CN102313902A - 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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CN102313902A
CN102313902A CN201110145900A CN201110145900A CN102313902A CN 102313902 A CN102313902 A CN 102313902A CN 201110145900 A CN201110145900 A CN 201110145900A CN 201110145900 A CN201110145900 A CN 201110145900A CN 102313902 A CN102313902 A CN 102313902A
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chebyshev
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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 seism processing field, more particularly to prestack depth migration method.
Background technology
The emphasis of current seismic exploration is construction complexity or the violent area of velocity variations.Accurate subsurface structure imaging can not be obtained using conventional migration processing means in these areas, pre-stack depth migration adapts to construct the accurately image in the complicated or violent area of velocity variations, is complex structural area accurately image most efficient method.
Depth migration technology starts from the seventies, has larger development in theoretical and method, is widely used the nineties eighties.It is most importantly the calculating of wave field extrapolation operator in pre-stack depth migration, the conventional method for calculating wave field extrapolation operator has at present：Kirchhoff integration methods, phase shift method, space-frequency domain finite difference calculus, split-step Forrier method, Fourier finite difference calculus and general screen method based on ray theory.
Since 1990s, the wave field propagation phase screen method for being widely used in the fields such as acoustics, electromagnetism is incorporated into reflection seismology, wave equation migration and wave-field simulation for seismic reflection wave field.Phase-screen method is more accurate to high frequency or the description of the wave field of narrow angular spread, it is adaptable to the weak cross directional variations of weak scattering and underground medium space, while also having high computational efficiency and the adaptivity to medium character spatial variations.But the dominant frequency of seismic wave is relatively low, when seismic wave field in the strong cross directional variations medium in space is propagated and is described, larger error occurs in the method.In order to accurately describe the propagation of seismic wave in strong cross directional variations medium, people have developed general screen method on the basis of phase screen is theoretical.General screen method not only maintains the separation split in step Fourier methods between Phase Coordinates variable and avoids coupling between the coordinate variable in this space of physics and its dual spaces coordinate variable, with efficiently with being easily achieved the characteristics of, and the seismic wave propagated for wide angle has at a relatively high description precision, also there is good adaptability to the strong cross directional variations medium of speed.
The approximate general screen operators approximate with Rytov of Born of scattered wave field in current wave field propagation is all first approximation propagation operator.Above-mentioned wave field propagation operator is relatively low to the description precision of strong cross directional variations medium medium wave propagation.In order to improve the deviation angle of wave field propagation, the present invention carries out polynomial expansion to single square of dispersion equation of one way wave equation by Chebyshev polynomials, derives the high-order expression formula of generalized screen propagator.
The content of the invention
More effectively to overcome drawbacks described above present in phase screen pre-stack depth migration, the present invention seeks to by carrying out higher-order expansion to dispersion equation, so as to improve the description precision to seismic wave.
It is an object of the invention to provide one kind on the basis of general screen wave field propagation operator order is improved, to improve the description precision to strong cross directional variations medium medium wave propagation.This method step is as follows：
First, medium velocity is
Figure 2011101459001100002DEST_PATH_IMAGE001
The dispersion relation formula of accurate upgoing wave be：
Figure 2011101459001100002DEST_PATH_IMAGE002
                      (1)
(1) in formula：
Figure 2011101459001100002DEST_PATH_IMAGE003
For angular frequency (Hz)；
Figure 2011101459001100002DEST_PATH_IMAGE004
For
Figure 2011101459001100002DEST_PATH_IMAGE005
The wave number (/m) in direction；
Figure 2011101459001100002DEST_PATH_IMAGE006
For
Figure 2011101459001100002DEST_PATH_IMAGE007
The wave number (/m) in direction；
Figure 622708DEST_PATH_IMAGE001
For medium velocity (m/s).In background velocity
Figure 2011101459001100002DEST_PATH_IMAGE008
In dispersion equation be：
                     (2)
(2) in formula：
Figure 978996DEST_PATH_IMAGE003
For angular frequency (Hz)；
Figure 2011101459001100002DEST_PATH_IMAGE010
For
Figure 539159DEST_PATH_IMAGE005
The background wave number (/m) in direction；
Figure 59002DEST_PATH_IMAGE006
For
Figure 532096DEST_PATH_IMAGE007
The wave number (/m) in direction；
Figure 10351DEST_PATH_IMAGE008
For background velocity (m/s).(1) formula substitution (2) formula is obtained：
Figure DEST_PATH_IMAGE011
                           (3)
(3) in formula：
Figure 60740DEST_PATH_IMAGE003
For angular frequency (Hz)；
Figure 649853DEST_PATH_IMAGE010
For
Figure 708945DEST_PATH_IMAGE005
The wave number (/m) in direction； 
Figure 358101DEST_PATH_IMAGE008
For medium velocity (m/s)；For medium velocity (m/s).Assuming that
Figure 2011101459001100002DEST_PATH_IMAGE012
, and, then radical sign can be turned in formula (3)：
 
