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distinction between these two kinds of prediction and the mathematical for-
mulation is the same in both cases.
In order to predict a gravity anomaly at P ,wemusthaveinformation
about the gravity anomaly function. The values observed at certain points
are the most important information. In addition, we need some information
on the form of the anomaly function. If the gravity measurements are very
dense, then the continuity or “smoothness” of the function is sucient - for
instance, for linear interpolation. Otherwise we may try to use statistical
information on the general structure of the gravity anomalies. Here we must
consider two kinds of statistical correlation: the autocorrelation - the corre-
lation between each other - of gravity anomalies and the correlation of the
gravity anomalies with height .
Correlation with height will for the moment be disregarded; Sect. 9.7
will be devoted to this topic. The autocorrelation is characterized by the
covariance function considered in Sect. 9.2.
Mathematically, the purpose of prediction is to find a function of the
observed gravity anomalies ∆ g 1 , g 2 , ..., g n
in such a way that the un-
known anomaly ∆ g P
at P is approximated by the function
= F (∆ g 1 , g 2 ,..., g n ) .
g P
(9-40)
Here ∆ g i denotes the value of ∆ g at a point i , not a spherical harmonic! In
practice, only linear functions of the ∆ g i are used. If we denote the predicted
value of ∆ g P
by ∆ g P , such a linear prediction has the form
n
g P
= α P 1 g 1 + α P 2 g 2 + ... + α Pn g n
α Pi g i .
(9-41)
i =1
The coecients α Pi depend only on the relative position of P and the grav-
ity stations 1 , 2 , ..., n ; they are independent of the ∆ g i . Depending on the
way we choose these coecients, we obtain different interpolation or extrap-
olation methods. Here are some examples.
Geometric interpolation
The “gravity anomaly surface”, as represented by a gravity anomaly map,
may be approximated by a polyhedron by dividing the area into triangles
whose corners are formed by the gravity stations and passing a plane through
the three corners of each triangle (Fig. 9.3). This is approximately what is
done in constructing the contour lines of a gravity anomaly map by means
of graphical interpolation.
Analytically, this interpolation may be formulated as follows. Let point
P be situated inside a triangle with corners 1, 2, 3 (Fig. 9.3). To each point
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