Lateral shift of the hyperbola top

Figure 21 shows a point scatterer below a dipping interface. As usual there is a higher velocity below.

 

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Figure 21 Rays emerging from a point scatterer beneath a velocity wedge (left). Travel-time curve (right). The slope at a is the negative of that at b. The midpoint between a and b is at the top of the h>0 curve.

This is a simple prototype for many lateral-velocity-variation problems. Surface arrival times will be roughly hyperbolic with distortion because of the velocity jump at the interface. The minimum travel time (hyperboloid top) has been displaced from its usual location directly above the point scatterer. Observe that

• At minimum time, the ray emerges going straight up.
• Minimum time is on the high-velocity side of the point scatterer.
• Minimum time is displaced further from the scatterer as offset increases.

The travel-time curve is roughly hyperbolic, but the asymptote on the left side gives the velocity of the medium on the left side, and the asymptote on the right gives the velocity on the right.

Let T(x) denote the travel time from the point scatterer to the surface point x. The travel time for a constant-offset section is then $t(y) \ =$ $T(y+h) \,+\, T(y-h)$.To find the earliest arrival, set $dt/dy\,=\,0$.This proves that the slope at a on Figure 21 is the negative of the slope at b. This shows why the displacement of the top of the hyperboloid from the scatterer increases with offset.

Lateral velocity variation causes hyperbolas to lose their symmetry. Computationally, it is the lens term that tilts hyperbolas, causing their tops to move laterally.