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Application of the CGG method in Velocity-stack inversion

In this section, the CGG method is tested on a velocity-stack inversion, which is useful not only for velocity analysis but also for various data processing. The applications of the velocity-stack inversion include separating multiples from the signal in the velocity domain  Kostov and Nichols (1995); Lumley et al. (1995), multiple-attenuation techniques using parabolic Radon transforms  Herrmann et al. (2000); Kabir and Marfurt (1999), missing-trace interpolation in the CMP domain Ji (1994), and so on. In these applications, the velocity-stack panels obtained by inversion are usually required to be as spiky and sparse as possible. Then the hyperbolic events represented by the isolated peaks in the velocity-stack panel are more easily distinguished from the rest of the noise.

The conventional velocity stack is performed by summing or estimating semblance  Taner and Koehler (1969) along the various hyperbolas in a CMP gather, resulting in a velocity-stack panel. Ideally a hyperbola in a CMP gather should be mapped onto a point in a velocity-stack panel. Summation along a hyperbola, or hyperbolic Radon transform (HRT), does not give such resolution. To obtain a velocity-stack panel with better resolution, Thorson and Claerbout 1985 formulated it as an inverse problem in which the velocity domain is the unknown space. If we find an operator $\bold H$ that transforms a point in a model space (velocity-stack panel), $\bold m$, into a hyperbola in data space (CMP gather), $\bold d$,
\bold d = \bold H \bold m ,\end{displaymath} (10)
and also find its adjoint operator $\bold H^T$,we can pose the velocity stack problem as an inverse problem. Inverse theory helps us to find a velocity-stack panel which synthesizes a given CMP gather via the operator $\bold H$.The usual process is to implement the inverse as the minimization of a least-squares problem and calculate the solution by solving the normal equation:
\bold H^T \bold H \bold m = \bold H^T \bold d.\end{displaymath} (11)
Since the number of equations and unknowns may be large, an iterative least-squares solver such as CG is usually preferred to solving the normal equation directly.

The least-squares solution has some attributes that may be undesirable. If the model space is overdetermined and has bursty noise in data, the least-squares solutions usually will be spread over all the possible solutions. Other methods may be more useful if we desire a parsimonious representation. To obtain a more robust solution, Nichols 1994 used the IRLS method for L1-norm minimization, and Guitton and Symes  2003 used the L-BFGS method for Huber-norm minimization. Another possibility is the CGG method proposed in the preceding section. In the next subsections the results of the CGG method for the velocity-stack inversion are compared with the results of conventional LS and L1-norm IRLS.