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When doing velocity analysis, general practice is to measure moveout
from a relatively sparse set of CRP gathers.
Kirchhoff depth migration is the preferred construction method
because it can produce the sparse set of CRP gathers without needing to image
the entire volume.
In addition, if our Green's function table is constructed correctly,
Kirchhoff methods do not
suffer from the velocity approximations needed by frequency domain methods.
Kirchhoff methods also have some deficiencies. The most glaring weakness
of Kirchhoff methods is the difficulty in constructing the Green's function
table.
To construct an accurate Green's function table we must account for,
and weight correctly, the multiple arrivals that occur in complex geology.
Calculating and accounting for multiple arrivals adds significantly to both
coding complexity and memory requirements. As a result,
a single arrival is often all that is used. Eikonal methods
Fomel (1997); Podvin and Lecomte (1991); Vidale (1990); van Trier and Symes (1991)
can efficiently
produce first arrivals, but in areas of complex geology the first arrival
is not always the most important arrival Audebert et al. (1997).
Nichols (1994)
proposed a band-limited method that
gave the maximum amplitude arrival, but the method is computationally
impractical in 3-D. As a result, people usually go to expensive ray based
methods but still face the difficult tasks of choosing the most
important arrival and correctly and efficiently interpolate the
traveltime field Sava and Biondi (1997).

The most computationally
attractive alternative to Kirchhoff methods is frequency
domain downward continuation.
Downward continuation has its own weaknesses.
Its primary weakness is speed. Downward continuation
can not be target oriented, so full volume imaging is required.
In addition, frequency domains methods in their purest form
can not handle lateral variations in velocity.
By migrating with multiple velocities and applying a space domain correction
to the wavefield,
they can do a fairly good job handling lateral variations
(this migration is normally
referred to as PSPI, Phase-shift plus interpolation)Ristow and Ruhl (1993).
Finally, downward continuation focuses the wavefield towards zero offset,
making conventional moveout analysis impossible.

We can create CRP
gathers where moveout analysis is possible by changing our imaging condition
Clayton and Stolt (1981); Prucha et al. (1999).
Given a wave-field
we follow the normal procedure of downward continuing the data
and extracting the image at the surface *z*=0 and zero time. Instead
of
extracting the image at zero offset, we note that reflection
angle can be evaluated by the differential equation:

| |
(5) |

where *z* is the depth, *x*_{h} is half-offset.
The topic of this paper is not migration, but tomography. The tomography
method could be applied with either Kirchhoff or PSPI. For us,
PSPI proved to be a more attractive choice. A 2-D and 3-D PSPI algorithm
was already available, where
a Kirchhoff approach would have required the coding of the migration algorithm
along with a suitable traveltime computation method.

** Next:** Characterizing moveout errors
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Stanford Exploration Project

4/28/2000