Downward continuation

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

$$u( t ,z) = u( t,z=0) \ast \delta( t+z/v)$$ (3)

$$U(\omega,z) = U(\omega,z=0) e^{-i\omega z/v}$$ (4)

Sign conventions must be attended to, and that is explained more fully in chapter .