基于偏微分方程的定向插值在偏移成像中的应用

对于射线类偏移成像来说,求解射线追踪系统中所涉及的属性值不在网格节点上的插值计算问题是一个非常重要的环节,它影响到求解走时、路径和振幅信息的计算效率和精度,进而影响到整个偏移成像的质量和效率.本研究根据速度模型的空间梯度特点,考虑被插值点处速度的梯度在横向和纵向的分布特征,构建基于速度梯度空间变化的偏微分方程算法,将近几年发展起来的基于偏微分方程的定向插值算法引入到射线类偏移成像当中,实现射线追踪当中涉及的属性值不在网格节点上的插值计算.由于偏微分方程法本身固有的特性(局部特征不变性、解的唯一性和线性叠加性),因此,该算法可以实现不破坏原始速度模型空间梯度结构的非网格节点属性的插值计算.通过在常用的速度模型上的插值计算对比、不同速度模型上射线路径对比分析以及复杂介质模型上最后的偏移成像结果分析可以得出,应用基于速度梯度构建的偏微分方程插值算法在进行插值计算的过程当中可以实现不破坏原始速度模型空间速度梯度结构的属性计算,同时应用该算法可以最终提高射线类偏移成像的质量.

Directional interpolation of the velocity model based on partial differential equations used in migration imaging

Interpolation of velocity and velocity derivative at non-grid nodes of velocity model is an important part of ray-type migration imaging,which affects the efficiency and accuracy of ray tracing and the quality and efficiency of whole migration imaging.According to the characteristics of the velocity model and considering the gradient effect of the velocity at interpolated points,partial differential equation algorithm has been constructed based on the minimum velocity gradient of the interpolated points along the x-direction and the z-direction of velocity model.In this work,the directional interpolation algorithm based on such partial differential equations in recent years is introduced into ray type migration imaging to achieve the velocity and velocity derivative interpolation of non-grid nodes.Because the partial differential equation method has linear superposition characteristic,uniqueness of the model solution,the local features of retention,the PDE interpolation algorithm can be used to solve the velocity and velocity derivative interpolation problem of non-grid nodes based on the spatial structure of the original velocity model.Through interpolation of the Marmousi and Sigsbee velocity models and comparison of ray paths,interpolation results and migration imaging results of rough and steep slope rugged seabed velocity models,it can be concluded that the application of partial differential equation interpolation to velocity models can better maintain the edge characteristics of the velocity models and improve the quality of migration imaging.