阅读说明：本技术 一种基于时域高阶有限差分法的高精度波场数值模拟方法 (High-precision wave field numerical simulation method based on time domain high-order finite difference method ) 是由 梁晨曦 张固澜 段景 李勇 罗一梁 何承杰 罗帆 杜皓 曾梦 詹熠宗 王佳 于 2021-03-24 设计创作，主要内容包括：本发明公开了一种基于时域高阶有限差分法的高精度波场数值模拟方法,包括以下步骤：S1.在波场方程的基础上,对空间2M阶展开,发展得到的传统时间二阶空间高阶的有限差分方法；S2.基于均匀网格有限差分方法以及Taylor公式,对二阶时间导数进行N阶精度的展开,得到奇数时刻时间高阶差分求解公式,并将奇数时刻时间高阶有限差分表示；S3.基于均匀网格有限差分方法以及Taylor公式,对二阶时间导数进行N阶精度的展开,得到偶数时刻时间高阶差分求解公式,并将偶数时刻时间高阶有限差分表示；S4.对波场模拟奇数时刻和偶数时刻有限差分方法进行奇偶加权处理。本发明在保证高稳定性的同时有效的压制了频散,可提供精确稳定的波场模拟结果。(The invention discloses a high-precision wave field numerical simulation method based on a time domain high-order finite difference method, which comprises the following steps of: s1, on the basis of a wave field equation, a space 2M order is expanded, and an obtained traditional time second-order space high-order finite difference method is developed; s2, based on a uniform grid finite difference method and a Taylor formula, carrying out N-order precision expansion on a second-order time derivative to obtain an odd-number moment time high-order difference solving formula, and expressing the odd-number moment time high-order finite difference; s3, based on a uniform grid finite difference method and a Taylor formula, carrying out N-order precision expansion on the second-order time derivative to obtain an even-number moment time high-order difference solving formula, and expressing the even-number moment time high-order finite difference; and S4, carrying out odd-even weighting processing on the finite difference method for simulating odd-numbered time and even-numbered time of the wave field. The invention effectively suppresses frequency dispersion while ensuring high stability and can provide accurate and stable wave field simulation results.)

1. A high-precision wave field numerical simulation method based on a time domain high-order finite difference method is characterized by comprising the following steps: the method comprises the following steps:

s1, on the basis of a wave field equation, a space 2M order is expanded, and an obtained traditional time second-order space high-order finite difference method is developed;

s2, based on a uniform grid finite difference method and a Taylor formula, carrying out N-order precision expansion on a second-order time derivative to obtain an odd-number moment time high-order difference solving formula, and expressing the odd-number moment time high-order finite difference;

s3, based on a uniform grid finite difference method and a Taylor formula, carrying out N-order precision expansion on the second-order time derivative to obtain an even-number moment time high-order difference solving formula, and expressing the even-number moment time high-order finite difference;

and S4, carrying out odd-even weighting processing on the finite difference method for simulating odd-numbered time and even-numbered time of the wave field.

2. The time-domain higher-order finite difference method-based high-precision wave field numerical simulation method according to claim 1, wherein: the step S1 includes:

s101, giving a wave equation

Where v is the propagation velocity, p is the pressure wavefield, t is the propagation time, and x and z represent the spatial x-axis and z-axis, respectively;

s102, by expanding the spatial 2M order, the traditional finite difference method of the temporal second order and the spatial high order can be expressed as follows:

wherein gamma is_mIs the coefficient of the spatial difference that is,is the discretized wavefield at time t, spatial location (x, z), Δ x, Δ z are discrete spatial intervals, Δ t is a discrete temporal interval,is the velocity at the discrete point (x, z).

3. The time-domain higher-order finite difference method-based high-precision wave field numerical simulation method according to claim 1, wherein: the step S2 includes:

s201, expressing the odd time high-order finite difference as:

wherein the content of the first and second substances,

λ_nthe index N is 1,2,3, N, which indicates the high order difference coefficient in time at odd time instants.

