Directional interpolation of the velocity model based on partial differential equations used in migration imaging
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摘要:
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.
Abstract: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.
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图 1
基于速度梯度的偏微分方程插值示意图
Figure 1.
Schematic diagram of partial differential equations interpolation
图 2
Marmousi速度模型以及应用不同插值算法插值后的速度模型
Figure 2.
Original Marmousi velocity model and its interpolation results using cubic and PDE algorithms
图 3
(a) 原速度模型局部区域放大；(b) 卷积插值后局部区域放大；(c) 偏微分方程插值后区域局部放大
Figure 3.
(a) Partially enlarged area of original Marmousi model; (b) Partially enlarged red area of interpolated Marmousi model used cubic; (c) Partially enlarged red area of interpolated Marmousi model used PDE
图 4
Sigsbee模型以及应用不同插值算法插值后的模型
Figure 4.
Original Sigsbee velocity model and its interpolation results using cubic and PDE algorithms
图 5
(a) 海底起伏速度模型图；(b) 射线路径示意图，黑色实线为二维三次卷积插值射线路径，虚线为偏微分方程插值射线路径
Figure 5.
(a) Velocity model of seafloor relief; (b) Sketch of ray paths. The solid black line is from two-dimensional cubic convolution interpolation, and dotted line is from PDE interpolation
图 6
(a) 倾斜海底层速度模型；(b) 射线路径示意图，黑色实线为二维三次卷积插值射线路径，虚线为偏微分方程插值射线路径
Figure 6.
(a) Velocity model of inclined seafloor layer; (b) Sketch of ray paths. The solid black line is from cubic convolution interpolation, and the dotted line is from PDE interpolation
图 7
(a) Marmousi速度模型图；(b) 射线路径示意图，黑色实线为二维三次卷积插值射线路径，虚线为偏微分方程插值射线路径
Figure 7.
(a) Marmousi velocity model; (b) Sketch of ray paths. The solid black line is form two-dimensional cubic convolution interpolation, and the dotted line is from PDE interpolation
图 8
(a) 复杂的大陡坡速度模型；(b) 卷积类插值光滑处理30次后大陡坡速度模型偏移结果；(c) PDE插值光滑处理30次后大陡坡速度模型偏移结果
Figure 8.
(a) Velocity model of big complex steep slope; (b) Migration results of the velocity model after smoothing with two-dimensional cubic convolution interpolation; (c) Migration results of the velocity model after smoothing with PDE interpolation
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Alvarez L, Lions P L, Morel J M. 1992. Image selective smoothing and edge detection by nonlinear diffusion. Ⅱ. SIAM Journal on Numerical Analysis, 29(3): 845-866. doi: 10.1137/0729052
Catté F, Lions P L, Morel J M, et al. 1992. Image selective smoothing and edge detection by nonlinear diffusion. SIAM Journal on Numerical Analysis, 29(1): 182-193. doi: 10.1137/0729012
Di Q Y, Tian F, Suo Y, et al. 2021. Linkage of deep lithospheric structures to intraplate earthquakes: A perspective from multi-source and multi-scale geophysical data in the South China Block. Earth-Science Reviews, 214(1): 103504, doi:10.1016/j.earscirev.2021.103504.
Gao C, Sun J G, Qi P, et al. 2015. 2-D Gaussian-beam migration of common-shot records in time domain. Chinese Journal of Geophysics (in Chinese), 58(4): 1333-1340, doi:10.6038/cjg20150420.
Gao Z H, Sun J G, Sun X, et al. 2017. A fast algorithm for depth migration by the Gaussian beam summation method. Journal of Geophysics and Engineering, 14(1): 173-183. doi: 10.1088/1742-2140/aa52dd
Han F X. 2009. On some computational problems in wavefront construction method[Ph. D. thesis] (in Chinese). Changchun: Jilin University.
Han F X, Sun J G, Yang H. 2008a. Ray-Tracing of wavefront construction by bicubic convolution interpolation. Journal of Jilin University (Earth Science Edition) (in Chinese), 38(2): 336-340, 346. http://en.cnki.com.cn/Article_en/CJFDTOTAL-CCDZ200802028.htm
Han F X, Sun J G, Yang H. 2008b. Interpolation algorithms in wavefront construction. Chinese Journal of Computational Physics (in Chinese), 25(2): 197-202. http://www.researchgate.net/publication/292657649_Interpolation_algorithms_in_wavefront_construction
Han J G, Wang Y, Chen C X, et al. 2017. Multiwave Gaussian beam prestack depth migration for 2D anisotropic media. Progress in Geophysics (in Chinese), 32(3): 1130-1139, doi:10.6038/pg20170324.
