CN107589446B - The tomography velocity modeling method of wave path is calculated using Gaussian beam - Google Patents
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Abstract
A kind of tomography velocity modeling method calculating wave path using Gaussian beam, comprising the following steps: step 1: deviating to obtain angle domain common image gathers by Gaussian beam, and it is poor to pick up remaining time using trace gather；Step 2: the expression formula of wave path chromatography kernel function is established using the Born approximation of wave equation；Step 3: calculating background wave field Green's function using Gaussian beam propagation operator；Step 4: the expression formula calculating computed tomography kernel function based on Green's function and chromatography kernel function.Method of the invention is suitable for complicated structure, stability with higher and modeling accuracy.
Description
Technical Field
The invention relates to the field of seismic data velocity modeling in oil and gas exploration and development, in particular to a tomography velocity modeling method for calculating a wave path by utilizing a Gaussian beam, which can be used for processing seismic data in oil geophysical exploration.
Background
In high and steep construction areas in south and west of China, underground construction is complex, the velocity transverse change of seismic waves is large, and seismic data imaging and velocity modeling have great challenges. The Gaussian beam prestack depth migration technology is not only suitable for the structural imaging of a complex area, but also suitable for being matched with the tomography velocity inversion to carry out velocity updating iteration. Compared with the rapid development of the offset method, the development of the velocity modeling method is relatively delayed, and the requirement of the imaging method on the velocity model cannot be met. At present, the time-lapse tomography inversion based on rays is the most widely applied speed updating method, and can provide a relatively accurate background speed model in simple construction, but the spatial resolution is not enough and is not sensitive to a low-speed body. Under the condition of a complex model, the problem of caustic is easy to appear in conventional ray propagation, and because the propagation process only considers grid points on rays, a constructed chromatographic matrix is very sparse and ill-conditioned, so that the numerical solution is unstable. The other velocity modeling method, full waveform inversion method, theoretically provides the inversion result with the highest resolution, but is limited to the perfect assumption of data, so that various problems exist in practical application. Therefore, a velocity modeling method having high stability and modeling accuracy for a complex configuration is desired.
Disclosure of Invention
The invention aims to provide a tomography velocity modeling method utilizing Gaussian beam calculation wave paths, which has higher stability and modeling precision when being suitable for complex construction.
The invention adopts the following solution:
a tomographic velocity modeling method for computing a wave path using a gaussian beam, comprising the steps of:
step 1: obtaining an angle domain common imaging point gather through Gaussian beam migration, and picking up a residual time difference by using the gather;
step 2: establishing an expression of a wave path chromatography kernel function by using Born approximation of a wave equation;
and step 3: calculating a background wave field Green function by utilizing a Gaussian beam propagation operator;
and 4, step 4: calculating the chromatography kernel function based on the Green function and the expression of the chromatography kernel function;
and 5: a chromatographic equation as shown in the following formula (1) was constructed:
∫KΔsdr＝Δd (1)
in the formula (1), a right-end term Δ d represents a residual time difference, a left-end term matrix K represents a chromatography kernel function, and a left-end term Δ s is a reciprocal velocity update quantity to be solved.
Preferably, the expression of the tomographic kernel function is represented by the background wave field green's function.
Preferably, the expression of the chromatographic kernel function is expressed by the following formula (4):
wherein,the kernel function at the end of the shot point is represented,representing the kernel function at the end of the demodulator probe, ω representing the circular frequency, v0Representing background velocity, pSAnd pRSlowness vectors, G, representing the shot and demodulator point starts, respectivelySGreen's function, G, representing the departure of a shot point to an imaging pointRGreen's function, G, representing the departure of a demodulator point to an imaging point0Green's function, G, representing the departure of a shot point to a demodulator probe0Represents G0Conjugate function of (a), xsRepresenting the coordinates of the shot from which the background wavefield is generated, x representing the coordinates of the imaging point, and y representing x from the shotsThe spatial point coordinates, Im, through which the straight line to the imaging point x passes represent the imaginary part.
Preferably, the background wave field green's function is expressed by the following equation (9):
where x represents the imaging point coordinates and y represents the from shot point xsThe spatial point coordinates through which a straight line to the imaging point x passes, ω represents the circle frequency, p represents the slowness vector of the gaussian beam,representing the initial incidence angle of the Gaussian beam on the earth surface, u (y, x, p, omega) representing the Gaussian beam expressed by rectangular coordinate system parameters, phi representing the initial amplitude coefficient, P(s) and Q(s) representing the parameters of the dynamic ray tracing, v(s) representing the velocity of a vertical point on a central ray, tau(s) representing the travel time along the central ray, and n representing the distance from a point in space to the central ray.
