一种基于高效迭代解法的频率域全波形反演

Full waveform inversion in the frequency domain with an efficient iterative solver

巨大的计算量是制约全波形反演(FWI)生产实用化的难题之一.为此,本文提出了一种高效的波场迭代解法,将其应用于频率域常密度声波方程FWI,并给出了详细的反演流程.通过建立用于波场迭代的目标函数,推导相应梯度、步长公式,新方法将反演中波场正传和残差波场反传过程转化为无约束优化问题,从理论上分析了新方法的计算效率显著高于常规FWI.在数值试验中,本文方法通过几次迭代便能获得高精度的正传、残差反传波场,收敛速度明显高于未经预处理的GMRES方法.进一步引入高效编码策略,新方法的计算时间约为常规编码FWI的1/8,与理论分析结果吻合(波场迭代次数为8,模型未知量个数约为7万),且波场迭代次数为6时,反演效果已与常规编码FWI相近.

The huge computational complexity is one of the major challenges in the application of full waveform inversion（FWI）technology.The most computationally expensive part of FWI is a forward modeling engine that numerically simulates wavefields.In this paper,an efficient iterative solver is proposed for wavefield continuation and applied to frequency-domain acoustic fullwaveform inversion with constant density.We reformulate the forward propagation of the source wavefield and the back propagation of the residual wavefield as unconstrained optimization problems.We also derive calculating formula for gradient and step-length.Theoreticall analysis indicates that the computational efficiency of the new method is significantly higher than that of the conventional FWI.Numerical examples prove that our method can obtain high-precision wavefields with several iterations and improve convergence rate compared with original GMRES method.With highefficiency source encoding technology,the calculation time of the new method is only about 1/8 of that of the conventional FWI,which is consistent with the theoretical analysis（The number of wavefield iteration is 8,and the number of unknown parameters is about 70 thousands）.Andwhen the number of wavefield iteration is 6,the inversion results of the new method is basically the same as that of the conventional FWI.