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There are three steps to implement plane-wave migration in tilted coordinates
for one plane-wave source.
First, the source and receiver wavefields are rotated to the new
coordinates. Second, for each frequency, source and receiver wavefields
are extrapolated into the subsurface and the image is generated
by cross-correlating the two wavefields in the new coordinates. Third,
the image is rotated back to the original Cartesian coordinates after stacking
the images of all the frequencies. In the first step, we rotate a 2D dataset for
each frequency, this cost is trivial compared to the wavefield extrapolation.
In the third step, we rotate the 3D image back to Cartesian coordinates after
stacking all frequencies, the cost is also very small.
To further reduce the cost of rotations,
the plane-wave sources with similar propagation directions share
the same tilted coordinate system. This reduces the rotation number in
the third step.
Figure illustrates how to sample the
ray parameters *p*_{x} and *p*_{y} and
how to put plane-wave sources together to share a coordinate system.
In Figure
we sample
(*p*_{x},*p*_{y}) in Cartesian coordinates, where *p*_{x},*p*_{y} are defined as:

| |
(21) |

| (22) |

where *p*_{max} is defined by the maximum take-off angle and *dp*_{x}, *dp*_{y} are the sampling of the ray parameters.
In Figure , each dot represents a plane-wave source with a ray parameter pair (*p*_{x},*p*_{y}).
Given the surface velocity, we calculate the azimuth angle
and take-off angle from (*p*_{x},*p*_{y}) and vice versa
using equations 16 and 17.
Therefore, we can divide the whole area into cells
using as the coordinate system.
Figure shows the cells in the coordinates
.All the plane-wave sources whose ray parameter pair (*p*_{x},*p*_{y}) fall in a cell share the same coordinate system.
The dots in the smallest circle in Figure represent plane-wave sources whose take-off angle is smaller than .
For those plane-wave sources,
we extrapolate its source and receiver wavefields in Cartesian coordinates.
For all plane-wave sources whose ray parameter pair (*p*_{x},*p*_{y}) is in a cell, we use their average
take-off angle and azimuth angle to design the coordinate system for the migration.

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Stanford Exploration Project

5/6/2007