S202, based on a uniform grid finite difference method and a Taylor formula, carrying out N-order precision expansion on a second-order time derivative to obtain an odd-moment time high-order difference solving formula:

then the wavefield at time t + Δ t is:

4. the time-domain higher-order finite difference method-based high-precision wave field numerical simulation method according to claim 1, wherein: the step S3 includes:

s301, representing the time high-order finite difference at even time as:

μ_na difference coefficient representing a temporal high order at an even time, with a subscript N being 1,2, 3.., N;

s302, based on a uniform grid finite difference method and a Taylor formula, carrying out N-order precision expansion on a second-order time derivative to obtain an even-number moment time high-order difference solving formula:

then the wavefield at time t + Δ t is obtained as:

5. the time-domain higher-order finite difference method-based high-precision wave field numerical simulation method according to claim 1, wherein: the step S4 includes:

carrying out odd-even weighting processing on the finite difference method for simulating odd time and even time of the wave field to obtain:

wherein the content of the first and second substances,

β_n＝ηλ_n+(1-η)μ_n,n＝2,3,...,N，β_nrepresenting the parity weight.

Technical Field

The invention relates to the technical field of seismic exploration, in particular to a high-precision wave field numerical simulation method based on a time domain high-order finite difference method.

Background

Wave equation numerical simulation has been widely applied to various aspects such as seismic forward modeling, reverse time migration, full waveform inversion and the like, and currently, methods for wave equation numerical simulation mainly include a pseudo-spectral method based on fourier transform, a finite element method based on irregular grid subdivision and a finite difference method based on difference approximation. The pseudo-spectrum method adopts Fourier transform to calculate a spatial partial derivative, so that numerical value dispersion caused by spatial dispersion can be suppressed; the finite element method can accurately simulate various irregular boundaries, so that step diffraction caused by rectangular mesh subdivision is avoided. Both methods require extensive calculations. In contrast, the finite difference method is widely applied to seismic forward research because of high computational efficiency, small required memory and simple implementation.

Because of the dispersion of the grid, the problem of dispersion is generated from two aspects of time and space, and for a long time, the method for solving the spatial dispersion caused by the grid dispersion is more, and at present, the space reaches the 2M-order precision, but in the problem of time precision, an improvement space exists at any time.

Disclosure of Invention

The invention aims to overcome the defects of the prior art and provides a high-precision wave field numerical simulation method based on a time domain high-order finite difference method, which provides a Taylor formula and a parity weighting algorithm based solution difference coefficient aiming at frequency dispersion caused by insufficient time precision, effectively suppresses the frequency dispersion while ensuring high stability and can provide a precise and stable wave field simulation result.

The purpose of the invention is realized by the following technical scheme: a high-precision wave field numerical simulation method based on a time domain high-order finite difference method comprises the following steps:

s1, on the basis of a wave field equation, a space 2M order is expanded, and an obtained traditional time second-order space high-order finite difference method is developed;

s2, based on a uniform grid finite difference method and a Taylor formula, carrying out N-order precision expansion on a second-order time derivative to obtain an odd-number moment time high-order difference solving formula, and expressing the odd-number moment time high-order finite difference;

s3, based on a uniform grid finite difference method and a Taylor formula, carrying out N-order precision expansion on the second-order time derivative to obtain an even-number moment time high-order difference solving formula, and expressing the even-number moment time high-order finite difference;

and S4, carrying out odd-even weighting processing on the finite difference method for simulating odd-numbered time and even-numbered time of the wave field.

The step S1 includes:

s101, giving a wave equation

Where v is the propagation velocity, p is the pressure wavefield, t is the propagation time, and x and z represent the spatial x-axis and z-axis, respectively;

s102, by expanding the spatial 2M order, the traditional finite difference method of the temporal second order and the spatial high order can be expressed as follows:

wherein gamma is_mIs the coefficient of the spatial difference that is,is the discretized wavefield at time t, spatial location (x, z), Δ x, Δ z are discrete spatial intervals, Δ t is a discrete temporal interval,is the velocity at the discrete point (x, z).

The step S2 includes:

s201, expressing the odd time high-order finite difference as:

wherein the content of the first and second substances,

λ_na difference coefficient representing the high order of time at an odd time instant, subscript N ═ 1,2, 3.., N;

s202, based on a uniform grid finite difference method and a Taylor formula, carrying out N-order precision expansion on a second-order time derivative to obtain an odd-moment time high-order difference solving formula:

then the wavefield at time t + Δ t is:

the step S3 includes:

s301, representing the time high-order finite difference at even time as:

μ_na difference coefficient representing a temporal high order at an even time, with a subscript N being 1,2, 3.., N;

s302, based on a uniform grid finite difference method and a Taylor formula, carrying out N-order precision expansion on a second-order time derivative to obtain an even-number moment time high-order difference solving formula:

then the wavefield at time t + Δ t is obtained as:

the step S4 includes:

carrying out odd-even weighting processing on the finite difference method for simulating odd time and even time of the wave field to obtain:

wherein the content of the first and second substances,

β_n＝ηλ_n+(1-η)μ_n,n＝1,2,...,N，β_nrepresenting the parity weight.

