Next: Limiting flattening model space
Up: R. Clapp: Moveout analysis
Previous: Flattening Review
There are two general approaches to calculating moveout parameters
using the flattening methodology. The first approach is to
perform parameter estimation in two phases. First, solve for the
nonlinear field, then construct a linear
problem to find the moveout parameters that best fit the
field.
The flattening algorithm provides a timeshift field
that is function of depth z, offset h, and CRP . As a first test
we want to estimate moveout of a volume migrated using downward continuation
migration. Biondi and Symes (2003) demonstrated that residual moveout can be approximated (assuming zero geologic dip)
as a function of angle and depth z through
 
(7) 
where is the moveout paremeter.
We can estimate as a global inverse problem. Defining
the above moveout equation above as mwe obtain the objective function Q,
 
(8) 
We can ensure spatial smoothness by introducing a roughener to the objective function to obtain,
 
(9) 
where is scaling parameter.
To test the methodology I migrated a line from a 3D North Sea dataset.
Figure displays two crosssections of the migrated
data (left) and the field (right) calculated from the volume.
A moveout field is then calculated from the
field using a conjugate gradient algorithm to minimize
equation 9. Figure shows the resulting
moveout field. The inversion approach has an additional advantage,
it easy to assess where the moveout parameterization effectively
described the time shifts and where it failed.
Figure shows the result of stacking the absolute value of the
residual over
the offset plane. Areas of high amplitude represent areas where a single
parameter did not accurately describe .
data
Figure 1 The left panel shows three crosssections of
the migrated image (depth, inline, angle). The right panel shows
the time shifts calculated from the volume.
rho1
Figure 2 The result of inverting for the moveout
parameter from the time shifts shown in the
right panel of Figure .
resid1
Figure 3 The spatial error fitting error associated
with the time shifts shown in Figure and the moveout parameter
shown in Figure .
Rather than solving for a single moveout parameter at each location,
we can solve for multiple moveout parameters simultaneously. To test this
approach I introduced a new operator that estimates
the moveout parameter by searching
for higher order moveout anomalies. For I chose an arbitrary moveout function,
 
(10) 
that attempts to see if a higher polynomial of the same form as to
help to describe the moveout. The optimization
goal of equation () becomes
 
(11) 
Figure shows the resulting (left) and (right)
fields. Note how similar the field is to the one in Figure ,
indicating that a twostage estimation approach would have yielded a similar result. Figure
shows the resulting residual. Note the decrease in some areas compared to Figure ,
but still showing areas where the moveout is significantly more complex.
rho2
Figure 4 The result of inverting for both (left panel) and (right
panel). Note the similarity to the single parameter estimation shown in Figure .
resid2
Figure 5 The fitting error associated with the two parameter fitting shown
in Figure .

 
The methodology of this section assumed that the field was accurate. The nonlinear
nature means this assumption is problematic, particularly when we are far from the correct solution.
In the context of the moveout problem, this means we are far from flat the defacto starting guess.
Next: Limiting flattening model space
Up: R. Clapp: Moveout analysis
Previous: Flattening Review
Stanford Exploration Project
5/6/2007