Getting pre-stack time migration travel times from the single square root operator
Liu Guo-Feng1,2, Liu Hong1, Li Bo1,2, and Meng Xiao-Hong2
1. Key Laboratory of Petroleum Geophysics, Institute of Geology and Geophysics, CAS, Beijing 100029, China.
2. State Key Laboratory of Geological Processes and Mineral Resources Geo-detection Laboratory of the Ministry of Education, Beijing 100083, China.
Abstract Improving the focusing capability of pre-stack time migration allows the imaged section to reflect structural characteristics, depth, and interface shape and it is a key step for the preparation of the initial depth migration velocity model. The traditional symmetrical travel time equation is derived based on the assumption of a layered model. It is difficult to achieve the desired effect of focusing in media with strong lateral variation. The non-symmetrical travel time equation based on Lie algebra and a pseudo-differential operator contains a lateral velocity derivative which can improve the focusing capability even in strongly lateral variable media and also the computation precision of the weight coefficients for relative amplitude preservation. Compared with the symmetrical methods, the non-symmetrical method is more effective. In this paper, we describe several key steps of non-symmetric pre-stack travel time calculation and present some test results using synthetic and real data.
This research was supported by the National Basic Research Program of China (Grant No. 2007CB209603), Key Project of the National Natural Science Foundation (Grant No. 40830424), State Key Laboratory of Geological Processes and Mineral Resources Geo-detection Laboratory of the Ministry of Education for their sponsorship (GPMR 200633, GDL0801).
Cite this article:
LIU Guo-Feng,LIU Hong,LI Bo et al. Getting pre-stack time migration travel times from the single square root operator[J]. APPLIED GEOPHYSICS, 2009, 6(2): 129-137.
[1]
Alkhalifah, T., 2000, The offset-midpoint traveltime pyramid in transversely isotropic media: Geophysics, 65(4), 1316 - 325.
[2]
Bleistein, N., 1999, Hagedoorn told us how to do Kirchhoff migration and inversion: The Leading Edge, 18, 918 - 927.
[3]
Claerbout, J. F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications, London.
[4]
Dong, C. H., and Zhang, J. F., 2009, Prestack time migration including surface topography: Chinese J. Geophys (in Chinese), 52(1), 239 - 244.
[5]
Docherty, P., 1991, A brief comparison of some Kirchhoff integral equation as for migration and inversion: Geophysics, 56(5), 1164 - 1169.
[6]
Feng, K., 1985, On difference schemes and asymplectic geometry: Symp Diff Geometry and Diff Equations: Science Press, Beijing, China.
[7]
Gray, S. H., 1997, True-amplitude seismic migration: A comparison of three approaches: Geophysics, 62(3), 929 - 936.
[8]
Hall, B. C., 2003, Lie group, Lie algebras, and representations, An elementary Introduction: World Book Publishing Company, Beijing.
[9]
Hubral, P., 1975, Time migration-some ray theoretical aspects: Geophysical Prospecting, 25, 738 - 745.
[10]
Iserles, A., 2002, Lie-group methods: Cambridge University Press, London, UK.
[11]
Li, S, X., 2001, High frequency approximation of wave equation and in geometry: Science Press, China.
[12]
Liu, H., Wang, X. M., and Zeng, R., 2007b, Symbol description to integral solution of one2way wave operator: Progress in Geophysics (in Chinese), 22(2), 463 - 471.
[13]
Liu, H., Yuan, J. H., Chen, J. B., and Shou, H., 2006, Theory of large step wavefield depth extrapolation: Chinese J. Geophys. (in Chinese), 49(6), 1779 - 1793.
[14]
Magnus, M., 1954, On the exponential solution of differential equations for a linear operator: Communications of Pure and Applied Mathematics, 2, 649 - 673.
[15]
Qi, M. Y., Xu, C. J., and Wang, W. K., 2005, Introduction of modern partial differential equations: Wuhan University Press, Wuhan, China.
Schleicher, J., Martin, T., and Hubral, P., 2007, Seismic true-amplitude imaging: Society of Exploration Geophysicists, Tulsa.
[18]
Sun, C. W., Wang, H. W., and Martinez, R. D., 2002, Optimized 6th order NMO correction for long-offset seismic data: Ann. Internat. Mtg., Soc. Expl. Geophys, Expanded Abstracts, 2313 - 2316.
[19]
Wang, X. M., 2008, Computational geometry and parallel implementation of integral migration: PhD Thesis, Institute of Geology & Geophysics, Chinese Academy of Sciences, Beijing.
[20]
Yilmaz, O. Z., 2001, Seismic data analysis: processing, inversion, and interpretation of seismic data: Society of Exploration Geophysicists, Tulsa.
[21]
Yuan, J. H., 2006, Research on theory and key techniques of prediction about stratigraphic filtering in complex condition: PhD Thesis, Institute of Geology & Geophysics, Chinese Academy of Sciences, Beijing.
[22]
Zhang, Y., and Zhang, G. Q., 2005, Theory of true-amplitude one-way wave equations and true-amplitude common-shot migration: Geophysics, 70(2), E1 - E10.
[23]
Zhang, Y., Zhang, G. Q., and Bleistein, N., 2003, True amplitude wave equation migration arising from true amplitude one-way wave equations: Inverse Problems, 19, 1113 - 1138.
[24]
Zhang, Y., 2007, Explicit marching method for reverse-time migration: Ann. Internat. Mtg., Soc. Expl. Geophys, Expanded Abstracts, 2300 - 2303.
2009, Vol. 6 
