The slope at a is the negative of that at b . Surface arrival times will be roughly hyperbolic with distortion because of the velocity jump at the interface. 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. The travel time for a constant-offset section is then .To find the earliest arrival, set .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.