基于横向导数的走时计算方法及其在叠前时间偏移中的应用
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摘要
在Kirchhoff积分叠前偏移方法中,需要反复使用两点之间的射线走时,现有走时计算方法的计算量(或存储量)受到了计算条件的限制。针对此,基于时间空间域到频率波数域和向量场到指数流形上的正反变换,提出了计算单程波算子旁轴走时的简便公式,将走时表示成空间变量(地面点到地下相点的水平距离)的多项式,将频率波数域单平方根算子表示成波数的多项式,运用Lie代数积分、指数映射和鞍点法将走时多项式的系数与单平方根算子的系数联系起来,运用单平方根算子的系数计算走时多项式的系数。该展开式与通常的旁轴展开式相比,增加了关于空间坐标的非对称项。在进行Kirchhoff积分叠前偏移时,先将走时多项式的系数计算好并存储起来(这些系数大约10个);在处理地震数据时,再利用这些系数"现算"走时,用于偏移成像。给出了Lie代数积分多项式、指数映射多项式计算的Magnus方法,该方法利用根树结构从低阶展开计算高阶展开;利用数值模拟方法,将有限差分法的结果与该公式(非对称)和对称公式的结果进行了对比,结果表明其比对称的走时公式有更高的精度。
In pre-stack time migration by Kirchhoff integral,the travel time of the two points are repeatedly used.However,the computa- tion amount(or storage cost)of conventional travel time computa- tion method is limited by computer conditions.In order to solve the problem,We proposed a simple formula for computing paraxial travel time of single-way wave operator.The formula is based on the forward and inverse transform between time-space domain to frequency-wavenumber domain and from vector field to exponential manifold.The travel time are expressed as polynomials of the hori- zontal offset between the two points,and the single-square-root operator in frequency-wavenumber domain are expressed as polyno- mials of wavenumber.Coefficients of travel time polynomials and that of single-square-root operator are related each other and calcu- lated by Lie algebraic integrand,exponential map and the saddle- point method.Comparing to conventional expansion,our paraxial travel time expansion increases asymmetric terms of the horizontal offset in the expression.In Kirchhoff migration,the coefficients of travel time polynomials are calculated and stored before seismic da- ta input(number of the coefficients are about 10).Travel times are calculated via the coefficients during Kirchhoff stacking of seismic data.A Magnus method calculated from Lie algebraic integrand polynomials,exponent mapping polynomials is given,which uses the tree-root scheme to compute high order terms based on lower order terms in the expansion method.The numerical comparison between our formulas,the results of finite difference method,sym- metry formula and our formula are compared by numerical model- ing.The results show that our method has higher accuracy than symmetry travel time formula.
In pre-stack time migration by Kirchhoff integral,the travel time of the two points are repeatedly used.However,the computa- tion amount(or storage cost)of conventional travel time computa- tion method is limited by computer conditions.In order to solve the problem,We proposed a simple formula for computing paraxial travel time of single-way wave operator.The formula is based on the forward and inverse transform between time-space domain to frequency-wavenumber domain and from vector field to exponential manifold.The travel time are expressed as polynomials of the hori- zontal offset between the two points,and the single-square-root operator in frequency-wavenumber domain are expressed as polyno- mials of wavenumber.Coefficients of travel time polynomials and that of single-square-root operator are related each other and calcu- lated by Lie algebraic integrand,exponential map and the saddle- point method.Comparing to conventional expansion,our paraxial travel time expansion increases asymmetric terms of the horizontal offset in the expression.In Kirchhoff migration,the coefficients of travel time polynomials are calculated and stored before seismic da- ta input(number of the coefficients are about 10).Travel times are calculated via the coefficients during Kirchhoff stacking of seismic data.A Magnus method calculated from Lie algebraic integrand polynomials,exponent mapping polynomials is given,which uses the tree-root scheme to compute high order terms based on lower order terms in the expansion method.The numerical comparison between our formulas,the results of finite difference method,sym- metry formula and our formula are compared by numerical model- ing.The results show that our method has higher accuracy than symmetry travel time formula.
