


 Conjugate guided gradient (CGG) method for robust inversion and its application to velocitystack inversion  

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Another way to modify the gradient direction
is to modify the gradient vector after the gradient is computed
from a given residual.
Since the gradient vector is in the model space,
any modification of the gradient vector imposes
some constraint in the model space.
If we know some characteristics of the solution
which can be expressed in terms of weighting in the solution space,
we can use the weight to redirect the gradient vector by applying the weight to it.
Again, by keeping the forward operator unchanged,
we don't need to recompute the residual when the weight has changed.
This algorithm can be implemented as shown in Algorithm 4.
Even though the model weighting has
different meaning from from residual weighting in the inversion result,
the analyses are similar in both cases.
As we redefined the contribution of each residual element
by weighting it with the absolute value of itself to some power,
we can do the same thing with each model element in the solution,

(10) 
where is a real number that depends on the problem we wish to solve.
If the operator used in the inversion is close to unitary,
the solution obtained after the first iteration already
closely approximates the real solution.
Therefore, weighting the gradient
with some power of the absolute value of the previous
iteration means that we downweight the importance of small model values
and improve the fit to the data by emphasizing model components
that already have large values.



 Conjugate guided gradient (CGG) method for robust inversion and its application to velocitystack inversion  

Next: CGG with residual and
Up: ConjugateGuidedGradient (CGG) method
Previous: CGG with residual weight
20110626