Figure 2011101459001100002DEST_PATH_IMAGE014
                       (4)
Secondly, willIt is launched into Chebyshev polynomials：
                                 (5)
(5) in formula：N is the exponent number of expansion；
Figure DEST_PATH_IMAGE017
For Chebyshev coefficient；
Figure 2011101459001100002DEST_PATH_IMAGE018
For Chebyshev polynomials,,
Figure 665171DEST_PATH_IMAGE018
With following relational expression：
Figure DEST_PATH_IMAGE019
       
Figure 2011101459001100002DEST_PATH_IMAGE020
(6)
Figure DEST_PATH_IMAGE021
       
Figure DEST_PATH_IMAGE022
 (7)
(6) in formula, (7) formula：
Figure DEST_PATH_IMAGE023
For Chebyshev's node,
Figure DEST_PATH_IMAGE024
Figure DEST_PATH_IMAGE025
；
Finally, deploy (4) formula with Chebyshev polynomials, obtain the multinomial 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
；
Figure DEST_PATH_IMAGE027
For Chebyshev coefficient；
Figure DEST_PATH_IMAGE028
For Chebyshev polynomials；
Figure 305284DEST_PATH_IMAGE029
For the exponent number of expansion.
The beneficial effects of the present invention are deployed to approach the dispersion equation of upgoing wave by Chebyshev polynomials, obtain the form of the Chebyshev polynomials expansion of upgoing wave dispersion equation, make it have and accurately describe wave field communication process in strong cross directional variations 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.
Brief description of the drawings
Fig. 1 is the relative error of the multinomial of Chebyshev expansion, the multinomial of Taylor expansion and exact value
Figure DEST_PATH_IMAGE030
With angle
Figure DEST_PATH_IMAGE031
The curve map of change
The abscissa of the figure is angle（θ）, ordinate is relative error magnitudes Er (θ)；Chebyshev polynomials expansion and the relative error of exact value are smaller than what Taylor polynomial deployed as can be seen from Figure, therefore the high precision of the ratio of precision Taylor polynomial expansion of Chebyshev polynomials expansion；
Fig. 2 is two-dimentional SEG/EAGE rate patterns figure
The abscissa of the figure is the distance of section horizontal direction, and ordinate is the depth of section；The speed field parameters of the model are：Horizontal sampled point is 1290, and longitudinal sampled point is 300, and the horizontal sampling interval is 30m, and longitudinal sampling interval is 30m, as can be seen from FIG. in the middle of have the salt dome body of a high speed；
Fig. 3 is two-dimentional SEG/EAGE In A Salt-dome Models Born single orders operator pre-stack depth migration profile
Abscissa is the distance of section horizontal direction, and ordinate is the depth of section；As can be seen from the figure the wave field below the profile of the salt dome body of its high speed, its high speed salt dome body is more in disorder, unintelligible；
Fig. 4 is two-dimentional SEG/EAGE In A Salt-dome Models Chebyshev polynomials expansion high-order general screen operator pre-stack depth migration profile
The abscissa of the figure is the distance of section horizontal direction, and ordinate is the depth of section；High speed salt dome body is clear-cut as can be seen from Figure, and the wave field below high speed body is clear.
Embodiment
Main realization principle, embodiment of technical scheme etc. are described in detail below according to accompanying drawing：
In order to compare Chebyshev polynomials, the precision of Taylor polynomial approximation by polynomi-als, it is assumed that
Figure DEST_PATH_IMAGE032
, then formula (4) can turn to：
Figure DEST_PATH_IMAGE033
                    (9)
(9) in formula：
Figure DEST_PATH_IMAGE034
For the direction of propagation of ripple.ThenThe rank expansion of Chebyshev 6 and the relative error of exact value be：
Figure DEST_PATH_IMAGE035
     (10)
(10) in formula：For the direction of propagation of ripple.The rank expansion of Taylor 6 and the relative error of exact value be：
Figure DEST_PATH_IMAGE036
        (11)
(11) in formula：For the direction of propagation of ripple.Fig. 1 is the relative error of the multinomial, the multinomial of Taylor expansion and exact value of Chebyshev expansion
Figure 46407DEST_PATH_IMAGE030
With angle
Figure 203106DEST_PATH_IMAGE031
The curve map of change.(8) formula is substituted into wave field extrapolation equation：
Figure DEST_PATH_IMAGE037
                   (12)
(12) in formula：
Figure DEST_PATH_IMAGE038
