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DOWNWARD CONTINUATION

Given a vertically upcoming plane wave at the earth's surface, say $u(t,x,z=0)=u(t) {\rm const}(x)$,and an assumption that the earth's velocity is vertically stratified, i.e. v=v(z), we can presume that the upcoming wave down in the earth is simply time-shifted from what we see on the surface. (This assumes no multiple reflections.) Time shifting can be represented as a linear operator in the time domain by representing it as convolution with an impulse function. In the frequency domain, time shifting is simply multiplying by a complex exponential. This is expressed as
\begin{eqnarray}
u( t ,z) &=& u( t,z=0) \ast \delta( t+z/v) \\ U(\omega,z) &=& U(\omega,z=0) \ e^{-i\omega z/v}\end{eqnarray} (3)
(4)
Sign conventions must be attended to, and that is explained more fully in chapter [*].



 
next up previous print clean
Next: Continuation of a dipping Up: Downward continuation Previous: Migration derived from downward
Stanford Exploration Project
12/26/2000