Preferably, the initial amplitude coefficient is expressed by the following formula (10):
wherein, ω isrDenotes the frequency of the reference circle, w0Denotes the initial width of the gaussian beam and v (x) denotes the velocity of the perpendicular point on the central ray.
Preferably, the first and second electrodes are formed of a metal,
wherein,
where α denotes the angle of the central ray propagation direction with the z-axis, which points in a vertically downward direction.
Preferably, the formula (6) is numerically solved using the longgutta method, and the initial value of the formula (6) is calculated by the following formula (8):
wherein, ω isrDenotes the frequency of the reference circle, w0Representing the initial width, v, of the Gaussian beaminitialRepresenting the velocity of the surface exit point.
Preferably, the tomographic kernel function is calculated by the following formula (11):
wherein,the kernel function at the end of the shot point is represented,representing the kernel function at the end of the demodulator probe, ω representing the circular frequency, v0Representing background velocity, pSAnd pRRepresenting slowness vectors of a shot point departure and a demodulator probe departure respectively, u representing a Gaussian beam function, u x representing a conjugate function of u, xsRepresenting the coordinates of the shot from which the background wavefield is generated, x representing the coordinates of the imaging point, and y representing x from the shotsThe spatial point coordinates through which the straight line to the imaging point x passes,representing the initial angle of incidence, Φ, of the Gaussian beam at the surface0ΦRPhi s respectively represent initial amplitude coefficients corresponding to a background field from a shot point to a demodulator probe, a disturbance field from the demodulator probe and a disturbance field from the shot point, and Im represents an imaginary part.
Compared with the prior art, the method has the advantages that the Gaussian beam operator is used for calculating the chromatographic wave path to replace the conventional ray for back projection, the advantages of flexibility and stability of the conventional ray chromatography technology are kept, meanwhile, the accuracy of the inversion operator is improved, and the spatial resolution of the conventional ray chromatography inversion is improved. In addition, the calculation stability of the chromatographic inversion can be improved by constructing an inversion matrix with smaller morbidity by using a Gaussian beam operator. Finally, the accuracy of the complicated structure speed chromatography modeling is improved.
Drawings
The above and other objects, features and advantages of the present disclosure will become more apparent by describing in more detail exemplary embodiments thereof with reference to the attached drawings.
FIG. 1 shows a flowchart of a tomography velocity modeling method using Gaussian beam computation wave paths in accordance with an exemplary embodiment;
FIG. 2 shows a single frequency Gaussian beam described by equation (5) in an exemplary embodiment;
FIG. 3 illustrates a Green's function represented by a plurality of Gaussian beam integrals as described by equation (9) in an exemplary embodiment;
FIGS. 4a and 4b show the tomographic kernel function of equation (11) at 20Hz and 40Hz, respectively, in an exemplary embodiment;
FIG. 5 shows a theoretical model in an exemplary embodiment, where the ordinate represents depth and the abscissa represents distance;
FIG. 6 illustrates an arbitrary set of shot traces extracted by the theoretical model in an exemplary embodiment, wherein the ordinate represents time samples and the abscissa represents the number of traces;
FIG. 7 shows an initial model in an exemplary embodiment, where the ordinate represents depth and the abscissa represents distance;
FIG. 8 shows an initial model offset profile in an exemplary embodiment, where the ordinate represents depth and the abscissa represents distance;
FIG. 9 shows the result of a Gaussian beam tomosynthesis update of the initial model of FIG. 7 in an exemplary embodiment, where the ordinate represents depth and the abscissa represents distance;
FIG. 10 shows an offset profile using the Gaussian beam tomographic inversion model of FIG. 9, where the ordinate represents depth and the abscissa represents distance;
FIG. 11 shows the results of a conventional tomosynthesis update of the initial model of FIG. 7, wherein the ordinate represents depth and the abscissa represents distance;
FIG. 12 shows a Gaussian beam shift profile using the updated model of FIG. 11, where the ordinate represents depth and the abscissa represents distance;
13a, 13b, 13c show offset imaging gathers corresponding to the initial model of FIG. 7, the Gaussian beam tomosynthesis update model of FIG. 9, and the conventional tomosynthesis update model of FIG. 11, respectively, with the ordinate representing depth and the abscissa representing angle.
Detailed Description
Preferred embodiments of the present disclosure will be described in more detail below with reference to the accompanying drawings. While the preferred embodiments of the present disclosure are shown in the drawings, it should be understood that the present disclosure may be embodied in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the disclosure to those skilled in the art.
