Spall may be a significant secondary source of seismic waves from underground explosions. The proper representation of spall as a seismic source is important for forward and inverse modeling of explosions for yield estimation and discrimination studies. We present a new derivation of a widely used point force representation for spall, which is based on a horizontal tension crack model. The derivation clarifies the relationship between point force and moment tensor representations of the tension crack. For wavelengths long compared with spall depth, the two representations are equivalent, and the moment tensor time history is proportional to the doubly integrated time history of the point force. Numerical experiments verify that, for regional seismic phases, this equivalence is valid for all frequencies for which the point-source (long wavelength) approximation is valid. Further analysis shows that the moment tensor and point force representations retain their validity for nonplanar spall surfaces, provided that the average dip of the surface is small. The equivalency of the two representations implies that a singular inverse problem will result from attempts to infer simultaneously the spectra of both of these source terms from seismic waveforms. If the spall moment tensor alone is estimated by inversion of waveform data, the inferred numerical values of its components will depend inversely upon the source depth that is assumed in the inversion formalism.

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