Statistical methods in physical geodesy - Physical Geodesy - page 352

Zero anomaly
If there are no gravity measurements in a large area - for instance, on the
oceans -, then the estimate

∆ g P ≡

0 (9-46)

is used in this area. In this trivial case all α Pi are zero.
If all known gravity stations are far away, and if we know of nothing
better, then this primitive extrapolation method is applied, although the
accuracy is poor. At best, this method may work with isostatic anomalies.

None of these three methods gives optimum accuracy. In the next section we
investigate the accuracy of the general prediction formula (9-41) and find
those coecients α Pi that yield the most accurate results.

9.5 Accuracy of prediction methods

In order to compare the various possible methods of prediction, to determine
their range of applicability, and to find the most accurate method, we must
evaluate their accuracy.
Consider the general case of Eq. (9-41). The correct gravity anomaly at
P is ∆ g P , the predicted value is

= n
i =1

∆ g P

α Pi ∆ g i . (9-47)

The difference is the error ε P of prediction,

=∆ g P − ∆ g P ε P =∆ g P − α Pi ∆ g i . (9-48)

i

By squaring we find

= ∆ g P − α Pi ∆ g i ∆ g P − α Pk ∆ g k

ε 2

P

i k

(9-49)

2
i

α Pi ∆ g P ∆ g i +
i

=∆ g 2

P −

α Pi α Pk ∆ g i ∆ g k .

k

Let us now form the average M of this formula over the area considered
(either a limited region or the whole earth). Then we have from (9-6),

M { ∆ g i ∆ g k } = C ( ik ) ≡ C ik ,

M { ∆ g P ∆ g i } = C ( Pi ) ≡ C Pi , (9-50)

∆ g 2 M {

P }

= C (0) ≡ C 0 .