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.
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
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.
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.