Geoscience Reference
In-Depth Information
The surface harmonics are given by
Hdσ,
H cos ψdσ
1
4 π
3
4 π
H 0 =
H 1 =
(2-278)
σ
σ
according to equation (1-89). Hence, we find from (2-277), on expressing H
by Poisson's integral (2-274) and substituting the integrals (2-278) for H 0
and H 1 , the basic formula
r 2
cos ψ Hdσ.
R 2
1
4 π
1
r
3 R
r 2
H P
=
(2-279)
l 3
σ
The reason for this modification of Poisson's integral is that the formulas
of physical geodesy are simpler if the functions involved do not contain har-
monics of degrees zero and one. It is therefore convenient to split off these
terms. This is done automatically by the modified Poisson integral (2-279).
We now apply these formulas to the gravity anomalies outside the earth.
Equation (2-272) yields at once
R
r
n +1
r g =
( n − 1) T n ( ϑ, λ ) .
(2-280)
n =0
Just as T n ( ϑ, λ ) is a Laplace surface harmonic, so is ( n
1) T n .Consequently,
r g , considered as a function in space, can be expanded into a series of
spherical harmonics and is, therefore, a harmonic function . Hence, we can
apply Poisson's formula to r g , getting
r 2
cos ψ R gdσ
R 2
R
4 π
1
r
3 R
r 2
r g P
=
(2-281)
l 3
σ
or
r 2
cos ψ gdσ.
R 2
4 πr
− R 2
l 3
1
r
3 R
r 2
g P
=
(2-282)
σ
This is the formula for the computation of gravity anomalies outside the
earth from surface gravity anomalies, or for the upward continuation of grav-
ity anomalies .
Finally, we discuss the exact meaning of the gravity anomaly δg P outside
the earth. We start with a convenient definition. The level surfaces of the
actual gravity potential, the surfaces
W = constant ,
(2-283)
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