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

Jun Ji

jun@sep.stanford.edu

ABSTRACT

This paper proposes a modified conjugate-gradient (CG) method, called the conjugate-guided-gradient (CGG) method, as an alternative iterative inversion method that is robust and easily manageable. The CG method for solving least-squares (LS) (i.e. L²-norm minimization) problems can be modified to solve for a different norm or different minimization criteria by guiding the gradient vector appropriately. The guiding can be achieved by iteratively weighting either the residual vector or the gradient vector during iteration steps. Weighting the residual vector can guide the solution to the minimum L^p-norm solution, and weighting the gradient vector can guide the solution to one constrained by a priori information imposed in the model space. In both cases, the minimum solutions are found in a least-squares sense along the gradient direction guided by the weights. Therefore, the solution found by the CGG method can be interpreted as the LS solution located in the guided gradient direction. I applied the CGG method to the velocity stack inversion, and the results suggest that the CGG method gives a far more robust model estimation than the standard L²-norm solution, with results comparable to, or better than, an L¹-norm IRLS (Iteratively Reweighted LS) solution.

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 L^p-norm solutions that are deduced from a maximum-likelihood formulation. This formulation allows the design of various statistical inversion solutions. Among the various L^p-norm solutions, the L¹-norm solution is more robust than the L²-norm solution, being less sensitive to spiky, high-amplitude noise Claerbout and Muir (1973); Scales et al. (1988); Scales and Gersztenkorn (1987); Taylor et al. (1979). However, the implementation of the algorithm to find L¹-norm solutions is not a trivial task. Iterative inversion algorithms called IRLS (Iteratively Reweighted LS) Gersztenkorn et al. (1986); Scales et al. (1988) are a good choice for solving L^p-norm minimization problems for $1\le p \le2$.

If the number of unknown model values is very large, LS problems are often solved by iterative solvers like the popular conjugate-gradient (CG) method. IRLS approaches for nonlinear inversion can be adapted to solve linear inverse problems by modifying the CG method  Claerbout (2004); Darche (1989); Nichols (1994).

This paper introduces a way to modify the CG method so that it can not only handle the general L^p-norm problem but also impose a priori constraints on the solution space. This method is called conjugate-guided-gradient (CGG), and is achieved by guiding the gradient vector during the iteration step. 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. Finally I test the proposed CGG method on velocity-stack inversions with both noisy synthetic data and real data. I compare the results of the CGG method with conventional LS and L₁-norm IRLS results.