Modern views on the determination of the figure of the earth - Physical Geodesy - page 320

density are concerned, the geoidal undulation N as obtained by this method
is as accurate as the orthometric height.
As a matter of fact, the gravity anomaly ∆ g in this method refers to
ground level; it is the difference between gravity at ground and normal grav-
ity at the telluroid. Instead of using directly this free-air anomaly, we may
also use other gravity anomalies - for instance, the isostatic anomaly in the
sense of Sect. 8.9.
To repeat a simple but fundamental principle: ∆ g, δg, ξ, η, ζ as obtained
by Molodensky's theory primarily always refer to the physical earth's sur-
face and not to sea level!

8.11 A first balance

The new methods described in this chapter are primarily intended for the
determination of the physical surface of the earth, but they are also well
suited for the determination of the geoid (Sect. 8.10). Their essential fea-
ture is that the gravity anomalies now refer to the ground, whether we deal
with free-air anomalies or with isostatic or other similarly reduced gravity
anomalies (Sect. 8.9).
The immediate result is the height anomaly ζ , the separation between
the geopotential and the corresponding spheropotential surface at ground
level. By plotting the height anomalies above the ellipsoid, we get the quasi-
geoid. This geoid-like surface has no physical significance, but it furnishes a
convenient visualization of the height anomalies. By plotting the orthometric
height from the earth's surface vertically downward, we obtain the geoid.
It is instructive to compare the geoid and the quasigeoid. The geoidal
undulation N and ζ , the undulation of the quasigeoid, are related by (8-
111), or

g − γ

H = H ∗ − N − ζ = H. (8-113)

γ

The term g γ is approximately equal to the Bouguer anomaly; this may

be seen by using (4-32) for γ together with

−

1
2

∂γ
∂h H.

= γ

γ − (8-114)

The quantity γ in the denominator can be replaced by our usual constant
γ 0 . Since the Bouguer anomaly is rather insensitive to local topographic
irregularities, the coecient is locally constant so that there is approximately
a linear relation between ζ and the local irregularities of the height H .In
other words, the quasigeoid mirrors the topography (Fig. 8.12).