Guided waves in three-dimensional structures, Geophys

A new formulation for the propagation of surface waves in three-dimensionally varying media is developed in terms of modal interactions. A variety of assumptions can be made about the nature of the modal ¢eld: a single set of reference modes, a set of local modes for the structure beneath a point, or a set of local modes for a laterally varying reference structure. Each modal contribution is represented locally as a spectrum of plane waves propagating in di¡erent directions in the horizontal plane. The in£uence of 3-D structure is included by allowing coupling between di¡erent modal branches and propagation directions. For anisotropic models, with allowance for attenuation, the treatment leads to a set of coupled 2-D partial di¡erential equations for the weight functions for di¡erent modal orders. The representation of the guided wave¢eld requires the inclusion of a full set of modes, so that, even for isotropic models, both Love and Rayleigh modes appear as di¡erent polarization states of the modal spectrum. The coupling equations describe the interaction between the di¡erent polarizations induced by the presence of the 3-D structure. The level of lateral variation within the 3-D model is not required to be small. Horizontal refraction or re£ection of the surface wave¢eld can be included by allowing for transfer between modes travelling in di¡erent directions. Approximate forms of the coupled equation system can be employed when the level of heterogeneity is small, for example the coupling between the fundamental mode and higher modes can often be neglected, or forward propagation can be emphasized by restricting the interaction to a limited band of plane waves covering the expected direction of propagation. Key words: 3-D structure, anisotropy, surface waves.