Embedded Boundary Method를 이용한 불규칙한 지형에서의 감쇠 파동장 기반 초동주시 계산

Damped Wave Equation-based Traveltime Calculation using Embedded Boundary Method for Irregular Topography

Abstract

감쇠 파동장 기반 초동주시 계산은 기존 파선추적법의 단점을 보완하는 초동주시 계산기법이다. 이 기법은 주파수 영역 감쇠 파동장을 구해야 하기 때문에 수치 모델링이 필수적이다. 하지만 육상 탄성파 탐사 자료를 모사할 경우 불규칙한 지형을 고려해야 하기 때문에 계산 효율이 좋은 유한 차분법을 적용하기 어렵다. 불규칙한 지형을 정확하게 모사하기 위해 유한 요소법을 사용할 경우 계산량이 크게 증가한다는 단점이 있다. 이 논문에서는 이러한 문제점을 극복하기 위하여 embedded boundary method (EBM)를 유한 차분법에 적용하여 불규칙한 지형에서 초동주시를 계산하였다. 제안한 초동주시 계산기법의 정확도와 효율성을 확인하기 위하여 유한 요소법과 비교하였다. 수치 실험 결과 EBM을 적용한 초동주시 계산기법이 유한 요소법을 이용한 방법과 대등한 정확성을 보였고, 계산 효율은 더 향상되는 것을 확인할 수 있었다.

The first-arrival traveltime calculation method based on the damped wave equation overcomes the shortcomings of ray-tracing methods. Since this algorithm needs to solve the damped wave equation, numerical modeling is essential. However, it is not desirable to use the finite-difference method (FDM), which has good computational efficiency, for simulating the land seismic data because of irregular topography. Thus, the finite-element method (FEM) which requires higher computational cost than FDM has been used to correctly describe the irregular topography. In this study, we computed first-arrival traveltimes in an irregular topographic model using FDM incorporating embedded boundary method (EBM) to overcome this problem. To verify the accuracy and efficiency of the proposed algorithm, we compared our results with those of FEM. As a result, the proposed method using EBM not only provided the same accuracy as the FEM but also showed the improved computational efficiency.

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 (16.7^o). 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.