Because of the importance of the point-scatterer model,
we will go to considerable lengths to visualize the functional dependence
among *t*, *z*, *x*, *s*, and *g* in equation (1).
This picture is more difficult--by one dimension--than is
the conic section of the exploding-reflector geometry.

To begin with,
suppose that the first square root in (1) is constant
because everything in it is held constant.
This leaves the familiar hyperbola in (*g*,*t*)-space,
except that a constant has been added to the time.
Suppose instead that the other square root is constant.
This likewise leaves a hyperbola in (*s*,*t*)-space.
In (*s*,*g*)-space, travel time is a function of *s* plus a function of *g*.
I think of this as one coat hanger, which is
parallel to the *s*-axis, being hung from another coat hanger,
which is parallel to the *g*-axis.

A view of the traveltime pyramid on the (*s*,*g*)-plane
or the (*y*,*h*)-plane is shown in Figure 2a.

Figure 2

Notice that a cut through the pyramid at
large *t* is a square, the corners of which have been smoothed.
At very large *t*,
a constant value of *t* is the square contoured in (*s*,*g*)-space,
as in Figure 2b.
Algebraically, the squareness becomes evident for a point reflector
near the surface, say, .Then (1) becomes

(2) |

More interesting and less obvious are the curves
on common-midpoint gathers and constant-offset sections.
Recall the definition that the midpoint between the shot and geophone is *y*.
Also recall that *h* is half the horizontal offset
from the shot to the geophone.

(3) | ||

(4) |

For rays that are near the vertical, the traveltime curves are far from the hyperbola asymptotes. Then the square roots in (1) may be expanded in Taylor series, giving a parabola of revolution. This describes the eroded peak of the pyramid.

12/26/2000