Hill N R. 1990. Gaussian beam migration. Geophysics, 55(11): 1416-1428. doi: 10.1190/1.1442788
Hill N R. 2001. Prestack Gaussian-beam depth migration. Geophysics, 66(4): 1240-1250. doi: 10.1190/1.1487071
Jiang H, Moloney C. 2002a. A new direction adaptive scheme for image interpolation.//Proceedings. International Conference on Image Processing. Rochester, NY, USA, USA: IEEE.
Jiang H, Moloney C. 2002b. Error concealment using a diffusion based method.//Proceedings. International Conference on Image Processing. Rochester, NY, USA, USA: IEEE.
Keys R. 1981. Cubic convolution interpolation for digital image processing. IEEE Transactions on Acoustics, Speech, and Signal Processing, 29(6): 1153-1160. doi: 10.1109/TASSP.1981.1163711
Munk W H. 1974. Sound channel in an exponentially stratified ocean, with application to SOFAR. The Journal of the Acoustical Society of America, 55(2): 220-226. doi: 10.1121/1.1914492
Perona P, Malik J. 1990. Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(7): 629-639. doi: 10.1109/34.56205
Wang D R, Yang Z H. 1990. Numerical Approximation Introduction (in Chinese). Beijing: Higher Education Press, 314-338.
Wu J Y. 2009. Research on partial differential equation-based methods for image processing[Ph. D. thesis] (in Chinese). Beijing: Beijing Jiaotong University.
Xiao Y N. 2005 Adaptive digital image interpolation and its application[Master's thesis] (in Chinese). Chongqing: Chongqing University.
Yang S S, Huang J P, Li Z C, et al. 2015. The review of imaging methods based on gaussian beam. Progress in Geophysics (in Chinese), 30(3): 1235-1242, doi:10.6038/pg20150332.
Yuan M L, Huang J P, Deng C, et al. 2016. Gaussian-beam based diagonal Hessian for amplitude-preserved imaging. Progress in Geophysics (in Chinese), 31(3): 1268-1273, doi:10.6038/pg20160346.
Yue Y B. 2011. Study on Gaussian beam migration methods in complex medium[Ph. D. thesis] (in Chinese). Qingdao: China University of Petroleum (East China).
高成, 孙建国, 齐鹏等. 2015. 2D共炮时间域高斯波束偏移. 地球物理学报, 58(4): 1333-1340, doi:10.6038/cjg20150420. http://www.geophy.cn//CN/abstract/abstract11395.shtml
韩复兴. 2009. 论波前构建法中的几个计算问题[博士论文]. 长春: 吉林大学.
韩复兴, 孙建国, 杨昊. 2008a. 基于二维三次卷积插值算法的波前构建射线追踪. 吉林大学学报(地球科学版), 38(2): 336-340, 346. https://www.cnki.com.cn/Article/CJFDTOTAL-CCDZ200802028.htm
韩复兴, 孙建国, 杨昊. 2008b. 不同插值算法在波前构建射线追踪中的应用与对比. 计算物理, 25(2): 197-202. https://www.cnki.com.cn/Article/CJFDTOTAL-JSWL200802013.htm
韩建光, 王赟, 陈传绪等. 2017. 二维各向异性多波高斯束叠前深度偏移. 地球物理学进展, 32(3): 1130-1139, doi:10.6038/pg20170324.
王德人, 杨忠华. 1990. 数值逼近引论. 北京: 高等教育出版社, 314-338.
仵冀颖. 2009. 偏微分方程在图像处理中应用的研究[博士论文]. 北京: 北京交通大学.
肖义男. 2005. 数字图像自适应插值技术及其应用研究[硕士论文]. 重庆: 重庆大学.
杨珊珊, 黄建平, 李振春等. 2015. 高斯束成像方法研究进展综述. 地球物理学进展, 30(3): 1235-1242, doi:10.6038/pg20150332.
袁茂林, 黄建平, 邓聪等. 2016. 基于高斯束的对角Hessian在保幅成像中的应用. 地球物理学进展, 31(3): 1268-1273, doi:10.6038/pg20160346.
岳玉波. 2011. 复杂介质高斯束偏移成像方法研究[博士论文]. 青岛: 中国石油大学(华东).
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