CN102313902B - Depth displacement method before generalized screen overlapping based on Chebyshev expansion - Google Patents
Depth displacement method before generalized screen overlapping based on Chebyshev expansion Download PDFInfo
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Abstract
The invention provides a method for solving a high-order generalized screen operator by utilizing Chebyshev polynomial in order to improve the description accuracy of wave propagation in a strong transverse velocity change medium, thereby improving a depth displacement method before generalized screen overlapping and realizing the imaging of a steep inclined structure and regions with violent transverse velocity changes.
Description
Technical field
The present invention relates to the seismic data process field, particularly relate to prestack depth migration method.
Background technology
The emphasis of current seismic prospecting is complex structure or velocity variations violent area.Adopt conventional migration processing means can not obtain accurate underground structure imaging in these areas, pre-stack depth migration can adapt to the accurately image in complex structure or velocity variations violent district, is the effective method of complex structural area accurately image.
The depth shift technology starts from the seventies, and had larger development the eighties on theoretical and method, and be widely used the nineties.Most importantly the calculating of wave field extrapolation operator in pre-stack depth migration, the method for calculating the wave field extrapolation operator commonly used has at present: based on kirchhoff integral method, phase shift method, space-frequency field method of finite difference, split-step Forrier method, Fourier method of finite difference and the general screen method of ray theory.
Since the nineties in 20th century, the wave field propagation phase screen method that is widely used in the fields such as acoustics, electromagnetics is incorporated in reflection seismology, is used for wave equation migration and the wave-field simulation of seismic reflection wave field.It is more accurate that phase-screen method is described the wave field of high frequency or narrow angular spread, is applicable to the weak horizontal change of weak scattering and underground medium space, also has simultaneously high counting yield and to the adaptivity of medium character spatial variations.But the dominant frequency of seismic event is lower, to the space tyrannical in change medium seismic wave field propagate when being described, larger error can appear in this method.Tyrannical to changing earthquake wave propagation in medium in order to describe accurately, people have been developed the general screen method on the basis of phase place screen theory.The general screen method has not only kept splitting in step Fourier method the separation between the Phase Coordinates variable and has avoided the coordinate variable in this space of physics and the coupling between its dual space coordinate variable, make it to have efficient characteristics with being easy to realize, and have quite high description precision for the seismic event of wide angle propagation, also have good adaptability to speed is tyrannical to changing medium.
The Born of the scattered wave field during current wave field is propagated general screen operator approximate and that Rytov is approximate is all the first approximation propagation operator.Above-mentioned wave field propagation operator is lower to tyrannical description precision to changing medium medium wave propagation.The deviation angle of propagating in order to improve wave field, the present invention carries out polynomial expansion by Chebyshev polynomials to single square of dispersion equation of one way wave equation, derives the high-order expression formula of general screen propagation operator.
Summary of the invention
For more effectively overcoming the defects that exists in phase place screen pre-stack depth migration, the present invention seeks to by dispersion equation is carried out higher-order expansion, thereby improve the description precision to seismic event.
The purpose of this invention is to provide on a kind of basis improving general screen wave field propagation operator order, improve tyrannical description precision to changing medium medium wave propagation.The method step is as follows:
(1)
(1) in formula:
Be angular frequency (Hz);
For
The wave number of direction (/m);
For
The wave number of direction (/m);
Be medium velocity (m/s).In the background velocity field
In dispersion equation be:
(2) in formula:
Be angular frequency (Hz);
For
The background wave number of direction (/m);
For
The wave number of direction (/m);
Be background velocity (m/s).(1) formula substitution (2) formula is got:
(3) in formula:
Be angular frequency (Hz);
For
The wave number of direction (/m);
Be medium velocity (m/s);
Be medium velocity (m/s).Suppose
, and
, in formula (3), radical sign can turn to:
Secondly, will
Be launched into Chebyshev polynomials:
(5) in formula: N is the exponent number of expansion;
Be Chebyshev coefficient;
Be Chebyshev polynomials,
,
Have following relational expression:
At last, launch (4) formula with Chebyshev polynomials, obtain the polynomial expression of the Chebyshev polynomials expansion of (4) formula.(4) the Chebyshev polynomials expansion of formula is as shown in (8) formula:
(8) in formula:
Be Chebyshev coefficient;
Be Chebyshev polynomials;
Be the exponent number that launches.