引文
1 Wiggins W.Kirchhoff integral extrapolation and mi- gration of nonplanar data[J].Geophysics,1984,49(8): 1239～1248
2 Dix C H.Seismic velocities from surface measurements [J].Geophysics,1955,20(1):68～86
3李幼铭．知识创新工程重大项目在大庆油田的技术成??果要览[J]．地球物理学进展，2003，18(1)：5～18
4李幼铭．对“油储”项目在大庆实施获得成果的概述及认识[J]．地球物理学进展，2006，21(1)：135～142
5李幼铭，刘洪，吴永刚，等．我国陆相油储地球物理研究历程回顾与认识[J]．地球物理学进展，2007，22(4)：1280～1284
6刘洪，李幼铭．油气勘探二次创业的油储地球物理方法研究回顾[J]，石油与天然气地质，2008，29(5)：648～653
7刘洪，袁江华，陈景波，等．大步长波场深度延拓的理论[J]．地球物理学报，2006，49(6)：1779～1793
8刘洪，袁江华，勾永峰，等．地震逆散射波场和算子的谱分解[J]．地球物理学报，2007，50(1)：240～247
9刘洪，刘国峰，武威，等．三维波动方程逆散射的基础理论研究[J]．石油物探，2007，46(6)：569～581
10王秀闽．积分偏移的计算几何与并行实现[D]．北京：中国科学院地质与地球物理研究所，2008
11刘洪，王秀闽，曾锐，等．单程波算子积分解的象征表示[J]．地球物理学进展，2007，22(2)：463～471
12 Cordes H O.The technique of pseudo-differential oper- ators[M].Cambridge:Cambridge University Press, 1995.1～382
13齐民有，徐超江，王维克．现代偏微分方程引论[M]．武汉：武汉大学出版社，2005．1～337
14 Feng K.On difference schemes and symplectic geome- try[A].In:Feng ed.Symp diff geometry and diff equa- tions[C].Beijing:Science Press,1985.42～58
15 Iserles A,Munthe-Kaas H,Nfrstt S,et al.Lie-group methods[J].Acta Numerica,2000,9(1):215～263
16 Celledoni E,Iserles A.Methods for the approximation of the matrix exponential in a Lie-algebraic setting[J].IMA Journal of Numerical Analysis,2001,21(2):463～488
17 Magnus W.Exponetial solution of differential equa- tions for a linear operator[J].Communications on Pure and Applied Mathematics,1954(7):649～673
18 McLachlan R I,Quispel G R W.Splitting methods[J]. Acta Numerica,2002,11(1):341～434
19刘洪，何利，刘国锋，等．地层滤波公式的李代数积分证明[A]．见：金翔龙，秦蕴珊，朱日祥，等编．中国地质地??球物理研究进展——庆贺刘光鼎院士80华诞[C]．北京：科学出版社，2008．412～422
20李世雄．波动方程的高频近似与辛几何[M]．北京：科学出版社，2001．1～141
21秦孟兆，陈景波．Maslov渐近理论与辛几何算法[J]．地球物理学报，2000，43(4)：522～533
22高亮，李幼铭，陈旭荣，等．地震射线辛几何算法初探[J]．地球物理学报，2000，43(3)：402～410
23尹峰，顾本立，李幼铭．二维走时反演及波动方程反演成像的研究[A]．见：中国地球物理学会，编．勘探地球物理北京(89)国际讨论会论文摘要[C]．北京：地质出版社，1989．270～273
24 Chapman C H,Drommond R.Body-wave seismograms in inhomogeneous media using Maslov asymptotic the- ory[J].Bulletin of the Seismological Society of Ameri- ca,1982(72):S227～S317
25 Hubral P.Time migration:some ray theoretical aspects [J].Geophysical Prospecting,1975,25(5):738～745
26 Larner K L,Hatton L,Gibson B S,et al.Depth migra- tion of imaged time section[J].Geophysics,1981,46 (5):734～750
27 Shragge J C.Riemarmian wavefield extrapolation:non- orthogonal coordinate systems[J].Geophysics,2008, 73(2):T11～T21
28 Iversen E,Tygel M.Image-ray tracing for joint 3D seismic velocity estimation and time-to-depth conver- sion[J].Geophysics,2008,73(3):S99～S114
29 Sava P,Fomel S.Riemannian wavefield extrapolation [J].Geophysics,2005,70(3):T45～T56
30 Lynn W S,Claerbout J F.Velocity estimation in later- ally varying media[J].Geophysics,1982,47(6): 884～897
31 Wenzel F.Approximate traveltimes in laterally inho- mogeneous media[J].Geophysics,1988,53(1): 129～130
32 Castle R J.Wave-equation migration in the presence of lateral velocity variations[J].Geophysics,1982,47(7): 1001～1011
