声波方程频率域有限元参数反演

FINITE ELEMENT INVERSION OF THE COEFFICIENTS OF ACOUSTIC EQUATION IN FREQUENCY DOMAIN

推导出频率域有限元声波正演方程 .为了消除边界反射 ,将Clayton Engquist旁轴波动方程吸收边界条件引入频率域 ,并对有限元刚度矩阵和质量矩阵进行压缩存储 ,利用广义共轭梯度法求解有限元方程获得正演解 .在此基础上 ,推导出在某一频率下波场数据残差δ U与单元物性参数修改量δλ之间关系的Jacobi矩阵 ,反演方法允许利用地面二维炮集全波场资料与给出初始模型参数的正演值的差值δ U ,迭代求得δλ.由于计算机内存的限制 ,方法计算不允许有过多数目的未知数个数 ,因此还提出了对同一介质物性单元的Jacobi矩阵元素进行压缩组装的措施 ,从而使反演的未知量个数减少 ,结合采用共轭梯度迭代法 ,使得只需利用有效波频段的少数一些频率即可进行迭代反演 .正演和反演理论模型的数值模拟结果表明方法是有效的 .

Finite element forward equation for acoustic media is deduced. In order to eliminate boundary reflection, the absorbing boundary conditions of Clayton Engquist paraxial wave equation are introduced to frequency domain. The stiffness matrix and mass matrix of finite element are compressed for storing. The forward solutions are obtained by using conjugate gradient algorithm. On these bases, we deduce Jacobi matrix between wave field residual data δ and element material coefficient adjusted value δ λ in a certain frequency. Using the difference δ between surface 2D shot data and theoretical forward data we can obtain, the adjusted value δ λ . Because of the limitation of the computer's storage, the greater number of unknowns is not permitted. The measure of compressing and assembling the element Jacobi matrix coefficients in same medium is also proposed. By using this method, the number of unknowns in inversion is reduced. Combined with conjugate gradient algorithm, only a few frequencies in useful wave domain are needed in inversion. Some results of forward and inversion theoretical numerical models are given to prove its effectiveness.