Gravity reduction - Physical Geodesy - page 147

The change of potential, δW , is for the Bouguer reduction expressed by

δW B = U T (3-82)

and for the topographic-isostatic reduction by

δW TI = U T − U C , (3-83)

the subscripts of the potential U corresponding to those of the attraction A
used in the preceding sections.
For the practical determination of U T and U C , the template technique,
as expressed in (3-32), may again be used (at least, conceptually):
U = ∆ U, (3-84)

where the relevant formulas are the first equation of (3-21), (3-9), (3-12),
and (3-15). The point U refers to is always the point P 0 at sea level (Fig. 3.1).
For U T we use U 0 , see (3-12), with b = H and density 0 (see Fig. 3.12). For
U C in the continental case, we use U e , see (3-9), with the following values:
Pratt-Hayford,

H
D 0 ;

b = c = d, density (3-85)

Airy-Heiskanen,

b = t, c = t + T,

density 1

− 0 .

(3-86)

The corresponding considerations for the oceanic case are left as an exercise
for the reader.
The indirect effect with Bouguer anomalies is very large, of the or-
der of ten times the geoidal undulation itself. See the map at the end of
Helmert (1884: Tafel I), where the maximum value is 440 m! The reason is
that the earth is in general topographic-isostatically compensated. There-
fore, the Bouguer anomalies cannot be used for the determination of the
geoid.
With topographic-isostatic gravity anomalies, as might be expected, the
indirect effect is smaller than N , of the order of 10 m. It is necessary, how-
ever, to compute the indirect effect δN I carefully, using exactly the same
topographic-isostatic model as for the gravity reductions.
Furthermore, before applying Stokes' formula, the topographic-isostatic
gravity anomalies must be reduced from the geoid to the cogeoid. This is
done by a simple free-air reduction, using (3-26), by adding to ∆ g I

the

correction

δ =+0 . 3086 δN [mgal] , (3-87)