Comparative study of the free-surface boundary condition in two-dimensional finite-difference elastic wave field simulation

2011 ◽  
Vol 8 (2) ◽  
pp. 275-286 ◽  
Author(s):  
Haiqiang Lan ◽  
Zhongjie Zhang
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Finite Difference ◽  
Elastic Wave ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Field Simulation ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition ◽  
Wave Field Simulation

scholarly journals An improved vacuum formulation for 2D finite-difference modeling of Rayleigh waves including surface topography and internal discontinuities

Geophysics ◽  
2012 ◽  
Vol 77 (1) ◽  
pp. T1-T9 ◽  
Author(s):  
Chong Zeng ◽  
Jianghai Xia ◽  
Richard D. Miller ◽  
Georgios P. Tsoflias
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Finite Difference ◽  
Surface Topography ◽  
Rayleigh Waves ◽  
Surface Boundary ◽  
Near Surface ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition ◽  
Grid Nodes

Rayleigh waves are generated along the free surface and their propagation can be strongly influenced by surface topography. Modeling of Rayleigh waves in the near surface in the presence of topography is fundamental to the study of surface waves in environmental and engineering geophysics. For simulation of Rayleigh waves, the traction-free boundary condition needs to be satisfied on the free surface. A vacuum formulation naturally incorporates surface topography in finite-difference (FD) modeling by treating the surface grid nodes as the internal grid nodes. However, the conventional vacuum formulation does not completely fulfill the free-surface boundary condition and becomes unstable for modeling using high-order FD operators. We developed a stable vacuum formulation that fully satisfies the free-surface boundary condition by choosing an appropriate combination of the staggered-grid form and a parameter-averaging scheme. The elastic parameters on the topographic free surface are updated with exactly the same treatment as internal grid nodes. The improved vacuum formulation can accurately and stably simulate Rayleigh waves along the topographic surface for homogeneous and heterogeneous elastic models with high Poisson’s ratios ([Formula: see text]). This method requires fewer grid points per wavelength than the stress-image-based methods. Internal discontinuities in a model can be handled without modification of the algorithm. Only minor changes are required to implement the improved vacuum formulation in existing 2D FD modeling codes.

scholarly journals A stable free‐surface boundary condition for two‐dimensional elastic finite‐difference wave simulation

Geophysics ◽  
1986 ◽  
Vol 51 (12) ◽  
pp. 2247-2249 ◽  
Author(s):  
John E. Vidale ◽  
Robert W. Clayton
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Finite Difference ◽  
Difference Approximation ◽  
Surface Boundary ◽  
Stability Range ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition ◽  
The One ◽  
Lateral Variations

Two of the persistent problems in finite‐difference solutions of the elastic wave equation are the limited stability range of the free‐surface boundary condition and the boundary condition’s treatment of lateral variations in velocity and density. The centered‐difference approximation presented by Alterman and Karal (1968), for example, remains stable only for β/α greater than 0.30, where β and α are the shear [Formula: see text] and compressional [Formula: see text] wave velocities. The one‐sided approximation (Alterman and Rotenberg, 1969) and composed approximation (Ilan et al., 1975) have similar restrictions. The revised‐composed approximation of Ilan and Loewenthal (1976) overcomes this restriction, but cannot handle laterally varying media properly.

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