3D Example - San Francisco Bay area Topography

Using the elevation map of the San Francisco Bay area illustrated in Figure , a second test was conducted to assess the applicability of the method in 3D. The maximum surface topographic relief is approximately 800 m; however, the elevation gradients and topographic wavelengths are significantly smaller and longer than the 2D test illustrated in Figure . The flat subsurface datum is set a depth of 8000 m.

 

Bay.2D Figure 4 Elevation map of the San Francisco Bay area used in 3D testing.

Figure  presents the slices through the 3D potential function results. The top panel shows a depth slice at approximately zero elevation, whereas the lower two panels show slices along Easting (bottom left) and Southing (bottom right) directions.

 

Pot3D
Figure 5 3D potential function for San Francisco Bay area topography. a) depth slice of the potential function at approximately zero elevation; b) vertical slice along Southing 20 km; and c) vertical slice along Easting 20 km. Note that the potential function is smoother than the previous example indicating less coordinate system focusing.

These profiles illustrate a PF that is smoother than the previous example.

Figure  shows the coordinate system generated along the same two slices shown in the panels b) and c) of

 

Ray.slice
Figure 6 Cross-section of the 3-D coordinate system developed for the San Francisco Bay area topography. Left-hand panel: vertical slice along along Southing 20 km; and right-hand panel: vertical slice along Easting 20 km.

Figure . The generated coordinate system much smoother than in the previous example, as expected from the smoothness of the PF. Figure  presents a perspective view of the ray-traced coordinate system results.

 

Rays3D
Figure 7 Perspective view of the ray-traced coordinate system developed from potential function in Figure .

The coordinate system rays are fairly straight, except in the region beneath topographic highs. Another way to visualize the ray coordinate system is to examine how the topography 'heals itself' at various $\tau$ steps. Figure  illustrates this for the $\tau_0$ (top left), $\tau_{N/3}$ (top right), $\tau_{2N/3}$ (bottom left) and $\tau_N$ (bottom right) surfaces, where N is the total number of extrapolation steps.

 

Rays3D2
Figure 8 Illustration of topographic coordinate system healing through examination of single-extrapolation step elevation differences (in km). Top left: step $\tau_0$; top right: step $\tau_{N/3}$; bottom left: step $\tau_{2N/3}$; and bottom right: step $\tau_N$. Sidebar shows the elevation differences (in km), where the greyscale has clipped according to the peak elevation difference at $\tau_0$.

The sidebars show the elevation difference between the lowest and highest points of each equipotential surface. The greyscale intensity has been clipped according to the maximum elevation difference at $\tau_0$.