Loudspeakers and Backward Light Cones

What would happen if the geophones were replaced by seismic sinks? In this case the backpropagated wavefield is given by:

 

$$s(\bf {r},t) \sum_g \sum_{t_s} p(\bf {r}_g,t_s)g_a(\bf {r},t\vert\bf {r}_g,t_s),$$  = 

$$~\sum_g \sum_{t_s} p(\bf {r}_g,t_s)\delta(t-t_s+\vert\bf {r}-\bf {r}_g\vert/c)/r,$$  =

$$~\sum_g p(\bf {r}_g,t+\vert\bf {r}-\bf {r}_g\vert/c)/r.$$  = (3.12)

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 t₀, these cones superimpose earlier along the t=t₀ plane to give a converging semi-circular 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

 

$$\tilde S(\bf {r},\omega ) \sum_g \tilde P(\bf {r}_g,\omega )e^{i\omega \vert\bf {r}-\bf {r}_g\vert/c}/r,$$  =  (3.13)

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

 

$$\tilde S(\bf {r},\omega ) \sum_g \tilde P(\bf {r}_g,\omega )e^{-i\omega \vert\bf {r}-\bf {r}_g\vert/c}/r.$$  =  (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.