Next: Backpropagated Residuals Direct Waves
Up: Physical Interpretation
Previous: Loudspeakers and Forward Light
What would happen if the geophones were replaced by seismic sinks?
In this case the backpropagated wavefield is given by:
Similar to the loudspeaker example,
equation 3.13 says that backward light cones are afixed
to the points on the hyperbola, and at some given listening time t_{0},
these cones superimpose earlier along the t=t_{0} plane to give a converging
semicircular wavefront, as shown by the dotted
lines in Figure 3.7. This wavefront
is coincident with that for an upgoing wavefield
emanating from the buried point scatterer.
To compare the backpropagated field to the forward propagated field
a Fourier transform in time to equation 3.13 to get

= 

(3.13) 
and a Fourier transform in time is applied to equation 3.14 to get

= 

(3.14) 
The difference between the backward and forward operations is that the
sign of the
extrapolation operator is opposite to one another.
Figure 3.7:
Same as previous figure except each loudspeaker is
an acausal, rather than causal, point source. Hence, the tips of backward
light cones are afixed to the hyperbola in the z=0 plane and their
backward wavefronts superimpose to describe a converging wavefront.
These converging wavefronts reconstruct the earlier wavefronts and converge to the source of the error in the
misfit residual. The slowness field is updated at this convergence
point to correct the error in the slowness model.

Next: Backpropagated Residuals Direct Waves
Up: Physical Interpretation
Previous: Loudspeakers and Forward Light
Gerard Schuster
19980729