A comparative study on the stress image and adaptive parameter-modified methods for implementing free surface boundary conditions in elastic wave numerical modeling

The stress image method is a simple and effective approach for implementing the free surface boundary conditions in elastic wave numerical modeling. This method assumes that the normal and shear stresses perpendicular to the free surface are antisymmetric with respect to the free surface. In this way, the values of the normal and shear stresses above the free surface can be updated. However, the stress image method is based on an intuitive viewpoint and lacks a physical foundation. The adaptive parameter-modified method is a recently proposed approach for implementing the free surface boundary conditions. Through adaptive modifications of the density and elastic parameters at the free surface, the implementation of the free surface boundary conditions is achieved. Based on the adaptive parameter-modified method, a new interpretation of the stress image method is used. We determine that the stress image method is equivalent to the adaptive parameter-modified method in terms of the staggered-grid finite-difference scheme for the elastic wave equation in displacement form. This result provides a physical foundation and explanation of the stress image method. Therefore, we can further develop the stress image method from a physical viewpoint. Numerical examples are also developed to perform the theoretical analysis.