3d beamlet migration in vti medium3d beamlet migration in vti medium a local cosine basis element...

3D Beamlet migration in VTI medium Jian Mao*, Ru-Shan Wu, University of California at Santa Cruz; Biaolong Hua, Paul Williamson, TOTAL E&P INC at Houston Summary Beamlet migration is a wave equation based migration approach. The use of local background velocity and local perturbations results in a two-scale decomposition of beamlet propagators: a background propagator for large-scale structures and a local phase screen correction for small-scale local perturbations. The beamlet propagator can handle strong lateral velocity variations with improved accuracy. Local cosine basis beamlet propagator was proposed and developed recently. It has been successfully applied to poststack and prestack depth migration for media with large lateral velocity variations. Here we extend the beamlet propagator into transversely isotropic medium with vertical symmetric axis (VTI) medium. We tested impulse responses, a synthetic dataset and a field dataset. The results demonstrated the validity of the VTI beamlet propagator. Introduction In most seismic processing and interpretation cases, we usually assume that a medium is isotropy. In isotropic medium, velocity is the same regardless the direction of measurement. However, anisotropy exists in many subsurface media (Thomsen, 1986), and the velocity depends on direction. If we use conventional methods to deal with the wave propagation in anisotropic media, it may result in large errors, which can lead to lower resolution and misplaced images of subsurface targets. Therefore, we have to consider anisotropy if we want to achieve higher-resolution and provide the details required for production-geophysics. In the recent years, a lot of efforts have been made to develop migration algorithms for anisotropic media. A number of approaches have been proposed to solve the anisotropic problem, including Gaussian beam methods and wave equation based methods (Alkhalifah, 1995; Ball, 1995; Alkhalifah, 1998; Rousseau and Hoop, 2001). Among these methods, wave equation based methods (one-way methods or full-wave based reverse time migration) can give much more accurate results than Kirchhoff migration. Zhang et al. (Zhang et al., 2005) propose a prestack depth migration method based on a one-way wave equation. They extend the Fourier finite-difference method (FFD) to VTI media. By introducing a reference velocity model at each depth during wave field extrapolation, the method can handle the strong lateral velocity variation. Beamlet migration approach is based on local reference velocities and local perturbation (Wu et al., 2000).

Migration methods based on this theory have been developed using the Gabor-Daubechies frame (GDF) (Chen et al., 2002, 2006) and local cosine bases (LCB) (Wang and Wu, 2002; Luo and Wu, 2003; Wang et al., 2003; Luo et al., 2004; Wu et al., 2008). In these methods, the wavefields are localized spatially with local windows and directionally with local wavenumbers. The wavefield at each depth is propagated with beamlet propagators (sparse propagator matrices), followed by local perturbation corrections. These methods can provide good imaging results when compared to traditional wave-equation based methods. The non-orthogonal Gabor-Daubechies frame decomposition needs at least a redundancy ratio of 4 in the wavefield representation, which results in large computational cost. Since the local cosine transform is orthogonal and has a fast algorithm, the wavefield decomposition and extrapolation using LCB beamlet have good computation efficiency. In this work, we first derived the 3D beamlet propagator for VTI medium by using the dispersion relationship in VTI medium. Then we show some numerical examples for both synthetic data and field data. The numerical results demonstrated the capability of VTI beamlet propagators. Theory and Methods The frequency-space-domain wavefield can be decomposed into beamlets, which provides us the localized information in both space and direction simultaneously. Beamlet transform uses translated window for the spatial localization and harmonic modulations for the directional localization. Both orthogonal bases (e.g., local cosine bases (LCB)) (Wu et al., 2008) and tight frames (e.g., Gabor-Daubechies frame) (Chen et al., 2006; Wu and Chen, 2006) were introduced into the beamlet decomposition. For the consideration of computational efficiency, we prefer orthogonal bases, which have no redundancy and have efficient decomposition and reconstruction. The LCB beamlet migration has been successfully tested through 3D models (Luo and Wu, 2003; Wang et al., 2003). Beamlet propagator can handle models with large velocity variations (e.g. salt models) because of the local perturbation approximation (Wu et al., 2000), which results in better image quality. On the other hand, the efficiency of beamlet propagator is comparable with other wave-equation based one-way propagators (e.g. Generalized screen propagator (de Hoop et al., 2000)). As a result, we extend 3D beamlet propagator to VTI case here. 1.Two dimensional local cosine bases

