CN108563802B - Method for improving numerical simulation precision of seismic converted waves - Google Patents
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Abstract
The invention discloses a method for improving the numerical simulation precision of seismic converted waves, which comprises the following steps: setting an original elastic wave fluctuation equation; introducing transverse wave components Ss and Sp to obtain a modified elastic wave fluctuation equation; and (3) solving the corrected elastic wave equation after equation equivalence is proved through mathematical formula derivation to obtain an Ss wave field and an Sp wave field, namely a converted wave field and a longitudinal wave field. The invention obtains P, S wave field completely separated, the wave field is clean, and there is no interference between longitudinal wave and transverse wave. The amplitude and the phase of the P, S wave after separation are kept unchanged, which is beneficial to further research on P, S propagation characteristics.
Description
Technical Field
The invention belongs to the technical field of seismic numerical simulation, and particularly relates to a method for improving the numerical simulation precision of seismic converted waves.
Background
The traditional elastic wave field obtained based on the numerical simulation of the elastic wave fluctuation equation contains longitudinal waves (P waves) and transverse waves (S waves), the two wave fields are separated before the P waves or the S waves are processed, and the process is difficult to ensure that the wave fields of the P waves and the S waves are not distorted and can also hardly ensure that the two wave fields can be completely separated. In recent years, a seabed node technology characterized by four-component reception is gradually developed in the field of earthquake, especially, the application of converted waves is more and more concerned by geophysicists, and the converted waves have very important significance for searching fractured hydrocarbon reservoirs, estimating lithology and the like.
At present, the elastic wave simulation can only obtain a mixed elastic wave field, including P waves and S waves, and the separation of longitudinal waves and transverse waves by a traditional gradient rotation operator is difficult to ensure that the wave field after separation has no amplitude and phase distortion.
Disclosure of Invention
In view of the above, the present invention provides a method for improving the accuracy of numerical simulation of seismic converted waves.
In order to solve the technical problem, the invention discloses a method for improving the numerical simulation precision of seismic converted waves, which comprises the following steps:
step 1: setting an original elastic wave fluctuation equation which is composed of a formula (1-1), a formula (1-2) and a formula (1-3),
wherein x, y and z represent three directions of space, u, v and w represent components of the elastic wave field in the three directions of x, y and z, and t represents propagation time; vp represents a longitudinal wave propagation velocity, and Vs represents a transverse wave propagation velocity;
step 2: introducing transverse wave components Ss and Sp into the original elastic wave fluctuation equation in the step 1 to obtain a modified elastic wave fluctuation equation;
introduction of variables Sp, Ss
Sp＝(UP、VP、WP)(2)
Ss＝(Us，Vs，Ws)(3)
Wherein, Up, Vp, Wp represent the components of P wave in X, Y, Z directions, Us, Vs, Ws represent the components of S wave in X, Y, Z directions;
substituting the formulas (2) and (3) into the original elastic wave fluctuation equation in the step one to obtain a corrected elastic wave fluctuation equation; the corrected elastic wave fluctuation equation consists of a formula (4-1) and a formula (4-9); the formula (4-1) -the formula (4-9) forms an equation set (4);
U＝UP+US(4-1)
V＝VP+VS(4-2)
W＝WP+WS(4-3)
and step 3: and (3) solving the corrected elastic wave equation in the step (2) after equation equivalence is proved through mathematical formula derivation to obtain expressions of the Ss wave field and the Sp wave field, and realizing the expressions of the Ss wave field and the Sp wave field, namely a converted wave field and a longitudinal wave field, through a program.
