Geoscience Reference
In-Depth Information
T
n
(
ϑ, λ
) is Laplace's surface harmonic of degree
n
. On the geoid, which as a
spherical approximation corresponds to the sphere
r
=
R
,wehaveformally
T
=
T
(
R, ϑ, λ
)=
∞
T
n
(
ϑ, λ
)
.
(2-268)
n
=0
We need not be concerned with questions of convergence here. Differentiating
the series (2-267) with respect to
r
, we find
(
n
+1)
R
r
n
+1
∞
∂T
∂r
=
1
r
δg
=
−
T
n
(
ϑ, λ
)
.
(2-269)
n
=0
On the geoid, where
r
=
R
, this becomes
∞
∂T
∂r
1
R
δg
=
−
=
(
n
+1)
T
n
(
ϑ, λ
)
.
(2-270)
n
=0
These series express the gravity disturbance in terms of spherical harmonics.
The equivalent of (2-263) outside the earth is
∂T
∂r
−
2
r
∆
g
=
−
T.
(2-271)
Its exact meaning will be discussed at the end of the following section. The
substitution of (2-269) and (2-267) into this equation yields
1)
R
r
n
+1
∞
∆
g
=
1
r
(
n
−
T
n
(
ϑ, λ
)
.
(2-272)
n
=0
On the geoid, this becomes
∞
1
R
∆
g
=
(
n
−
1)
T
n
(
ϑ, λ
)
.
(2-273)
n
=0
This is the spherical-harmonic expansion of the gravity anomaly.
Note that even if the anomalous potential
T
contains a first-degree spher-
ical term
T
1
(
ϑ, λ
), it will in the expression for ∆
g
be multiplied by the factor
1
1 = 0, so that ∆
g
can never have a first-degree spherical harmonic - even
if
T
has one.
−
