Geoscience Reference
In-Depth Information
The definition of the
gravity anomaly
∆
g
and the
gravity disturbance δg
has, on the earth's surface, the same form as in the classical case of geoid
and sea level:
∂T
∂h
+
1
∂γ
∂h
T,
∆
g
=
g
P
−
γ
Q
=
−
(8-32)
γ
∂T
∂h
.
(8-33)
The gravity disturbance
δg
has become practically important only through
GPS, since
h
, the ellipsoidal height of
P
, can be measured using GPS and
hence
γ
P
, the normal gravity
γ
at
P
,canbedetermined.
As usual, Bruns' formula applies at
P
0
(classical geoid height
N
)and
P
(Molodensky height anomaly
ζ
) as well:
N
=
T
(
P
0
)
γ
δg
=
g
P
−
γ
P
−
=
,
(8-34)
ζ
=
T
(
P
)
γ
,
(8-35)
with some approximate value for
γ
such as
γ
45
◦
. Equation (8-32) can be
reformulated as the
boundary conditions for the Molodensky problem
∂T
∂h
−
∂γ
∂h
T
+∆
g
=0
,
1
γ
(8-36)
cf. (2-251), and
for the GPS problem
, cf. (2-252),
∂T
∂h
+
δg
=0
.
(8-37)
These two boundary conditions apply at the surface
S
(Molodensky) and at
sea level as well.
Finally we introduce the spherical approximation, disregarding the flat-
tening
f
in the equations (which are linear relations between small quanti-
ties).
Note:
The spherical approximation is a formal operation (disregarding
f
in small
ellipsoidal
quantities) and does not mean using a “reference sphere”
instead of a reference ellipsoid in any geometrical sense (Moritz 1980 a: p. 15).
This would imply geoidal heights on the order of 20 km!
Then (8-36) and (8-37) reduce to
∂T
∂r
+
2
r
T
+∆
g
=0
,
(8-38)
∂T
∂r
+
δg
=0
.
(8-39)
These equations, for the Molodensky and the GPS problem, are valid both
at sea level (classical) and at
S
(Molodensky).
