Potentials and displacements for two theoretical seismic sources

Theoretical, P, S V, and SH displacement potentials and displacements for a double couple or point shear dislocation source and for a 'mixedquadrupole' source at any arbitrary orientation in an isotropic homogeneous elastic space are expressed as multiple integral and derivative operations on the source history in the time domain and their algebraic equivalent in the frequency domain. These sources have the same angle orientation functions , which are given explicitly. The double couple and 'mixed quadrupole' are both quadrupole sources but, unlike the double couple, the P and S waves from a 'mixed quadrupole' have different source histories. Analytic displacements are obtained using as examples the Ohnaka shear dislocation history for a double couple and the Randall and Archambeau tectonic release histories for 'mixed quadrupole ' sources. The displacement fields are investigated numerically, in order to establish a criterion for estimating the minimum range for applying far-field theory results to the total displacement field. The chosen criterion is the ratio of the far-field peak amplitude, which is a function of source rise or duration time, to the static displacement, which is a near-field phenomenon . The proposed criterion is found to be conservative as to the minimum range for the farfield , predicted ( 1 /R) dependence of the total field peak amplitude, but quite satisfactory for time domain estimates of moment and corner frequency based on far-field theory .