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)