© 2012 SEG DOI http://dx.doi.org/10.1190/segam2012-0572.1SEG Las Vegas 2012 Annual Meeting Page 1

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3D Beamlet migration in VTI medium

A local cosine basis element can be specified by its spatial position nx , interval (the nominal length of the window) 1n n nL x x , and the wavenumber index m ( 0, , 1m M , M denotes the sample point number of each interval; 0, , 1n N , N denotes the total interval number) as follows

( ) 2 coscmn n m nn

b x B x x xL

, (1)

where ( )cmnb x denote the local cosine bases. The local

horizontal wavenumber is defined by 12

mn

mL

. The

expression nB x denotes a bell function (see Appendix A for detailed definition), which is smooth and supported in the compact interval 1[ , ]n nx x for

1n nx x , with , as the left and right overlapping radius, respectively. For the 3D case, the two dimensional local cosine bases function is composed by two one dimensional local cosine functions as follow:

1 2 1 2

1 1 21 2

1 2 1 1 2 2

1 2

,

,

( , )

2 , cos cos

m m n n

m n m n

n n m n m nn n

b x y

b x b y

B x y x x x yL L

(2) where

1 1,m nx and 2 2,m ny are the local horizontal

wavenumber and the local window position along x-axis and y-axis respectively.

1 2,n nL L are the interval length

along x-axis and y-axis. The expression 1 2,

,n nB x y is the two dimensional bell function. Then the decomposition coefficients of a 2D function ,f x y by the two dimensional local cosine transform can be calculated by the inner product as follow

1 2 1 2, ,

ˆ , , ,jl mn m m n nf f x y b x y . (3)

Similar with the 2D case, we use fast local cosine transform instead of calculating the inner product of equation (3) directly. In practice, we do local cosine transform along x-axis first and then along y-axis. In the Appendix, we give more details about local cosine transform and its fast implementation. 2.Beamlet propagation in the 3D case The wave field at depth z can be decomposed into beamlets with windows along the x-axis and y-axis

1 2 1 2 1 2 1 2

1 1 2 2

1 2 1 2 1 2 1 2

1 1 2 2

, ,

,

( , , ) ( , )

, ( , ) ( , )

ˆ ( , , , ) ( , )

z

m m n n m m n nn m n m

z n n m m m m n nn m n m

u x y z u x y

u b x y b x y

u x y b x y

(4)

where 1 2 1 2,

( , )m m n nb x y are the two dimensional beamlet

atoms, 1 2 1 2

ˆ ( , , , )z n n m mu x y are the coefficients of the

decomposition locate at space locus 1 2,n nx y and wavenumber locus 1 2,m m . The wavefield at depth z z can be extrapolated by the background propagator (0)P and perturbation operator (1)P as follow

1 2 1 2 1 2 1 2

1 1 2 2 1 2

1 2 1 2 1 2 1 2 1 2 1 2

1 2

(1), ,

(0),

( , )

( , ) ( , )

ˆ ( , , , )

z z

j j l l l l n nl j l j n n

j j l l m m n n z n n m mm m

u x yb x y P x y

P u x y

. (5)

The background propagator (0)P can be calculated as 3.Beamlet propagator in VTI medium For three dimensional acoustic wave equation, the dispersion relation for isotropic medium can be expressed as follow

2 2 20 x yk , (6)

Where 00

kV

( is circular frequency and 0V is

velocity), x and y are horizontal wavenumber along x-axis and y-axis respectively, is vertical wavenumber. In VTI medium, the dispersion relation is formulated as follow

2 2 2

0 2 2 2 21

1x x y

x x y

QVk

Q V

, (7)

