求解波动方程的任意差分精细积分法
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摘要
在地震偏移成像技术中 ,常常要反向外推波场 ,因此涉及到波动方程的数值求解问题。本文提出了一种求解偏微分方程的新的半解析方法———任意差分精细积分 (ADPI)法。其大体思路是 :空间域上作坐标离散 ,但不采用古典差分法的等分离散方式 ,而是一种相对自由的、任意的离散法 ;时间域上则采用子域精细积分的方法 ,既保留了精细积分法的高精度 ,又克服了工作量大、占用存储大等缺点。该方法具有精度高、带宽小、稳定性好等多项突出优点 ,并且可以灵活处理各类边界条件。本文从简单一维、较普遍一维、以及二维 3种情形讨论该方法对波动方程的具体应用 ,结合实际的算例 ,详细分析了各种算法的可行性和精度特点。
Wave equation migration is often applied to solve seismic imaging problems. Usually, the finite difference method is used to obtain the numerical solution of the wave equation. The arbitrary difference precise integration (ADPI) method is discussed and would be applied in the seismic migration.The ADPI method has its own distinctive idea. When discretizing coordinates in space domain, it employs a relatively unrestrained form instead of the one used by the conventional finite difference method. Moreover, in time domain it adopts the way of subdomain precise integration method. As a result, it not only takes the merits of high precision and narrow bandwidth, but also can process various of boundary conditions and describe the feature of inhomogeneous medium better. Numerical results show the benefit of the presented algorithm by using the ADPI method.
Wave equation migration is often applied to solve seismic imaging problems. Usually, the finite difference method is used to obtain the numerical solution of the wave equation. The arbitrary difference precise integration (ADPI) method is discussed and would be applied in the seismic migration.The ADPI method has its own distinctive idea. When discretizing coordinates in space domain, it employs a relatively unrestrained form instead of the one used by the conventional finite difference method. Moreover, in time domain it adopts the way of subdomain precise integration method. As a result, it not only takes the merits of high precision and narrow bandwidth, but also can process various of boundary conditions and describe the feature of inhomogeneous medium better. Numerical results show the benefit of the presented algorithm by using the ADPI method.
引文
强士中等,1999,任意差分精细积分法及数值稳定性分析,应用数学和力学,20(3),256～262。
马在田,1989,地震成像技术———有限差分法偏移,35～86,北京:石油工业出版社。
钟万勰,1994,结构动力方程的精细时程积分法,大连理工大学学报,34(2),131～136。
钟万勰,1995,子域精细积分及偏微分方程数值解,计算结构力学及其应用,12(3),253～260。
SheriffRE&GeldartLP著,初英等译,1999,勘探地震学,第二版,408～423,北京:石油工业出版社。
马在田,1989,地震成像技术———有限差分法偏移,35～86,北京:石油工业出版社。
钟万勰,1994,结构动力方程的精细时程积分法,大连理工大学学报,34(2),131～136。
钟万勰,1995,子域精细积分及偏微分方程数值解,计算结构力学及其应用,12(3),253～260。
SheriffRE&GeldartLP著,初英等译,1999,勘探地震学,第二版,408～423,北京:石油工业出版社。
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