CN114357831A - Non-grid generalized finite difference forward modeling method, device, storage medium and equipment - Google Patents
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Abstract
The invention relates to a mesh-free generalized finite difference forward modeling method, a device, a storage medium and equipment, wherein the method comprises the following steps: deriving a non-grid generalized finite difference format of the frequency domain scalar wave from a two-dimensional frequency domain scalar wave equation, and performing forward modeling by using the non-grid generalized finite difference format of the frequency domain scalar wave; adding boundary conditions to a two-dimensional frequency domain scalar wave equation to obtain a matrix equation of a grid-free generalized finite difference format of the frequency domain scalar wave; obtaining single-frequency wave field values of different frequencies by solving a matrix equation of a frequency domain scalar wave meshless generalized finite difference format, and obtaining a wave field value of a time domain through Fourier inverse transformation, wherein the wave field value of the time domain is a forward simulation result. The invention can improve the precision and efficiency of forward simulation.
Description
Technical Field
The invention relates to the technical field of geophysical exploration, in particular to a grid-free generalized finite difference forward modeling method, a device, a storage medium and equipment.
Background
With the development of the times, oil and gas resources are continuously consumed, the amount of the oil and gas resources is continuously reduced, the demand of the countries in the world for energy is increased day by day, and non-renewable traditional energy sources such as petroleum and coal are still the main part of energy structures of the countries and are irreplaceable in the aspects of political and economic development of the countries, strategic safety of the countries and the like. With the efficient development and utilization of domestic oil and gas exploration, in the past decades, oil and gas reservoirs with low exploration difficulty have been basically explored and exploited, and the current oil and gas development is directed to deep water, deep layers, complex areas and unconventional oil and gas. The oil and gas exploration in the areas is difficult, the inversion work is difficult, and the important factor is the accuracy and the efficiency of forward simulation, so that the improvement of the accuracy and the efficiency of the forward simulation is necessary.
Disclosure of Invention
In view of the above problems, an object of the present invention is to provide a method, an apparatus, a storage medium, and a device for forward modeling of frequency domain scalar wave equation without grid generalized finite difference, which can improve the accuracy and efficiency of forward modeling.
In order to achieve the purpose, the invention adopts the following technical scheme:
the invention relates to a grid-free generalized finite difference forward modeling method, which comprises the following steps:
deriving a non-grid generalized finite difference format of the frequency domain scalar wave from a two-dimensional frequency domain scalar wave equation, and performing forward modeling by using the non-grid generalized finite difference format of the frequency domain scalar wave;
adding boundary conditions to a two-dimensional frequency domain scalar wave equation to obtain a matrix equation of a grid-free generalized finite difference format of the frequency domain scalar wave;
obtaining single-frequency wave field values of different frequencies by solving a matrix equation of a frequency domain scalar wave meshless generalized finite difference format, and obtaining a wave field value of a time domain through Fourier inverse transformation, wherein the wave field value of the time domain is a forward simulation result.
Preferably, the two-dimensional frequency domain scalar wave equation is as follows:
in the formula, u is a frequency domain wave field value; x is the lateral spatial position; z is the longitudinal spatial position; omega is angular frequency; v is the medium velocity; s (omega) is a seismic source item; delta (x-x)s)δ(z-zs) Is the seismic source location;
defining a residual function B (u):
in the formula u0The wave field value is the central node; u. ofjIs the value of the wave field of the surrounding nodes, hjIs the horizontal distance from the jth node around to the central node, and hj＝xj-x0；kjIs the vertical distance from the center node, and kj＝zj-z0；ωjThe weight function represents the influence degree of the surrounding nodes on the central node; x is the lateral spatial position; z is the longitudinal spatial position; n is the finite difference node number; j is the node serial number;
taking the second order partial differential equation, let D be the partial derivative of all terms that may occur at the center node:
solving the partial derivative of the residual function to the partial derivative vector, and making the derivative value zero to obtain a matrix equation, solving the equation to obtain the value of each order of partial derivative, and finally solving the spatial partial derivative term in the equation (1) to obtain a matrix equation with the following form:
in the formula (I), the compound is shown in the specification,
ξj,ηjthe coefficient value of each wave field value of the surrounding nodes is obtained.
Preferably, the grid-free generalized finite difference forward modeling method for the frequency domain scalar wave is as follows:
in the grid-free generalized finite difference forward modeling method, preferably, the boundary condition added to the two-dimensional frequency domain scalar wave equation is a PML boundary condition; the two-dimensional frequency domain scalar wave equation containing the PML boundary condition is as follows:
in formula (6):
in the formula (f)0The dominant frequency of the source wavelet; PML is the thickness of the absorbing layer; lxAnd lzRespectively from the node in the absorption layer to the edge of the truncated modelDistance of the boundary in horizontal and vertical directions; a is an empirical constant, typically taken to be 1.79; i is an imaginary unit.
