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

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Most iterative solvers for the LS problem search the minimum solution on a line or a plane in the solution space. In the CG algorithm, not a line, but rather a plane is searched. A plane is made from an arbitrary linear combination of two vectors. One vector is chosen to be the gradient vector. The other vector is chosen to be the previous descent step vector. Following Claerbout (1992), a conjugate-gradient algorithm for the LS solution can be summarized as shown in Algorithm 1.

In Algorithm 1, the represents a convergence check
such as the tolerance of residual vector ,
a maximum number of iteration, and so on.
The subroutine cgstep() updates model and residual
using the previous iteration descent vector in the conjugate space
, where is the iteration step,
and the conjugate gradient vector
.
The update step size is determined by minimizing
the quadrature function composed from
(the conjugate gradient)
and
(the previous iteration descent vector in the conjugate space)
as follows Claerbout (1992):

Notice that the gradient vector ( ) in the CG method for LS solution is the gradient of the squared residual and is determined by taking the derivative of the squared residual (i.e. the -norm of the residual, ) with respect to the model :

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

2011-06-26