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reduction according to the Pratt-Hayford system as the depth of compen-
sation D goes to zero. This is sometimes useful.
Again we have
A T + A C + F, (3-97)
where A C = A is now to be computed using the second equation of
(3-19) with c = H P
g H = g
and κ = H ; H P
is the height of the station P and H
the height of the compartment.
The indirect effect is
U C . (3-98)
The potential U C = U is to be computed using the first equation of
(3-19) with κ = H as before, but c = 0 since it refers to the geoidal point
P 0 . The corresponding δN is very small, amounting to about 1 m per 3 km of
average elevation. It may, therefore, usually be neglected so that the cogeoid
of the condensation reduction practically coincides with the actual geoid.
Even the “direct effect”, −A T + A C , can usually be neglected, as the
attraction of the Helmert layer nearly compensates that of the topography.
There remains
δW = U T
g H = g + F,
(3-99)
that is, the simple free-air reduction. In this sense, the simple free-air reduc-
tion may be considered as giving approximate boundary values at the geoid ,
to be used in Stokes' formula. To the same degree of approximation, the
“free-air cogeoid” coincides with the actual geoid.
Hence, the free-air anomalies
g F
= g + F − γ
(3-100)
may be considered as approximations of “condensation anomalies”
g H = g H
γ.
(3-101)
The many facets of free-air reduction
This is one of the most basic, most dicult, and most fascinating topics of
physical geodesy. In fact, the free-air anomaly means several conceptually
different but related concepts.
1. The term F above has been seen to be part of every gravity reduction
rather than a full-fledged gravity reduction itself.
2. Approximately, free-air anomalies may be identified with Helmert's
condensation anomalies as we have seen above.
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