Mixed staggered grid finite difference numerical simulation and reverse time migration of velocity-stress acoustic equation
HU Ziduo1,2, LIU Wei1,2, SONG Jiawen3, ZENG Qingcai4, TIAN Yancan1,2, HAN Linghe1,2
1. PetroChina Research Institute of Petroleum Exploration & Development-Northwest, Lanzhou, Gansu 730020, China; 2. PetroChina Key Laboratory of Reservoir Description, Lanzhou, Gansu 730020, China; 3. GRI, BGP Inc., CNPC, Zhuozhou, Hebei 072750, China; 4. PetroChina Research Institute of Petroleum Exploration and Development, Beijing 100083, China
Abstract：The conventional staggered grid finite difference method (C-SFD) is widely used to simulate the velocity-stress acoustic equation. Although the spatial difference operator can reach the order of 2M difference accuracy (M represents the number of sets of axial grid points equidistant from the center point of the spatial difference operator), the discretized difference acoustic equation only has the second-order difference accuracy, resulting in low simulation accuracy and poor stability. In this paper, the axial grid points and off-axial grid points are used to construct the spatial difference operator to approximate the first-order spatial partial derivative, and a mixed staggered grid finite difference method (M-SFD) suitable for the velocity-stress acoustic equation simulation is constructed. Based on the time-space dispersion relationship and Taylor series expansion, the difference coefficient solution equations are established, and the analytical solution of the difference coefficient is derived. The discretized difference acoustic equation given by M-SFD can reach 4th-order, 6th-order, 8th-order, or even any even order difference accuracy. The dispersion analysis shows that under the condition of specific Courant condition number value (such as Courant Condition number r=0.3), C-SFD cannot control the numerical dispersion error within 1% by adjusting the value of M. By adjusting the values of M and N(N represents the number of sets of off-axial grid points equidistant from the center point of the spatial difference operator), M-SFD can basically control the dispersion error within 1‰ or even within 0.1‰. Stability analysis shows that when M values are the same, the stability of M-SFD is stronger than that of C-SFD. Numerical simulation examples show that when the computational efficiency is almost the same, M-SFD can more effectively suppress numerical dispersion than C-SFD, resulting in higher simulation accuracy. M-SFD can also use a larger time sampling interval than C-SFD to achieve higher computational efficiency while holding higher simulation accuracy. This paper further extends M-SFD to inverse time migration. As a wave field propagation operator in inverse time migration, M-SFD can effectively eliminate imaging artifacts caused by numerical dispersion, thereby improving the imaging accuracy and resolution of deep structures.
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