A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations

Discussion

Note that in our method the inversion is applied directly for slowness-squared $\mathbf{w}$ (and thus velocity), and the update $\delta \mathbf{w}$ will incorporate dependency of velocities throughout the domain, as physically honored by image rays. Previous methods (Cameron et al., 2008; Iversen and Tygel, 2008; Cameron et al., 2007) that rely on time-stepping in $t_0$ account for such dependencies only locally. Moreover, regularization provides a convenient and effective way of handling ill-posedness of the underlying PDEs. For these reasons, the proposed method should be numerically robust compared to the previous approaches.

A 3-D extension of the proposed method is straightforward. Instead of a scalar $x_0$ we need to handle both inline and crossline coordinates $\mathbf{x_0} = (x_0, y_0)$. Consequently, the geometrical spreading in 3-D becomes a matrix $\mathbf{Q}$, whose determinant is related to the generalized Dix velocity $v_d (t_0,\mathbf{x_0})$ and interval velocity $v (z,\mathbf{x})$ (Cameron et al., 2007). Starting from this relationship, an iterative time-to-depth conversion can be established by following procedures similar to the ones described in this paper.

The main limitation of our approach is the failure of underlying theory at caustics, which in turn limits either the depth or the extent of lateral velocity variation of the model. A possible solution is to divide the original domain into several depth intervals, then apply velocity estimation and redatuming from one interval to another recursively (Li and Fomel, 2013; Bevc, 1997). By doing so, image-ray crossing may not develop within each interval thanks to a limited depth.

Another issue is the handling of in-flow boundaries other than the earth surface. In the synthetic examples, we avoided this problem by limiting the interval velocity model within image ray coverage. Alternatively, we could pad the input $v_d$ laterally. The padding should simply replicate the original boundaries. By doing so, image rays at the new left and right boundaries must run straightly downward as the media there are of $v(z)$ type. Although we could not expect the inversion to fix the in-flow boundary, the resulting errors should be local around that boundary.

Finally, in our approach time-migration velocity not only determines the prior model but also drives the inversion. Consequently, errors in time-migration velocity have a direct influence on the accuracy of inverted model. Note that usually the picking of time-migration velocity is carried out with certain smoothing. Combined with regularization in the time-to-depth conversion, the resulting interval velocity model may contain limited fine-scale features and high velocity contrasts. In this regard, we suggest our method as an efficient estimation of an initial guess for subsequent depth-imaging velocity model refinements.

A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations