Downward continuation

Vertical exaggeration example

To examine questions of vertical exaggeration and spatial resolution we consider a line of point scatters along a $45^\circ$ dipping line in $(x , z)$-space. We impose a linear velocity gradient such as that typically found in the Gulf of Mexico, i.e.  $v(z)=v_0+\alpha z$ with $\alpha=1/2 s^{-1}$. Viewing our point scatterers as a function of traveltime depth, $\tau = 2\int_0^z dz/v(z)$ in Figure 7.11 we see, as expected, that the points, although separated by equal intervals in $x$, are separated by shorter time intervals with increasing depth. The points are uniformly separated along a straight line in $(x , z)$-space, but they are nonuniformly separated along a curved line in $(x,\tau)$-space. The curve is steeper near the earth's surface where $v(z)$ yields the greatest vertical exaggeration. Here the vertical exaggeration is about unity (no exageration) but deeper the vertical exaggeration is less than unity (horizontal exaggeration).

sagmod Figure 11. Points along a 45 degree slope as seen as a function of traveltime depth.

Applying the points spray out into hyperboloids (like hyperbolas, but not exactly) shown in Figure 7.12. The obvious feature of this synthetic data is that the hyperboloids appear to have different asymptotes.

sagdat Figure 12. The points of Figure 7.11 diffracted into hyperboloids.

In fact, there are no asymptotes because an asymptote is a ray going horizontal at a more-or-less constant depth, which will not happen in this model because the velocity increases steadily with depth.

(I should get energetic and overlay these hyperboloids on top of the exact hyperbolas of the Kirchhoff method, to see if there are perceptible traveltime differences.)

Downward continuation