Geoscience Reference
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and postulate that in approximating
T
(
P
)by
f
(
P
), the observations (10-4)
will be reproduced exactly. The condition for this is
q
b
k
ϕ
k
(
P
i
)=
T
(
P
i
)=
f
i
,
(10-5)
k
=1
which may be written as a system of linear equations
q
A
ik
b
k
=
f
i
with
A
ik
=
ϕ
k
(
P
i
)
(10-6)
k
=1
or in matrix notation
Ab
=
f
.
(10-7)
Ifthesquarematrix
A
is regular, then the coecients
b
k
are uniquely de-
termined by
b
=
A
−
1
f
.
(10-8)
This model is suitable, for instance, for a determination of the geoid by
satellite altimetry, since this method, rather directly, yields geoidal heights
N
i
and hence, by Bruns' theorem (2-236),
T
(
P
i
)=
γ
i
N
i
.Fortheastro-
geodetic geoid determination, we must generalize this model, which leads us
to collocation.
Collocation
Here we wish to reproduce, by means of the approximation (10-2),
q
mea-
sured values which again are assumed to be errorless (this assumption is
not essential and will be dropped later). These measured values are assumed
to be linear functionals
L
1
T,L
2
T,...,L
q
T
of the anomalous potential
T
.
“Linear functional” means nothing else than a quantity
LT
that depends
linearly on
T
but need not be an ordinary function but may, say, also con-
tain a differentiation or an integral; essentially, it is the same as a “linear
operator”.
In fact, deflections of the vertical,
1
γ
∂T
∂x
,
1
γ
∂T
∂y
,
ξ
=
−
η
=
−
(10-9)
but also gravity anomalies,
∂T
∂z
−
2
R
T,
∆
g
=
−
(10-10)
and gravity disturbances
∂T
∂z
δg
=
−
(10-11)