Further, the derivation of the mathematical formula in step 3 proves that the equation is equivalent to:
substituting the (4-4) th formula and the (4-7) th formula in the modified elastic wave fluctuation equation into the (4-1) th formula,
the above equation is formula (5), and formula (5) is fully equivalent to formula (1-1) in equation set (1);
similarly, an equation set obtained by substituting the (4-5) th equation and the (4-8) th equation in the modified elastic wave fluctuation equation into the (4-2) th equation is completely equivalent to the (1-2) th equation in the original elastic wave fluctuation equation: substituting the (4-6) th formula and the (4-9) th formula in the modified elastic wave fluctuation equation into the (4-2) th formula to obtain an equation set which is completely equivalent to the (1-1) th formula in the original elastic wave fluctuation equation, namely the modified elastic wave fluctuation equation is equivalent to the original elastic wave fluctuation equation;
then, it turns out that the introduced variables Sp and Ss can represent the compressional wave field and the shear wave field, respectively:
because the second derivative of the Sp rotation is 0, the first derivative of the Sp rotation is deduced to be a constant, the Sp rotation is a linear function or a constant, and the Sp rotation is obtained according to the condition of the initial zero solution of the wave equation, wherein the Sp rotation is 0, namely Sp is a non-rotation field, and Sp is a longitudinal wave; likewise, Ss is proven to be a shear wave.
Further, the step 3 of solving the corrected elastic wave equation to obtain the Ss wave field and the Sp wave field specifically includes: solving equations (4-1) - (4-9) by finite difference method to obtain the expressions of P wave and S wave, space 6 order:
wherein ui represents a P wave, uj represents an S wave, and DH represents a grid length; and realizing the expression of P wave and S wave by a program to obtain a converted wave shear wave field and a converted wave longitudinal wave field.
Compared with the prior art, the invention can obtain the following technical effects:
1) the invention obtains P, S wave field completely separated, the wave field is clean, and there is no interference between longitudinal wave and transverse wave.
2) The amplitude and the phase of the P, S wave after separation are kept unchanged, which is beneficial to further research on P, S propagation characteristics.
3) The invention accelerates based on the GPU and improves the simulation efficiency.
Of course, it is not necessary for any one product in which the invention is practiced to achieve all of the above-described technical effects simultaneously.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the invention and not to limit the invention. In the drawings:
FIG. 1 is a diagram of a longitudinal wave velocity model according to the present invention;
FIG. 2 is a diagram of a shear wave velocity model according to the present invention;
FIG. 3 is a snapshot of the longitudinal wave field at time 0.5s according to the present invention;
FIG. 4 is a fast graph of the converted wave field at 0.5s according to the present invention;
FIG. 5 is a P-wave single shot record diagram of the present invention;
fig. 6 is a diagram of an S-wave single shot record of the present invention.
Detailed Description
The following embodiments are described in detail with reference to the accompanying drawings, so that how to implement the technical features of the present invention to solve the technical problems and achieve the technical effects can be fully understood and implemented.
The idea of the invention is that: the invention provides an equivalent elastic wave equation, which takes P wave and S wave as variables, obtains a completely separated converted wave field by solving the wave field equation and simultaneously obtains a longitudinal wave field. The method can play an important role in the aspects of elastic wave theory research, converted wave field characteristic analysis, converted wave processing explanation and the like.
The theoretical basis of the converted wave numerical simulation is that through the replacement of variables, transverse wave components Ss and Sp are introduced into an original elastic wave fluctuation equation, and after the equation equivalence is proved through the derivation of a mathematical formula, a corrected elastic fluctuation equation is solved to obtain an Ss wave field and an Sp wave field, namely a converted wave field and a longitudinal wave field. The specific mathematical derivation process is as follows:
wherein x, y and z represent three directions of space, u, v and w represent components of the elastic wave field in the three directions of x, y and z, and t represents propagation time; vp represents a longitudinal wave propagation velocity, and Vs represents a transverse wave propagation velocity.
