Seismic Source Directivity from Doppler Effect Analysis, Part I: Theory - Caldeira, B (2004)
The directivity effects, a characteristic of finiteness seismic sources, are generated by the rupture in preferential directions. Those effects are manifested through different cadencies in the seismological measures from azimuthally distributed stations. The apparent durations are expressed as (e.g. Aki and Richards, 1980), (1), where L, v, c and ??are, respectively, the fault length, the rupture velocity, the wave velocity and the angle between rupture direction and ray. This time duration can be measured directly from waveform or indirectly from Relative Source Time Function (RSTF). Equation (1) is deduced from a simple source model (Haskell model) that considers unidirectional uniform rupture propagation and a homogeneous elastic isotropic media. If we consider a more general propagation model, with spherical concentric layers, we obtain (2), where p is the ray parameter and the earth radius. Similar equation can be obtained through physical considerations about a model composed by a sequence of subevents unilater- ally distributed along a line (Doppler Effect). Based on the same considerations we can do a more detailed analysis through (3), where is the time interval between 2 identified pulses in the rupture referential and j indicate the number of station. Based on this theory, we have developed a computational code DIRDOP (DIRectivity DOPpler effect) which determines the rupture direction and velocity from pulse durations observed in waveforms or RSTF. We used this code to analyse recent major seismic events including the unilateral 23 June, 1999 Arequipa (Peru, Mw=8.2) earthquake and the bilateral 21 May 2003 Boumerdes (Algeria, Mw=6.7) earthquake amongst others. The results are similar to those obtained by other methods.