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Pointscatterer geometry is a good starting point to discuss convertedwaves
prestack commonazimuth migration. The equation for the travel time is the sum
of a downgoing travel path with Pvelocity (v_{p}) and an upcoming
travel path with Svelocity (v_{s}),
 
(1) 
where and represent the source and receiver vector locations
and is the pointscatterer subsurface position.
Commonazimuth migration is a wavefieldbased, downwardcontinuation algorithm.
The algorithm is based on a recursive solution of the oneway wave equation
Claerbout (1985). The basic continuation step used to compute the wavefield at depth
from the wavefield at depth z can be expressed in the frequencywavenumber
domain as follows:
 
(2) 
After each depthpropagation step, the propagated wavefield is equivalent to
the data that would have been recorded if all sources and receivers were placed at the new
depth level Schultz and Sherwood (1980).
The wavefields are propagated with two different velocities, a Pvelocity for the
downgoing wavefield and an Svelocity for the upcoming wavefield.
The basic downward continuation for converted waves is performed by applying the
DoubleSquareRoot (DSR) equation:
 
(3) 
or in midpointoffset coordinates,
 
(4) 
The commonazimuth downwardcontinuation operator takes advantage of the reduced dimensionality
of the data space, which results from using a commonazimuth resorting of the data.
Rosales and Biondi (2004) discuss how to do this resorting for convertedwave
data.
The general continuation operator can then be expressed as follows Biondi and Palacharla (1996):
 

 
 (5) 
Since commonazimuth data is independent of k_{yh}, the integral can be
pulled inside and analytically approximated by the stationaryphase method Bleinstein (1984).
The application of the stationaryphase method is based on a highfrequency approximation.
By geometrical means we derive the
stationarypath approximation for converted waves.
The expression for comes from substituting the stationarypath approximation into
the expression for the full DSR of equation (4):
 
(6) 
where
 
(7) 
Next: Impulse response
Up: Rosales and Biondi: PSCOMAZ
Previous: Introduction
Stanford Exploration Project
5/3/2005