Nearly Optimal Acoustic and Elastic Boundary Conditions

Seismic Modeling needs effective boundary conditions to model the free surface and to limit the size of the computational model. The typical three dimensional seismic model is large and would exceed the size of available computer memories without some effective boundary on the sides and bottom of the model. The reflection coefficient of the surface with the air water/land interface is considered strong given the vast differences in densities. The finite nature of the sides and bottom cause the real problem. Numerous numerical approaches have been proposed for creating these artificial boundaries. Among them are the one-way wave equations and the sponge damping zones. First order hyperbolic wave equations have waves moving in one direction. To truly model the propagation of acoustic waves and all the internal reflections the second order wave equation is needed. However, at the boundary edge what we need is a transmission of energy out of the model and no reflection of energy back into the model. Because the finite model is truncated at the boundary we will not generate any of the energy which ought to really be reflected back into the model region from the missing external regions. This is a source of error in our modeling process, an omission error. We have to expect that the effective region of interest is far enough inside the model that the recorded surface or well data is sufficiently accurate in representing reflection data. One-way wave equations result from factoring the full wave equation into two first order systems. This was the technique of Reynolds as documented in his 1976 paper. He factored the wave equation and computed results using two forms of his scheme. His second form was more successful and presented without comment. The sponge or damping zone was presented by Cerjan in 1982 and he showed a scheme that computed a gently tapered a set of weights that were applied to the edges of model wave fields. In this scheme the formula was presented without any rigorous development. By comparison of these two methods it is possible developed a mathematical relationship between the sponge numerics and the one way angle dependent wave equation as developed by Higdon. Further, by using an energy measure it can be shown that the two parameter space of the sponge method has one or more local minima. Using these local minima parameters the effectiveness of the sponge method is enhanced to the point where the boundary effects are reduced to the noise level of the data. Results for 2D and 3D acoustic and 2D elastic wave codes are used to demonstrate the benefits of this nearly optimal sponge boundary condition.