For
Figure 879813DEST_PATH_IMAGE007
The wave number (/m) in direction；
Figure DEST_PATH_IMAGE039
For the sampling number of depth direction；For in the depth direction
Figure 269599DEST_PATH_IMAGE029
The depth (m) of individual point；
Figure 291387DEST_PATH_IMAGE003
For frequency (Hz)；
Figure DEST_PATH_IMAGE041
For
Figure 424297DEST_PATH_IMAGE005
The wave number in direction；
Figure DEST_PATH_IMAGE042
For the step size (m) of depth direction；
Figure DEST_PATH_IMAGE043
For in depthWave field；
Figure DEST_PATH_IMAGE045
For in depth
Figure 589349DEST_PATH_IMAGE040
Wave field.Then have：
Figure DEST_PATH_IMAGE046
  (13)
(13) in formula：
Figure DEST_PATH_IMAGE047
For imaginary number,；
Figure 122224DEST_PATH_IMAGE038
For
Figure 667475DEST_PATH_IMAGE007
The wave number (/m) in direction；For the sampling number of depth direction；
Figure 25131DEST_PATH_IMAGE040
For in the depth direction
Figure 606154DEST_PATH_IMAGE029
The depth (m) of individual point；
Figure 955095DEST_PATH_IMAGE003
For frequency (Hz)；
Figure 672384DEST_PATH_IMAGE010
For in background velocity
Figure 84299DEST_PATH_IMAGE008
'sThe background velocity of wave (/m) in direction；
Figure 836409DEST_PATH_IMAGE042
For the step size (m) of depth direction；
Figure 145555DEST_PATH_IMAGE008
For background velocity (m/s)；
Figure 459862DEST_PATH_IMAGE001
For medium velocity (m/s)；
Figure DEST_PATH_IMAGE049
For the exponent number of Chebyshev expansion；
Figure DEST_PATH_IMAGE050
For Chebyshev coefficient；
Figure DEST_PATH_IMAGE051
For Chebyshev polynomials；For in depth
Figure 145369DEST_PATH_IMAGE044
Wave field；
Figure 774934DEST_PATH_IMAGE045
For in depth
Figure 602686DEST_PATH_IMAGE040
Wave field.First approximation is done to second exponential term of (13) formula：
Figure DEST_PATH_IMAGE052
                    (14)
(14) in formula：
Figure 770231DEST_PATH_IMAGE047
For imaginary number,
Figure 470858DEST_PATH_IMAGE048
；
Figure 813984DEST_PATH_IMAGE010
For in background velocity's
Figure 203563DEST_PATH_IMAGE005
The background velocity of wave (/m) in direction；
Figure 832515DEST_PATH_IMAGE042
For the step size (m) of depth direction；
Figure 967830DEST_PATH_IMAGE049
For the exponent number of Chebyshev expansion；
Figure 591578DEST_PATH_IMAGE050
For Chebyshev coefficient；For Chebyshev polynomials.Formula (14) is substituted into formula (13) and inversefouriertransform is done to frequency-spatial domain：
 (15)
(15) in formula：
Figure DEST_PATH_IMAGE054
For horizontal direction；
Figure 282203DEST_PATH_IMAGE047
For imaginary number,
Figure 599921DEST_PATH_IMAGE048
；
Figure 394570DEST_PATH_IMAGE040
For in the depth direction
Figure 636720DEST_PATH_IMAGE029
The depth (m) of individual point；
Figure 11070DEST_PATH_IMAGE003
For frequency (Hz)；
Figure 652136DEST_PATH_IMAGE010
For in background velocity
Figure 89458DEST_PATH_IMAGE008
's
Figure 815974DEST_PATH_IMAGE005
The background velocity of wave (/m) in direction；
Figure 790752DEST_PATH_IMAGE042
For the step size (m) of depth direction；
Figure 20745DEST_PATH_IMAGE008
For background velocity (m/s)；
Figure 894548DEST_PATH_IMAGE001
For medium velocity (m/s)；For the exponent number of Chebyshev expansion；
Figure 824512DEST_PATH_IMAGE050
For Chebyshev coefficient；
Figure 189240DEST_PATH_IMAGE051
For Chebyshev polynomials；ForInversefouriertransform on direction；
Figure DEST_PATH_IMAGE056
For in depth
Figure 449899DEST_PATH_IMAGE044
Wave field；
Figure DEST_PATH_IMAGE057
For in depth
Figure 156693DEST_PATH_IMAGE040
Wave field.
In order to examine migration imaging effect, the high-order general screen operator deployed using Chebyshev polynomials carries out pre-stack depth migration imaging to two-dimentional SEG/EAGE models.The speed field parameters of the model are that horizontal sampled point is 1290, and longitudinal sampled point is 300, and the horizontal sampling interval is 30m, and longitudinal sampling interval is 30m.Fig. 2 is the rate pattern of Marmousi models.Comparison diagram 3, Fig. 4, carrying out pre-stack depth migration single order operator pre-stack depth migration more approximate than conventional Born using the high-order general screen operator of the Chebyshev polynomials expansion in the present invention has more preferable imaging effect, salt dome structure form is apparent, tectonic boundary becomes apparent from, wave field particularly below salt dome is apparent, and structural configuration becomes apparent from.

