Equivalent anisotropic first-order pseudo-acoustic wave equations modeling and its efficiency analysis
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摘要: 为克服各向异性弹性波动方程正演模拟的局限，本文研究了各向异性介质拟声波方程的交错网格有限差分数值解法.首先，从VTI介质胡克定律和qP-qSV波频散关系两种思路出发，通过声假设近似，给出了两种不同形式的VTI介质一阶拟声波方程，并通过引入波场的伪速度分量，推导了一种新的VTI介质一阶应力-速度方程，并通过旋转坐标系将其推广到TTI介质中；其次，构造了一阶拟声波方程的交错网格高阶有限差分格式，并推导了相应的PML边界条件；最后，对本文方法中固有的qSV人为干扰波的产生机制和压制方法进行了简单讨论.数值结果表明：3种一阶拟声波方程在运动学和动力学上是等价的，相对于各向异性弹性波正演模拟，其节省了内存，提高了计算效率；各向异性因素会影响反射波旅行时和振幅等波场特征，在后续的处理、反演和解释中不可忽略；VTI介质HESS模型的逆时偏移结果也验证了本文方法的合理性.Abstract: In order to overcome the limitations of anisotropic elastic wave equations modeling, this paper studied anisotropic pseudo-acoustic wave modeling by using staggered-grid finite-difference method. Starting from two different ideas, i.e., Hooke’s law and qP-qSV dispersion relation for VTI media, two first-order pseudo-acoustic wave equations in two forms for VTI media are given. Meanwhile, a new VTI first-order stress-velocity equations is derived through introducing the pseudo-velocity components of the wavefields, and then they are generalized to TTI media through the rotated coordinate system. Then the staggered-gird high-order finite-difference forms of the first-order pseudo-acoustic wave equations and the corresponding PML boundary condition are deduced. Finally, the inherent qSV-wave artifact generating mechanism and suppression method are discussed briefly. Numerical results show that the three first-order pseudo-acoustic wave equations are equivalent in kinematics and dynamics. Compared with anisotropic elastic wave modeling, they can save computational memory and enhance the efficiency. Anisotropic factor can affect the travel-time and amplitude of the reflection wave so we cannot ignore them in subsequent processing，inversion and interpretation. The reverse-time migration results of VTI-HESS model also validate the method proposed in this paper.
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图 1 交错网格及波场分量位置示意图
Figure 1. Schematic diagram of staggered-grids and the location of wavefield
图 2 波场快照对比(t=400 ms)
(a)VTI介质弹性波方程正演得到的垂直应力分量波场；(b)用式(2)正演得到的波场q；(c)用式(6)正演得到的波场p；(d)用式(7)正演得到的波场p
Figure 2. Comparison of wavefield snapshots (t=400 ms)
(a)Wavefield of vertical stress component modeled by elastic wave equation in VTI media;(b)Wavefield q modeled by equation(2);(c)Wavefield p modeled by equation(6);(d) Wavefield p modeled by equation(7)
图 3 应用PML边界条件后的波场快照
(a)qP波，t=600 ms；(b)qSV波，t=2 000 ms
Figure 3. Wavefield snapshots with PML boundary condition
(a)qP wave，t=600 ms；(b)qSV wave, t=2 000 ms
图 4 设置震源环前后各向异性介质拟声波正演的波场快照对比(t=400 ms)
(a)VTI介质(设置震源环前)；(b)VTI介质(设置震源环后)；(c)TTI介质(设置震源环前，θ=45°)；(d)TTI介质(设置震源环后，θ=45°)
Figure 4. Wavefield snapshots comparison of anisotropic pseudo-acoustic wave modeling without and with source box (t=400 ms)
(a)VTI media (without source box)；(b)VTI media (with source box)；(c)TTI media (without source box, θ=45°)；(d)TTI media (with source box, θ=45°)
