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It is often convenient to arrange the calculation of a wave
to remove the effect of overall translation,
thereby making the wave appear to ``stand still.''
It is easy enough to introduce
the time shift t_{0} of a vertically propagating wave
in a hypothetical medium of velocity , namely,
 
(67) 
Instead of solving equations in coordinates (t,x,z)
we could solve them in the socalled retarded coordinates
(t',x',z') where
t'=tt_{0}(z), x'=x and z'=z.
(For more details, see IEI sections 2.52.7.)
A time delay t_{0} in
the time domain corresponds to multiplication
by in the domain.
Thus, the wave pressure P is related
to the timeshifted mathematical variable Q by
 
(68) 
which is a generalization of equation (55)
to depthvariable velocity.
(Equations () and () apply in both x and
k_{x}space).
Differentiating with respect to z gives
 
(69) 
 (70) 
Next, substitute () and ()
into Table .4
to obtain the retarded equations in Table .5.
Table:
Retarded form of phaseshift equations.




zero 

























general 
diffraction 
+ thin lens 



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Stanford Exploration Project
12/26/2000