Geoscience Reference
In-Depth Information
Let the masses outside the geoid be removed or moved inside the geoid, as
described in Sect. 8.2, and consider the effect of this procedure on quantities
referred to the ground
.
We denote the changes in potential and in gravity by
δW
and
δg
;then
the new values at ground will be
W
c
=
W
−
δW ,
(8-95)
g
c
=
g
−
δg .
(It is clear that
δg
here is not the gravity disturbance!) The disturbing
potential
T
=
W − U
becomes
T
c
=
T
−
δW .
(8-96)
The physical surface
S
as such will remain unchanged, but the telluroid Σ will
change, because its points
Q
are defined by
U
Q
=
W
P
, and the potential
W
at
any surface point
P
will be affected by the mass displacements according to
(8-95). The distance
QQ
c
between the original telluroid Σ and the changed
telluroid Σ
c
(Fig. 8.9) is given by
QQ
c
=
δW
γ
(8-97)
according to Bruns' theorem. This is identical with the variation of the height
anomaly
ζ
,sothat
ζ
c
=
δW
γ
δζ
=
ζ
−
.
(8-98)
Normal gravity
γ
on the telluroid Σ becomes on the changed telluroid Σ
c
γ
c
=
γ
+
∂γ
∂h
δζ
=
γ
+
1
∂γ
∂h
δW ,
(8-99)
γ
P
c
³
³
earth's surface S
Q
c
±³
Q
c
changed telluroid
telluroid
ellipsoid E
Fig. 8.9. Telluroid before and after gravity reduction, Σ and Σ
c