The invention also provides a grid-free generalized finite difference forward modeling device, which comprises:
the first processing unit is used for deducing a grid-free generalized finite difference format of the frequency domain scalar wave from a two-dimensional frequency domain scalar wave equation so as to realize forward modeling;
the second processing unit is used for adding boundary conditions to the two-dimensional frequency domain scalar wave equation to obtain a matrix equation of the frequency domain scalar wave grid-free generalized finite difference format;
and the third processing unit is used for solving a matrix equation of a frequency domain scalar wave meshless generalized finite difference format to obtain single-frequency wave field values of different frequencies, and obtaining a wave field value of a time domain through Fourier inverse transformation, wherein the wave field value of the time domain is a forward simulation result.
The present invention also provides a computer storage medium having a computer program stored thereon, which, when being executed by a processor, implements the above-mentioned gridless generalized finite difference forward modeling method steps.
The invention also provides computer equipment which comprises a memory, a processor and a computer program stored on the memory and capable of running on the processor, wherein the processor realizes the steps of the gridless generalized finite difference forward modeling method when executing the computer program.
Due to the adoption of the technical scheme, the invention has the following advantages:
the method is a non-grid method, the partial derivative of an unknown parameter is expressed as the linear combination of the surrounding node values, the nodes in the model are distributed discretely and are not mutually connected, and the limit of the grid is broken through;
the invention can flexibly select the calculation frequency band; the frequency components are independent of each other; the method is suitable for multi-source simulation and seismic wave simulation of a medium related to frequency.
Drawings
FIG. 1 is a 30Hz single-frequency slice obtained by forward modeling of a frequency domain generalized finite difference method;
FIG. 2 is a 150ms wavefield snapshot from a frequency domain forward evolution;
FIG. 3 is a 30Hz single-frequency slice obtained by forward modeling of non-uniform nodes by a frequency domain generalized finite difference method;
FIG. 4 is a diagram of a Marmousi model;
FIG. 5 is a 20Hz single-frequency slice obtained by forward modeling of a Marmousi generalized finite difference method;
FIG. 6 is a 780ms wave field snapshot of a Marmousi model frequency domain forward simulation;
FIG. 7 shows that all reference signs in a forward modeling seismic record chart of a Marmousi model frequency domain generalized finite difference method are marked.
Detailed Description
The preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings so that the objects, features and advantages of the invention can be more clearly understood. It should be understood that the embodiments shown in the drawings are not intended to limit the scope of the present invention, but are merely intended to illustrate the spirit of the technical solution of the present invention.
The invention provides a grid-free generalized finite difference forward modeling method which can establish a suitable field node distribution form according to different geologic body models and has stronger adaptability to complex models. The forward modeling of the frequency domain generalized finite difference method not only has the advantages of the forward modeling of the frequency domain, but also is not limited by a regular grid, has certain flexibility, and has high precision and efficiency.
The invention provides a grid-free generalized finite difference forward modeling method, which comprises the following steps:
1) deriving a non-grid generalized finite difference format of the frequency domain scalar wave from a two-dimensional frequency domain scalar wave equation, and performing forward modeling by using the non-grid generalized finite difference format of the frequency domain scalar wave;
the two-dimensional frequency domain scalar wave equation is as follows:
in the formula, u isA frequency domain wave field value; x is the lateral spatial position; z is the longitudinal spatial position; omega is angular frequency; v is the medium velocity; s (omega) is a seismic source item; delta (x-x)s)δ(z-zs) Is the source location.
The generalized finite difference method is based on multivariate function Taylor expansion and weighted least square method, each order partial derivative of a central node is expressed as linear combination of surrounding points, the function value is subjected to binary Taylor expansion at the central node, each item above the second order is saved, and a residual function B (u) is defined:
in the formula u0The wave field value is the central node; u. ofjWave field values of surrounding nodes are obtained; h isjIs the horizontal distance from the jth node around to the central node, and hj＝xj-x0；kjIs the vertical distance from the center node, and kj＝zj-z0；ωjThe weight function represents the influence degree of the surrounding nodes on the central node; x is the lateral spatial position; z is the longitudinal spatial position; n is the finite difference node number; j is the node sequence number.