Claims (2)

1. a kind of general screen prestack depth migration method based on Chebyshev expansion, this method includes：
Step 1 carries out polynomial expansion to the Exact Equation of the upgoing wave of frequency-wave-number domain, ask for deploying polynomial coefficient using Chebyshev polynomials expansion, make the multinomial formula that it deploys and the approximation ratio highest of the dispersion equation of accurate upgoing wave, ask for the polynomial each term coefficient of expansion；
Step 2 obtains Chebyshev expansion multinomial coefficient, as shown in formula (1)：
Figure 2011101459001100001DEST_PATH_IMAGE002
    (1)
In formula (1)：
Figure 2011101459001100001DEST_PATH_IMAGE004
,For angular frequency (Hz),
Figure 2011101459001100001DEST_PATH_IMAGE008
For
Figure 2011101459001100001DEST_PATH_IMAGE010
The wave number (/m) in direction,
Figure DEST_PATH_IMAGE012
For medium velocity (m/s),
Figure DEST_PATH_IMAGE014
For medium velocity (m/s),
Figure DEST_PATH_IMAGE016
For the exponent number of expansion,
Figure DEST_PATH_IMAGE018
For Chebyshev coefficient,
Figure DEST_PATH_IMAGE020
For Chebyshev polynomials；
Step 3 substitutes into formula (1) first approximation of wave field extrapolation equation and utilization index, obtains the wave field extrapolation equation of Chebyshev polynomials expansion；
Step 4 reads in two-dimentional SEG/EGEA model datas, and Fourier analysis is carried out to data；
Step 5 is in frequency-wave-number domain, on each step size, the high-order general screen operator downward continuation wave field deployed using Chebyshev polynomials；
The wave field of next depth is the result after a upper depth wave field extrapolation during step 6 continuation, and the result progress inversefouriertransform after continuation is overlapped into imaging to frequency-spatial domain, imaging results are exported.
2. according to the method described in claim 1, it is characterised in that：
High-order general screen operator is asked for using Chebyshev polynomials expansion；During polynomial expansion, Chebyshev polynomials expansion is carried out to the dispersion equation of frequency-wave-number domain upgoing wave, so that the multinomial of expansion preferably approaches the dispersion equation of upgoing wave, the factor of medium velocity change is take into account during expansion simultaneously, improves and precision is described to the wave field of strong lateral speed change medium.
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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN110646839A (en) * 2018-06-27 2020-01-03 中国石油化工股份有限公司 Seismic wave simulation method and system
CN111812708A (en) * 2019-04-11 2020-10-23 中国石油天然气股份有限公司 Seismic wave imaging method and device

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

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

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN110646839A (en) * 2018-06-27 2020-01-03 中国石油化工股份有限公司 Seismic wave simulation method and system
CN111812708A (en) * 2019-04-11 2020-10-23 中国石油天然气股份有限公司 Seismic wave imaging method and device

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