Conjugate guided gradient (CGG) method for robust inversion and its application to velocity-stack inversion |
(1) |
(2) |
(3) |
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):
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Conjugate guided gradient (CGG) method for robust inversion and its application to velocity-stack inversion |