地球物理学报  2013, Vol. 56 Issue (3): 953-960 PDF

Incident angle field estimation: a one-way propagator approach
ZHANG Jian-Feng, ZHANG Hui, LIU Li-Nong
Key Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China
Abstract: We present a robust method to estimate the incident angle field based on the one-way propagator. The employment of monochromatic wave field is to reduce the extrapolation time dramatically and ensure the generation of the incident angle field in the whole imaging space. This incident angle field estimation method has high computational efficiency and can be applied in getting angle gather in one-way wave equation depth migration scheme. Comparison with depth migration, the cost of incident angle field estimation can be omitted. The proposed algorithm is free of smoothness of velocity model and hence can avoid large incident angle variation with small velocity perturbation compared to the traditional ray tracing or travel-time gradient picking method. This means the proposed angle field estimation method is more suitable for real velocity model and match well with the one-way wave equation prestack depth migration scheme. Computation of incident angle fields of the 2-D, 3-D layered and complicated structured models show the high computational efficiency, stability and precision of the proposed algorithm..
Key words: One-way wave equation      Incident angle      Angle gather
1 引言

2 基于单程波算子的入射角计算理论

 (1)

 (2)

 (3)

 (4)

 (5)

 (6)

 (7)

 (8)

x趋于零, 式(8)仍可得到稳定的结果, 而当0.6 < x≤1.2时, 式(8)的近似有很好的精度.利用式(8), 我们发展了稳定的相除算法, 即

 (9)

 (10)

3 基于单程波方法的入射角计算流程

 图 1 基于单程波深度延拓的入射角计算流程图 Fig. 1 Flow chart of angle field estimation based on the one-way wave equation wavefield extrapolation

4 数值算例 4.1 二维模型数值算例

 图 2 均匀模型入射角度分布及其与理论值对比分析 (a)均匀模型入射角度图；(b)图(a)中不同深度横向位置角度与理论值对比，图中实线为理论值，虚线为本文方法计算值. Fig. 2 Comparisons of theoretical and estimated angles on 2D homogeneous model (a) Angle field of a 2D homogeneous model; (b) Comparisons of theoretical (illustrated by the solid line) and estimated (illustrated by the dashed line) angles at different depth.

 图 3 海上复杂地质模型(a)及其入射角计算结果分布(b) Fig. 3 (a) A fairly complicated model and (b) the estimated angle field with one-way wave equation based scheme
 图 4 本文计算入射角方法流程稳定性测试 (a)加入速度扰动前后的速度曲线对比，图中实线和虚线分别对应加入速度扰动前后的速度；(b)加入速度扰动前后本文方法计算的入射角度对比，图中实线和虚线分别对应加入速度扰动前后的入射角度. Fig. 4 Stability test of the proposed one-way wave equation based angle field estimation scheme (a) Comparison of velocity values without (illustrated by the solid line) andwith (illustrated by the dashed line) randomvelocity perturbation; (b) Comparison of estimated angles without (illustrated by the solid line) and with (illustrated by the dashed line) random velocity perturbation.

Marmousi模型   为进一步验证本文方法流程对复杂横向速度变化的适应能力, 将其应用于SEG Marmousi模型, 该模型存在断层、背斜、尖灭等复杂构造, 同时具有较强的横向变速.图 5显示了其速度模型.应用该方法计算炮点位于不同横向位置时模型的入射角分布, 为更好地分析角度的合理性, 选择较有代表性的位置将角度的等值线和速度模型叠合起来示于图 6.

 图 5 Marmousi速度模型 Fig. 5 Marmousi velocity model

 图 6 震源位于不同位置时入射角等值线和速度模型 叠合图: (a) CDPl00，(b) CDP400，(c) CDP650 Fig. 6 The incident angle contour line underlying velocity model with the source at different position i. e. CDP100 (a), CDP400 (b) and CDP 650(c), respectively
4.2 三维模型数值算例

 图 7 不同位置纵切面的理论值与计算值等值线对比分析：图中实线为理论值，虚线为本文方法计算值 Fig. 7 Comparisons of theoretical (illustrated by the solid line) and estimated (illustrated by the dashed line) angle field sections at (a) offset 0 mat inline direction, (b) offset 0 mat crossline direction, (c) offset 1680m at inline direction and (d) offset 1680m at crossline direction, respectively (a) Offset 0m inline; (b) Offset 0m crossline; (c) Offset 1680m inline; (d) Offset 1680m crossline.
 图 8 不同深度水平切片上理论角度与计算角度的等值线对比分析 图中实线为理论值，虚线为本文方法计算值. Fig. 8 Comparisons between the contour lines at different depth slices of theoretical (illustrated by the solid line) and estimated (illustrated by the dashed line) angles (a) Slice at the depth of 1 km; (b) Slice at the depth of 3 km; (c) Slice at the depth of 5 km.

5 结论

 [1] Wapenaar K, Goudswaard J, van Wijngaarden A J. Multi-angle, multi-scale inversion of migrated seismic data. The Leading Edge , 1999, 18(8): 928-932. DOI:10.1190/1.1438409 [2] Berkhout A J, Ongkeihong L, Volker A W F, et al. Comprehensive assessment of seismic acquisition geometries by focal beams-Part 1:Theoretical considerations. Geophysics , 2001, 66(3): 911-917. DOI:10.1190/1.1444981 [3] 张辉, 王成祥, 张剑锋. 基于单程波方程的角度域照明分析. 地球物理学报 , 2009, 52(6): 1606–1614. Zhang H, Wang C X, Zhang J F. One-way wave equation based illumination analysis in angle domain. Chinese J. Geophys. (in Chinese) , 2009, 52(6): 1606-1614. [4] de Bruin C G M, Wapenaar C P A, Berkhout A J. Angle-dependent reflectivity by means of prestack migration. Geophysics , 1990, 55(9): 1223-1234. DOI:10.1190/1.1442938 [5] Zhang Y, Xu S, Bleistein N, et al. True-amplitude, angle-domain, common-image gathers from one-way wave-equation migrations. Geophysics , 2007, 72(1): S49-S58. DOI:10.1190/1.2399371 [6] Biondi B, Symes W W. Angle-domain common-image gathers for migration velocity analysis by wavefield-continuation imaging. Geophysics , 2004, 69(5): 1283-1298. DOI:10.1190/1.1801945 [7] Kuhl H, Sacchi M D. Least-squares wave-equation migration for AVP/AVA inversion. Geophysics , 2003, 68(1): 262-273. DOI:10.1190/1.1543212 [8] Červeny V. Seismic Ray Theory. Cambridge: Cambridge University Press, 2001 . [9] Claerbout J F. Toward a unified theory of reflector mapping. Geophysics , 1971, 36(3): 467-481. DOI:10.1190/1.1440185 [10] Liu L N, Zhang J F. 3D wavefield extrapolation with optimum split-step Fourier method. Geophysics , 2006, 71(3): T95-T108. DOI:10.1190/1.2197493 [11] Zhang J F, Liu L N. Optimum split-step Fourier 3D depth migration:Developments and practical aspects. Geophysics , 2007, 72(3): S167-S175. DOI:10.1190/1.2715658 [12] 张剑锋, 卢宝坤, 刘礼农. 波动方程深度偏移的频率相关变步长延拓方法. 地球物理学报 , 2008, 51(1): 222–228. Zhang J F, Lu B K, Liu L N. Frequency-dependent varying-step depth extrapolation scheme for wave equation based migration. Chinese J. Geophys. (in Chinese) , 2008, 51(1): 222-228.