An adaptive free-surface expression for three-dimensional finite-difference frequency-domain modelling of elastic wave

Finite-difference frequency-domain modelling of seismic wave propagation is attractive for its efficient solution of multisource problems, and this is crucial for full-waveform inversion and seismic imaging, especially in the three-dimensional seismic problem. However, implementing the free surface in the finite-difference method is nontrivial. Based on an average medium method and the limit theorem, we present an adaptive free-surface expression to describe the behaviour of wavefields at the free surface, and no extra work for the free-surface boundary condition is needed. Essentially, the proposed free-surface expression is a modification of density and constitutive relation at the free surface. In comparison with a direct difference approximate method of the free-surface boundary condition, this adaptive free-surface expression can produce more accurate and stable results for a broad range of Poisson's ratio. In addition, this expression has a good performance in handling the lateral variation of Poisson's ratio adaptively and without instability.