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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.