分层坐标变换法起伏自由地表弹性波叠前逆时偏移

传统有限差分方法在处理起伏地表时存在一些困难,而坐标变换法可将起伏地表映射为水平地表以克服此缺点.但同时,地下构造被变换得更加复杂,导致了波传播和成像的不准确.本文提出了一种分层的坐标变换方法,并应用到了弹性波逆时偏移中,此方法既可以克服起伏地表的影响,又可以不破坏地下构造.波场正向延拓、逆时延拓和分离是在辅助坐标系下完成的,而成像是在笛卡尔坐标系下完成的.通过对简单起伏模型和中原起伏模型的试算证明了本文提出方法的准确性.同时,对两种极端起伏地层高程不准确的情况进行测试可以看出:分层坐标变换逆时偏移方法的成像效果远好于传统坐标变换方法.

Elastic wave modeling and pre-stack reverse time migration of irregular free-surface based on layered mapping method

The traditional Finite Difference(FD)method has some difficulties in calculating synthetic waveforms for the velocity with an irregular surface, while the mapping method is verified to be able to overcome such disadvantage by transforming the irregular surface into horizontal one. However, the interfaces below the surface are also transformed to more complex ones, which are possible to generate the inaccuracy during wave propagation and imaging. To solve these problems from the irregular surface velocity model, we propose a new layered mapping operator for the FD, and then extend it to develop a new elastic wave forward modeling and reverse time migration(RTM)method.The basic idea of the layered mapping method is to discrete each layer into curved grids, and then mapping them to rectangular grids, which can transform the irregular borders to horizontal ones. When the curved grids are transformed to rectangular ones, the differential equations will transform from the physical domain to the calculation domain at the same time. During the process of elastic RTM, the layered mapping method should be used both in the forward propagation and reverse time propagation of the wave field on the source and receiver sides, respectively. The input initial velocity is the one after the same mapping process. Wave field forward continuation, reverse time continuation and separation are implemented in an auxiliary system coordinate while imaging in Cartesian coordinates. A velocity model with four layers with an irregular surface is used to calculate the forward modeling synthetic waveforms. By comparing the snapshots from the traditional mapping method with the layered mapping method proposed in this paper, we can find that our methods can treat the irregular surface well, while the results from the traditional mapping method may generate some artifacts. The snapshots after separating the wave field are displayed, which shows that P-wave and S-wave are separated clearly. Then the elastic RTM using the layered mapping method is tested. The imaging results suggest that the PP and PS images are both very accurate, and the events are very clear with high SNR. At last, a complex surface topography model is used to test the adaptability and robustness of the method. The imaging results show that the energy of the PS image is weaker than that of the PP image, especially in deep layer, while the imaging range of the PS image is wider than that of the PP image. Deep reflection interfaces are imaged very well with high SNR. Compared with the traditional mapping RTM, the imaging results using the layered mapping method are much better in accuracy and SNR. In practice, the elevation near the surface may not be obtained accurately, so two extreme inaccurate cases are used to test the flexibility of the layered mapping method, which suggests that when the extreme inaccurate elevations are used, the transformation results are much better than that using the traditional mapping method and the images are also better.We proposed a new layered mapping operator which is further applied to the elastic reverse time migration method for an irregular surface. After the numeric synthetic tests for several standard velocity models, we can draw the conclusions as follows:(1)The layered mapping method has the advantage of traditional finite difference methods(fast computing speed and low memory), and can overcome the disadvantage in treating the irregular surface.(2)We extend the layered mapping method to the elastic reverse time migration for the irregular surface. This method images better near the surface and in the deep subsurface.(3)The layered mapping method can be used only in the region near the surface, while in the deep subsurface using rectangular grids.(4)Even if the elevations of interfaces near the surface cannot be obtained accurately, the imaging results are also better.(5)The layered mapping RTM method proposed in this paper can be easily extended to FWI, LSRTM and some other methods that need huge computation cost and memory storage. However, the proposed method has a tougher stable condition, and the computational efficiency is still low as traditional elastic RTM. So in the following study, the algorithm should be improved to increase the stability and GPU technology and dual-variable algorithm will be applied to improve the computational efficiency.