Despite the broad use of the heterogeneous finite-difference (FD) method for seismic waveform modeling, accurate treatment of material discontinuities inside the grid cells has been a serious problem for many years. One possible way to solve this problem is to introduce effective grid elastic moduli and densities (effective parameters) calculated by the volume harmonic averaging of elastic moduli and volume arithmetic averaging of densities in grid cells. This scheme enables us to place a material discontinuity in an arbitrary position in the spatial grids. Standard Earth models have made a significant contribution to synthetic seismogram calculations with a variety of numerical procedures such as the FD method. For the FD computation of seismic waveform with these models, we must first ensure accurate treatment of material discontinuities in the radius (or depth). The present paper introduces a FORTRAN subroutine ACE which calculates effective parameters analytically for an arbitrary spatial region in either the radius or depth direction for four major standard Earth models, namely, the PREM, IASP91, SP6, and AK135. This program is intended for all FD users who are concerned with seismic wave simulation for these models.
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