Gravity field outside the earth - Physical Geodesy - page 246

P

H 1

A

H

h 1

level of F
W= 1

h

F

sea level

P 0 A 0

W= 0

Fig. 6.3. Reduction to sea level and to the level of F

level anomalies ∆ g . Or we may reduce to any other level surface W = W 1 ,
for instance, to that passing through F (Fig. 6.3), using h 1 instead of h in
(6-58). Then we should also use H 1 , rather than H , in (6-57). For large-
scale purposes, reduction to sea level appears to be preferable. Probably such
a reduction will attain a considerable amount only in exceptional cases so
that it can usually be neglected and H in the formulas of Sects. 6.3 and 6.4
may be taken as the height of P above sea level or above ground. See also
Sect. 8.6.

Computation of the gravity vector
After computing the components δg r ,δg ϕ ,δg λ by numerical integration, we
may transform them into Cartesian coordinates δg x ,δg y ,δg z

with respect

to the global coordinate system.
We may go via ellipsoidal-harmonic coordinates according to Sect. 6.2.
For the small quantities δg u ,δg β ,δg λ , we may apply the spherical approxi-
mation, neglecting a relative error of the order of the flattening. If the flat-
tening is neglected, then the ellipsoidal-harmonic coordinates u, β, λ reduce
to the spherical coordinates r, ϕ, λ so that as a spherical approximation

δg u = δg r ,

β = δg ϕ ,

(6-59)

δg λ being rigorously the same in both systems. Thus, δg r ,δg ϕ ,δg λ may also
be considered as the components of δ g in ellipsoidal-harmonic coordinates.
Then we have

g u = γ u + δg r , β = γ β + δg ϕ , λ = δg λ ; (6-60)

and g x ,g y ,g z are obtained by (6-17), the components of g replacing the cor-
responding components of γ . It is evident that the spherical approximation