beamlet migration using local cosine basis with shifting...
TRANSCRIPT
Beamlet migration using local cosine basis with shifting windows Mingqiu Luo*, Ru-Shan Wu, Xiao-Bi Xie Institute of Geophysics and Planetary Physics, University of California, Santa Cruz, CA, 95064
Summary For beamlet migration methods, the wave fields normally are statically and regularly windowed, and the wave field in each window is propagated with beamlet propagators based on local cosine basis (LCB) or Gabor-Daubechies frame (GDF), followed by local perturbations. These methods can provide pretty good imaging results when compared to traditional methods. However, with the LCB beamlet method, some artifacts are present at steeply inclined interface of large velocity contracts. The artifacts can be greatly reduced by dynamically shifting windows in the LCB beamlet method. The numerical results on 2D and 3D models will show the improvement in the imaging quality. Introduction Wu et al. (2000) proposed beamlet migration methods based on local reference velocity and local perturbation theory. The migration methods with Gabor-Daubechies frame (GDF) and local cosine basis (LCB) have been presented by Wu and Chen (2001), Wang and Wu (2002), and Luo and Wu (2003). In these methods, the wave fields are statically and regularly windowed, and the wave field in each window is propagated with beamlet propagators, followed by local perturbation. These methods can provide good imaging results when compared to traditional methods. However, with LCB beamlet method, some artifacts are present in the area with steeply inclined interface of large velocity contract. In this paper, we first give a brief description for the LCB beamlet method and then discuss the concepts and method of dynamically shifting windows. The improvement in image quality will be illustrated through numerical examples. beamlet migration Generally, in frequency-space (f-x) domain, the scalar equation can be written as,
0),,,()],,(/[ 22222 =ωω+∂+∂+∂ zyxuzyxvzyx . (1)
where, u stands wave field, ),,( zyxv stands velocity
function. The wave field at depth z can be decomposed into beamlets with windows along the x-axis and y-axis,
)2(),(),,,(ˆ
),(),(~
),,(),(
yxbyxu
yxbyxbyxuyxu
mnqpn m p q
qmpnz
mnqpn m p q
mnqpzz
∑∑∑∑
∑∑∑∑
γξ=
><=
where ),( yxbmnqp are the decomposition atoms,
),,,(ˆ qmpnz yxu γξ are the coefficients of the decomposition
atoms locate at space locus ),( pn yx and wavenumber locus
),( qm γξ , and there is,
γ∆=γξ∆=ξ∆=∆= qmypyxnx qmpn ,,, .
For a local beam, we can introduce a local perturbation theory to account for the interaction between the beamlets and heterogeneities. Then the wave field at depth zz ∆+ can be calculated as
)3(),,,(ˆ
),(
),(),,,(ˆ),(
0,
),(
∑∑
∑∑∑∑∑∑
∑∑∑∑
⋅
=
=
∆∆
∆+
m qqmpnzmqnpjlir
n p
zyxki
l j i rjlir
mnqpn m p q
qmpnzzz
yxuP
eyxb
yxayxuyxu
np
γξ
γξ
where ),,(),,(),,( 20
22 zyxkzyxkzyxk pnnp −=∆ .
Local cosine basis propagator For local cosine basis, the atoms can be written as
)()(),( yxyxb qpmnmnpq ψψ= . (4)
Here ]/))(2
1(cos[)(/2)( nnnnmn LxxmxbLx −+π=ψ ,
where nnn xxL −= +1 is the nominal length of the window
and )(xbn is the bell (window) function.
We can obtain the local cosine beamlet propagator:
)5()]()()][()([
)]()()][()([
81
0000
0000
)]()([0,
qqqq
mmmm
ziyyxxi
pnmqnpjlir
bbbb
bbbb
eeddLL
P npprnl
γγγγγγγγξξξξξξξξ
γξπ
ζγξ
+−+−−++−⋅+−+−−++−⋅
= ∆−+−∫ ∫
where )(),( 2220 γ+ξ−=ζ pnnp yxk .
Shifting windows Normally, the wave fields are statically and regularly windowed with the LCB beamlet method, which is known as normal windows as shown in Figure 1(a). The wave field in
SEG Int'l Exposition and 74th Annual Meeting * Denver, Colorado * 10-15 October 2004
Beamlet migration using local cosine basis with shifting windows
each window is propagated with a local cosine basis propagator, followed by local perturbations, which can be regarded as first and second approximations, respectively. Therefore, for a very large lateral velocity variation within a single window, the propagation of the wave field in the window also has similar, but smaller errors on wide angle waves to those of the Split-Step Fourier (SSF) method (Stoffa, et al., 1990). Actually, these errors are so small that can be omitted for most cases. But for steeply inclined interface of large lateral velocity contrast as shown in figure 1(a), the errors which exist in the windows A, B, and C can be accumulated with depth and produce noticeable artifacts in the imaging results. In order to reduce the error accumulation with the depth, the row of windows is dynamically shifted in a random distance at every depth, which is known as shifting windows, as shown in figure 1(b). Because the velocity contrast is removed in window B, the wave field propagation is now correct in window B, the errors generated in windows A and C can not accumulate enough to affect the final imaging results.