The objective of tomographic velocity modeling is to construct a tomographic equation as shown in equation (1) below:
∫KΔsdr＝Δd (1)
in the formula (1), a right-end term Δ d represents the residual time difference of the imaging gather, a left-end term matrix K represents a tomographic kernel function, which represents a back-projection path and weight, and a left-end term Δ s is the slowness (inverse velocity) update quantity to be solved. The innovation of the method is that a chromatography kernel function K is obtained, and a chromatography equation is constructed on the basis of the chromatography kernel function K, so that the slowness (inverse speed) updating amount can be solved.
To construct the tomographic equation shown in equation (1), an exemplary embodiment of the present invention is implemented according to the following process:
(1) solving the right-end term Δ d of the chromatographic equation: extracting a common imaging point gather of an angle domain by utilizing Gaussian beam migration, and picking up a residual time difference by utilizing the gather to serve as a right-end term delta d of a chromatographic equation, wherein the step is similar to the conventional tomography;
(2) the chromatographic kernel function K is obtained in three steps: firstly, establishing an expression of a wave path chromatography kernel function K by using Born approximation of a wave equation; then, calculating a background wave field Green function by using a Gaussian beam propagation operator; and finally, calculating the chromatography kernel function K by using the expression of the Green function and the chromatography kernel function K.
A tomographic velocity modeling method using a gaussian beam to calculate a wave path according to an exemplary embodiment is described in detail below with reference to fig. 1, including the steps of:
step 1: obtaining angle domain common imaging point gather through Gaussian beam deviation, and picking up residual time difference by using the gather
And extracting an angle domain common imaging point gather through Gaussian beam offset, and picking up the residual time difference by using the gather. This step is similar to conventional tomographic imaging methods and will not be described herein.
Step 2: establishing an expression of a wave path tomographic kernel function K using Born approximation of the wave equation
According to scattering theory, the Born scattering field can be expressed as the following equation (2):
wherein S is0Representing a background wavefield; v. of0Representing the background velocity; k is a radical of0Represents the background wavenumber; x is the number ofsSource point (i.e., shot) coordinates representing the generation of the background wavefield; x represents the imaging point coordinates, and Δ S represents the scatter field from the shot point to the imaging point; y denotes the scattered field from shot xsThe spatial point coordinates through which the straight line to the imaging point x passes; Δ v (y) represents a velocity perturbation of the spatial point y, the fringe field Δ S being caused by the velocity perturbation Δ v; gsRepresenting a Green function from a shot point to an imaging point; p is a radical ofSA slowness vector representing a shot point; ω represents the circle frequency.
Substituting equation (2) into the travel time disturbance can obtain the relationship between the imaging domain travel time disturbance and the velocity disturbance, as shown in equation (3) below:
wherein, Δ tGBMRepresenting the residual time difference picked up in a Gaussian beam offset angle domain common imaging point gather; the number of the theta's is,respectively representing the field angle and the azimuth angle in the track set; gSGreen's function, G, representing the departure of a shot point to an imaging pointRGreen's function, G, representing the departure of a demodulator point to an imaging point0Green's function, G, representing the departure of a shot point to a demodulator probeS、GR、G0Jointly forming a background wave field Green function G; g0Represents G0Im denotes the imaginary part.
Equation (3) represents the relationship between imaging field travel time disturbance and velocity disturbance, which is represented by the background wave field Green's function G (including G)S、GRAnd G0) And (4) showing.
Therefore, the expression of the frequency domain wave path tomographic kernel function K can be expressed as the following formula (4):
obviously, the formula (4) expresses that the tomographic kernel function K includes two components, i.e., the shot-end kernel functionAnd a demodulator probe end kernel functionThe expression of the tomographic kernel K is represented by the background wave field green function G.
And step 3: computing background wave field Green function by using Gaussian beam propagation operator
The single-frequency gaussian beam can be expressed as the following formula (5):
wherein u (s, n, ω) represents a monochromatic gaussian beam, s represents the central ray propagation arc length, n represents the distance of a point in space from the central ray, ω represents the circle frequency, τ represents the travel time along the central ray, v represents the velocity of a perpendicular point on the central ray, and P and Q represent parameters found by dynamic ray tracing, wherein:
wherein,
α is the angle between the propagation direction of the central ray and the z-axis, which points in the vertical downward direction.
Fig. 2 shows an example of a single-frequency gaussian beam represented by formula (5).
