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

Discussion

In the conventional NMO processing, one needs to scan over a range of possible velocities and pick the appropriate velocity trend from semblance maxima. Therefore, the cost of velocity scanning is roughly proportional to the number of scanned velocities $N_{V}$ times the input data size. Anisotropic velocity analysis is performed by simultaneously scanning two (or more) parameters. Consequently, the number of trial velocities/parameters squares, which increases the computational time dramatically. In oriented processing, the effective anisotropy parameters turn into data attributes according to equations 16-18. These parameters are directly mapped from the slope field $R$ to the correct zero-slope/offset traveltime $\tau _0$ . The cost of local slope estimation with plane-wave destruction method is proportional to the data size times the number of estimation iterations $N_{I}$ times the 2D filter size $N_{F}$ . Typically $N_{I}=10$ and $N_{F}=6,$ which roughly correspond to scanning $N_{V}=60$ velocities. However, unlike semblance analysis, this cost does not increase if we are estimating one, two or more parameters. The cost of the semblance scan becomes even more prohibitive when processing wide-azimuth data. The computational advantages of our approach are encouraging especially with respect to multi-azimuth processing and orthorhombic velocity analysis, where time processing is controlled by at least five parameters (Tsvankin, 2006).

Automation, in addition to speed, is another clear advantage of the slope-based processing. Slope estimation provided by plane-wave destruction represents an automated approach to velocity analysis. It may require a limited user-interaction in choosing input parameters. A user-supplied initial guess for the slope field can accelerate the nonlinear optimization , thereby providing a more reliable estimate for the slopes. The smoothness of the output slopes is controlled by shaping regularization; the length of a 2-D triangular smoothing filter controls smoothness along $p$ and $\tau$ direction in the $\tau$ -$p$  transformed CMP data. If the input seismic data are not regularly and properly sampled in space, as often happens in wide-azimuth acquisition, the $\tau$ -$p$  transform may add to the data coherent-noise artifacts. This can affect the final result of PWD slope estimation. Thus, if the seismic $\tau$ -$p$  data are noisy, increasing the length of smoothing filters can help in achieving a more stable solution, despite some loss in resolution. In contrast, for high SNR data, less smoothing yields better-resolved slope fields.

All the equations we have developed in this paper hold for S-wave data as long as we use two parameters S-wave phase-velocity approximation (Stovas, 2009). The combination of the results from P-wave and S-wave processing may enable a retrieval of all the elastic parameters needed to build an initial VTI anisotropic model suitable for depth processing.

The application of the proposed method is also limited by the underlying assumption of vertical variation of the velocity model with the horizontal symmetry plane. In principle, the method can handle limited lateral variation of the velocity. Therefore, it can be used for dense anisotropic moveout analysis at the early stages of processing.

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