应用球型差分模板的低秩有限差分法纯qP波逆时偏移

Pure qP⁃wave reverse time migration using low rank finite difference method with spherical difference stencil

各向异性介质拟声波方程正演模拟及逆时偏移存在伪横波干扰及数值不稳定现象,成像质量不佳;同时计算效率也是各向异性逆时偏移的重要影响因素。为此,提出一种兼具成像质量和计算效率的纯qP波逆时偏移成像方法。首先,从波动方程伪解析解和各向异性精确频散关系出发,分别构建了VTI和TTI介质纯qP波伪解析解波场延拓公式,避免了推导显式波动方程,无需对频散关系做平方处理,由此消除了波场模拟存在的伪横波干扰;然后,通过设计三维球型或二维圆形差分模板,并借助低秩有限差分法求解与模型相适应的差分系数,推导了基于低秩有限差分的纯qP波波场延拓公式,在时间方向延拓时,无需进行多次Fourier变换,降低了计算复杂度;最后,在此基础上,利用GPU并行计算正、反向地震波场,提高了逆时偏移的成像效率。典型二维和三维模型试算结果表明:该方法能够消除伪横波干扰,保证计算过程稳定,在二维和三维各向异性介质中都具有较强的适应性和较高的计算效率。

Pseudo shear⁃wave artifacts and numerical instability exist in forward modeling and reverse time migration of quasi⁃acoustic equations for anisotropic media,producing poor imaging quality.Meanwhile,computational efficiency is an important factor in anisotropic reverse time migration.Therefore,a pure qP⁃wave reverse time migration method is proposed,where both imaging quality and computational efficiency are taken into account.Firstly,based on the pseudo⁃analytic solution of the wave equation and the exact anisotropic dispersion relation,the wave field continuation formula of the pseudo⁃analytic solution of pure qP⁃wave in VTI and TTI media is constructed respectively,which avoids the derivation of the explicit wave equation and does not need to square the dispersion relation,thus eliminating the pseudo⁃shear wave artifacts existing in the wave field simulation.Then,by designing a three⁃dimensional(3D)spherical or two⁃dimensional(2D)circular difference stencil and solving the diffe⁃rence coefficients appropriate to the model with the help of the low⁃rank finite difference method,a pure qP⁃wave field continuation formula based on low⁃rank finite difference is derived,which reduces the computational complexity by eliminating the need for multiple Fourier transforms during continuation along the time direction.On this basis,the GPU is used to calculate the forward and backward seismic wave fields in parallel,which improves the imaging efficiency of the reverse time migration.The experimental results of typical 2D and 3D models show that the proposed method can eliminate the pseudo shear⁃wave artifacts,ensure the stability of the calculation process,and have strong adaptability and high computational efficiency in both 2D and 3D anisotropic media.