a transversely isotropic medium, comprising the following steps:

[0035] S1: Obtain the elastic properties of the transversely isotropic medium in the test environment;

[0036] In some embodiments of the present application, the elastic properties of the transversely isotropic medium are fourth-order tensors , and 1≤i≤3, 1≤j≤3, 1≤k≤3, 1≤l≤3.

[0037] 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 that describes 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:

[0038] The two indices, i and j, represent the direction of the force. In three-dimensional space, they can take values ​​of 1 (x-direction), 2 (y-direction), or 3 (z-direction), indicating the direction along which the force is applied.

[0039] The two indices k and l represent the direction of the strain or displacement. Similarly, they can also take the values ​​1, 2, or 3, indicating the direction in which the medium is strained or displaced.

[0040] 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.

[0041] 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 strain in the x and y directions.

[0042] S2: Select m phase velocity directions of seismic waves and calculate the group velocity component and group velocity direction of the seismic wave corresponding to each phase velocity direction using the Christoffel equation;

[0043] In some embodiments of the present application, the Christoffel equation is:

[0044] ;(1)

[0045] In the formula, det represents the determinant calculation, c is the phase velocity, 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.

[0046] 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.

[0047] Transversely isotropic (TI) media is a widely used anisotropic medium in which the phase velocity and group velocity are only related to five independent elastic parameters. When the medium has a vertical symmetry axis ( ), then the expression of the Christoffel matrix is:

[0048] ;(2)

[0049] in, represents the stress generated in the e-th direction and the strain in the f-th direction, and n1, n2 and n3 represent the three components of the phase velocity direction respectively. Used to refer to the formula (2) 、 、 、……、 、 The e-th direction is the same as the fourth-order tensor The j-th direction and the f-th direction in the fourth-order tensor are the same The kth direction in .

[0050] S3: Substitute the group velocity components and group velocity directions of the seismic waves obtained in S2 into the polynomial fitting formula to obtain the group velocity components of the seismic waves corresponding to different group velocity directions;

[0051] For each phase velocity direction, the polynomial fitting method formula is:

[0052] ;(8)

[0053] 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, p <m。

[0054] Substituting formula (2) into formula (1) yields:

[0055] (3)

[0056] The above formula can be rewritten as

[0057] (4)

[0058] From this, we can obtain the phase velocities (qP, qSV, qSH). qP refers to the quasi-longitudinal wave (or P wave), qSV refers to the quasi-vertical shear wave (or SV wave), and qSH refers to the quasi-horizontal shear wave (or SH wave).

[0059] The group velocity component can be obtained based on the relationship between phase velocity and group velocity:

[0060] (5)

[0061] The group velocity component can be used to obtain the magnitude and direction of the group velocity:

[0062] (6)

[0063] (7)

[0064] 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. Therefore, it is speculated 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).

[0065] According to formula (4), the phase velocity c is a fourth-order function of the phase velocity direction n.

[0066] From formula (5), we can know that the group velocity component and phase velocity direction n qThe relationship between them is a function of no more than fourth order. From formula (6), we can see that the relationship between the group velocity v and the phase velocity direction is also a function of no more than fourth order, and the two have the same order.

[0067] In some embodiments of the present application, the order p of the polynomial fitting formula is 4.

[0068] 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:

[0069] 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 squares method. The formula is as follows:

[0070] ;

[0071] 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;

[0072] 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;

[0073] 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.

[0074] Figure 2 This figure compares the calculation results of the seismic group velocity distribution in transversely isotropic media provided by the embodiment of the present application with those of a conventional method for calculating seismic group velocity distribution. The fitted black line represents the calculation results of the seismic group velocity distribution in transversely isotropic media, and the red line represents the results of the conventional method. In the N1 and N3 directions, the results of the two methods are essentially identical. Therefore, the calculation method for the seismic group velocity distribution in transversely isotropic media provided by the present application is not only simple but also accurate.