Where 0z

kV

, xV and zV are horizontal and vertical

phase velocity respectively, with 1 2x zV V . 1

1 2Q

and

1 2

, where and are Thomsen

parameter (Thomsen, 1986). In isotropic medium, 0 , 0 , 0 , 1Q and x zV V . Equation (7) can also be

expressed as follow


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2 2

20 2 2

20

1 2

1 2

x y

x y

k

k

. (8)

With this new dispersion relation, we can derive the beamlet propagator for VTI case similarly. The VTI beamlet propagator still can be spitted into two parts: the background propagator (0)P and perturbation operator

(1)P . The perturbation operator (1)P stays the same as that in isotropic case. The background propagator (0)P can be expressed as

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

21

1 1 1 1

1 1

1 1 1 1

1 1

2 2

(0),

(0)

2

/ 2 / 20 0

/ 2 / 20 0

/ 20

( , , , ; , , , )

1 14 (2 )

ˆ ˆ( ) ( )

ˆ ˆ( ) ( )

ˆ (

j n j n

m n m n

j n

j j l l m m n n

l l j j n n m m

x yn n

i L i Lx j x j

i L i Lx m x m

i Ly

P

P x y x y

d dL L

e b e b

e b e b

e b

2 2

2 2

2 2 2 2

2 2

1, 21 1 2 2

/ 20

/ 2 / 20 0

( ) ( )

ˆ) ( )

ˆ ˆ( ) ( )

j n

m n m n

x l n y l n n n

i Lj y j

i L i Ly m y m

i x x i y y i z

e b

e b e b

e e e

. (9)

Where

1 2

2 22

1, 2 0 2 2

20

1 2,

1 2

x yn n n n

x y

k x y

k

is the

local vertical wavenumber. Numerical examples 1. Impulse response

Figure 1 Impulse response using FFD propagator in VTI medium

Figure 2 Impulse response using LCB propagator in VTI medium

First, we calculated the impulse response for a homogeneous medium, in which the vertical phase velocity is 2000 m/s, and the Thomsen parameters 0.3 ,

0.2 . Figure 1 and Figure 2 are the impulse responses generated by FFD propagator and LCB propagator. We see that the FFD result has big errors along the diagonal directions due to the split-step wide angle corrections. 2. Synthetic data Then we tested the synthetic VTI model with steep structures. Figure 3 shows the velocity map. The anisotropy parameter is set in different layers from the top to bottom. Figure 4 gives the poststack migration images using conventional isotropic LCB propagator and VTI LCB propagator. We can see the improvement of the migration image using VTI propagators, especially for the position of the steep events and the structures in deep area.

Figure 3 Velocity map of steep model


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Figure 4 poststack migration image with isotropic LCB (upper)

and VTI LCB (lower) propagators 3. Field data

Figure 5 Poststack migration image using isotropic LCB

propagator

Figure 6 Poststack migration image using VTI LCB propagator

Finally, we use our VTI propagator on a real dataset. Figure 5 and Figure 6 are the poststack migration images using isotropic and VTI LCB propagator respectively. We see the continuity for some structures are improved a lot after using VTI propagator. Conclusions We derived 3D beamlet propagator for VTI medium. As FFD propagator has the anisotropic correction in the wide angle correction part, it results in inaccuracy by using the split-step implementation. The impulse response shows very clear difference compared with FFD method. Both the synthetic data and field data tests show the validity of the VTI beamlet propagator. For the future work, we will extend the beamlet propagator to TTI medium. Acknowledgement We’d like to thank TOTAL E&P USA give us the opportunity to present this work. We thank Huimin Guan and Fuchun Gao in TOTAL R&T group, Yaofeng He in Exxon Mobil, Jun Cao in Conoco Philips and Xiaobi Xie from UC Santa Cruz for helpful discussions. We also appreciate the HPC help from Terrence Liao and Saber Feki in TOTAL E&P USA.


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http://dx.doi.org/10.1190/segam2012-0572.1 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2012 Annual International Meeting, SEG, Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES

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3d beamlet migration in vti medium3d beamlet migration in vti medium a local cosine basis element...

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