Least-square inversion with inexact adjoints. Method of conjugate directions: A tutorial

Example 2: Velocity transform

The next test example is the velocity transform inversion with a CMP gather from the Mobil AVO dataset (Lumley et al., 1994; Lumley, 1994; Nichols, 1994). I use Jon Claerbout's veltran program (Claerbout, 1995b) for anti-aliased velocity transform with rho-filter preconditioning and compare three different pairs of operators for inversion. The first pair is the CMP stacking operator with the ``migration'' weighting function $\left(w = {{\left(t_0/t\right)} \over \sqrt{t}}\right)$ and its adjoint. The second pair is the ``pseudo-unitary'' velocity transform with the weighting proportional to $\sqrt{\vert s x\vert}$, where $x$ is the offset and $s$ is the slowness. These two pairs were used in the velocity transform inversion with the iterative conjugate-gradient solver. The third pair uses the weight proportional to $\vert x\vert$ for CMP stacking and $\vert s\vert$ for the reverse operator. Since these two operators are not exact adjoints, it is appropriate to apply the method of conjugate directions for inversion. The convergence of the three different inversions is compared in Figure 4. We can see that the third method reduces the least-square residual error, though it has a smaller effect than that of the pseudo-unitary weighting in comparison with the uniform one. The results of inversion after 10 conjugate-gradient iterations are plotted in Figures 5 and 6, which are to be compared with the analogous results of Lumley (1994) and Nichols (1994).

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Figure 4. Comparison of convergence of the iterative velocity transform inversion. The left plot compares conjugate-gradient inversion with unweighted (uniformly weighted) and pseudo-unitary operators. The right plot compares pseudo-unitary conjugate-gradient and weighted conjugate-direction inversion.

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Figure 5. Input CMP gather (left) and its velocity transform counterpart (right) after 10 iterations of conjugate-direction inversion.

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Figure 6. The modeled CMP gather (left) and the residual data (right) plotted at the same scale.

Least-square inversion with inexact adjoints. Method of conjugate directions: A tutorial