Accuracy of the explicit planar free-surface boundary condition implemented in a fourth-order staggered-grid velocity-stress finite-difference scheme

We compute the accuracy of two implementations of the explicit planar free-surface boundary condition for 3D fourth-order velocity-stress staggered-grid finite differences, 1/2 grid apart vertically, in a uniform half-space. Due to the staggered grid, the closest distance between the free surface and some wave-field components for both implementations is 1/2-grid spacing, Overall, the differences in accuracy of the two implementations are small. When compared to a reflectivity solution computed at the staggered positions closest to the surface, the total misfit for all three components of the wave field is generally found to be larger for the free surface colocated with the normal stresses, compared to that for the free surface colocated with the xz and yz stresses. However, this trend is reversed when compared to the reflectivity solution exactly at the free surface (the misfit encountered in staggered-grid modeling). When the wave field is averaged across the free surface, thereby centering the staggered wave field exactly on the free surface, the free-surface condition colocated with the xz and yz stresses generates the smallest total misfit for increasing epicentral distance. For an epicentral distance/hypocentral depth of 10, the total misfit of this condition is about 15% smaller than that for the condition colocated with the normal stresses, mainly controlled by the misfit on the Rayleigh wave.