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For far-field wave-motion response to a point dynamic dislocation source, the temporal and spatial features of the source mechanism are characterized, respectively, by two factors, i.e., a source time function for dislocation growth and a combination of nine couples of impulse forces that is equivalent to the final dislocation. The mathematical representation for each of the couples, referred to as spatial couples, is a couple of impulses acting in opposing directions with an infinitesimal separation distance or, in the limit, by the derivative of the impulse with respect to the separation-distance parameter. This study proposes a temporal-spatial pulse representation for the nine couples, referred to as temporal-spatial couples, and subsequently for the dislocation source modeling. Each temporal-spatial couple consists of two impulses acting in opposite directions with both an infinitesimal separation distance and an infinitesimal time delay. By examining dynamite source modeling, this study shows that the proposed representation can intrinsically integrate the spatial and temporal features of the dislocation sources from the response point of view. This study also shows an example of a point, shear-slip seismic source modeling using traditional and proposed pulse representations for far-field wave motion. Discussion is finally provided for the implications of the proposed representation in broad applications.

Seismic waves in a layered half-space with lateral inhomogeneities, generated by a buried seismic dislocation source, are investigated in these two consecutive papers. In the first paper, the problem is formulated and a corresponding approach to solve the problem is provided. Specifically, the elastic parameters in the laterally inhomogeneous layer, such as P and S wave speeds and density, are separated by the mean and the deviation parts. The mean part is constant while the deviation part, which is much smaller compared to the mean part, is a function of lateral coordinates. Using the first-order perturbation approach, it is shown that the total wave field may be obtained as a superposition of the mean wave field and the scattered wave field. The mean wave field is obtainable as a response solution for a perfectly layered half-space (without lateral inhomogeneities) subjected to a buried seismic dislocation source. The scattered wave field is obtained as a response solution for the same layered half-space as used in the mean wave field, but is subjected to the equivalent fictitious distributed body forces that mathematically replace the lateral inhomogeneities. These fictitious body forces have the same effects as the existence of lateral inhomogeneities and can be evaluated as a function of the inhomogeneity parameters and the mean wave fleld. The explicit expressions for the responses in both the mean and the scattered wave fields are derived with the aid of the integral transform approach and wave propagation analysis.

Seismic wave scattering representation for the layered half-space with lateral inhomogeneities subjected to a seismic dislocation source has been formulated in the companion paper with the use of first-order perturbation (Born-type approximation) technique. The total wave field is obtained as a superposition of the mean and the scattered wave fields, which are generated, respectively, by a series of double couples of body forces equivalent to the seismic dislocation source and by fictitious body forces equivalent to the existence of the lateral inhomogeneities in the layered half-space. The responses in both the mean and the scattered wave fields are found with the aid of an integral transform technique and wave propagation analysis. The characteristics of the scattered waves and their effects on the mean waves or corresponding induced ground and/or underground mean responses are investigated in this paper. In particular, coupling phenomena between P-SV and SH waves and wave number shifting effects between the mean and the scattered wave responses are presented in detail. With the lateral inhomogeneities being assumed as a homogeneous random field, a qualitative analysis is provided for estimating the effects of the lateral inhomogeneities on the ground motion, which is related to a fundamental issue: whether a real earth medium can or cannot be approximately considered as a laterally homogeneous layer. The effects of the lateral inhomogeneities on the ground motion time history are also presented as a quantitative analysis. Finally, a numerical example is carried out for illustration purposes.