Gravity Calculations over Bodies with more Complex Shapes
Although it is possible to derive analytic expressions for the computation of the
gravitational acceleration over additional bodies with simple shapes (cylinders,
slabs, etc.), we already have enough information to describe a general scheme
for computing gravity anomalies over bodies with these and more complex shapes.
The basis for this computation lies in the
approximation of a complex body as
a distribution of point masses.
Previously, we derived the vertical component of the gravitational
acceleration due to a point mass with mass m as
We can approximate the body with complex shape as a distribution of point masses. The gravitational
attraction of the body is then nothing more than the sum of the gravitational attractions of
all of the individual point masses as illustrated below.
In mathematical notation, this sum can be written as
where z represents the depth of burial of each point mass, d represents the horizontal
position of each point mass, and x represents the horizontal position of the observation point.
Only the first three terms have been written in this equation. There is,
in actuality, one term in this
expression for each point mass. If there are N point masses, this equation can be written more
compactly as
For more detailed information on the computation of gravity anomalies over complex two- and three-
dimensional shapes look at the following references.
- Talwani, Worzel, and Landisman, Rapid Gravity Computations for Two-Dimensional
Bodies with Application to the Mendocino Submarine Fraction Zone, Journal
Geophysical Research, 64, 49-59, 1959.
- Talwani, Manik, and Ewing, Rapid Computation of Gravitational Attraction
of Three-Dimensional Bodies of Arbitrary Shape, Geophysics, 25, 203-225,
1960.
These are quite old! but these basic techniques underly modern modelling software
packages such as those available from Encom
Technology or Geosoft.
This is the final screen in the Gravity Notes package. The next
page is another self-assessment quiz.
