Loudspeakers and Forward Light Cones

If the geophones on the surface z=0 are transformed into loudspeakers, where the seismograms denoted by $p(\bf {r}_g,t_s)$ (where $\bf {r}_g=[x,z=0]$) are used as the time history of the loudspeaker's causal signal, then Huygen's principle says that the resulting wavefield $s(\bf {r},t)$ is a linear superposition of weighted point sources given by:

 

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

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

$$~\sum_g p(\bf {r}_g, t-\vert\bf {r}-\bf {r}_g\vert/c)/r ,$$  = (3.11)

where $r =\vert\bf {r}-\bf {r}_g\vert$. Musgrave (1970) restates Huygen's principle as "Given a wavefront at some time t₀, consider every point on the wavefront as a secondary point of disturbance initiated at t₀.". Consistent with this definition, Equation 3.13 says that the forward light cones have their tips fixed to points on the hyperbola, and at some later listening time, their associated wavefronts superimpose to give an expanding semi-circular wavefront, as shown by the dotted semi-circle in Figure 3.6. This wavefront is that for a downgoing wavefield reflected from the free-surface at z=0.

  

Figure 3.6: Seismic waves recorded by geophones on the z=0 plane are denoted by the dashed hyperbola, and emanate from "past wavefronts" that emerge from the buried point source at *. The geophones can be then be turned into causal loudspeakers with the seismic traces as time histories of the loudspeaker's signal. These loudspeakers generate the downgoing future wavefront denoted by dots.