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. 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 L-norm solution,
with results comparable to, or better than, an ^{2}L-norm
IRLS (Iteratively Reweighted LS) solution.
^{1} |

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 *L ^{1}*-norm solution is more robust than the

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 *L _{1}*-norm IRLS results.

- CG method for LS and IRLS Inversion
- Conjugate Guided Gradient(CGG) method
- Application of the CGG method in Velocity-stack inversion
- : REFERENCES
- About this document ...

5/23/2004