DOWNWARD CONTINUATION AND IMAGING

By now you are probably wondering what good are downward-continued seismic waves. The answer is that once we know the wavefields inside the earth we can do many seismic data analysis tasks. The obvious one is migration. We can do it better than Kirchhoff. Given the upcoming wave at the earth surface $\hbox{Wave}(t,x,z=0)$, pushing it down to all z in the earth gives us $\hbox{Wave}(t,x,z)$.The exploding reflector idea is that the source is on the reflectors at t=0. Thus we construct images with  

$$\hbox{Image}\ ( x , z )\ \eq\ \ \hbox{Wave}\ ( t=0 , x , z )$$ (21)

There is a curious symmetry between the input and output of the imaging process:

t = 0

all$\ z$

Diffraction

is sometimes regarded as the natural process that creates and enlarges hyperboloids. Migration

is the computer process that does the reverse.

Compared to the Kirchhoff method, downward continuation methods offer several advantages: (1) we handle v(z) media exactly without calculating any ray paths; (2) we get the correct amplitude versus angle; (3) we automatically include various phase shifts and wave phenomena; (4) interpolation and aliasing artifacts are much reduced.