Geoscience Reference
In-Depth Information
where ∆
g
is assumed to be known at every point of the geoid, then we see that
a linear combination of
T
and
∂T/∂n
is given upon that surface. According
to Sect. 1.13, the determination of
T
is, therefore, a
third boundary-value
problem of potential theory
. If it is solved for
T
, then the geoidal height,
which is the most important geometric quantity in physical geodesy, can be
computed by Bruns' formula (2-237).
Therefore, we may say that the basic problem of physical geodesy, the
determination of the geoid from gravity measurements, is essentially a third
boundary-value problem of potential theory.
2.13
Spherical approximation and expansion of the
disturbing potential in spherical harmonics
The reference ellipsoid deviates from a sphere only by quantities of the order
of the flattening,
f
.
=3
·
10
−
3
. Therefore, if we treat the reference ellipsoid
as a sphere in equations relating quantities of the anomalous field, this may
cause a relative error of the same order. This error is usually permissible in
N, T,
∆
g, δg
, etc. For instance, the absolute effect of this relative error on
the geoidal height is of the order of 3
10
−
3
N
;since
N
hardly exceeds 100 m,
this error can usually be expected to be less than 1 m.
As a spherical approximation, we have
·
γ
=
GM
r
2
∂γ
∂h
=
∂γ
2
GM
r
3
1
γ
∂γ
∂h
=
2
r
.
,
=
−
,
−
(2-255)
∂r
We introduce a mean radius
R
of the earth. It is often defined as the radius of
a sphere that has the same volume as the earth ellipsoid; from the condition
4
3
πR
3
=
3
πa
2
b,
(2-256)
we get
√
a
2
b.
3
R
=
(2-257)
In a similar way, we may define a mean value of gravity,
γ
0
,asnormalgravity
at latitude
ϕ
=45
◦
(Moritz 1980b: p. 403). Numerical values of about
R
= 6371 km
,
0
= 980
.
6 gal
(2-258)
are usual. Then
1
γ
∂γ
∂h
=
2
R
,
−
(2-259)
∂γ
∂h
=
−
2
γ
0
R
.
