Generalized Born scattering of elastic waves in 3-D media

It is well known that when a seismic wave propagates through an elastic medium with gradients in the parameters which describe it (e.g. slowness and density), energy is scattered from the incident wave generating low-frequency partial reflections. Many approximate solutions to the wave equation, e.g. geometrical ray theory (GRT), Maslov theory and Gaussian beams, do not model these signals. The problem of describing partial reflections in 1-D media has been extensively studied in the seismic literature and considerable progress has been made using iterative techniques based on WKBJ, Airy or Langer type ansätze. In this paper we derive a first-order scattering formalism to describe partial reflections in 3-D media. The correction term describing the scattered energy is developed as a volume integral over terms dependent upon the first spatial derivatives (gradients) of the parameters describing the medium and the solution. The relationship we derive could, in principle, be used as the basis for an iterative scheme but the computational expense, particularly for elastic media, will usually prohibit this approach. The result we obtain is closely related to the usual Born approximation, but differs in that the scattering term is not derived from a perturbation to a background model, but rather from the error in an approximate Green's function. We examine analytically the relationship between the results produced by the new formalism and the usual Born approximation for a medium which has no long-wavelength heterogeneities. We show that in such a case the two methods agree approximately as expected, but that in a media with heterogeneities of all wavelengths the new gradient scattering formalism is superior. We establish analytically the connection between the formalism developed here and the iterative approach based on the WKBJ solution which has been used previously in 1-D media. Numerical examples are shown to illustrate the examples discussed.     

Scattering of waves in elastic media containing passive and active cracks

Scattering of wavefields in a 3-D medium that includes passive and/or active structures, is numerically solved by using the boundary integral equation method (BIEM). The passive structures are velocity anomalies that generate scattered waves upon incidence, and the active structures contain endogenous fracture sources, which are dynamically triggered by the dynamic load due to the incident waves. Simple models are adopted to represent these structures: passive cracks act as scatterers and active cracks as fracture sources. We form cracks using circular boundaries, which consist of many boundary elements. Scattering of elastic waves by the boundaries of passive cracks is treated as an exterior problem in BIEM. In the case of active cracks, both the exterior and interior problems need to be solved, because elastic waves are generated by fracturing with stress drop, and the growing crack boundaries scatter the incident waves from the outside of the cracks. The passive cracks and/or active cracks are randomly distributed in an infinite homogeneous elastic medium. Calculations of the complete waveform considering a single scatter show that the active crack has weak influence on the attenuation of first arrivals but strong influence on the amplitudes of coda waves, as compared with those due to the passive crack. In the active structures, multiple scattering between cracks and the waves triggered by fracturing strongly affect the amplitudes of first arrivals and coda waves. Compared to the case of the passive structures, the attenuation of initial phase is weak and the coda amplitudes decrease slowly.