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I address the problem of cycle skipping
by employing linearized image perturbations.
If we define , we can write a
discrete version of the image perturbation
using a Taylor series expansion of psrm as
| |
(69) |

where the ' sign denotes differentiation relative to the velocity
ratio parameter . For the image perturbations computed
with storm.differential, I use the name
*linearized image perturbations* .
dip graphically illustrates this procedure.
**WEP2.imag
**

Figure 6
Comparison of image perturbations obtained as a difference
between two migrated images (b) and as the result of the forward
WEMVA operator applied to the known slowness perturbation (c).
Panel (a) depicts the background image corresponding to the
background slowness.
Since the slowness perturbation is large (),
the image perturbations in panels (b) and (c) are
different from each-other.

The linearized prestack Stolt residual migration
operator can be
computed analytically, as described in Appendix C.
With this operator, I can compute linearized image perturbations
in two steps. First, I run residual migration for a large range of velocity
ratios and pick at every image point the ratio which
maximizes flatness of the gathers.
Then, I apply the operator in storm.differential
to the background image and scale the result with the picked
.

**WEP2.rays
**

Figure 7
Comparison of slowness backprojections using the WEMVA operator
applied to image perturbations computed as a difference
between two migrated images (b,d) and as the result of the
forward WEMVA operator applied to a known slowness perturbation
(c,e).
Panel (a) depicts the background image corresponding to the
background slowness.
Since the slowness perturbation is large (),
the image perturbations in panels (b) and (c) and the
fat rays in panels (d) and (e) are different from
each-other.
Panel (d) shows the typical behavior associated with the
breakdown of the Born approximation.

The linearized image perturbations approximate the
non-linear image perturbations caused by arbitrary velocity model changes.
They are based on the gradient of the image
change relative to a velocity model change, and are
less restrictive than the Born approximation limits.

WEP2.raan shows how the linearized image perturbation
methodology applies to the synthetic example used earlier in this
section. All panels are similar to the ones in
WEP1.rays and WEP2.rays, except that
the left panels (b and d) correspond to linearized image perturbations,
instead of simple image perturbations.
Again, I compare image and slowness perturbations
with the ideal perturbations obtained
by the forward WEMVA operator (c and e).
Both the image and slowness perturbations are identical
in shape and magnitude.

**WEP2.raan
**

Figure 9
Panels (a) and (c) correspond to
Cartesian coordinates, and
panels (b) and (d) correspond to
ray coordinates.
Velocity model with an overlay of the
ray coordinate system initiated by a D
source at the surface (a);
image obtained by downward continuation in
Cartesian coordinates with the
equation (c);
velocity model with an overlay of the
ray coordinate system (b);
image obtained by wavefield extrapolation in
ray coordinates with the
equation (d).espite the fact that the slowness perturbation is large
(), the image perturbations in panels (b) and (c)
and the fat rays in panels (d) and (e) are practically identical,
both in shape and in magnitude.

** Next:** WEMVA sensitivity kernels
** Up:** Wave-equation migration velocity analysis
** Previous:** Cycle skipping in image
Stanford Exploration Project

11/4/2004