Asymptotic pseudounitary stacking operators |

Velocity transform is another form of hyperbolic stacking with the summation path

where corresponds to the offset, is the stacking slowness, and is the estimated zero-offset traveltime. Hyperbolic stacking is routinely applied for scanning velocity analysis in common-midpoint stacking. Velocity transform inversion has proved to be a powerful tool for data interpolation and amplitude-preserving multiple suppression (Lumley et al., 1995; Thorson, 1984; Ji, 1995).

Solving equation (69) for , we find that the asymptotic inverse and adjoint operators have the elliptic summation path

(70) |

The weighting functions of the asymptotic pseudo-unitary velocity transform are found using formulas (28) and (29) to have the form

The factor for pseudo-unitary velocity transform weighting has been discovered empirically by Claerbout (1995).

Figure 4 shows the output of a numerical test of the least-squares velocity transform inversion using a CMP gather from the Mobil AVO dataset (Lumley et al., 1995). The input CMP gather (shown in the left plot of Figure 5) is inverted using an iterative conjugate-gradient method and two different weighting scheme: the uniform weighting and the asymptotic pseudo-unitary weights (71-72). Analogously to Figure 1, the iterative convergence is measured by the least-squares norm of the data residual error at different iterations. Figure 4 shows that the pseudo-unitary weighting provides a noticeably faster convergence at the first three iterations. At later iterations, the residual errors of the two methods are very close to each other. The use of a pseudo-unitary weighting will be justified in this case if only three iterations are practically affordable. The results of inversion after 10 conjugate-gradient iterations are plotted in Figures 5 and 6. The right plot in Figure 5 shows the output of the velocity transform inversion: an optimized velocity scan. Figure 6 shows the corresponding modeled CMP gather and the residual error. The error is negligible which indicates a successful inversion.

cgiter
Comparison of convergence of the
iterative velocity transform inversion. The dashed line corresponds
to the unweighted (uniformly weighted) operator. The solid line
corresponds to the asymptotic pseudo-unitary operator. The latter
provides a faster convergence at early iterations.
Figure 4. | |
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dircvv
Input CMP gather (left) and its
velocity transform counterpart (right) after 10 iterations of
iterative least-squares inversion.
Figure 5. |
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dirrst
The modeled CMP gather (left) and
the residual error (right) plotted at the same scale.
Figure 6. |
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Asymptotic pseudounitary stacking operators |

2013-03-03