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B
best-fitting ellipsoid
for region AB
A
geoid
mean earth ellipsoid
Fig. 5.20. A locally best-fitting ellipsoid and the mean earth ellipsoid
Hayford computed the international ellipsoid as ellipsoid that best fits the
isostatically reduced vertical deflections in the United States. Rapp (1963)
made an interesting recomputation.
Please note: Don't use formula (5-151) in spite of its historical impor-
tance: the determination of local best-fitting ellipsoids is hopelessly obsolete
now!
The previously described method is impaired by unknown density anoma-
lies and by the lack of complete isostatic compensation. Therefore, it is better
to go still one step further and subtract the gravimetrically computed values
ξ g g from the astrogeodetic deflections ξ a a . Then the minimum condition
( ξ a
η g ) 2 = minimum
ξ g ) 2 +( η a
(5-152)
results. Thus, we may say that Hayford's method is equivalent to the use of
(5-152), the gravimetric values ξ g g being approximated by deflections that
represent the effect of topography and of its isostatic compensation only. If
the isostatic compensation were complete, and if we had perfect knowledge
of the density above the geoid, both methods would give exactly the same
result if applied properly.
Equivalence of different definitions of the earth ellipsoid
It is quite remarkable that the minimum definitions (5-148) or (5-149) and
a similar definition due to Rudzki, using the condition
(∆ g ) 2 = minimum ,
(5-153)
σ
yield results which, to the usual spherical approximation, are identical with
each other and with the physical definition in terms of M , W 0 , C
A ,and
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