Gravity reduction - Physical Geodesy - page 137

3. Restore the removed masses to their former position, and add alge-
braically their attraction to g at Q .

The purpose of this slightly complicated but logically clear procedure
is that in step 2 the free-air gradient can be used. If we here replace the
actual free-air gradient by the normal gradient ∂γ/∂h , then the error will
presumably be smaller than in using (3-43).
Note that the free-air gradient can also be accurately computed alterna-
tively by (2-394); the gravity anomalies ∆ g in this formula are the gravity
anomalies obtained after performing step 2, that is, Bouguer anomalies re-
ferred to the lower point Q .
The effect of the masses above Q (steps 1 and 3) may be computed, e.g.,
by means of some kind of template or computer procedure for numerical
three-dimensional integration. If the terrain correction is neglected and only
the infinite Bouguer plate between P and Q of the normal density =
2 . 67 g cm − 3 is taken into account, then we obtain with the steps numbered
as above:

gravity measured at P g P

1. remove Bouguer plate − 0 . 1119 ( H P − H Q )

2. free-air reduction from P to Q +0 . 3086 ( H P − H Q )

3. restore Bouguer plate − 0 . 1119 ( H P − H Q )

together: gravity at Q

g Q = g P

+0 . 0848 ( H P −

H Q ) .

(3-46)
This is the same as (3-45), which is, thus, confirmed independently. We see
now that the use of (3-43) or (3-45) amounts to replacing the terrain with
a Bouguer plate.
Finally, we note that the reduction of Poincare and Prey, abbreviated
as Prey reduction , yields the actual gravity which would be measured inside
the earth if this were possible. Its purpose is, thus, completely different from
the purpose of the other gravity reductions which give boundary values at
the geoid.
It cannot be directly used for the determination of the geoid but is needed
to obtain orthometric heights as will be discussed in Sect. 4.3. Actual gravity
g 0 at a geoidal point P 0 is related to Bouguer gravity g B , Eq. (3-38), by

g 0 = g B − A T,P 0 .

(3-47)

It is obtained by subtracting from g B the attraction A T,P 0 of the topographic
masses on P 0 , which corresponds to restoring the topography after the free-
air reduction of Bouguer gravity from P to P 0 .