Gravity reduction - Physical Geodesy - page 148

δN in meters. This correction δ is the indirect effect on gravity ;itisofthe
order of 3 mgal.
Now the topographic-isostatic gravity anomalies refer strictly to the co-
geoid. The application of Stokes' formula gives N c , which according to (3-80)
is to be corrected by the indirect effect δN to give the undulation N of the
actual geoid.

Deflections of the vertical
The indirect effect on the deflections of the vertical is, in agreement with
Eqs. (2-377), given by

1
R

∂δN
∂ϕ

δξ = − ,

(3-88)

1
R cos ϕ

∂δN
∂λ

δη = − .

The indirect effect is essentially identical with the so-called topographic-
isostatic deflection of the vertical (Heiskanen and Vening Meinesz 1958:
pp. 252-255).
The topographic-isostatic reduction as such is very much alive, however.
It is practically the only gravity reduction used for geoid determination at
the present time (with the possible exception of free-air reduction, which is
acasebyitself).
The last purely gravimetric geoid, before the advent of satellites, was the
Columbus Geoid (Heiskanen 1957).

3.8 The inversion reduction of Rudzki

It is possible to find a gravity reduction where the indirect effect is zero.
This is done by shifting the topographic masses into the interior of the geoid
in such a way that

U C = U T . (3-89)

Then

δW = U T − U C =0 . (3-90)

This procedure was given by M. P. Rudzki in 1905. For the present purpose,
we may consider the geoid to be a sphere of radius R (Fig. 3.13). Let the
mass element dm at Q be replaced by a mass element dm at a certain point
Q inside the geoid situated on the same radius vector. The potential due to