Equation (6) is a system of first order ordinary differential equations, and can be solved numerically by using the longge stoke method. The initial values of the system are shown in equation (8):
wherein, ω isrDenotes the frequency of the reference circle, w0Denotes the initial width of the Gaussian beam (the initial Gaussian beam is a plane wave), vinitialRepresenting the velocity of the surface exit point. The calculation of P and Q from equations (6) to (8) is entered into equation (5), and the integration is used to obtain the green function G of the background wave field, as shown in equation (9) below:
where p represents the slowness vector of the Gaussian beam,representing the initial incidence angle of the Gaussian beam on the earth surface, u (y, x, p, omega) representing the Gaussian beam represented by rectangular coordinate system parameters, phi representing the initial amplitude coefficient, P(s) and Q(s) representing the parameters of dynamic ray tracing, which can be obtained by calculation according to the formula (6), v(s) representing the velocity of a vertical point on a central ray, tau(s) representing the travel time of the central ray, and n representing a point in space to a middle pointThe distance of the cardiac ray.
Φ is the initial amplitude coefficient of the above-mentioned superposition integral equation (9), which can be obtained by comparison with an analytical green's function, as shown in the following equation (10):
fig. 3 shows an example of a green's function represented by a plurality of gaussian beam integrals as described in equation (9).
And 4, step 4: calculating a chromatographic kernel function K by using a green function and a chromatographic kernel function expression
By substituting the green function G calculated by equation (9) into equation (4), the gaussian beam-based tomographic kernel function K can be calculated, as shown by equation (11) below:
wherein u represents a gaussian beam function determined by equation (5), u represents a conjugate function of u, Φ0And phiRΦ s represents the initial amplitude coefficient corresponding to the background field from the shot to the demodulator probe, the perturbation field from the demodulator probe, and the perturbation field from the shot, respectively, and is determined by equation (10).
FIGS. 4a and 4b show the chromatographic kernel function described by equation (11) at 20Hz and 40Hz, respectively.
Finally, according to the residual time difference obtained in the step 1 and the chromatographic kernel function of the formula (11), the slowness (inverse speed) update quantity can be solved based on the formula (1).
Application example
To illustrate the effectiveness of the present invention, a theoretical model (as shown in fig. 5, wherein the ordinate represents depth and the abscissa represents distance) is designed in this example, and the model comprises 731 grids (CDP number) in the horizontal direction, the grid spacing is 10m, and 550 grids in the vertical direction, the grid spacing is 10 m. The model is used for subsequent forward modeling and verification of the migration effect.
Based on the theoretical model forward modeling of fig. 5, any 3 shot gathers (as shown in fig. 6) are extracted, in fig. 6, the ordinate represents time sample points, the abscissa represents the number of tracks, and the sampling interval is 1 ms.
An iso-gradient model (as shown in fig. 7) is designed as an initial model for gaussian beam migration and angle gathers are output. In fig. 7, the ordinate represents depth, and the abscissa represents distance.
FIG. 8 shows a Gaussian beam offset profile using the initial model of FIG. 7, where the ordinate represents depth and the abscissa represents distance. It can be seen that the diffraction of the initial model does not converge and the imaging depth does not match.
Fig. 9 shows the result of a gaussian beam tomosynthesis update of the initial model of fig. 7 by a method according to an exemplary embodiment of the present invention, wherein the ordinate represents depth and the abscissa represents distance.
FIG. 10 shows an offset profile using the Gaussian beam tomographic inversion model of FIG. 9, where the ordinate represents depth and the abscissa represents distance. It can be seen from fig. 10 that diffracted waves converge, a reflection interface can be returned to a correct depth position, and the in-phase axis focusing at a cutting position is better, which indicates that the gaussian beam tomography inversion is superior to the conventional ray tomography method in a region with severe change of transverse velocity.
FIG. 11 shows the results of a conventional tomosynthesis update of the initial model of FIG. 7, where the ordinate represents depth and the abscissa represents distance.
FIG. 12 shows a Gaussian beam shift profile using the updated model of FIG. 11, where the ordinate represents depth and the abscissa represents distance. Comparing the shift results of fig. 10 and 12, it can be seen that the shift results of the conventional ray tomographic update model are slightly inferior to those of the gaussian beam tomographic update model.
13a, 13b, 13c show offset imaging gathers corresponding to the initial model of FIG. 7, the Gaussian beam tomosynthesis update model of FIG. 9, and the conventional tomosynthesis update model of FIG. 11, respectively, with the ordinate representing depth and the abscissa representing angle. Through comparison, the upwarp phenomenon of the in-phase axis of the initial angle gather shows that the initial speed is small, and the velocity model obtained through chromatographic inversion is well leveled after the deviation of the angle gather. Compared with the conventional ray chromatography, the angle gather extracted by the Gaussian beam chromatography is closer to the angle gather extracted by the real velocity model.
