【摘要】 地震波场模拟和偏移成像等有限差分隐格式算法中的重要环节 ,是实现亥姆霍兹算子表示矩阵H的快速求逆运算 .在螺旋边界条件下 ,H具有Toeplitz结构的正定厄密矩阵 ,其快速求逆可由谱法LU分解实现 .本文分析了谱法LU分解对提高计算速度的原理及特点 ,并着重讨论了在不同类型的介质模型中 ,采用谱法分解矩阵H时带来的数值误差、误差的分布及其对波场计算的影响 .研究结果表明 ,对均匀介质而言 ,矩阵H各列具有相同的非零元素分布 ,谱法LU分解的误差在吸收边界条件下 ,不影响波场模拟和成像计算 ;但对于非均匀介质模型 ,矩阵H各列具有不同的非零元素分布 ,谱法LU分解的误差随介质不均匀性程度的增大而增大 ,势必影响非均匀介质中波场计算 .在波场模拟和成像等有限差分隐格式算法中 ,采用谱法LU分解完成矩阵求逆时 ,必须考虑到并尽量减少该方法的误差对波场计算的影响 .更多还原

【Abstract】 The inversion of Helmholtz matrix, a finite-difference representation of Helmholtz operator, plays a critical role in the numerical methods of seismic modeling and 3D depth migration. The Helmholtz matrix has a Toeplitz structure with a helical boundary condition. It is easy to be inversed by an efficient LU decomposition method, a spectral factorization method (SF method). In this paper we discuss the efficiency of SF method in LU decomposition, the errors of SF method in decomposing the matrix H of several different velocity models, errors distribution and their effects on seismic imaging. The result in this paper indicates that for constant velocity model all the arrays of matrix H have same nonzero values, and the errors brought by SF method will not affect the computation of wave fields if an absorb boundary condition is used; for non-constant velocity model, each array in matrix H has different nonzero values, and the error increases with variability degree of the velocity model, which are fatal to wave fields calculation. Therefore, once a SF method is used to calculate the wave field in seismic modeling and imaging, the errors given by this method should be considered to ensure a correct wave fields, which will be discussed in another paper.更多还原