Formula (1) represents the original elastic wave fluctuation equation, including formula (1-1), formula (1-2), and formula (1-3);
setting a longitudinal wave velocity model, wherein the velocity in the velocity model represents Vp, and as shown in fig. 1, the velocity model is divided into 3 layers, from top to bottom, the first layer is seawater with a velocity of 1500m/s, the second layer is 2000m/s, and the third layer is 2500 m/s; a wave detection point is on the surface, a shot point is on the surface, and an actual seismic data acquisition mode is simulated;
setting a transverse wave velocity model, wherein the velocity in the velocity model represents Vs, and as shown in FIG. 2, the transverse wave velocity of each layer is obtained by converting longitudinal wave velocity through a geological empirical formula, the velocity model is divided into 3 layers, from top to bottom, the velocity of the first layer is 866m/s, the velocity of the second layer is 1155m/s, and the velocity of the third layer is 1443 m/s; the demodulator probe is on the surface, and the shot point is on the surface;
step 2, introducing transverse wave components Ss and Sp to obtain a modified elastic wave fluctuation equation which is an equation set (4):
SP＝(UP，VP，WP)(2)
Ss＝(Us，Vs，Ws)(3)
wherein, Up, Vp, Wp represent the components of P wave in x, y, z directions, respectively, and Us, Vs, Ws represent the components of S wave in x, y, z directions, respectively.
Bringing the formulas (2) and (3) into the formula (1) to obtain the following equation set (4), wherein the equation (4) includes the formula (4-1) to the formula (4-9):
U＝UP+US(4-1)
V＝VP+VS(4-2)
W＝WP+WS(4-3)
wherein, Up, Vp, Wp represent the components of P wave in x, y, z directions, Us, Vs, Ws represent the components of S wave in x, y, z directions; x, y and z represent three directions of space, u, v and w represent components of the elastic wave field in the three directions of x, y and z, and t represents propagation time; vp represents a longitudinal wave propagation velocity, and Vs represents a transverse wave propagation velocity.
Step 3, solving the corrected elastic wave equation after equation equivalence is proved through mathematical formula derivation to obtain an Ss wave field and an Sp wave field, namely a converted wave field and a longitudinal wave field:
first, equation (1) and equation (4) prove to be equivalent.
Substituting the (4-4) th and (4-7) th equations in equation (4) into the (4-1) th equation, we obtain:
wherein x, y and z represent three directions of space, u, v and w represent components of the elastic wave field in the three directions of x, y and z, and t represents propagation time. Vp represents a longitudinal wave propagation velocity, and Vs represents a transverse wave propagation velocity. Up and Us respectively represent the longitudinal wave field separation and the transverse wave field component on the x component;
it is easy to find that the formula (5) is completely equivalent to the (1-1) th formula in the equation set (1). The equation set (1) herein refers to equation (1), which represents the original elastic wave fluctuation equation.
Similarly, the equation set obtained by substituting the (4-5) th and (4-8) th equations in the equation set (4) into the (4-2) th equation is completely equivalent to the (1-2) th equation in the equation set (1): the equation set obtained by substituting the (4-6) th expression and the (4-9) th expression in the equation set (4) into the (4-2) th expression is completely equivalent to the (1-1) th expression in the equation set (1), that is, the equation set (4) is equivalent to the equation (1).
Then, it turns out that the introduced variables Sp and Ss can represent the compressional wave field and the shear wave field, respectively:
Since the second derivative of the Sp rotation is 0, it can be deduced that the first derivative of the Sp rotation is constant, and the Sp rotation is a linear function or constant, which can be obtained according to the condition of the initial zero solution of the wave equation, the Sp rotation is 0, i.e., Sp is a non-rotation field, and Sp is a longitudinal wave. Likewise, Ss may also be demonstrated as a shear wave.
Example 1 takes a two-dimensional case as an example, and solves the problem by a finite difference method.
A method for improving the numerical simulation precision of seismic converted waves is carried out according to the following steps:
U＝UP+US(4-1)
W＝WP+WS(4-3)
step 2: solving the equation by a finite difference method to obtain expressions (6-order space) of P wave and S wave:
wherein i, j represents coordinates in x, y directions of spatial grid points involved in calculation, corresponding to i, j in the context, and a (n) represents a difference coefficient; ui denotes a P wave, uj denotes an S wave, and DH denotes a distance of the adjacent mesh involved in the calculation.
