Gravity field of the earth - Physical Geodesy - page 90

Accordingly, the difference in magnitude is the gravity disturbance

δg = g P − γ P . (2-232)

The difference in direction - i.e., the deflection of the vertical - is the same
as before, since the directions of γ P and γ Q coincide virtually.
The gravity disturbance is conceptually even simpler than the gravity
anomaly, but it has not been that important in terrestrial geodesy. The
significance of the gravity anomaly is that it is given directly: the gravity
g is measured on the geoid (or reduced to it), see Chap. 3, and the normal
gravity γ is computed for the ellipsoid.

A very important remark
So far, for historical reasons, much more gravity anomalies ∆ g are available
and are being processed than gravity disturbances δg . By GPS, however, the
point P is determined rather than Q . Therefore, in future, we may expect
that δg will become more important than ∆ g .
However, mirroring the present state of practice of physical geodesy, we
continue mainly to work with ∆ g . Most statements about ∆ g will also apply
for δg , with obvious modifications, such as with Molodensky's corrections
(see Chap. 8), and Stokes' formula will be replaced by Koch's formula (see
below in this chapter).

Relations
There are several basic mathematical relations between the quantities just
defined. Since

= U Q + ∂U
∂n

U P N = U Q − γN, (2-233)

Q

we have

W P = U P + T P = U Q − γN + T P . (2-234)

Because

W P = U Q = W 0 , (2-235)

we find

T = γN (2-236)

(where we have omitted the subscript P on the left-hand side) or

N = T

γ .

(2-237)

This is the famous Bruns formula , which relates the geoidal undulation to
the disturbing potential.