Conjugate guided gradient (CGG) method for robust inversion and its application to velocity-stack inversion

INTRODUCTION

The inverse problem has received considerable attention in various geophysical applications. One of the most popular inverse solutions is the least squares (LS) solution. The LS solution is a member of a family of generalized $\ell^p$-norm solutions that are deduced from a maximum-likelihood formulation. This formulation allows the design of various statistical inversion solutions. Among the various $\ell^p$-norm solutions, the $\ell ^1$-norm solution is more robust than the $\ell^2$-norm solution, because it is less sensitive to spiky, high-amplitude noise Scales et al. (1988); Scales and Gersztenkorn (1987); Claerbout and Muir (1973); Taylor et al. (1979). In order to take advantages of both $\ell^2$ and $\ell ^1$ norm solutions, hybrid $\ell^1/\ell^2$ - norm solutions are also tried Huber (1973); Guiiton and Symes (2003); Bube and Langan (1997). However, the implementation of the algorithm to find $\ell ^1$-norm solutions is not a trivial task, which uses linear programming techniques Taylor et al. (1979) and needs a large quantity of computer memory. Iterative inversion algorithm called Iteratively Reweighted Least Squares (IRLS) method Scales and Gersztenkorn (1987); Scales et al. (1988); Bube and Langan (1997); Gersztenkorn et al. (1986) is a good choice for solving $\ell^p$-norm minimization problems for $1\le p \le2$. The IRLS approach which was originally developed for nonlinear inversion can be adapted to solve linear inverse problems in $\ell^p$-norm sense by modifying the iterative inversion method such as conjugate gradient (CG) method  Claerbout (2004); Darche (1989); Nichols (1994).

The $\ell^p$-norm minimizing IRLS inversion can be used to any inversion problem whose required properties are the robustness to spiky noise and the parsimony of the model, and the velocity-stack inversion is one of them. The velocity-stack inversion is useful not only for velocity analysis but also for various data processing applications. The applications of the velocity-stack inversion include non-hyperbolic noise removal in CMP gathers Guiiton and Symes (2003); Nichols (1994), multiple-removal  Lumley et al. (1995); Hampson (1986); Foster and Mosher (1992); Kabir and Marfurt (1999); Herrmann et al. (2000); Thorson and Claerbout (1985); Kostov and Nichols (1995) and missing offset reconstruction Ji (1994); Sacchi and Ulrych (1995), 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.

This paper introduces a modification of the conventional CG method for solving LS problem so as to be robust and produce a parsimonious model estimation. The modified CG method is called Conjugate Guided Gradient (CGG) method. The modification of the CG method is performed by guiding the gradient vector during the iteration steps. Guiding the gradient vector is achieved by iteratively reweighting either the residual vector or the gradient vector during iteration steps like IRLS (Iteratively Reweighted Least Squares) method does. The weighting of the residual vector makes the CGG method robust and the weighting of the gradient vector makes the CGG method produce a parsimonious model estimation. In the first section, I review the conventional CG method for solving LS problems and show how the IRLS approach differs from the standard LS approach. Next, I explain the CGG method and contrast it with both LS and IRLS methods. Finally the proposed CGG method is tested on velocity-stack inversions with both synthetic and real data and the results of the CGG method are compared with conventional LS and $\ell ^1$-norm IRLS results.

Conjugate guided gradient (CGG) method for robust inversion and its application to velocity-stack inversion