Assume that velocity varies nicely with depth in such a way that the T(X) curve for P (surface to surface) has the simple analytical form T = 2(X + 1)1/2 − 2. Now consider the phase PP for this...

Assume that velocity varies nicely with depth in such a way that the T(X) curve

for P (surface to surface) has the simple analytical form T = 2(X + 1)1/2 − 2.

Now consider the phase PP for this velocity structure, containing two P legs

with a surface bounce point between source and receiver. Using the above T(X)

function, show that for the travel time to be stationary with respect to changes in the

bouncepoint position, the bouncepoint must occur at the midpoint between source

and receiver. Show that this is a maximum time point. (Note: Consider only bouncepoints within the vertical plane connecting source and receiver. The midpoint is a

minimum time point for perturbations perpendicular to the source–receiver plane,

and thus the PP bouncepoint on a two-dimensional surface is actually a minimax

or saddle point.)