Taking the second order partial differential equation, let D be the partial derivative of all terms that may occur at the center node:
solving the partial derivative of the residual function to the partial derivative vector, and making the derivative value zero to obtain a matrix equation, solving the equation to obtain the value of each order of partial derivative, and finally solving the spatial partial derivative term in the equation (1) to obtain a matrix equation with the following form:
in the formula (I), the compound is shown in the specification,
ξj,ηjthe coefficient value of each wave field value of the surrounding nodes is obtained.
The frequency domain scalar wave meshless generalized finite difference format is as follows:
2) adding PML boundary conditions to a two-dimensional frequency domain scalar wave equation to obtain a matrix equation of a grid-free generalized finite difference format of the frequency domain scalar wave;
the two-dimensional frequency domain scalar wave equation (i.e., matrix equation) including the PML boundary condition is as follows:
in formula (6):
in the formula (f)0The dominant frequency of the source wavelet; PML is the thickness of the absorbing layer; lxAnd lzRespectively the distances from the node in the absorption layer to the boundary of the truncation model in the horizontal direction and the vertical direction; a is an empirical constant, typically taken to be 1.79; i is an imaginary unit.
It should be noted that, the forward modeling uses a finite model space, which inevitably generates boundary reflection, so that a boundary condition needs to be added to absorb the boundary reflection, and the PML boundary condition is one of the most commonly used boundary conditions.
3) Obtaining single-frequency wave field values of different frequencies by solving a matrix equation of a frequency domain scalar wave meshless generalized finite difference format, and obtaining a wave field value of a time domain through Fourier inverse transformation, wherein the wave field value of the time domain is a forward simulation result.
Equation (6) can be summarized as a large matrix solving problem, the left term of the equation is a coefficient term and a wave field value, the right term of the equation is a seismic source, and when the frequency is given, the wave field value of the corresponding frequency slice can be solved.
It should be noted that, when solving the matrix equation of the frequency domain scalar wave meshless generalized finite difference format in the meshless generalized finite difference forward modeling method, node dispersion is performed in a solving area, discrete nodes can be uniform or non-uniform, and each partial derivative value of each discrete point is represented as a linear combination of surrounding points in a corresponding node cloud and is not limited by a regular mesh. The grid-free generalized finite difference forward modeling method provided by the invention is matched with non-uniform node distribution, can perform relatively accurate forward modeling on a complex medium model, reduces the calculation time, and is beneficial to the implementation of the follow-up work of seismic exploration.
The invention also provides a grid-free generalized finite difference forward modeling device, which comprises:
the first processing unit is used for deducing a grid-free generalized finite difference format of the frequency domain scalar wave from a two-dimensional frequency domain scalar wave equation so as to realize forward modeling;
the second processing unit is used for adding boundary conditions to the two-dimensional frequency domain scalar wave equation to obtain a matrix equation of the frequency domain scalar wave grid-free generalized finite difference format;
and the third processing unit is used for solving a matrix equation of the frequency domain scalar wave meshless generalized finite difference format to obtain single-frequency wave field values of different frequencies, and obtaining a wave field value of a time domain through Fourier inverse transformation, wherein the wave field value of the time domain is a forward simulation result.
The present invention also provides a computer storage medium having a computer program stored thereon, which, when being executed by a processor, implements the above-mentioned gridless generalized finite difference forward modeling method steps.
The invention also provides computer equipment which comprises a memory, a processor and a computer program stored on the memory and capable of running on the processor, wherein the processor realizes the steps of the gridless generalized finite difference forward modeling method when executing the computer program.
Example 1:
the specific technical scheme is described by a uniform medium model and a two-dimensional Marmousi model:
the first step is as follows: and (3) deducing a frequency domain scalar wave equation generalized finite difference format:
firstly, a two-dimensional frequency domain scalar wave equation is given:
the spatial partial derivative term in equation (1) has the following form:
in the formula
The generalized finite difference format of the frequency domain scalar wave equation is as follows:
the second step is that: adding boundary conditions:
forward modeling uses truncated models and therefore requires the addition of boundary conditions to absorb boundary reflections, PML boundary conditions being one of the most commonly used. The frequency domain scalar wave equation containing the PML absorption boundary condition is as follows:
the third step: solving of matrix equations
A large-scale matrix equation can be obtained by a frequency domain scalar wave equation containing PML absorption boundary conditions, the left end of the matrix equation comprises coefficient terms and wave field values, the right end of the matrix equation is a seismic source term, and the solution of the matrix equation is the wave field value of a certain frequency slice. After each single-frequency wave field value is obtained through calculation, Fourier inversion is carried out, and then the wave field value of the time domain can be obtained. The size of the uniform medium model is 2000m multiplied by 2000m, the dominant frequency of the wavelet is 20Hz, and the model speed is 3000 m/s. Fig. 1 and 2 show 300ms wavefield snapshots of a computed frequency domain 30Hz single frequency slice and inverse transform to time domain, respectively. Fig. 3 adopts non-uniform node distribution, and random disturbance is performed on uniform nodes on the basis of the uniform nodes, and it can be seen from the figure that under the condition of non-uniform node distribution, a generalized finite difference method can still obtain a better numerical simulation result, and the feasibility and the practicability of the non-grid generalized finite difference method are proved. Fig. 5, 6 and 7 are results obtained by forward modeling of the Marmousi model in the frequency domain, and it can be seen from the figures that the forward modeling results are better and can adapt to complex models.
