Geoscience Reference
In-Depth Information
The local covariance function of gravity anomalies
C
(
P, Q
)givenby
Tscherning-Rapp can be defined as
∞
n
−
1
2)(
n
+
B
)
s
n
+2
P
n
(cos
ψ
)
.
C
(
P, Q
)=
A
(9-13)
(
n
−
n
=
N
+1
Modeling the covariance function means in practice fitting the empirically
determined covariance function (through its three essential parameters: the
variance
C
0
, the correlation length
ξ
and the variance of the horizontal
gradient
G
0
) to the covariance function model. Hence the four parameters
A, B, N
and
s
are to be determined through this fitting procedure. A simple
fitting of the empirical covariance function was done using the COVAXN-
subroutine (Tscherning 1976).
The essential parameters of the empirical covariance parameters for 2489
gravity stations in Austria are 740.47 mgal
2
for the variance
C
0
and 43.5 km
for the correlation length
ψ
1
. The value of the variance for the horizontal
gradient
G
0
was roughly estimated as 100 E
2
(note that E indicates the
Eotvos unit, where 1 E = 10
−
9
s
−
2
).
With a fixed value
B
= 24, the following Tscherning-Rapp covariance
function model parameters were fitted:
s
=0
.
997 065
,A
= 746
.
002 mgal
2
and
N
= 76. The parameters were used for the astrogeodetic, the gravimetric
as well as the combined geoid solution.” (End of quotation.)
The Tscherning-Rapp model can be summed to get closed expressions.
Its popularity is due to its comprehensiveness: there are expressions for co-
variances of various quantities derived by covariance propagation (Sect. 10.1),
and to its flexibility since it contains several parameters which can be given
various numerical values.
Remark.
The spherical-harmonic expression of the covariance function
is considered in Sect. 9.3. The theory of global and local covariance func-
tions is described in great detail in Moritz (1980 a: Sects. 22 and 23). The
three essential parameters of a local covariance function (variance
C
0
, corre-
lation length
ξ
, and curvature parameter
G
0
) are also defined there. Funda-
mental numerical studies on local covariance functions have been made by
Kraiger (1987, 1988).
9.3
Expansion of the covariance function
in spherical harmonics
The more or less complicated integral formulas of physical geodesy frequently
take on a much simpler form if they are rewritten in terms of spherical
harmonics. A good example is Stokes' formula (see Sect. 2.15).
