Multipole radiation of seismic waves from explosions in nonspherical cavities and its application to signal identification
Authors: Ben-Menahem Ari , Mikhailov Oleg 
Published: October 2005

A new method, named ‘‘spherical mapping approximation’’ (SMA), is developed for the evaluation of displacement fields of body waves and surface waves from explosions in nonspherical cavities embedded in elastic media. Under SMA, the explosion-generated stress distribution on the surface of an arbitrary cavity is mapped onto the surface of an equivalent virtual spherical cavity having the volume of the true cavity. The analytical results express the displacement field in terms of a multipole double-sum expansion of spherical eigenvectors with coefficients in the form of a finite Legendre transform of the components of the normal vector of the cavity boundary. These ‘‘cavity integrals’’ can be evaluated exactly for spheroidal and cylindrical inclusions. In the long-wave far-field approximation, symmetric finite cavities are shown to be equivalent to a linear combination of point dipoles directed along the principal cavity axes.
The ensuing radiation patterns yield, in general, four-lobe patterns for S waves, two-lobe patterns for P waves, and single to two-lobe Rayleigh-wave pattern, independent of the details of the cavities’ shape. However, all radiation patterns are modulated by a frequency-dependent ‘‘cavity factor’’ that embodies the boundary conditions on the cavity surface. Moreover, it is shown that the radiation pattern for P waves from a nonspherical symmetrical cavity in the long-wave far-field approximation is always dipolar. Since the radiation pattern of radiated P waves from a standard earthquake is always quadrupolar, the cavity explosion behaves like a non-double-couple earthquake. Thus the examination of the deviatoric moment tensor of a given seismic event enables one, in principle, to state whether it is a standard earthquake or perhaps (if the S-wave pattern is quadrupolar) an evasion of the test-ban treaty. It is shown that SMA can be easily fitted to a multilayered elastic half-space if the equivalent source is placed in one of the layers. In that case, Love waves will be generated in addition to P, S, and Rayleigh waves. Displacement patterns for body and surface waves are calculated for spheroidal and cylindrical cavities for a wide range of aspects ratios and corresponding aperture angles, exhibiting the whole gamut of cavity shapes from a line source to a disk.
The moments of the equivalent dipoles are shown to depend on the corresponding cavity integrals, the elastic constants of the medium in the neighborhood of the source, and the initial energy injection. All nonspherical cavities generate strong shear waves except for special aperture angles at which a spherical P wave is generated, unaccompanied by S waves. The wave-spectra of body waves (surface waves) exhibit a corner frequency (peak frequency) at a wavelength equal to the radius of the equivalent sphere. This enables one to deduce the size of the cavity from the spectrum of its far-field displacement signals, provided that the explosion is fully decoupled and that the interaction of the shock wave with the medium occurred in the elastic regime. The results of the present research are applicable to the detection and identification of seismic signals from clandestine underground nuclear explosions.