Cheops' pyramid

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.

 

cheop
Figure 2 Left is a picture of the traveltime pyramid of equation ((1)) for fixed x and z. The darkened lines are constant-offset sections. Right is a cross section through the pyramid for large t (or small z). (Ottolini)

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, $z \to 0$.Then (1) becomes  

$$v\,t\ \eq \ \vert s \ -\ x \vert\ \ +\ \ \vert g \ -\ x \vert$$ (2)

The center of the square is located at (s,g) = (x,x). Taking travel time t to increase downward from the horizontal plane of (s,g)-space, the square contour is like a horizontal slice through the Egyptian pyramid of Cheops. To walk around the pyramid at a constant altitude is to walk around a square. Alternately, the altitude change of a traverse over g (or s) at constant s (or g) is simply a constant plus an absolute-value function.

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.

$$\begin{eqnarray} y\ \ \ \ &=&\ \ \ \ {g \ +\ s \over 2 } \\ h\ \ \ \ &=&\ \ \ \ {g \ -\ s \over 2 }\end{eqnarray}$$ (3) (4)

A traverse of y at constant h is shown in Figure 2. At the highest elevation on the traverse, you are walking along a flat horizontal step like the flat-topped hyperboloids of Figure 8. Some erosion to smooth the top and edges of the pyramid gives a model for nonzero reflector depth.

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.