Beneficial effect of the present invention is, launches to approach the dispersion equation of upward traveling wave by Chebyshev polynomials, obtains the form that the Chebyshev polynomials of upward traveling wave dispersion equation are launched, and makes it have an accurate description tyrannical to changing wave field communication process in medium.Therefore the general screen pre-stack depth migration of Chebyshev polynomials expansion is better than the general screen prestack depth migration method of conventional Taylor expansion.
Description of drawings
Fig. 1 is the polynomial expression of Chebyshev expansion, the polynomial expression of Taylor expansion and the relative error of exact value
With angle
The curve map that changes
The horizontal ordinate of this figure is angle (θ), and ordinate is relative error value Er (θ); As can be seen from Figure Chebyshev polynomials launch with the relative error of exact value than Taylor polynomial launch little, therefore the precision that the ratio of precision Taylor polynomial that Chebyshev polynomials are launched launches is high;
Fig. 2 is two-dimentional SEG/EAGE rate pattern figure
The horizontal ordinate of this figure is the distance of section horizontal direction, and ordinate is the degree of depth of section; The velocity field parameter of this model is: horizontal sampled point is 1290, and vertically sampled point is 300, and laterally sampling interval is 30m, and vertically sampling interval is 30m, as can be seen from FIG. the middle salt dome body that a high speed is arranged;
Fig. 3 is two-dimentional SEG/EAGE In A Salt-dome Model Born single order operator pre-stack depth migration sectional view
Horizontal ordinate is the distance of section horizontal direction, and ordinate is the degree of depth of section; The as can be seen from the figure profile of its salt dome body at a high speed, the wave field of its high speed salt dome body below is more in disorder, unintelligible;
Fig. 4 is that two-dimentional SEG/EAGE In A Salt-dome Model Chebyshev polynomials are launched high-order general screen operator pre-stack depth migration sectional view
The horizontal ordinate of this figure is the distance of section horizontal direction, and ordinate is the degree of depth of section; The wave field of the below of high speed salt dome body clear-cut, and high speed body as can be seen from Figure is clear.
Embodiment
The below realizes that to the main of technical scheme of the present invention principle, embodiment etc. are described in detail with reference to the accompanying drawings:
For the precision that comparison Chebyshev polynomials, Taylor polynomial polynomial expression approach, suppose
, formula (4) can turn to:
(9) in formula:
Be direction of wave travel.
Chebyshev's 6 rank expansions and the relative error of exact value be:
(10) in formula:
Be direction of wave travel.
Taylor's 6 rank expansions and the relative error of exact value be:
(11)
(11) in formula:
Be direction of wave travel.Fig. 1 is the polynomial expression of Chebyshev expansion, the polynomial expression of Taylor expansion and the relative error of exact value
With angle
The curve map that changes.With (8) formula substitution wave field extrapolation equation:
(12) in formula:
For
The wave number of direction (/m);
Sampling number for depth direction;
For on depth direction
The degree of depth of individual point (m);
Be frequency (Hz);
For
The wave number of direction;
Step size (m) for depth direction;
For in the degree of depth
Wave field;
For in the degree of depth
Wave field.Have:
(13)
(13) in formula:
Be imaginary number,
For
The wave number of direction (/m);
Sampling number for depth direction;
For on depth direction
The degree of depth of individual point (m);
Be frequency (Hz);
For at background velocity
The background velocity of wave of direction (/m);
Step size (m) for depth direction;
Be background velocity (m/s);
Be medium velocity (m/s);
Exponent number for Chebyshev expansion;
Be Chebyshev coefficient;
Be Chebyshev polynomials;
For in the degree of depth
Wave field;
For in the degree of depth
Wave field.Second exponential term to (13) formula done first approximation:
(14) in formula:
Be imaginary number,
For at background velocity
The background velocity of wave of direction (/m);
Step size (m) for depth direction;
Exponent number for Chebyshev expansion;
Be Chebyshev coefficient;
Be Chebyshev polynomials.With formula (14) substitution formula (13) and do inversefouriertransform to frequency-spatial domain:
(15) in formula:
Be horizontal direction;
Be imaginary number,
For on depth direction
The degree of depth of individual point (m);
Be frequency (Hz);
For at background velocity
The background velocity of wave of direction (/m);
Step size (m) for depth direction;
Be background velocity (m/s);
Be medium velocity (m/s);
Exponent number for Chebyshev expansion;
Be Chebyshev coefficient;
Be Chebyshev polynomials;
For
Inversefouriertransform on direction;
For in the degree of depth
Wave field;
For in the degree of depth
Wave field.