[0075] Some embodiments of the present application further provide a system for calculating the group velocity distribution of seismic waves in a transversely isotropic medium, which executes any of the above-mentioned methods for calculating the group velocity distribution of seismic waves in a transversely isotropic medium, such as Figure 3As shown, the system includes an elastic medium acquisition module, a coefficient determination parameter acquisition module, and a calculation module;

[0076] An elastic medium acquisition module is used to obtain the elastic properties of the transversely isotropic medium in the test environment;

[0077] 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 wave corresponding to each phase velocity direction through the Christoffel equation, and obtain the parameters required for coefficient determination;

[0078] 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 different group velocity directions of the seismic waves;

[0079] Some embodiments of the present application further provide an electronic device, the electronic device comprising:

[0080] processor and memory;

[0081] 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.

[0082] Figure 3 This is a 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 .

[0083] 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.

[0084] Memory 502 may include one or more computer program products, which 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. Non-volatile memory may include, for example, read-only memory (ROM), a hard disk, flash memory, etc. One or more computer program instructions may be stored on a computer-readable storage medium, and processor 501 may execute the program instructions to implement the method for calculating the seismic wave group velocity distribution in a transversely isotropic medium described above in any embodiment of the present application and / or other desired functions. Various contents such as initial external parameters and thresholds may also be stored in the computer-readable storage medium.

[0085] 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 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, including prompts, etc. The output device 504 may include, for example, a display, a speaker, a printer, a communication network, and remote output devices connected thereto.

[0086] 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, the electronic device 500 may further include any other appropriate components according to specific application conditions.

[0087] 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.

[0088] The computer program product may be written in any combination of one or more programming languages ​​to implement the program code for performing the operations of the embodiments of the present application, including object-oriented programming languages ​​such as Java, C++, and conventional procedural programming languages ​​such as "C" or similar programming languages. The program code may be executed entirely on the user's computing device, partially on the user's computing device, as a standalone software package, partially on the user's computing device and partially on a remote computing device, or entirely on a remote computing device or server.

[0089] 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 above.

[0090] The computer-readable storage medium may be any combination of one or more readable media. The readable medium may be a readable signal medium or a readable storage medium. The readable storage medium may include, for example, but is not limited to, an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, device, or device, or any combination thereof. More specific examples (a 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 thereof.

[0091] It should be noted that the terms used in this application are only for describing specific embodiments and are not intended to limit 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 and may also include the plural. The terms "comprise", "include" or any other variants 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 further restrictions, the elements defined by the sentence "comprise a..." do not exclude the presence of other identical elements in the process, method or device including the elements.

[0092] It should also be noted that the terms "center", "up", "down", "left", "right", "vertical", "horizontal", "inside", "outside", etc., indicating orientations or positional relationships, are based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing this 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 this 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 an indirect connection through an intermediate medium, or it can be a communication between the internal parts of 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.

[0093] 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 them. 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 m phase velocity directions of seismic waves and calculate the group velocity component and group velocity direction of the seismic wave corresponding to each phase velocity direction using the Christoffel equation; S3: Using the group velocity components and group velocity directions of the seismic waves obtained in S2, the group velocity components corresponding to different group velocity directions of the seismic waves are obtained by 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, p <m。 2. The method for calculating the seismic 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 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 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, 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 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 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-th direction and the strain in the f-th direction, and n1, n2 and n3 represent the three components of the phase velocity direction respectively.

7. The method for calculating the seismic group velocity distribution in a transversely isotropic medium according to claim 2, wherein: In S3, the group velocity components and group velocity directions of the seismic waves obtained in S2 are substituted into the polynomial fitting formula to obtain the group velocity components of the seismic waves 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 group velocity direction to be measured, and obtain the group velocity component corresponding to the group velocity direction to be measured according to the group velocity direction to be measured and the polynomial fitting formula of the determined coefficients.

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