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

$$\begin{eqnarray} u( t ,z) &=& u( t,z=0) \ast \delta( t+z/v) \\ U(\omega,z) &=& U(\omega,z=0) \ e^{-i\omega z/v}\end{eqnarray}$$ (9) (10)