Model Indeterminancy
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, &Delta&rho in the form
R and &Delta&rho 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 &Delta&rho, compare
these to those that were observed, and then vary R and &Delta&rho until the
predicted acceleration matches the observed acceleration.
This sounds simple enough, but there is a significant problem: there are an infinite number
of combinations of R and &Delta&rho that produce exactly the same gravitational
acceleration! For example, lets assume that we have found values for R and &Delta&rho
that fit our observations such that
Any other combination of values for R and &Delta&rho will also fit the observations as
long as R cubed times &Delta&rho equals 2.5. Examples of the gravity observations produced
by four of these solutions is 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 Interminancy, 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.