Geoscience Reference
In-Depth Information
for finite-dimensional vectors
a
and matrix
L
using the usual summation
over two equal indices, and
N
i
=
L
i
∆
g
P
leading to
cov(
N
P
,N
Q
)=
L
i
L
j
cov(∆
g
P
,
∆
g
Q
)
,
(10-27)
where
N
i
denotes the geoidal height at point
i
and ∆
g
is the gravity anomaly
at point
P
,and
L
denotes the Stokes formula. Explicit expressions are found
in Moritz (1980 a: Sect. 15).
In this statistical interpretation, we take the kernel function
K
(
P, Q
)
as the covariance function
C
(
P, Q
). Then
f
(
P
) is an optimal estimate (in
the sense of least variance) for the anomalous potential
T
and hence for
the height anomaly
ζ
=
T/γ
, on the basis of arbitrary measurement data.
For geoid determination in mountainous areas, relevant terrestrial measure-
ment data primarily are
ξ, η
,and∆
g
. The covariances
C
ik
and
C
Pi
are
given by known analytical expressions, see Tscherning and Rapp (1974) or
Moritz (1980 a: Sect. 15). A general computer program for collocation is
described in Sunkel (1980).
Least-squares collocation may easily be generalized to observational data
affected by random errors; systematic effects may also be taken into consid-
eration. In addition to the estimated quantities (
f
in our present case) we
may also compute their standard error by a formula similar to (10-24). A
comprehensive presentation of a least-squares collocation may be found in
Moritz (1980 a).
You cannot learn collocation from this slight chapter only!
Harmonicity of the covariance functions.
In three-dimensional space, the covariance functions, being kernel functions
and their linear functional transformations
L
, are always harmonic. If we
have (9-25),
C
(
ψ
)=
∞
c
n
P
n
(cos
ψ
)
(10-28)
n
=2
on the sphere, then in space there will be
R
2
rr
n
+2
C
(
r, r
,ψ
)=
∞
c
n
P
n
(cos
ψ
)
(10-29)
n
=2
(Moritz 1980 a: Sect. 23, Eq. (32-1)). The point
P
(
r, θ, λ
) is the computation
point, and
Q
(
r
,θ
,λ
) is a current data point;
ψ
is the spherical distance
between (
θ, λ
)and(
θ
,λ
), and
R
is the mean radius of the earth. The de-
pendence on
r
is given by the factor
r
−
(
n
+2)
(10-30)
