三维一阶速度-应力声波方程全波形反演

基于时间域三维一阶速度—应力声波方程，采用基于摄动理论的伴随状态法，推导出时间域三维一阶速度—应力声波方程的伴随方程及相应的纵波速度的梯度计算公式，并采用交错网格有限差分法计算正向传播波场及逆时外推的伴随波场，进而计算出纵波速度的梯度，在此基础上采用共轭梯度法更新纵波速度模型，实现了基于一阶速度—应力声波方程的三维全波形反演方法。由于基于一阶速度—应力声波方程，该方法能够方便地采用交错网格有限差分法求解。由于基于GPU在时间域实现，波场正向传播及逆时外推时较直接、快速。模型测试结果证明了方法的可行性和有效性。

Three-dimensional full waveform inversion based on the first-order velocity-stress acoustic wave equation

We propose in this paper a three-dimensional full waveform inversion based on the first-order velocity-stress acoustic wave equation.First based on the time-domain three-dimensional first-order velocity-stress acoustic wave equation,an adjoint equation of the time-domain three-dimensional first-order velocity-stress acoustic wave equation and a corresponding gradient equation of P-wave velocity are derived with the adjoint-state method based on the perturbation theory.Then the staggered grid finite difference method is adopted to obtain the synthetic forward wavefield and the backward adjoint wavefield in the time domain.After that the gradient of the P-wave velocity is calculated.On this basis,the conjugate gradient method is used to update P-wave velocity model.Finally the 3D full waveform inversion method based on the first-order velocity-stress acoustic wave equation is implemented.Because this method is based on the first-order velocity-stress acoustic wave equation,the staggered grid finite difference method can be easily applied,and the synthetic wavefield and adjoint wavefield are directly and quickly extrapolated based on GPU in the time domain.The feasibility and effectiveness of the proposed method are demonstrated by model tests.