Step 3, realizing the discrete expression in the step 2 through a program to obtain a converted wave transverse wave field and a converted wave longitudinal wave field, and specifically comprising the following steps:
step 3.1, for the elastic wave equation in two-dimensional isotropic medium
Where x, y, and z represent three directions in space, λ and μ represent lame constants, τ zz, τ xx, and τ xz represent components of stress, and vx and vz represent components of velocity components.
Second order approximation of time
vx, vz denote the components of the velocity component, Δ t denotes the sampling rate, and m denotes the order taken after taylor expansion.
Subtracting the two equations to obtain an approximate difference equation of 2M order
vx represents the component of the velocity component and Δ t represents the sampling rate.
In the above n-pair equation, the equal-sign sides of the nth pair equation are simultaneously multiplied by the coefficient cn (n), and then subtracted, and all the equations are summed to obtain:
Substituting the above formula into (16) to obtain
Substituted into (15) to obtain
The same can be said for all discrete formats of stress-displacement equations:
wherein i, j, k respectively represent coordinates of three-dimensional space grid points in x, y, z directions, x, y, z represent three directions of space, λ μ represents a Lamei constant, τ zz, τ xx, τ xz represent components of stress, vx, vz represent components of velocity components, Δ t represents a sampling rate, N represents a difference order, Δ x, Δ z represent grid steps in the horizontal and vertical directions, and cn (N) is a coefficient.
Thus, a simulation result of the elastic wave was obtained.
And 3.2, extracting longitudinal and transverse waves from vx and vz components obtained by elastic wave simulation through formulas (7) and (8), wherein the derivation processes of the formulas (7) and (8) are as above.
Where i, j represents the coordinates of a certain grid point, a (n) represents the differential system ui represents the P wave, uj represents the S wave, and DH represents the grid length, i.e., Δ x, Δ z as mentioned in the above formula.
The converted wave extraction method in the current market obtains a longitudinal and transverse wave field by separating an elastic wave field by using a divergence rotation operator, the method hardly causes the amplitude and phase information of the wave field to be distorted, and is unfavorable for later data research, so that the invention provides a completely separated longitudinal and transverse wave data simulation method. Compared with the traditional divergence curl operator for separating longitudinal waves and transverse waves, the longitudinal waves and the transverse waves obtained by the method are completely separated from each other and do not interfere with each other, and the obtained longitudinal wave and transverse wave fields retain original amplitude and phase information, so that the method has great benefits on converted wave characteristic research, converted wave data processing and inversion method research.
Fig. 3 is a wave field snapshot of longitudinal waves, fig. 4 is a wave field snapshot of converted transverse waves, which is obtained from step 2 and step 3 of the embodiment, and through comparison of the two graphs, the longitudinal waves and the transverse waves can be found to be completely separated from each other and not to interfere with each other, thereby verifying the effectiveness of the invention.
The longitudinal wave record of fig. 5, fig. 6 is a shear wave record, which is obtained from step 2 and step 3 of the specific embodiment, fig. 3 and fig. 4 are by wave field snapshot comparison, fig. 5 and fig. 6 are by record comparison finally obtained, the P wave and the S wave are completely separated from each other, no converted wave is generated at the collar offset, the polarity of the converted wave is opposite on both sides of the shot point, which are the same as the actual phenomenon, and the effectiveness of the invention is proved again.
As can be seen from fig. 3 to 6, the two wave fields are completely independent wave fields, and there are no shear waves on the compressional wave recordings and no compressional waves on the compressional wave recordings. The single shot record is generally the final result of forward modeling, and from the single shot, the longitudinal and transverse waves are completely independent wave fields, so that the phenomenon that the transverse waves exist on the longitudinal wave field or the longitudinal waves exist on the transverse wave record due to incomplete separation of the longitudinal waves and the transverse waves in the traditional method can be avoided.
While the foregoing description shows and describes several preferred embodiments of the invention, it is to be understood, as noted above, that the invention is not limited to the forms disclosed herein, but is not to be construed as excluding other embodiments and is capable of use in various other combinations, modifications, and environments and is capable of changes within the scope of the inventive concept as expressed herein, commensurate with 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.
