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

CGG with residual and model weights guide

In the previous two subsections, we examined the meaning of weighting the residual vector and the gradient vector, respectively. Since applying the weighting in both residual space and model space is nothing but changing the direction of the descent for the solution search, the weighting is not limited either to residual or to model space. We can weight both the residual and the gradient as shown in Algorithm 5.

Again, Algorithm 5 is different from the conventional CG method (Algorithm 1) only in the step of gradient computation. Whether we modify the gradient in the residual sense or in the model sense, it changes only the gradient direction (i.e. the direction in which the solution is sought) and the solution is found in the least-squares sense in that direction. Therefore, the problem solved by the CGG method is a linear problem and the CGG algorithm always converges to a solution, which is different from the LS solution that is located along the original gradient direction. Notice that the CGG algorithm (Algorithm 5) is simpler than the IRLS algorithm (Algorithm 2), but the CGG method gives a similar solution as the one of the IRLS method, which are demonstrated with examples shown in the following section.

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