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 PDF

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CN102313902B
CN102313902B CN 201110145900 CN201110145900A CN102313902B CN 102313902 B CN102313902 B CN 102313902B CN 201110145900 CN201110145900 CN 201110145900 CN 201110145900 A CN201110145900 A CN 201110145900A CN 102313902 B CN102313902 B CN 102313902B
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expansion
chebyshev
wave
wave field
depth
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罗仁泽
黄元溢
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Southwest Petroleum University
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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

A kind of general screen prestack depth migration method based on Chebyshev expansion
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:
At first, medium velocity is
Figure 2011101459001100002DEST_PATH_IMAGE001
The dispersion relation formula of accurate upward traveling wave be:
(1)
(1) in formula:
Figure 2011101459001100002DEST_PATH_IMAGE003
Be angular frequency (Hz);
Figure 2011101459001100002DEST_PATH_IMAGE004
For
Figure 2011101459001100002DEST_PATH_IMAGE005
The wave number of direction (/m);
Figure 2011101459001100002DEST_PATH_IMAGE006
For The wave number of direction (/m);
Figure 622708DEST_PATH_IMAGE001
Be medium velocity (m/s).In the background velocity field
Figure 2011101459001100002DEST_PATH_IMAGE008
In dispersion equation be:
Figure 2011101459001100002DEST_PATH_IMAGE009
(2)
(2) in formula:
Figure 978996DEST_PATH_IMAGE003
Be angular frequency (Hz);
Figure 2011101459001100002DEST_PATH_IMAGE010
For
Figure 539159DEST_PATH_IMAGE005
The background wave number of direction (/m);
Figure 59002DEST_PATH_IMAGE006
For
Figure 532096DEST_PATH_IMAGE007
The wave number of direction (/m);
Figure 10351DEST_PATH_IMAGE008
Be background velocity (m/s).(1) formula substitution (2) formula is got:
Figure 2011101459001100002DEST_PATH_IMAGE011
(3)
(3) in formula:
Figure 60740DEST_PATH_IMAGE003
Be angular frequency (Hz);
Figure 649853DEST_PATH_IMAGE010
For
Figure 708945DEST_PATH_IMAGE005
The wave number of direction (/m); Be medium velocity (m/s); Be medium velocity (m/s).Suppose , and
Figure DEST_PATH_IMAGE013
, in formula (3), radical sign can turn to:
Figure 2011101459001100002DEST_PATH_IMAGE014
(4)
Secondly, will Be launched into Chebyshev polynomials:
Figure 2011101459001100002DEST_PATH_IMAGE016
(5)
(5) in formula: N is the exponent number of expansion;
Figure DEST_PATH_IMAGE017
Be Chebyshev coefficient;
Figure DEST_PATH_IMAGE018
Be Chebyshev polynomials,
Figure 748643DEST_PATH_IMAGE017
,
Figure 665171DEST_PATH_IMAGE018
Have following relational expression:
Figure DEST_PATH_IMAGE019
Figure DEST_PATH_IMAGE020
(6)
Figure DEST_PATH_IMAGE021
Figure DEST_PATH_IMAGE022
(7)
(6) in formula, (7) formula:
Figure DEST_PATH_IMAGE023
Be Chebyshev's node,
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:
Figure DEST_PATH_IMAGE026
(8)
(8) in formula:
Figure 624510DEST_PATH_IMAGE012
Be Chebyshev coefficient;
Figure DEST_PATH_IMAGE028
Be Chebyshev polynomials;
Figure 305284DEST_PATH_IMAGE029
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
Figure DEST_PATH_IMAGE030
With angle
Figure DEST_PATH_IMAGE031
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
Figure DEST_PATH_IMAGE032
, formula (4) can turn to:
Figure DEST_PATH_IMAGE033
(9)
(9) in formula: Be direction of wave travel.
Figure 915826DEST_PATH_IMAGE015
Chebyshev's 6 rank expansions and the relative error of exact value be:
Figure DEST_PATH_IMAGE035
(10)
(10) in formula:
Figure 339723DEST_PATH_IMAGE034
Be direction of wave travel.
Figure 799523DEST_PATH_IMAGE015
Taylor's 6 rank expansions and the relative error of exact value be:
(11)
(11) in formula:
Figure 311800DEST_PATH_IMAGE034
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
Figure 203106DEST_PATH_IMAGE031
The curve map that changes.With (8) formula substitution wave field extrapolation equation:
Figure DEST_PATH_IMAGE037
(12)
(12) in formula:
Figure DEST_PATH_IMAGE038
For
Figure 879813DEST_PATH_IMAGE007
The wave number of direction (/m);
