Geoscience Reference
In-Depth Information
are the equations of the geoid and the ellipsoid, where in general the con-
stants
W
0
and
U
0
are different. As in Sect. 2.12, we have, using Fig. 2.12,
W
P
=
U
Q
−
γN
+
T
, but now
U
Q
=
U
0
=
W
0
=
W
P
,sothat
γN
=
T
−
(
W
0
−
U
0
)
.
(2-345)
Denoting the difference between the potentials by
δW
=
W
0
− U
0
,
(2-346)
we obtain the following simple generalization of Bruns' formula:
N
=
T
−
δW
γ
.
(2-347)
We also need the extension of Eqs. (2-246) through (2-250). Those for-
mulas which contain
N
instead of
T
are easily seen to hold for an arbitrary
reference ellipsoid as well, but the transition from
N
to
T
is now effected by
means of (2-347). Hence, Eq. (2-247), i.e.,
∂T
∂h
+
∂γ
∆
g
=
−
∂h
N,
(2-348)
remains unchanged, but (2-248) becomes
∂T
∂h
+
1
∂γ
∂h
T
1
γ
∂γ
∂h
δW .
∆
g
=
−
−
(2-349)
γ
Therefore, the fundamental boundary condition is now
∂T
∂γ
∂γ
∂h
δW .
∂h
+
1
∂h
T
=∆
g
+
1
−
(2-350)
γ
γ
The spherical approximations of these equations are
N
=
T
−
δW
γ
0
(2-351)
and
∂T
∂r
−
2
R
T
+
2
R
δW
∆
g
=
−
(2-352)
and
∂T
∂r
−
2
R
T
=∆
g
2
R
δW .
−
−
(2-353)