The invention has the beneficial effects that: the wave field simulation uniform grid finite difference method is suitable for wave field simulation of wave equations of any medium, and provides a better method for suppressing frequency dispersion aiming at the frequency dispersion phenomenon caused by grid dispersion.

Drawings

FIG. 1 is a flow chart of a method of the present invention;

FIG. 2 is a schematic diagram of a comparison of wavefield snapshots of the present invention with other methods in an embodiment;

FIG. 3 is a schematic illustration of a seismic recording according to the invention in an embodiment.

Detailed Description

The technical solutions of the present invention are further described in detail below with reference to the accompanying drawings, but the scope of the present invention is not limited to the following.

As shown in fig. 1, a high-precision wave field numerical simulation method based on a time domain high-order finite difference method includes the following steps:

s1, on the basis of a wave field equation, a space 2M order is expanded, and an obtained traditional time second-order space high-order finite difference method is developed;

s2, based on a uniform grid finite difference method and a Taylor formula, carrying out N-order precision expansion on a second-order time derivative to obtain an odd-number moment time high-order difference solving formula, and expressing the odd-number moment time high-order finite difference;

s3, based on a uniform grid finite difference method and a Taylor formula, carrying out N-order precision expansion on the second-order time derivative to obtain an even-number moment time high-order difference solving formula, and expressing the even-number moment time high-order finite difference;

and S4, carrying out odd-even weighting processing on the finite difference method for simulating odd-numbered time and even-numbered time of the wave field.

Wherein the step S1 includes:

s101, giving a wave equation

Where v is the propagation velocity, p is the pressure wavefield, t is the propagation time, and x and z represent the spatial x-axis and z-axis, respectively;

s102, developing a traditional time second-order space high-order finite difference method by expanding a space 2M order:

wherein gamma is_mIs the coefficient of the spatial difference that is,is the discretized wavefield at time t, spatial location (x, z), Δ x, Δ z are the discrete spatial separation, Δ t is the discrete temporal separation;is the velocity at the discrete point (x, z);

for even time, we are timeThe expression is given:

carrying out Taylor expansion on each item of the differential coefficient, and solving the difference coefficient:

the wavefield at time t + Δ t may be obtained:

similarly, for odd moments:

the wavefield at time t + Δ t may be obtained

The parity weighting thereof can be given by:

as shown in fig. 2, in the embodiment of the present application, the size of the model grid used is 8M, 501 sampling points are used in total, the time sampling interval is 1ms, the method of the present invention uses time 4-order precision, the conventional method uses time 2-order precision, the main frequency is 25Hz, the speed is 2000M/s, and meanwhile, the spatial order is 2M ═ 16 uniform medium model, in fig. 2, (a) is a uniform medium conventional method 2ms wave field snapshot, (b) is a uniform medium conventional method 2ms wave field snapshot part, (c) is a uniform medium present invention method 2ms wave field snapshot, (d) is a uniform medium present invention method 2ms wave field snapshot part, which can be obtained by comparing the conventional time second-order method in fig. 2 with the time high-order method provided by the present invention, and under the same model parameters, the wave field obtained by the method of the present invention has higher precision

As shown in fig. 3, in the embodiment of the present application, a Marmousi model with nx × nz of 1001 × 701, a grid size of 10M, a time sampling interval of 1ms, a time 4-order precision, a main frequency of 16Hz, and a spatial order of 2M of 16 is used, where (a) is a medium parameter diagram of the Marmousi model, (b) is a Marmousi model ground seismic record, and (c) is a Marmousi model vsp record in fig. 3. As can be seen from FIG. 3, the method of the invention obtains a more ideal application effect for a complex marmousi model.

While the foregoing description shows and describes a preferred embodiment of the invention, it is to be understood, as noted above, that the invention is not limited to the form disclosed herein, but is not intended to be exhaustive or to exclude other embodiments and may be used in various other combinations, modifications, and environments and may be modified within the scope of the inventive concept described herein by the above teachings or the skill or knowledge of the relevant art. And that modifications and variations may be effected by those skilled in the art without departing from the spirit and scope of the invention as defined by the appended claims.