The above-described embodiment is only one embodiment of the present invention, and various modifications or changes based on the principles disclosed in the present invention will be readily apparent to those skilled in the art, and the present invention is not limited to the above-described embodiment of the present invention, and therefore, the foregoing description is only preferred and not limiting.
Claims (4)
1. A tomographic velocity modeling method for computing a wave path using a gaussian beam, comprising the steps of:
step 1: obtaining an angle domain common imaging point gather through Gaussian beam migration, and picking up a residual time difference by using the gather;
step 2: establishing an expression of a wave path chromatography kernel function by using Born approximation of a wave equation;
and step 3: calculating a background wave field Green function by utilizing a Gaussian beam propagation operator;
and 4, step 4: calculating the chromatography kernel function based on the Green function and the expression of the chromatography kernel function;
and 5: a chromatographic equation as shown in the following formula (1) was constructed:
∫KΔsdr＝Δd (1)
in the formula (1), a right-end term delta d represents the residual time difference, a left-end term matrix K represents a chromatography kernel function, and a left-end term delta s is the inverse velocity update quantity to be solved;
wherein the expression of the tomographic kernel function is represented by the background wave field green's function;
wherein the expression of the chromatography kernel is expressed by the following formula (4):
wherein,the kernel function at the end of the shot point is represented,representing the kernel function at the end of the demodulator probe, ω representing the circular frequency, v0Representing background velocity, pSAnd pRSlowness vectors, G, representing the shot and demodulator point starts, respectivelySGreen's function, G, representing the departure of a shot point to an imaging pointRGreen's function, G, representing the departure of a demodulator point to an imaging point0Green's function, G, representing the departure of a shot point to a demodulator probe0Represents G0Conjugate function of (a), xsRepresenting the coordinates of the shot from which the background wavefield is generated, x representing the coordinates of the imaging point, and y representing x from the shotsThe coordinates of spatial points through which a straight line to an imaging point x passes, Im represents an imaginary part;
wherein the background wave field green's function is expressed by the following equation (9):
where x represents the imaging point coordinates and y represents the from shot point xsCoordinate of a spatial point where a straight line to an imaging point x passes, ω represents a circular frequency, p represents a slowness vector of a Gaussian beam, θ represents an initial incident angle of the Gaussian beam on the earth surface, u (y, x, p, ω) represents the Gaussian beam represented by a rectangular coordinate system parameter, Φ represents an initial amplitude coefficient, p(s) and q(s) represent parameters of dynamic ray tracing, v(s) represents a velocity of a vertical point on a central ray, τ(s) represents a travel time along the central ray, and n represents a distance from a point in space to the central ray;
wherein the tomographic kernel function is calculated by the following formula (11):
wherein,the kernel function at the end of the shot point is represented,representing the kernel function at the end of the demodulator probe, ω representing the circular frequency, v0Representing background velocity, pSAnd pRRepresenting slowness vectors of a shot point departure and a demodulator probe departure respectively, u representing a Gaussian beam function, u x representing a conjugate function of u, xsRepresenting the coordinates of the shot from which the background wavefield is generated, x representing the coordinates of the imaging point, and y representing x from the shotsThe coordinates of the spatial points through which the straight line to the imaging point x passes, theta represents the initial angle of incidence of the Gaussian beam on the surface, phi0、ΦR、ΦsRespectively representing the backs from shot to geophoneAnd Im represents an imaginary part.
2. The tomographic velocity modeling method using gaussian beam computation wave path according to claim 1, wherein said initial amplitude coefficient is expressed by the following formula (10):
wherein, ω isrDenotes the frequency of the reference circle, w0Denotes the initial width of the gaussian beam and v (x) denotes the velocity of the perpendicular point on the central ray.
3. The tomographic velocity modeling method using gaussian beam computation wave paths as in claim 1, wherein:
wherein,
where α denotes the angle of the central ray propagation direction with the z-axis, which points in a vertically downward direction.
4. The tomographic velocity modeling method using gaussian beam computation wave path according to claim 3, wherein said equation (6) is numerically solved using the longkutta method, and an initial value of said equation (6) is calculated by the following equation (8):
wherein, ω isrDenotes the frequency of the reference circle, w0Representing the initial width, v, of the Gaussian beaminitialRepresenting the velocity of the surface exit point.
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