Fig. 1. Grid structures of FDM modeling scheme in irregular topography. (a) An example of Cartesian grid and a difference stencil for 9-point FDM. The black line indicates free surface with irregular topography. The green line indicates the assumed free surface in the standard FDM and the red points indicate the nodal points with free surface conditions. (b) An example of grid points used for EBM. The red line and points are related to extrapolation and the blue line and points are related to interpolation.
Fig. 2. Modeling results using 9-point FDM and FEM in a homogeneous velocity model with irregular topography. (a)-(d): Snapshots of FDM at 0.04 s, 0.28 s, 0.52 s and 0.76 s, respectively. (e)-(f): Snapshots of FEM at 0.04 s, 0.28 s, 0.52 s and 0.76 s, respectively. (i)-(l): Wavefields extracted from the snapshots along the horizontal line at the depth of source location (0.04 s, 0.28 s, 0.52 s and 0.76 s, respectively). The blue solid lines indicate FDM results and the orange dashed lines indicate FEM results.
Fig. 3. Homogeneous velocity model having a tilted flat surface (16.7o). The red box indicates enlarged grid structure for FDM.
Fig. 4. Calculated traveltimes in the velocity model shown in Fig. 3: The dashed contour lines indicate the traveltimes calculated by EBM and the solid contour lines indicate the analytical traveltimes.
Fig. 6. Comparison of the traveltimes calculated by EBM and FEMs with structured and unstructured meshes. (a) Homogeneous velocity model with a flat surface. Yellow star indicates the shot location. The p-wave velocity is 4000 m/s. (b) Traveltime error curves between analytical solution and respective numerical results. The blue, orange and green lines indicate the traveltime errors for EBM, FEM with structured mesh, and FEM with unstructured mesh, respectively.
Fig. 7. Modified 2D Canadian Foothills velocity model. The actual velocity of the air layer is 330 m/s, but it is represented as blue color (complementary color for red or orange) to clearly distinguish the free surface boundary.
Fig. 8. Results of traveltime calculation using the 2D Canadian Foothills velocity model shown in Fig. 7: (a) Contour map of traveltimes calculated by EBM. (b) Traveltime curves calculated with various source locations (1.5 km, 3.75 km, 6 km, 8.25 km and 10.5 km).
Fig. 9. Comparison of computational time between EBM (blue) and FEM with unstructured mesh (orange). The graphs show the elapsed times when the grid spacing are (a) 15 m and (b) 5 m, respectively. In the case of FEM, the grid spacing means the reference grid spacing for mesh generation.
Fig. 5. (a) Comparison of traveltime curves. The blue solid line indicates the traveltime curve computed by EBM, the red dotted line indicates staircase discretization and the orange dashed line is the analytical solution. (b) Traveltime residuals between analytical solutions and EBM results. (c) Traveltime residuals between analytical solutions and staircase discretization results.