图 6 单炮地震记录对比
(a,d)用式(2)正演得到的炮记录；(b,e)用式(6)正演得到的炮记录；(c,f)用式(7)正演得到的炮记录；(g)TTI介质拟声波方程正演得到的炮记录；(h)VTI介质弹性波方程正演得到的炮记录；(i)各向同性声波方程正演得到的炮记录.其中，(a)-(c)为设置震源环前的结果；(d)-(f)为设置震源环后的结果
Figure 6. Comparison of single shot seismic records
(a,d) Shot records modeled by equation (2)；(b,e) Shot records modeled by equation (6)；(c,f) Shot records modeled by equation (7);(g) Shot records modeled by TTI pseudo-acoustic wave equation；(h)Shot records modeled by VTI elastic wave equation；(i) Shot records modeled by isotropic acoustic wave equation. Figs.(a)-(c) are results without source box, and Figs.(d)-(f) are results with source box
图 7 单道地震信号对比
(a)从图6a，b，c中近偏移距(地面位置为1 200 m)处抽取的单道信号对比；(b,c)从图6d e，f中近偏移距(地面位置为1 200 m)和远偏移距(地面位置为200 m)处抽取的单道信号对比；(d,e)从图6f，h中近偏移距(地面位置为1 200 m)和远偏移距(地面位置为200 m)处抽取的单道信号对比；(f,g) 从图6f，g，i中近偏移距(地面位置为1 200 m)和远偏移距(地面位置为200 m)处抽取的单道信号对比
Figure 7. Comparison of single trace seismic signals
Fig.(a) is single trace signals comparison of near offset (distance=1 200 m) extracted from Figs.6a,b,c; Fig.(b) and (c) are single trace signals comparison of near offset (distance=1 200 m) and far offset (distance=200 m) extracted from Figs.6d,e,f; Figs.(d) and (e) are single trace signals comparison between near offset (distance=1 200 m) and far offset (distance=200 m) extracted from Figs.6f, h; Figs.(f) and (g) are single trace signals comparison between near offset(distance=1 200 m) and far offset (distance=200 m) extracted from Figs.6f, g, i
图 9 各向同性(a)和各向异性(b)介质逆时偏移剖面
Figure 9. Fig.9 Reverse-time migration sections for isotropic (a) and anisotropic (b) media
表 1 几种波动方程的计算效率对比
Table 1. Computational efficiency comparison of the several wave equations
方程类型 计算量 平均计算时间/s VTI一阶拟声波方程(2) ∂p/∂x,∂q/∂z,∂u/∂x,∂w/∂z 72 VTI一阶拟声波方程(6) ∂p/∂x,∂p/∂z,∂u/∂x,∂k/∂z,∂w/∂z,∂ζ/∂z 105 VTI一阶拟声波方程(7) ∂p/∂x,∂p/∂z,∂q/∂x,∂q/∂z,∂up/∂x,∂wq/∂z 103 各向同性声波方程 ∂p/∂x,∂p/∂z,∂u/∂x,∂w/∂z 70 VTI弹性波方程 ∂τxx/∂x,∂τxz/∂z,∂τxz/∂x,∂τzz/∂z,∂vx/∂x,∂vx/∂z,
∂vz/∂x,∂vz/∂z130 
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Du X, Fletcher R, Fowler P J. 2008. A new pseudo-acoustic wave equation for VTI media[C]//70th EAGE Conference and Exhibition. Rome: European Association of Geoscientists and Engineers: 1-5. Duveneck E, Milcik P, Bakker P M. 2008. Acoustic VTI wave equations and their application for anisotropic reverse-time migration[C]//SEG Technical Program Expanded Abstracts. Las Vegas: Society of Exploration Geophysicists: 2186-2190. Zhang H, Zhang G. 2009. Removing S-wave noise in TTI reverse time migration[C]//79th International Exposition and Annual Meeting, SEG. Houston: Society of Exploration Geophysicists: 1-5. Zhou H, Zhang G, Zhang Y. 2006. An anisotropic acoustic wave equation for VTI media[C]//68th EAGE Conference and Exhibition. Vienna: European Association of Geoscientists and Engineers: 1-5. -
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