Ph.D. student LAN Haiqiang and his teacher ZHANG Zhongjie simulate seismic wave fields in homogeneous and lateral heterogeneous models using six kinds of free-surface boundary condition approximate schemes and evaluate their stability and applicability by comparing with corresponding analytical solutions, and then quantitatively evaluate the accuracies of different approximate schemes from the misfit of the amplitude and phase between the numerical and analytical results.

Their results confirm that the composed scheme becomes unstable for the Vs/Vp ratio less than 0.57, and suggest that (1) the one-sided scheme is only accurate to first order and therefore introduces serious errors for the shorter wavelengths, other schemes are all of second-order precision; (2)the new composed, implicit and boundary-modified schemes are stable even when the Vs/Vp ratio is less than 0.2; (3) the implicit and boundary-modified schemes are able to deal with laterally varying (heterogeneous) free surface; (4) in the corresponding stability range, the one-sided scheme shows remarkable errors in both phase and amplitude compared to analytical solution (which means larger errors in travel-time and reflection strength), the other five approximate schemes show better performance in travel-time (phase) than strength (amplitude).

Lan and Zhang. Comparative study of free-surface boundary condition in two-dimensional finite-difference elastic wave-field simulation. Journal of Geophysics and Engineering, 2011, 8: 275–286（Download Here）

Figure. Accuracy of the investigated methods for simulation of the planar free surface of the homogeneous model. (Image by LAN)