Iterative inversion methods have been unsuccessful at inverting seismic data obtained from complicated earth models (e.g. the Marmousi model), the primary difficulty being the presence of numerous local minima in the objective function. The presence of local minima at all scales in the seismic inversion problem prevent iterative methods of inversion from attaining a reasonable degree of convergence to the neighborhood of the global minimum. The multigrid method is a technique that improves the performance of iterative inversion by decomposing the problem by scale. At long scales there are fewer local minima and those that remain are further apart from each other. Thus, at long scales iterative methods can get closer to the neighborhood of the global minimum. We apply the multigrid method to a subsampled, low‐frequency version of the Marmousi data set. Although issues of source estimation, source bandwidth, and noise are not treated, results show that iterative inversion methods perform much better when employed with a decomposition by scale. Furthermore, the method greatly reduces the computational burden of the inversion that will be of importance for 3-D extensions to the method.

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Implicit finite-difference implementations of the paraxial wave equation are widely used in industrial prestack and post-stack migration programs for imaging and velocity analysis. This type of implementation gives rise to numerical artifacts which, in general, do not degrade image quality but which do impede effective velocity analysis. This paper reviews the artifacts generated by the paraxial approximation and a post-extrapolation, spatially varying filtering scheme is described which completely eliminates these artifacts. The method is illustrated with numerous examples.

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