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 | Wave-equation migration velocity analysis by residual moveout fitting |  |
![[pdf]](icons/pdf.png) |
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The derivatives of 4
with respect to the vector of moveout
parameters is easily evaluated using the following expression:
![$\displaystyle {\frac {\partial {J}} {{\boldsymbol \mu}_{\vec x}} }' = {\frac {\...
...M_{\gamma }\left[\bar{{\boldsymbol \mu}}_{\vec x}\right]{\bf\bar I_{\gamma }} .$](img44.png) |
(A-8) |
The linear operator
can be represented as a
matrix.
The elements of this matrix are given by:
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(A-9) |
The first term (I)
is given by the depth-derivative of the image
after moveout.
This term can be numerically evaluated by
applying to the moved-out image
a finite-difference approximation of the first-derivative operator.
The second term (II) is different from zero
only when the spatial coordinate
of the image element
is the same as the coordinate corresponding
to the moveout parameter
.
When they do, and
for the choice of moveout parameters expressed in equation 7,
we have
.
The preceding expression simplifies
when the gradient is evaluated for
.
This simplifying condition is actually always fulfilled
unless the optimization algorithm includes inner iterations
for fitting the moveout parameters using a linearized approach.
Under this condition,
equation 8 becomes
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(A-10) |
and equation 9 becomes
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(A-11) |
 |
 |
 |
 | Wave-equation migration velocity analysis by residual moveout fitting |  |
![[pdf]](icons/pdf.png) |
Next: Derivatives of moveout parameters
Up: Gradient of the objective
Previous: Gradient of the objective
2010-11-26