A Fourth Order Accurate SH-Wave Staggered Grid Finite-difference Algorithm with Variable Grid Size and VGR-Stress Imaging Technique

Abstract

This article presents a new approach for the implementation of a planar-free surface boundary condition. It is based on a vertical grid-size reduction above the free surface during the explicit computation of a free surface boundary condition. This technique is very much similar to the well-known stress imaging technique. VGR-stress imaging technique name is proposed for this new free surface boundary condition (VGR stands for ‘vertical grid-size reduction’). To study the performance of the proposed VGR-stress imaging technique, it was implemented in a newly developed second order accurate in time and fourth-order accurate in space (2, 4) staggered grid SH-wave finite-difference (FD) algorithm with variable grid size. It was confirmed that the effective thickness (ETH) of first soil layer becomes less by one-half of vertical grid size than the assigned thickness (ATH), if stress imaging technique is used as a free surface boundary condition. The qualitative and quantitative results of various numerical experiments revealed that the proposed VGR-stress imaging technique is better than the stress imaging technique since it is free from the thickness discrepancy arising due to the use of images of stress components across the free surface. On the basis of iterative numerical experiments, it was confirmed that the stability condition for this FD scheme with variable grid size is $$ \frac{{V_{S} \Delta t}} {{\min (\Delta x,\Delta z)}} \le 0.71. $$ It was also inferred that at least five to six grid points per shortest wavelength are required to avoid the grid dispersion. The maximum grid-spacing ratio up to 12.5 or even more did not affect the accuracy of (2,4) SH-wave algorithm. The obtained reduction of 10.46 and 5.38 folds in the requirement of computational memory and time for a particular basin-edge model, as compared with the homogeneous grid size, reflects the efficacy of the new FD algorithm.