The effect of rupture velocity upon seismic first motions

Abstract

The effect of rupture velocity upon the first motion radiated by seismic sources is to multiply the radiation pattern by a scalar factor. This modulating function is F₁ = ζ/(1 − (ζ/c) cosψ) for unilateral dislocations and F₂ = 2ζ/(1 − (ζ/c)² cos²ψ) for bilateral dislocations; here ζ and c are the velocities of rupture and wave propagation respectively and ψ is the angle between the direction of rupture propagation and the radius vector to the observer. It is presumed that ζ > c. This result constitutes a correction to the first motion theories of Knopoff and Gilbert. The relation of the first motion to the complete response is shown by a sample calculation. It is concluded that the amplitude of the first half cycle of trace displacement on a seismogram is probably a valid measure of the first motion at least for major shallow-focus earthquakes.