Fast time-to-depth conversion and interval velocity estimation in the case of weak lateral variations

Discussion

In this study, we restrict our consideration to mild heterogeneity and only focus on the first-order perturbative effects from lateral velocity variations. The higher-order terms are important for the consideration of stronger variations. Furthermore, we emphasize that the proposed method utilize a laterally homogeneous background model $w_r(z)$ as the reference. The update from lateral heterogeneity comes entirely from the estimated first-order change $\Delta w (x,z)$ . When the considered medium deviates significantly from such assumption, for example, in the linear gradient model (equation 23), the proposed method will produce erroneous results and regular Dix-inverted velocity may represent a more feasible option.

An important underlying assumption of the proposed method involves well-behaved image rays with the absence of caustics, which in turn imposes the limits on the size and the degree of velocity variation in the model. Dividing the original model into several depth intervals to ensure an agreement with such assumption is a possible alternative (Li and Fomel, 2015).

Another possible issue to the proposed method concerns the direction of traveling image rays. Our algorithm assumes that the rays can only enter from the surface (in-flow boundary) and exit the model at the side edges or at the bottom edge. However, it is possible that parts of the model require in-flow image rays from the side edges (Figures 3 and 7). We avoid this complication by limiting our consideration of the results to the windowed part within the coverage of image rays.

Numerical implementation of the proposed algorithm involves taking derivatives in steps 4 and 7. To mitigate the effects of possible sharp contrasts, we propose applying a smoothing filter. This is particularly important because the numerical artifacts will get accumulated to the later depth as the algorithm proceeds. We employ iterations of triangle smoothing when generating shown numerical examples.

The proposed method can be extended to 3D in a straightforward manner. The lateral coordinates $x_0$ and $x$ become vectors $\mathbf{x_0}=(x_0,y_0)$ and $\mathbf{x}=(x,y)$ for consideration of displacements in both in-line and cross-line directions. The geometrical spreading of an image ray becomes a matrix $\mathbf{Q}$ . Following the similar procedure as described in this paper, an efficient framework for 3D time-to-depth conversion and interval velocity estimation can be developed.

Conventionally, time-migration process relies on a hyperbolic summation curve, which is only approximately correct in general anisotropic media with lateral heterogeneity (Yilmaz, 2001; Alkhalifah, 1997; Black and Brzostowski, 1994). As proposed by Cameron et al. (2007) and employed in this study, additional consideration of the geometrical spreading of image rays can help mitigate the possible errors from the hyperbolic assumption and increase the range of applicability of time migration in laterally heterogeneous media. Recent studies by Dell et al. (2013) on the general expression of diffraction traveltime in anisotropic media and Sripanich et al. (2017) on the influence of lateral heterogeneity on the Taylor coefficients of the traveltime expansion shed some light on how the complexities from lateral heterogeneity and anisotropy can influence seismic traveltimes beyond the hyperbolic assumption and represent a further step forward towards the goal of making time-domain imaging more accurate and versatile.

Lastly, we point out that Alkhalifah et al. (2001) proposed a notable alternative approach to handle the effects of lateral heterogeneity and anisotropy in time-domain processing by recasting the problem in terms of vertical traveltime. This method allows for an application of the Dix inversion in laterally factorized media, where the the ratio between the NMO velocity and the vertical velocity of P waves remains relatively constant, at the expense of increased computational cost.

Fast time-to-depth conversion and interval velocity estimation in the case of weak lateral variations