Velocity continuation describes how a seismic image changes given a change in migration velocity. This description turns out to be a wave propagation process, in which images propagate along a velocity axis. In the anisotropic case, the velocity model is multi-parameter. Therefore, anisotropic image propagation is multi-dimensional. We extend time-domain velocity continuation to the 3D azimuthally anisotropic case. We use a three-parameter slowness model, which can be related to azimuthal variations in velocity, as well as their principal directions. This information is useful for fracture and reservoir characterization from seismic data. We provide synthetic diffraction imaging examples to illustrate the concept of azimuthal continuation and to analyze the impulse response of the 3D velocity continuation operator.
Velocity continuation, introduced by Fomel (1994, 2003b), provides a framework for describing how a seismic image changes given a change in the migration velocity model. Similar in concept to residual migration (Rothman et al., 1985) and cascaded migrations (Larner and Beasley, 1987), velocity continuation is a continuous formulation of the same process. Velocity continuation has found application in migration velocity analysis (Fomel, 2003a; Schleicher et al., 2008) and diffraction imaging (Fomel et al., 2007). Fomel (1994) and Hubral et al. (1996) point out that velocity continuation is a wave propagation process where, instead of wavefronts propagating as a function of time, images propagate as a function of migration velocity. Recent work has extended the concept to heterogeneous and anisotropic velocity models (Alkhalifah and Fomel, 1997; Adler, 2002; Iversen, 2006; Schleicher and Alexio, 2007; Duchkov and de Hoop, 2009). To account for anisotropy, the seismic velocity model must become multi-parameter. Consequentially, velocity continuation generalizes to a process of implementing image transformations caused by changes in multiple parameters rather than velocity alone. In 3D, azimuthal variation in velocity has been shown to be an indicator of preferentially aligned vertical fractures (Crampin, 1984), lateral heterogeneity (Levin, 1985), regional stress (Sicking et al., 2007), or a combination of these factors. Accounting for azimuthal variations in velocity results in better event focusing and improved imaging (Sicking and Nelan, 2008). With these benefits as motivation, we extend time-domain velocity continuation to 3D, accounting for the case of azimuthally variable migration velocity.
The theory of velocity continuation formulates the connection between the seismic velocity model and the seismic image as a wave propagation process. In doing so, the process can be implemented in the same variety of ways as seismic migration. Seismic migration in its many forms is commonly derived starting at the wave equation, which is broken into the eikonal and transport equations, and if necessary, a system of ray tracing equations. Velocity continuation is derived in the opposite order (Fomel, 2003b). Starting with a traveltime equation which describes the image, a corresponding kinematic equation is derived to describe how the image moves according to a change in imaging parameters. Subsequently, the kinematic equation is used to derive a corresponding wave equation, which describes the dynamic behavior of the image as a propagation through imaging parameter coordinates.