Gravity field of the earth - Physical Geodesy - page 88

n'

n

geoid
W= 0

P

N

g P

Q

reference
ellipsoid
U= 0

° °

Q

Fig. 2.12. Geoid and reference ellipsoid

T is called the anomalous potential ,or disturbing potential .Wecomparethe
geoid

W ( x, y, z )= W 0 (2-225)

with a reference ellipsoid

U ( x, y, z )= W 0 (2-226)
of the same potential U 0 = W 0 .Apoint P of the geoid is projected onto
the point Q of the ellipsoid by means of the ellipsoidal normal (Fig. 2.12).
The distance PQ between geoid and ellipsoid is called the geoidal height ,or
geoidal undulation , and is denoted by N . Unfortunately, there is a conflict of
notation here. Denoting both the normal radius of curvature of the ellipsoid
and the geoidal height by N is well established in geodetic literature. We
continue this practice, as there is little chance of confusion.
Consider now the gravity vector g at P and the normal gravity vector γ

at Q .The gravity anomaly vector ∆ g is defined as their difference:

∆ g = g P − γ Q . (2-227)

A vector is characterized by magnitude and direction. The difference in mag-
nitude is the gravity anomaly

∆ g = g P − γ Q ; (2-228)

the difference in direction is the deflection of the vertical .
The deflection of the vertical has two components, a north-south com-
ponent ξ and an east-west component η (Fig. 2.13). As the direction of the
vertical is directly defined by the astronomical coordinates latitude Φ and
longitude Λ, the components ξ and η can be expressed by them in a sim-
ple way. The actual astronomical coordinates of the geoidal point P ,which