Figure DEST_PATH_IMAGE039
Sampling number for depth direction;
Figure DEST_PATH_IMAGE040
For on depth direction
Figure 269599DEST_PATH_IMAGE029
The degree of depth of individual point (m);
Figure 291387DEST_PATH_IMAGE003
Be frequency (Hz);
Figure DEST_PATH_IMAGE041
For
Figure 424297DEST_PATH_IMAGE005
The wave number of direction;
Figure DEST_PATH_IMAGE042
Step size (m) for depth direction;
Figure DEST_PATH_IMAGE043
For in the degree of depth
Figure DEST_PATH_IMAGE044
Wave field;
Figure DEST_PATH_IMAGE045
For in the degree of depth Wave field.Have:
(13)
(13) in formula:
Figure DEST_PATH_IMAGE047
Be imaginary number,
Figure DEST_PATH_IMAGE048
Figure 122224DEST_PATH_IMAGE038
For
Figure 667475DEST_PATH_IMAGE007
The wave number of direction (/m); Sampling number for depth direction;
Figure 25131DEST_PATH_IMAGE040
For on depth direction The degree of depth of individual point (m);
Figure 955095DEST_PATH_IMAGE003
Be frequency (Hz);
Figure 672384DEST_PATH_IMAGE010
For at background velocity
Figure 152618DEST_PATH_IMAGE005
The background velocity of wave of direction (/m);
Figure 836409DEST_PATH_IMAGE042
Step size (m) for depth direction;
Figure 145555DEST_PATH_IMAGE008
Be background velocity (m/s);
Figure 459862DEST_PATH_IMAGE001
Be medium velocity (m/s);
Figure DEST_PATH_IMAGE049
Exponent number for Chebyshev expansion;
Figure DEST_PATH_IMAGE050
Be Chebyshev coefficient;
Figure DEST_PATH_IMAGE051
Be Chebyshev polynomials;
Figure 923466DEST_PATH_IMAGE043
For in the degree of depth
Figure 145369DEST_PATH_IMAGE044
Wave field;
Figure 774934DEST_PATH_IMAGE045
For in the degree of depth
Figure 602686DEST_PATH_IMAGE040
Wave field.Second exponential term to (13) formula done first approximation:
Figure DEST_PATH_IMAGE052
(14)
(14) in formula:
Figure 770231DEST_PATH_IMAGE047
Be imaginary number,
Figure 470858DEST_PATH_IMAGE048
Figure 813984DEST_PATH_IMAGE010
For at background velocity
Figure 938935DEST_PATH_IMAGE008
Figure 203563DEST_PATH_IMAGE005
The background velocity of wave of direction (/m);
Figure 832515DEST_PATH_IMAGE042
Step size (m) for depth direction;
Figure 967830DEST_PATH_IMAGE049
Exponent number for Chebyshev expansion;
Figure 591578DEST_PATH_IMAGE050
Be Chebyshev coefficient;
Figure 143170DEST_PATH_IMAGE051
Be Chebyshev polynomials.With formula (14) substitution formula (13) and do inversefouriertransform to frequency-spatial domain:
Figure DEST_PATH_IMAGE053
(15)
(15) in formula:
Figure DEST_PATH_IMAGE054
Be horizontal direction;
Figure 282203DEST_PATH_IMAGE047
Be imaginary number,
Figure 599921DEST_PATH_IMAGE048
For on depth direction
Figure 636720DEST_PATH_IMAGE029
The degree of depth of individual point (m);
Figure 11070DEST_PATH_IMAGE003
Be frequency (Hz);
Figure 652136DEST_PATH_IMAGE010
For at background velocity
Figure 89458DEST_PATH_IMAGE008
Figure 815974DEST_PATH_IMAGE005
The background velocity of wave of direction (/m);
Figure 790752DEST_PATH_IMAGE042
Step size (m) for depth direction;
Figure 20745DEST_PATH_IMAGE008
Be background velocity (m/s);
Figure 894548DEST_PATH_IMAGE001
Be medium velocity (m/s);
Figure 842781DEST_PATH_IMAGE049
Exponent number for Chebyshev expansion;
Figure 824512DEST_PATH_IMAGE050
Be Chebyshev coefficient;
Figure 189240DEST_PATH_IMAGE051
Be Chebyshev polynomials;
Figure DEST_PATH_IMAGE055
For Inversefouriertransform on direction; For in the degree of depth
Figure 449899DEST_PATH_IMAGE044
Wave field;
Figure DEST_PATH_IMAGE057
For in the degree of depth
Figure 156693DEST_PATH_IMAGE040
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):
f ( x ) = 1.00000154 - 0.5167624 x - 0.14011172 x 2 + 0.0575672 x 3 + 0.06914592 x 4
- 0.1964848 x 5 - 0.17172928 x 6 + Σ i = 7 ∞ c i T i - - - ( 1 )
In formula (1):
Figure FDA00002969639200013
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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Citations (3)

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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

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
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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