Accuracy of the Explicit Planar Free-Surface Boundary Condition Implemented in a Fourth-Order Staggered-Grid Velocity-Stress Finite-Difference Scheme

Abstract

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.

Imaging Near-surface Heterogeneities by Natural Migration of Back-scattered Surface Waves

“…For the numerical examples in this paper, source functions are bandpassed between 1 and 20 Hz, and the time interval is 0.3 ms. Synthetic ambient noise is generated by staggered-grid finite-difference simulations of the isotropic elastic wave equation (Virieux 1986) with a free-surface boundary condition (Gottschämmer & Olsen 2001).…”

Seismic Waves - Research and Analysis

“…The free surface matches the top side of the FD grid and the stress is forced to zero on the free surface (Gottschamer & Olsen, 2001). Alternatively, the method of image can be used to implement the free surface along a virtual plane located half a grid interval above the topside of the FD grid (Virieux, 1986).…”

Hybrid Broadband Ground-Motion Simulations: Combining Long-Period Deterministic Synthetics with High-Frequency Multiple S-to-S Backscattering

“…The Q values in the 3D velocity-density model are based on Q relations validated against data for the Northridge earthquake . Perfectly matched layers (Marcinkovich and Olsen, 2003) were used to implement absorbing boundary conditions on the sides and bottom of the grid, and a zerostress free-surface boundary condition was used at the top (Gottschämmer and Olsen, 2001). The code has been extensively validated for a wide range of problems, from simple point-sources in a half-space to complex ruptures on dipping faults in 3D crustal models (Day et al, 2008).…”