Seismic Source Radiation and Moment Tensor in the Time Domain

Abstract

Seismic waves in an elastic medium are produced by an inelastic deformation at the seismic source. Let us assume that the seismic source occupies the volume V, in which the body forces f are present, and in which the jump of displacement takes place on the surface Σ, composed of a positive side, Σ+, and a negative side, Σ−, with the unit vector, v, normal to the source surface and pointing outwards. If the jump of displacement on the surface Σ, when crossing it from Σ+ to Σ− is [u(ζ, τ)] at point ζ on the source and at time τ, then the n components of the displacement outside the source at a point x = x(x₁,x₂,x₃), can be expressed in the general form by the representation theorem (Aki and Richards, 1980), assuming that sources of traction are not allowed:

$$ {u_n}\left( {x,t} \right) = \int\limits_{ - \infty }^\infty {d\tau \int {\int\limits_V {\int {{f_p}\left( {\eta, \tau } \right){G_{np}}\left( {x,t - \tau; \eta, 0} \right)dV\left( \eta \right) + } } } } \int\limits_{ - \infty }^\infty {d\tau \int\limits_\Sigma {\int {\left[ {{u_i}\left( {\zeta, \tau } \right)} \right]{c_{ijpq}}{\nu_j}\partial {G_{np}}\left( {x,t - \tau; \zeta, 0} \right)/\partial {\zeta_q}d\Sigma \left( \zeta \right)} } } $$

(7.1)

In the above formula G is a Green function, and the tensor c _ijkl describes material constants. In the case of an isotropic body these values reduce to only two Lamé constants. In elastodynamics the Green function can be described as the displacement at the observation point x caused by the unit force applied at the source point ζ. The Green function is a tensor because forces and displacements are three dimensional vectors. The ‘np’ component of the Green function, G(x,t; ζ,τ), is a k-component of the displacement at the observation point x at moment t, caused by the unit force component n, applied at the source point ζ at moment τ.