Contribution of vertically travelling plane S-waves to dynamic and static displacements near a finite fault

Representing seismic waves numerically by, for example, the discrete wavenumber method, we normally decompose waves into P-SV and SH components. This scheme, however, cannot treat the exactly vertically travelling S-wave element (VTSE), which corresponds to kx=ky= 0, whereas VTSE exists except for a few cases of fault geometry. In order to deal with VTSE from a point source, a new S-wave potential must be introduced, so that its polarization is restricted on a horizontal plane. The polarization angle of VTSE is a function of the fault geometry represented by dip, rake and strike. In this study, we first point out that VTSE from a finite fault cannot be calculated even if we use the above new S-wave potential, because the terms originated from fault finiteness become zero if kx=ky= 0. In order to remedy this difficulty for a finite source, we define a new VTSE potential for a rectangular fault. The contribution of VTSE is large for displacements in a very low frequency range such as static displacement. The static displacement due to VTSE does not depend on a fault depth or station locations. When we take a small but finite value for kx and ky instead of zero, say dk= 10-5, we can obtain practically accurate results even without the VTSE potential, including static displacements that have not been discussed in detail yet. Following the above scheme, accurate seismograms with both dynamic and static components can be simulated simultaneously for a fault of any configuration and station location.