Claims (3)
1. A method for improving numerical simulation precision of seismic converted waves is characterized by comprising the following steps:
step 1: setting an original elastic wave fluctuation equation which is composed of a formula (1-1), a formula (1-2) and a formula (1-3),
wherein x, y and z represent three directions of space, u, v and w represent components of the elastic wave field in the three directions of x, y and z, and t represents propagation time; vp represents a longitudinal wave propagation velocity, and Vs represents a transverse wave propagation velocity;
step 2: introducing transverse wave components Ss and Sp into the original elastic wave fluctuation equation in the step 1 to obtain a modified elastic wave fluctuation equation;
introduction of variables Sp, Ss
Sp＝(UP、VP、WP)(2)
Ss＝(Us，Vs，Ws)(3)
Wherein, Up, Vp, Wp represent the components of P wave in X, Y, Z directions, Us, Vs, Ws represent the components of S wave in X, Y, Z directions;
substituting the formulas (2) and (3) into the original elastic wave fluctuation equation in the step one to obtain a corrected elastic wave fluctuation equation; the corrected elastic wave fluctuation equation consists of a formula (4-1) and a formula (4-9); the formula (4-1) -the formula (4-9) forms an equation set (4);
U＝UP+US(4-1)
V＝VP+VS(4-2)
W＝WP+WS(4-3)
and step 3: and (3) solving the corrected elastic wave equation in the step (2) after equation equivalence is proved through mathematical formula derivation to obtain expressions of the Ss wave field and the Sp wave field, and realizing the expressions of the Ss wave field and the Sp wave field, namely a converted wave field and a longitudinal wave field, through a program.
2. The method for improving the numerical simulation accuracy of the seismic converted waves according to claim 1, wherein the equation equivalence proved by the derivation of the mathematical formula in the step 3 is as follows:
substituting the (4-4) th formula and the (4-7) th formula in the modified elastic wave fluctuation equation into the (4-1) th formula,
the above equation is formula (5), and formula (5) is fully equivalent to formula (1-1) in equation set (1);
similarly, an equation set obtained by substituting the (4-5) th equation and the (4-8) th equation in the modified elastic wave fluctuation equation into the (4-2) th equation is completely equivalent to the (1-2) th equation in the original elastic wave fluctuation equation: substituting the (4-6) th formula and the (4-9) th formula in the modified elastic wave fluctuation equation into the (4-2) th formula to obtain an equation set which is completely equivalent to the (1-1) th formula in the original elastic wave fluctuation equation, namely the modified elastic wave fluctuation equation is equivalent to the original elastic wave fluctuation equation;
then, it turns out that the introduced variables Sp and Ss can represent the compressional wave field and the shear wave field, respectively:
wherein: i, j and k respectively represent coordinates of three-dimensional space grid points in x, y and z directions and represent rotation operators; because the second derivative of the Sp rotation is 0, the first derivative of the Sp rotation is deduced to be a constant, the Sp rotation is a linear function or a constant, and the Sp rotation is obtained according to the condition of the initial zero solution of the wave equation, wherein the Sp rotation is 0, namely Sp is a non-rotation field, and Sp is a longitudinal wave; likewise, Ss is proven to be a shear wave.
3. The method for improving the accuracy of numerical simulation of seismic converted waves according to claim 1, wherein the step 3 of solving the modified elastic wave equation to obtain the Ss wave field and the Sp wave field specifically comprises: solving equations (4-1) - (4-9) by finite difference method to obtain the expressions of P wave and S wave, space 6 order:
wherein i, j represents coordinates in x, y directions of spatial grid points involved in calculation, corresponding to i, j in the context, and a (n) represents a difference coefficient; ui represents a P wave, uj represents an S wave, and DH represents a grid length; and realizing the expression of P wave and S wave by a program to obtain a converted wave shear wave field and a converted wave longitudinal wave field.
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