In order to check the migration imaging effect, the high-order general screen operator that adopts Chebyshev polynomials to launch carries out pre-stack depth migration imaging to two-dimentional SEG/EAGE model.The velocity field parameter of this model is that horizontal sampled point is 1290, and vertically sampled point is 300, and laterally sampling interval is 30m, and vertically sampling interval is 30m.Fig. 2 is the rate pattern of Marmousi model.Comparison diagram 3, Fig. 4, the high-order general screen operator that adopts the Chebyshev polynomials in the present invention to launch carries out pre-stack depth migration and has better imaging effect than the approximate single order operator pre-stack depth migration of conventional Born, the salt dome structure form is more clear, tectonic boundary is more obvious, particularly the wave field of salt dome below is more clear, and structural feature is more obvious.
Claims (2)
1. general screen prestack depth migration method based on Chebyshev expansion, the method comprises:
The dispersion equation of the upward traveling wave of step 1 pair frequency-wavenumber domain carries out polynomial expansion, utilize Chebyshev polynomials to launch to ask for and launch polynomial coefficient, make the polynomial expression of its expansion the highest with the approximation ratio of the dispersion equation of accurate upward traveling wave, ask for and launch polynomial every coefficient;
Step 2 is obtained the Chebyshev expansion multinomial coefficient, shown in (1):
In formula (1):
ω is angular frequency (Hz), k
z0For the wave number of z direction (/m), c is the background velocity (m/s) of medium, v is medium velocity (m/s), the exponent number of i for launching, c
iBe Chebyshev coefficient, T
iBe Chebyshev polynomials;
Step 3 gets with the first approximation of formula (1) substitution wave field extrapolation equation and utilization index the wave field extrapolation equation that Chebyshev polynomials are launched;
Step 4 is read in two-dimentional SEG/EGEA model data, and data are carried out Fourier analysis;
Step 5 on each step size, adopts the high-order general screen operator downward continuation wave field of Chebyshev polynomials expansion in frequency-wavenumber domain;
In step 6 continuation process, the wave field of the next degree of depth is the result after a upper degree of depth wave field extrapolation, the result after continuation is carried out inversefouriertransform carry out stacking image to frequency-spatial domain, the output imaging results.
2. method according to claim 1 is characterized in that:
Utilize Chebyshev polynomials to launch to ask for high-order general screen operator; In the process of polynomial expansion, dispersion equation to frequency-wavenumber domain upward traveling wave carries out the Chebyshev polynomials expansion, make the polynomial expression of expansion better approach the dispersion equation of upward traveling wave, considered simultaneously the factor that medium velocity changes in the process of launching, improved tyrannical wave field to the velocity variations medium is described precision.
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CN101021568A (en) * | 2007-02-07 | 2007-08-22 | 匡斌 | Three-dimensional integral prestack depth migration method |
CN101839998A (en) * | 2009-03-18 | 2010-09-22 | 中国石油天然气集团公司 | High precision prestack depth migration method |
US20100256916A1 (en) * | 2009-04-03 | 2010-10-07 | Chevron U.S.A. Inc. | Method for target-oriented reverse time migration for prestack depth imaging |
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CN101021568A (en) * | 2007-02-07 | 2007-08-22 | 匡斌 | Three-dimensional integral prestack depth migration method |
CN101839998A (en) * | 2009-03-18 | 2010-09-22 | 中国石油天然气集团公司 | High precision prestack depth migration method |
US20100256916A1 (en) * | 2009-04-03 | 2010-10-07 | Chevron U.S.A. Inc. | Method for target-oriented reverse time migration for prestack depth imaging |
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