Finally, it should be noted that: the above examples are only intended to illustrate the technical solution of the present invention, but not to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.
Claims (7)
1. A mesh-free generalized finite difference forward modeling method is characterized by comprising the following steps:
deriving a non-grid generalized finite difference format of the frequency domain scalar wave from a two-dimensional frequency domain scalar wave equation, and performing forward modeling by using the non-grid generalized finite difference format of the frequency domain scalar wave;
adding boundary conditions to a two-dimensional frequency domain scalar wave equation to obtain a matrix equation of a grid-free generalized finite difference format of the frequency domain scalar wave;
obtaining single-frequency wave field values of different frequencies by solving a matrix equation of a frequency domain scalar wave meshless generalized finite difference format, and obtaining a wave field value of a time domain through Fourier inverse transformation, wherein the wave field value of the time domain is a forward simulation result.
2. The meshless generalized finite difference forward modeling method of claim 1, wherein the two-dimensional frequency domain scalar wave equation is:
in the formula, u is a frequency domain wave field value; x is the lateral spatial position; z is the longitudinal spatial position; omega is angular frequency; v is the medium velocity; s (omega) is a seismic source item; delta (x-x)s)δ(z-zs) Is the seismic source location;
defining a residual function B (u):
in the formula u0The wave field value is the central node; u. ofjIs the value of the wave field of the surrounding nodes, hjIs the horizontal distance from the jth node around to the central node, and hj＝xj-x0；kjIs the vertical distance from the center node, and kj＝zj-z0；ωjThe weight function represents the influence degree of the surrounding nodes on the central node; x is the lateral spatial position; z is the longitudinal spatial position; n is the finite difference node number; j is the node serial number;
taking the second order partial differential equation, let D be the partial derivative of all terms that may occur at the center node:
solving the partial derivative of the residual function to the partial derivative vector, and making the derivative value zero to obtain a matrix equation, solving the equation to obtain the value of each order of partial derivative, and finally solving the spatial partial derivative term in the equation (1) to obtain a matrix equation with the following form:
in the formula (I), the compound is shown in the specification,
ξj,ηjthe coefficient value of each wave field value of the surrounding nodes is obtained.
3. The meshless generalized finite difference forward modeling method according to claim 2, wherein the meshless generalized finite difference format of the frequency domain scalar wave is:
4. the meshless generalized finite difference forward modeling method of claim 1, wherein the boundary condition added to the two-dimensional frequency domain scalar wave equation is a PML boundary condition; the two-dimensional frequency domain scalar wave equation containing the PML boundary condition is as follows:
in formula (6):
in the formula (f)0Is the principal of the seismic source waveletFrequency; PML is the thickness of the absorbing layer; lxAnd lzRespectively the distances from the node in the absorption layer to the boundary of the truncation model in the horizontal direction and the vertical direction; a is an empirical constant, typically taken to be 1.79; i is an imaginary unit.
5. A mesh-free generalized finite difference forward modeling apparatus, comprising:
the first processing unit is used for deducing a grid-free generalized finite difference format of the frequency domain scalar wave from a two-dimensional frequency domain scalar wave equation so as to realize forward modeling;
the second processing unit is used for adding boundary conditions to the two-dimensional frequency domain scalar wave equation to obtain a matrix equation of the frequency domain scalar wave grid-free generalized finite difference format;
and the third processing unit is used for solving a matrix equation of a frequency domain scalar wave meshless generalized finite difference format to obtain single-frequency wave field values of different frequencies, and obtaining a wave field value of a time domain through Fourier inverse transformation, wherein the wave field value of the time domain is a forward simulation result.
6. A computer storage medium having a computer program stored thereon, wherein the computer program, when being executed by a processor, is adapted to carry out the gridless generalized finite difference forward method steps of any of claims 1-4.
7. A computer device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, wherein the processor implements the gridless generalized finite difference forward modeling method steps of any of claims 1-4 when executing the computer program.
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