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The original surface seismic data are usually shot gathers. A typical seismic shot gather
(the receiver wavefield of a shot at the surface) is
a five dimensional object: R(sy,sx,ry,rx,z=0,t),
where (sy,sx) is the source location,
(rx,ry) is the receiver location and t is the travel time.
After a Fourier transformation in t, we have the receiver wavefield in the frequency
domain , where is the angular frequency.
Each shot represents a real physical experiment.
The most straight forward way to obtain the image of the subsurface
is shot-profile migration, in which we obtain
the local image of each experiment independently and form the final image
of the subsurface by stacking all the local images.
A typical shot-profile migration algorithm includes two steps. First, source and receiver wavefields
are extrapolated into the subsurface using one-way wave equations. In isotropic media they are defined as follows:
where v=v(x,y,z) is the velocity of the media, is the source wavefield,
which is an impulse at the surface and is the receiver wavefield.
Second, the image is formed by cross-correlating
the source and receiver wavefields: