Extended common-image-point gathers for anisotropic wave-equation migration

Introduction In regions characterized by complex subsurface structure, wave-equation depth migration is a powerful tool for accurately imaging the earth’s interior. The quality of the final image greatly depends on the quality of the model which includes anisotropy parameters (Gray et al., 2001). In particular, it is important to construct subsurface velocity models using techniques that are consistent with the methods used for imaging. Generally speaking, there are two possible strategies for velocity estimation from surface seismic data in the context of wavefield-based imaging (Sava et al., 2010). One possibility is to formulate an objective function in the data space, prior to migration, by matching the recorded data with simulated data. Techniques in this category are known by the name of waveform inversion. Another possibility is to formulate an objective function in the image space, after migration, by measuring and correcting image features that indicate model inaccuracies. Techniques in this category are known as wave-equation migration velocity analysis (MVA). The key component for an MVA technique implemented in the image space is the analysis of image attributes which indicate inaccurate imaging. These attributes are often represented by image extensions, e.g. reflectivity as functions of angle or offset which exploit the semblance principle stating that images constructed for different seismic experiments are kinematically similar if the correct velocity is used. This property can be exploited for velocity model building by minimizing objective functions to optimize certain image attributes. For example, we can consider flatness or focusing measured on image gathers Symes (2009). The analysis of image attributes is more critical in anisotropic media, as the model is described by more than one parameter. In transversely isotropic (TI) with vertical symmetry axis (VTI) media, the acoustic problem can be described by three parameters (Alkhalifah and Tsvankin, 1995): the vertical velocity, the NMO velocity, and the anisotropy parameter η that relates the NMO velocity to the horizontal velocity. In this case, however, the recorded data contain information on only the NMO velocity and η (Alkhalifah and Tsvankin, 1995), and thus, the vertical velocity is not resolved. This fact holds for complex anisotropy, but at a lesser extent (Alkhalifah et al., 2001). However, since the stratification in the Earth subsurface is not always horizontal, we can expect the symmetry axis to have some deviation from the vertical especially around, for example, salt-body flanks. For TI media with a tilt in the axis of symmetry two additional parameters, θ and α that describe the tilt in 3-D, are needed to fully characterize acoustic wave propagation. These two parameters are often estimated by assuming that the tilt direction is normal to the medium structure or in the direction of the velocity gradient (Alkhalifah and Bednar, 2000; Audebert et al., 2006). Alkhalifah and Sava (2010) use this fact to develop equations to describe imaging in media in the which the tilt of the symmetry axis is normal to the reflector dip. An image obtained by wave-equation migration can be extracted from the reconstructed wavefields by the application of an extended imaging condition: