The correct kinematics for layered models were predicted
by the theory. An illustration of the raypaths for a single horizontal
layer is shown in Figure 1.
The cross-correlation picks out the times *T*_{QB}-*T*_{PA} and
*T*_{PAB}-*T*_{PA}=*T*_{AB}. These two times correspond to a `direct wave'
traveling directly
between the geophones with apparent slowness equal to that of the incoming
wave, and the reflection received at *B* if there was a source at *A*.

The theory has not yet been extended to Earth
models with lateral variations in velocity. It is important to establish if
the conjecture still holds for *v*(*x*,*z*) models; and if it does not, then
the limits of its applicability should be found and understood.

Incorporating a simple lateral velocity variation into
Figure 1 gives Figure 2. Again the
cross-correlation picks out two events. The velocity anomaly affects
*T*_{PA} and *T*_{PAB} equally so the kinematics of the reflection
event remain correct. However *T*_{QB} is not effect by the
velocity anomaly, so the kinematics of the `direct wave' are altered.
Although this figure does not disprove the conjecture for a *v*(*x*,*z*)
models, it does show that lateral variations may affect the direct wave
but not the reflection event.

rays1
Ray paths for a single plane wave incident from below a horizontal
layered model. Cross-correlation lag between events recorded at A
and B corresponds to reflection event in conventional seismogram.
Figure 1 |

rays2
Ray paths for a single plane wave incident from below a model with
lateral velocity anomaly. Time of reflection event is not affected
by the anomaly, but the `direct wave' is.
Figure 2 |

rays3
Ray paths for a single point source below a horizontal
layered model. Time of reflection event is not affected
by the location of the source, but the `direct wave' is.
Figure 3 |

11/11/1997