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I compute the propagation of waves in space in a
homogeneous medium by phase shifting in depth. Given a wavefield at
some depth *z*_{0}, the wavefield at another
depth, *z*, can be expressed as a phase shifted version of this
wavefield.
In general the operator is a matrix operator
but if the wavefield, *P* is expressed as a vector of amplitudes of
the different wavetypes the operator *A* is diagonal. The different
wavetypes are the eigensolutions of the Christoffel equation (see
appendix) so the operator is diagonalized. For each of the six
wavetypes the extrapolation in depth can be expressed as,
Thus, once I have calculated the wave amplitudes at the top of any layer I can
calculated the amplitudes at the bottom of the layer by phase shifting the
data. The value of *p*_{z} is the same for all frequencies but the phase
shift through the layer for each frequency is given by.

This calculation is different for each frequency but on a vector computer it
is fully vectorized over frequency.
If I am ``turning off'' propagation of a particular wavetype it is at this
stage that I do it. I can simply zero the amplitude of the wave instead of
applying the shifting operator.

** Next:** Calculation of reflection and
** Up:** TWO WAY PHASE SHIFT
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Stanford Exploration Project

12/18/1997