2 Dix C H.Seismic velocities from surface measurements [J].Geophysics,1955,20(1):68～86
3李幼铭．知识创新工程重大项目在大庆油田的技术成??果要览[J]．地球物理学进展，2003，18(1)：5～18
4李幼铭．对“油储”项目在大庆实施获得成果的概述及认识[J]．地球物理学进展，2006，21(1)：135～142
5李幼铭，刘洪，吴永刚，等．我国陆相油储地球物理研究历程回顾与认识[J]．地球物理学进展，2007，22(4)：1280～1284
6刘洪，李幼铭．油气勘探二次创业的油储地球物理方法研究回顾[J]，石油与天然气地质，2008，29(5)：648～653
7刘洪，袁江华，陈景波，等．大步长波场深度延拓的理论[J]．地球物理学报，2006，49(6)：1779～1793
8刘洪，袁江华，勾永峰，等．地震逆散射波场和算子的谱分解[J]．地球物理学报，2007，50(1)：240～247
9刘洪，刘国峰，武威，等．三维波动方程逆散射的基础理论研究[J]．石油物探，2007，46(6)：569～581
10王秀闽．积分偏移的计算几何与并行实现[D]．北京：中国科学院地质与地球物理研究所，2008
11刘洪，王秀闽，曾锐，等．单程波算子积分解的象征表示[J]．地球物理学进展，2007，22(2)：463～471
12 Cordes H O.The technique of pseudo-differential oper- ators[M].Cambridge:Cambridge University Press, 1995.1～382
13齐民有，徐超江，王维克．现代偏微分方程引论[M]．武汉：武汉大学出版社，2005．1～337
14 Feng K.On difference schemes and symplectic geome- try[A].In:Feng ed.Symp diff geometry and diff equa- tions[C].Beijing:Science Press,1985.42～58
15 Iserles A,Munthe-Kaas H,Nfrstt S,et al.Lie-group methods[J].Acta Numerica,2000,9(1):215～263
16 Celledoni E,Iserles A.Methods for the approximation of the matrix exponential in a Lie-algebraic setting[J].IMA Journal of Numerical Analysis,2001,21(2):463～488
17 Magnus W.Exponetial solution of differential equa- tions for a linear operator[J].Communications on Pure and Applied Mathematics,1954(7):649～673
18 McLachlan R I,Quispel G R W.Splitting methods[J]. Acta Numerica,2002,11(1):341～434
19刘洪，何利，刘国锋，等．地层滤波公式的李代数积分证明[A]．见：金翔龙，秦蕴珊，朱日祥，等编．中国地质地??球物理研究进展——庆贺刘光鼎院士80华诞[C]．北京：科学出版社，2008．412～422
20李世雄．波动方程的高频近似与辛几何[M]．北京：科学出版社，2001．1～141
21秦孟兆，陈景波．Maslov渐近理论与辛几何算法[J]．地球物理学报，2000，43(4)：522～533
22高亮，李幼铭，陈旭荣，等．地震射线辛几何算法初探[J]．地球物理学报，2000，43(3)：402～410
23尹峰，顾本立，李幼铭．二维走时反演及波动方程反演成像的研究[A]．见：中国地球物理学会，编．勘探地球物理北京(89)国际讨论会论文摘要[C]．北京：地质出版社，1989．270～273
24 Chapman C H,Drommond R.Body-wave seismograms in inhomogeneous media using Maslov asymptotic the- ory[J].Bulletin of the Seismological Society of Ameri- ca,1982(72):S227～S317
25 Hubral P.Time migration:some ray theoretical aspects [J].Geophysical Prospecting,1975,25(5):738～745
26 Larner K L,Hatton L,Gibson B S,et al.Depth migra- tion of imaged time section[J].Geophysics,1981,46 (5):734～750
27 Shragge J C.Riemarmian wavefield extrapolation:non- orthogonal coordinate systems[J].Geophysics,2008, 73(2):T11～T21
28 Iversen E,Tygel M.Image-ray tracing for joint 3D seismic velocity estimation and time-to-depth conver- sion[J].Geophysics,2008,73(3):S99～S114
29 Sava P,Fomel S.Riemannian wavefield extrapolation [J].Geophysics,2005,70(3):T45～T56
30 Lynn W S,Claerbout J F.Velocity estimation in later- ally varying media[J].Geophysics,1982,47(6): 884～897
31 Wenzel F.Approximate traveltimes in laterally inho- mogeneous media[J].Geophysics,1988,53(1): 129～130
32 Castle R J.Wave-equation migration in the presence of lateral velocity variations[J].Geophysics,1982,47(7): 1001～1011
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