Gravity field outside the earth - Physical Geodesy - page 245

that the ellipsoid is replaced by a sphere in any geometrical sense ;rather
it means that in the originally elliptical formulas the first and higher pow-
ers of the flattening are neglected, whereby they formally become spherical
formulas.
Since the gravity anomalies, etc., are referred to an ellipsoid, we must be
very careful in computing t , which enters into the formulas of Sect. 6.3. If
an exact sphere of radius R were used as a reference surface, then we should
have r = R + H ,where H is the elevation of the computation point above
the sphere. Actually, we use a reference ellipsoid; then we again have

R
R + H ,

r = R + H, t = (6-57)

but H is now the elevation above the ellipsoid (or, to a sucient accuracy,
above sea level ), the constant R = 6371 km being the earth's mean radius.
Thus, r as computed by (6-57) differs from the geocentric radius vector

r = x 2 + y 2 + z 2 . We have already mentioned that we may replace the
geocentric latitude ϕ by the ellipsoidal latitude ϕ , as far as T and δ g are
concerned - for instance, by putting ϕ = ϕ in (6-26) or (6-29).

Data
For all computations dealing with the external gravity field of the earth,
free-air gravity anomalies must be used for ∆ g , since all other types of grav-
ity anomalies correspond to some removal or transport of masses whereby
the external field is changed. If, in addition to ∆ g , deflections of the vertical
ξ, η (in the upward continuation) are used, then these quantities should be
computed from free-air anomalies. If, as usually done, the normal free-air
gradient ∂y/∂h . =0 . 3086 mgal/m is used for the free-air reduction, then the
free-air anomalies refer, strictly speaking, to the earth's physical surface (to
ground level) rather than to the geoid (to sea level). The N values com-
puted from them by Stokes' formula are height anomalies ζ , referring to the
ground, rather than heights of the actual geoid. However, this distinction is
insignificant and can be ignored in most cases, so that we may consider ∆ g
as sea-level anomalies (see Sect. 8.6).
If we cannot neglect this distinction, aiming at highest accuracy in high
and steep mountains for low altitudes H , then we may proceed as follows. We
reduce the free-air anomaly ∆ g from the ground point A to the corresponding
point A 0 at sea level (Fig. 6.3):

∂ ∆ g
∂h

∆ g harmonic =∆ g − h, (6-58)

and use the sea level anomaly ∆ g harmonic so obtained. The vertical gradient
∂ ∆ g/∂h may be computed by applying formula (2-394) using the ground-