基于螺旋线上谱因式分解的地震波场隐式辛算法

SEISMIC WAVE MODELING WITH IMPLICIT SYMPLECTIC METHOD BASED ON SPECTRAL FACTORIZATION ON HELIX

均匀介质、复杂各向同性介质和各向异性介质中的地震波传播过程 ,可用统一形式的标量声波方程描述 .考虑到在无损耗条件下 ,地震波方程描述了地震波场这一个无穷维的哈密顿体系随时间的演化过程 ,该过程为一个单参数连续辛变换 ,因而可以在其哈密顿形式表述下导出其辛格式 .与显式辛算法相比 ,隐式辛格式对应的隐式辛几何算法具有无条件稳定的特点 ,可以允许较大的计算步长 .但是由于隐式算法不可避免地面临高阶矩阵的求逆 ,其每一步的计算速度较慢 .为实现矩阵快速求逆 ,文中采用了螺旋边界条件下谱因式分解的方法 .在螺旋边界条件下 ,需要求逆的矩阵化为带状矩阵 ,而且其各列非零元素的位置和大小具有非常好的相似性 ,因而可以采用谱因式分解的方法实现快速LU分解 .文中采用二阶精度的隐式蛙跳辛格式和谱因式分解方法 ,计算了常速度、层状介质和Marmousi模型中的波场 .计算表明 。

Based on the common equation of seismic wave field and its Hamiltonian formalism, an implicit symplectic method (ISM) is proposed for seismic wave modeling in this paper. Combined with helix boundary conditions, the spectral factorization method is applied to give a fast way for the ISM modeling. The propagation of seismic wave can be described by the acoustic equation of a single scalar quantity in the constant, complex isotropy and anisotropy media. With the assumption of no dissipation, the scalar equation gives the symplectic evolution of seismic wave field, which is an infinite dimensional Hamiltonian system. Therefore, its Hamiltionian formalism and symplectic schemes can be given. Compared with explicit symplectic method (ESM), ISM is unconditionally stable method. Consequently, a long time step and a long time calculation are possible for ISM, which may be impossible for ESM in the same conditions. Unfortunately, it is unavoidably to calculate the inverse of large matrix in each step of ISM, which is time consuming process. Fortunately, however, the matrix has striped shape with a helical boundary conditions. And more, the nonzero units in each row of the matrix not only are located in the same relative positions but have similar values. Therefore, spectral factorization method can be used to get the inversion of the matrix in a very quick way. Based on spectral factorization method, an ISM scheme with second order in time is applied to get the wave fields in constant velocity, layer velocity and Marmousi velocity models. As the results indicates, with the help of spectral factorization, ISM is a quite good method in seismic wave modeling.