Figure 1, Comparison of normal windows and shifting windows Imaging results on 2D models Here, we compared imaging results with the LCB method using both normal windows and shifting windows with a 2D STEEP model (Bertrand Duquet, Institut Français du Pétrole), as shown in figure 2(a). It has 476 samples with an interval of 20m in the horizontal dimension and 326 samples with an interval of 10m in depth. The minimum velocity is 1500m/s and maximum velocity is 4000m/s. Figure 2(b) and figure 2(c)
are the imaging results by LCB method with normal windows and shifting windows respectively. And from the imaging results, we can see the image quality is greatly improved by the shifting windows.
0
100
200
300
0 100 200 300 400
(a) 2D steep velocity model
0
100
200
300
0 100 200 300 400
(b) Image from normal-window LCB method
0
100
200
300
0 100 200 300 400
(c) Image from shifting-window LCB method Figure 2, 2D steep velocity model and images by normal-window LCB method and shifting-window LCB method. We also applied the normal-window LCB method and the
SEG Int'l Exposition and 74th Annual Meeting * Denver, Colorado * 10-15 October 2004
Beamlet migration using local cosine basis with shifting windows
shifting-window LCB method to the 2D SEG/EAGE salt model, as shown in figure 3. The minimum velocity is 5000 feet/s and the maximum velocity is 14700feet/s. Three regions near the area of the steeply inclined interface of large lateral velocity contrast are labeled in figure 3(a) with the letters A, B and C. The image quality in these areas is greatly improved for the shifting-window LCB method, as shown in figure 3(b) and figure 3(c).
0
100
200
0 200 400 600 800 1000 1200
(a) 2D SEG/EAGE velocity model
0
100
200
0 200 400 600 800 1000 1200
(b) Image from normal-window LCB method
0
100
200
0 200 400 600 800 1000 1200
(c) Image from shifting-window LCB method
Figure 3, 2D SEG/EAGE salt and images by normal-window LCB method and shifting-window LCB method.
Imaging result on 3D SEG/EAGE salt model Here, the Fourier Finite Difference (FFD) method (Ristow and Rühl, 1994), the normal-window LCB method and the shifting-window LCB method are applied to 3D SEG/EAGE salt model, which has 250, 250 and 201 grid intervals in x, y and z directions, respectively. Figure 4 shows a horizontal slice and a vertical slice of the velocity model and their imaging results. The image quality at the areas denoted by arrows is also greatly improved with the shifting-window LCB method. Conclusion Dynamically shifting windows in a random distance at different depths during LCB beamlet migration can greatly reduce the artifacts generated by error accumulation at sharp and steeply inclined interface of strong velocity contrast, and improves the total image quality. Acknowledgements: The authors would like to acknowledge the support from WTOPI (Wavelet Transform On Propagation and Imaging for seismic exploration) Project, the DOE/Basic Energy Sciences project and the DOD/DTRA project at University of California, Santa Cruz. And thank Henri Calandra and Biaolong Hua for providing the STEEP synthetic data and helpful discussions. References
Luo M. and Wu R.S., 2003, 3D beamlet prestack depth migration using the local cosine basis propagator, Expanded Abstracts, SEG 73th Annual Meeting, 985-988.
Ristow, D. and Rühl, T., 1994, Fourier finite difference migration, Geophysics, 59, 1882-1893.
Stoffa, P.L., Fokkenma, J.T., de Luna Freire, R.M., and Kessinger, W.P., 1990, Split-step Fourier migration, Geophysics, 55, 410-421.
Wang, Y. and Wu, R.S., 2002, Beamlet prestack depth migration using local cosine basis propagator, Expanded Abstracts, SEG 72th Annual Meeting, 1340-1343.
Wu, R.S. Wang, Y. and Gao, J.H., 2000, Beamlet migration based on local perturbation theory, Expanded Abstracts, SEG 70th Annual Meeting, 1008-1011.
Wu, R.S. and Chen L., 2001, Beamlet migration using Gabor-Daubechies frame propagator, Expanded Abstracts, EAGE 63th Annual Meeting, 74-78.
A B
C
SEG Int'l Exposition and 74th Annual Meeting * Denver, Colorado * 10-15 October 2004
Figure 4, 3D SEG/EAGE salt model and images by FFD method, normal-window LCB method and shifting-window LCB method.
SEG Int'l Exposition and 74th Annual Meeting * Denver, Colorado * 10-15 October 2004