Model Indeterminacy
We have now derived the gravitational attraction associated with a simple spherical body. The
vertical component of this attraction was shown to be equal to:
Notice that our expression for the gravitational acceleration over a sphere contains a term
that describes the physical parameters of the spherical body; its radius, R, and its
density contrast, Dr, in the form
R and Dr are two of the parameters
describing the sphere that we would like to
be able to determine from our gravity observations (the third is the depth to the center
of the sphere z). That is, we would like to compute
predicted gravitational accelerations given estimates of R and
Dr, compare
these to those that were observed, and then vary R and Dr until the
predicted acceleration matches the observed acceleration.
This sounds simple enough, but there is a significant problem: there is an infinite number
of combinations of R and Dr that produce
exactly the same gravitational
acceleration! For example, let's assume that we have found values for
R and Dr
that fit our observations such that
Any other combination of values for R and Dr will also fit the observations as
long as R cubed times Dr equals 31.25.
Examples of the gravity observations produced
by four of these solutions are shown below.
Our inability to uniquely resolve parameters describing a model of the earth
from geophysical observations is not unique to the gravity method but is present
in all geophysical methods. This is referred to using a variety of expressions:
Model Indeterminacy, Model Equivalence, and Nonuniqueness to
name a few. No matter what it is called, it always means the same thing; a particular
geophysical method can not uniquely define the geologic structure underlying
the survey. Another way of thinking about this problem is to realize that a
model of the geologic structure can uniquely define the gravitational field
over the structure. The gravitational field, however, can not uniquely define
the geologic structure that produced it.
If this is the case, how do we determine which model is correct? To do this we must incorporate
additional observations on which to base our interpretation. These additional observations
presumably will limit the range of acceptable models we should consider when interpreting our
gravity observations. These observations could include geologic observations or
observations from different types of geophysical surveys.
