Analysis of image gathers in factorized VTI media

Abstract

Because events in image gathers generated after prestack depth migration are sensitive to the velocity field, they are often used in migration velocity analysis for isotropic media. Here, we present an analytic and numerical study of P-wave image gathers in transversely isotropic media with a vertical symmetry axis (VTI) and establish the conditions for flattening such events and positioning them at the true reflector depth. Application of the weak-anisotropy approximation leads to concise expressions for reflections in image gathers from homogeneous and factorized v(z) media in terms of the VTI parameters and the vertical velocity gradient k z. Flattening events in image gathers for any reflector dip requires accurate values of the zero-dip NMO velocity at the surface [V nmo (z = 0)], the gradient k z , and the anellipticity coefficient η. For a fixed error in V nmo and k z , the magnitude of residual moveout of events in image gathers decreases with dip, while the moveout caused by an error in η initially increases for moderate dips but then decreases as dips approach 90 •. Flat events in image gathers in VTI media, however, do not guarantee the correct depth scale of the model because reflector depth depends on the vertical migration velocity. For factorized v(x, z) media with a linear velocity variation in both the x-and z-directions, the moveout on image gathers is controlled by V nmo (x = z = 0), k z , η, and a combination of the horizontal velocity gradient k x and the Thomsen parameter δ (specifically, k x √ 1 + 2δ). If too large a value of any of these four quantities is used in migration, reflections in the image gathers curve downward (i.e., they are undercorrected; the inferred depth increases with offset), while a negative error results in overcorrection. Lateral heterogeneity tends to increase the sensitivity of moveout of events in image gathers to the parameter η, and errors in η may lead to measurable residual moveout of horizontal events in v(x, z) media even for offset-to-depth ratios close to unity. These results provide a basis for extending to VTI media conventional velocity analysis methods operating with image gathers. Although P-wave traveltimes alone cannot be used to separate anisotropy from lateral heterogeneity (i.e., k x is coupled to δ), moveout of events in image gathers does constrain the vertical gradient k z. Hence, it may be possible to build VTI velocity models in depth by supplementing reflection data with minimal a priori information, such as the vertical velocity at the top of the factorized VTI layer.

Analytic solution of ray-tracing equations for a linearly inhomogeneous and elliptically anisotropic velocity model

“…Such a model was used by Slawinski et al (2004) to consider the traveltime inversion for the three parameters (a, b, and χ ) that define this model (as shown in equation 4 below). Moreover, factorized media can be used for modeling the subsurface (Sarkar and Tsvankin, 2003). An exhaustive study on factorized media is presented byČervený (1989).…”

VTI anisotropic corrections and effective parameter estimation after isotropic prestack depth migration

“…In practice, this is not the case, especially when well data are not available. Earlier studies showed that if only surface P-wave data are available, it will be difficult if not impossible to resolve accurately the three quantities V z , , and ␦ ͑Le Stunff et al, 2001;Sarkar and Tsvankin, 2004͒. When relying on surface data only, we suggest using the near offsets in the IMO domain to perform a standard isotropic interval-velocity analysis ͑such as depth residual moveout and tomography͒ prior to the estimation of and ␦.…”

“…Application of isotropic prestack depth imaging in regions with significant anisotropy will result in poor image quality and wrong positioning of subsurface reflectors ͑Alkhalifah and Larner, 1994; Alkhalifah, 1997;Sarkar and Tsvankin, 2003͒. Two of the most common problems are ͑1͒ inaccurate imaging of steeply dipping reflectors and ͑2͒ less than optimal isotropic prestack depth migration ͑PSDM͒ results that do not tie wells.…”

Estimation of the anisotropy parameters from imaging moveout of diving wave in a factorized anisotropic medium

“…The factorized model (Sarkar and Tsvankin, 2003;Stovas, 2010) involves anisotropy and vertical heterogeneity. In our case, the factorized medium is defined as a transversely isotropic model with a vertical symmetry axis (VTI) under the acoustic approximation (Alkhalifah, 1998).…”

“…The factorized model, assuming vertical heterogeneity and constant anisotropy parameters, is useful in seismic data processing and modeling (Alkhalifah, 1995;Sarkar and Tsvankin, 2003). The raytracing problem in factorized models with elliptic anisotropy is discussed extensively by Rogister and Slawinski (2005).…”