Modern views on the determination of the figure of the earth - Physical Geodesy - page 314

so that the new gravity anomaly will be

γ + 1
γ ∂h δW

∂γ

∆ g c = g c − γ c =( g − δg ) − (8-100)

or

1
γ

∂γ
∂h δW .

∆ g c =∆ g − δg − (8-101)

The reduced gravity anomaly ∆ g c consists of the free-air anomaly (in
the Molodensky sense) ∆ g and two reductions:

1. the direct effect, − δg , of the shift of the outer masses on g ;and

2. the “indirect effect”,

1
γ

∂γ
∂h δW ,

− (8-102)

of this shift on γ , because of the change of the telluroid to which γ
refers.

Letusrepeatoncemorethatalltheseanomalies∆ g c

refer to the physical

surface of the earth, to “ground level”!
If the masses outside the geoid are completely removed, then ∆ g c is
a Bouguer anomaly; if the outer masses are shifted vertically downward
according to some isostatic hypothesis, then ∆ g c is an isostatic anomaly, etc.
In this way we may get a “ground equivalent” for each conventional gravity
reduction. The two are always related by analytical continuation. See below
for the isostatic anomalies; for analytical continuation see Sect. 8.6.
Now we may describe the determination of the height anomalies ζ in a
way that is similar to the corresponding procedure for the geoidal undula-
tions N of Sect. 8.2:

1. The masses outside the geoid are, by computation, removed entirely or
else moved inside the geoid; W and g change to W c and g c according

to (8-95).
2. The point at which normal gravity is computed is moved from the
ellipsoid upward to the telluroid point Q .
3. The indirect effect, the distance QQ c = δζ , is computed by (8-98).
4. The point to which normal gravity refers is now moved from the point
Q of the telluroid Σ to the point Q c

of the changed telluroid Σ c ,ac-

cording to (8-99).
5. The changed height anomalies ζ c are computed from the “reduced”
gravity anomalies ∆ g c (8-101) by any solution of Molodensky's prob-
lem, such as Eq. (8-57) or (8-68).