It is usually believed that angular aperture of seismic data should be at least 20°to allow estimation of the subsurface anisotropy. Although this is certainly true for reflection data, for which anisotropy parameters are inverted from the stacking velocities or the nonhyperbolic moveout, traveltimes of direct P-and S-waves recorded in typical downhole microseismic geometries make it possible to infer seismic anisotropy in angular apertures as narrow as about 10°. To ensure the uniqueness of such an inversion, it has to be performed in a local coordinate frame tailored to a particular data set. Because any narrow fan of vectors is naturally characterized by its average direction, we choose the axes of the local frame to coincide with the polarization vectors of three plane waves corresponding to such a direction. This choice results in a significant simplification of the conventional equations for the phase and group velocities in anisotropic media and makes it possible to predict which elements of the elastic stiffness tensor are constrained by the available data. We illustrate our approach on traveltime synthetics and then apply it to perforation-shot data recorded in a shale-gas field. Our case study indicates that isotropic velocity models are inadequate and accounting for seismic anisotropy is a prerequisite for building a physically sound model that explains the recorded traveltimes.
In reflection seismology one places point sources and point receivers on the earth's surface, forming an acquisition geometry. Each source generates acoustic waves in the subsurface, that are reflected where the medium properties vary discontinuously. The reflections that can be observed at the receivers are used to image these discontinuities or reflectors assuming a background medium. We analyze methods to circumvent the repeated imaging of reflectors under varying background media, or the repeated modelling of reflections under varying acquisition geometries. These methods involve the introduction of the notion of seismic continuation. Here, we develop the foundation of, and a comprehensive framework for seismic continuation while extending earlier approaches to allow for the formation of caustics. Traditionally, seismic continuation has been viewed from a geometrical (ray asymptotic) point of view; here, we introduce the notion of wave-equation continuation through the appearance of evolution equations.
Many processes in seismic data analysis and seismic imaging can be identified with solution operators of evolution equations. These include data downward continuation and velocity continuation. We have addressed the question of whether isochrons defined by imaging operators can be identified with wavefronts of solutions to an evolution equation. Rays associated with this equation then would provide a natural way of implementing prestack map migration. Assuming absence of caustics, we have developed constructive proof of the existence of a Hamiltonian describing propagation of isochrons in the context of common-offset depth migration. In the presence of caustics, one should recast to a sinking-survey migration framework. By manipulating the double-square-root operator, we obtain an evolution equation that describes sinking-survey migration as a propagation in two-way time with surface data being a source function. This formulation can be viewed as an extension of the exploding reflector concept from zero-offset to sinking-survey migration. The corresponding Hamiltonian describes propagation of extended isochrons (fronts with constant two-way time) connected by extended isochron rays. The term extended reflects the fact that two-way time propagation now takes place in high-dimensional space with the following coordinates: subsurface midpoint, subsurface offset, and depth. Extended isochron rays can be used in a natural manner for implementing sinking-survey migration in a map-migration fashion.
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