Velocity-independent $\tau$ -$p$ moveout in a horizontally-layered VTI medium

Introduction

macro model of the subsurface and remains one of the most labor-intensive and time-consuming procedures in the conventional approach to seismic data analysis (Yilmaz, 2000). In time-domain imaging, effective seismic velocities are picked from coherency scans. Moreover, anisotropic velocity model building requires more than just a single parameter scan for nonhyperbolic traveltime approximation (Alkhalifah and Tsvankin, 1995). This means that anisotropic velocity analysis is at least twice more computationally intensive than its traditional isotropic counterpart. Conventional human-aided velocity analysis takes up a significant part of the time needed to process seismic data. Even with semi-automatic picking software, this phase alone might take weeks or even months for modern 3D data sets. Several approaches have been proposed to automatize and simplify velocity analysis and traveltime picking procedures in order to reduce the time and manual work required for handpicked velocities (Siliqi et al., 2007; Lambaré et al., 2003; Lambaré, 2008). However, these tools still require significant manual inspection and editing for quality control.

The idea of using local event slopes estimated from prestack seismic data goes back to the work of Rieber (1936) and Riabinkin (1957). Several following papers outline the importance of local data slopes in seismic data processing, particularly the role that slope estimates play in the algorithm of stereotomography (Lambaré et al., 2003; Sword, 1987; Lambaré, 2008). The concept of velocity-independent time-domain imaging goes back to Ottolini (1983). Wolf et al. (2004) pointed out that it is possible to perform hyperbolic moveout velocity-analysis by estimating local data slopes in the prestack data domain using an automated method such as plane-wave destruction (Fomel, 2002). This methodology is attractive because it can be less time-consuming than the manual work required to handpick velocities.

By estimating local event slopes in prestack seismic reflection data, Fomel (2007b) demonstrated that it is possible to accomplish all common time-domain imaging tasks, from normal moveout to prestack time migration, without the need to estimate seismic velocities or other attributes. Local slopes contain complete information about the reflection geometry. Once they are estimated, seismic velocities and all the other moveout parameters turn into data attributes and become directly mappable from the prestack data domain into the time-migrated image domain. Fomel (2007b) focused on the isotropic prestack time processing and showed several results of oriented (slope-based) velocity analysis and imaging, both on synthetic and real data. Although he developed the mathematical framework for velocity-independent non-hyperbolic processing in the time-offset $t$ -$X$  domain, he did not provide examples to demonstrate its use and efficacy. Burnett and Fomel (2009a,b) extended the method to 3D elliptically anisotropic moveout corrections.

In this paper, we extend the concept of velocity-independent seismic processing to P-wave VTI data in the $\tau$ -$p$  or slant-stack domain obtained by Radon-transforming CMP data. We account for VTI anisotropy only, but the theory developed here should work for a general anisotropic horizontally-layered velocity model. We assume that each layer is laterally homogeneous with a horizontal symmetry plane and that the incidence plane represents a symmetry plane for the model as a whole so that wave propagation is two-dimensional. The $\tau$ -$p$  transform is the natural domain for anisotropic parameter estimation in layered media (Fomel, 2008; van der Baan and Kendall, 2002; Douma and van der Baan, 2008) because it allows for simpler and more accurate moveout modeling and inversion than the conventional methods applied in $t$ -$X$  domain. Since the horizontal slowness is preserved, each trace in $\tau$ -$p$  CMP gathers sees the contributions of rays that share the same segments of trajectory in the layers. Therefore, one can simply sum the contribution of each individual layer and obtain the overall $\tau$ -$p$   moveout signature. This makes modelling or ray tracing a linear procedure. Moreover, by literally subtracting all the unwanted layers, we can isolate the contribution of a specific layer and access directly its interval parameters without relying on the effective-parameter approximations as normally happens in $t$ -$X$  domain.

After $\tau$ -$p$  transform, seismic data are mapped to the slowness domain, where the reflection signature depends on the vertical component of phase slowness. Phase velocity is the natural parameter to work with in the case of anisotropic data, because explicit expressions exist for phase velocities in all the anisotropic media that display an horizontal symmetry plane. Unfortunately, exact expressions for $\tau$ -$p$  signatures are not always practical. Nevertheless, approximate expressions provide accurate traveltime predictions (Tsvankin et al., 2010).

After describing the advantages of processing anisotropic data in the $\tau$ -$p$  domain, we derive the oriented (slope-based) NMO equation that describes direct mapping from prestack data to zero-slope time (analogous to zero-offset time in $t$ -$X$  domain). We obtain the effective values of anisotropy parameters as data attributes derived from local slope and curvature estimates and directly mappable to the appropriate zero-slope time. Similarly to conventional $t$ -$X$ processing, several procedures applied in $\tau$ -$p$  domain rely on coherency analysis (Sil and Sen, 2008; van der Baan and Kendall, 2002) or traveltime picking plus inversion (Fowler et al., 2008; Wang and Tsvankin, 2009) to retrieve the anisotropy parameters. We believe that our procedure is more attractive because it is fully automated and less time-consuming than searching for the best-fit moveout trajectory through simultaneous two-parameter inversion or semblance scans.

Interval parameters as well as effective parameters can be regarded as data attributes obtained from local slopes. Unlike $t$ -$X$  domain, processing data in $\tau$ -$p$  offers two alternatives to conventional Dix (1955) inversion: stripping and Fowler's equations (Fowler et al., 2008). These relations can be considered as the VTI extension of the ``straightedge determination of interval velocity'' method proposed by Claerbout (1978). These three formulations for interval-parameter inversion require an estimate of the local-curvature field. To estimate curvature, we perform a numerical differentiation of the slope estimates. This procedure usually returns noisy and biased curvature values that affect parameter estimation, especially for those parameters that control long-spread/large-angle moveout.

Fowler's equations offer a solution to this problem. In these equations, the curvature dependence is absorbed by the zero-slope time that we can estimate by applying the predictive painting algorithm (Fomel, 2010). This approach does not involve any curvature estimation and represents a more robust way for obtaining the zero-slope time mapping field required (1) to automatically flatten or NMO correct the $\tau$ -$p$  CMP gathers (2) to retrieve interval parameters using the curvature-independent Fowler's equations. This last approach to data processing makes the anisotropy-parameter estimation closer to an imaging processing task. Its only requirement is the ability to extract the best local-slope field from the data.

Velocity-independent $\tau$ -$p$ moveout in a horizontally-layered VTI medium