Spherical Cap Complete Bouguer Correction: Traverse Examples

In this article, I want to provide some examples of computations performed by my Spherical Cap Complete Bouguer Correction software. If you have not yet done, make sure you read my processing methodology article to familiarize yourself with the approach.

Example 1 – Maximum expected error

The first question a geophysicist might ask is – “Is there a big difference between terrain corrections computed for a spherical cap Earth model and terrain corrections computed for a spherical plate (flat) Earth model?”.

Recall that the cap curves downward away from the gravity station with increasing radius and at the standard maximum radius of 166735 meters is 2161 meters below the station RL. Excess mass in the far-field that is below the station RL will increase the observed gravity rather than decrease it. This leads to an error in the terrain correction. How large is this error?

I ran a simulation for a single gravity observation that is surrounded by a circular mountain range. The range reaches its peak at a radius of 166735 meters. The first range simulates the Australian Alps and is 2000 meters high. The second simulates the Rocky Mountains and is 4000 meters high. The third simulates the Himalaya and is 8000 meters high. The width of the mountain ranges also increases with height. The datum level is 0m and the crustal density is 2.67 g/cc. I generated plate and cap models out to 166735 meters with four zones.

The first table shows the terrain correction computed for a spherical plate and the second table shows the terrain correction computed for a spherical cap. All units are in micrometers per second squared. In all cases, if you add the plate terrain correction then you are overestimating the terrain effect. The error caused by using a plate instead of a cap is summarized in the third table.

	Zone A	Zone B	Zone C	Zone D	Total
Alps	0	0	0	0.57	0.57
Rockies	0	0	0	4.89	4.89
Himalaya	0	0	0	46.09	46.09

	Zone A	Zone B	Zone C	Zone D	Total
Alps	0	0	0	-1.19	-1.190
Rockies	0	0	0	-2.14	-2.14
Himalaya	0	0	0	17.99	17.99

	Over-correction
Alps	1.78
Rockies	7.03
Himalaya	28.1

My conclusion is that the error is significant in all scenarios. On the other hand, for a local gravity survey, the effect is not likely to be detrimental to attempts to model local geological density variations because the difference is only generated by terrain that is more than 20 kilometers distant from the station. Consequently, the error is likely to vary smoothly across the survey and could be removed with a simple high pass filter. But then again, it’s nice to get things right!

Example 2 – Death Valley Lake

The software needs to be able to distinguish between offshore bathymetry that is below sea level and onshore terrain that is below sea level. This is achieved by supplying a land-water mask raster and querying this raster whenever the terrain is below sea level to make sure the location is offshore.

If you want, you can circumvent this rule to create models of lakes and compute the gravitational influence of the water in the lake. For example, if you fill Death Valley in California up to sea level then it creates a measurable gravitational anomaly of about 3 milligals.

Example 3 – Offshore to Onshore Traverse

This example is a traverse from the edge of the continental shelf to the coast and then inland over almost featureless terrain. The aim is to get an idea of the magnitude of bathymetric terrain corrections. The regional setting is shown below. The traverse runs SW to NE from the abyssal plain in South Australia across the continental slope and shelf and then onshore across largely flat terrain. A large spherical cap radius of 819.2 kilometers was used incorporating six zones. Water density was 1.025 g/cc and crustal density was 2.67 g/cc. The datum plane was set at -6000 meters, about 400 meters below the base of the abyssal plain in this region.

The traverse starts in deep ocean at 136 degrees East on the abyssal plain which lies almost 5600 meters below sea level. It then rises, with some significant irregularities, to the continental shelf at about 137.5 degrees East, which is never deeper than 100 meters. The shelf extends to about 139 degrees East and the traverse continues across largely flat terrain to 141 degrees East.

The profile stack shows the traverse from the SW to the NE. The bottom track map shows the regional setting with the traverse illustrated by the central black line.

The next profile (from the bottom) shows the model cross-section in the bottom axis and then the terrain correction in the top axis. In both cases the vertical scale is linear. The terrain correction for a spherical cap is the blue line and the terrain correction for a flat spherical plate is the black line, which is almost colinear at this scale. The second from the top profile shows the same data but with logarithmic vertical scales. In this profile, you can now see the gap between the spherical cap (blue) and spherical plate (black) terrain corrections. The top profile shows the Spherical Cap Complete Bouguer Correction in green. This is the sum of the contributions from crustal rock and ocean water. These two contributions are brown and blue curves, respectively.

The terrain correction is massive offshore, starting at 390 milligals and dropping quickly to 20 milligals at the edge of the continental shelf. It then steadily falls to about 10 milligals as it approaches the coast and then drops to 5 milligals onshore. The flat plate terrain correction is smaller, and this is noticeable onshore where the correction is only about 0.7 milligals – 4.3 milligals less than the cap correction.

Onshore, the effect of the water continues to decline slowly. By the end of the traverse, the cap terrain correction is about 1.5 milligals and the flat plate correction is about 0.15 milligals.

My conclusion is that the bathymetric contribution is significant, even up to 4 degrees away. Also, the flat plate correction significantly underestimates the magnitude of the correction (mass located deeper but closer horizontally creates a greater vertical gravitational contribution than mass that  is shallower but further away).

Example 4 – Mountain Range Traverse

This example is a traverse from deep inland across almost featureless terrain, then over the Australian Alps then then out over the continental shelf and to the abyssal plain. In this example we can see the magnitude of terrain corrections for the mountain range and also I want to use it to investigate the required radius of the spherical cap to properly incorporate the bathymetric terrain.

The regional setting is shown below. The traverse runs East to West from largely featureless terrain in South Australia across the Australian Alps and then out to sea. Water density was 1.025 g/cc and crustal density was 2.67 g/cc. The datum plane was set at -6000 meters, about 400 meters below the base of the abyssal plain in this region.

The profile stack shows the same information as in the previous example. The radius of the spherical cap was 819.2 kilometers. The features are similar, except that the mountains cause a significant terrain effect in this example of up to about 12 milligals.

The next profile illustrates what effect changing the radius of the spherical cap has on the terrain correction. I used three radii – 166.735 kilometers, 409.6 kilometers, and 819.2 kilometers. The terrain corrections for these are shown as the green, purple and brown curves. The vertical axis is logarithmic.

Firstly, you can see that using a larger radius lifts the magnitude of the correction across the entire profile. In the purple curve, you can see that the correction goes negative as you move away from the mountains, having already lost the effect of the bathymetry. The brown curve, using a standard radius of 166.735 kilometers, shows inflections and long-wavelength features that do not reflect reality. My conclusion is that you need to use a much larger radius than the standard to properly account for bathymetry effects. You can download a PDF of this article here.