Geoscience Reference
In-Depth Information
are such linear functionals of
T
; here,
x, y, z
denotes a local coordinate sys-
tem in which the
z
-axis is vertical upwards and the
x
-and
y
-axes are directed
towards north and east, and
R
= 6371 km is a mean radius of the earth.
Equation (10-9) is a consequence of equations such as (2-377), with
∂s
=
∂x
or
∂y
;normalgravity
γ
may be considered constant with respect to horizon-
tal derivation. Equation (10-10) is the well-known fundamental equation of
physical geodesy in spherical approximation (2-263). Equations (10-9) and
(10-10) refer to the earth's surface.
To repeat, by saying that deflections of the vertical and gravity distur-
bances and anomalies are linear functionals of
T
, we simply indicate the fact
that
ξ, η, δg,
∆
g
depend on
T
by the expressions (10-9) and (10-10), which
clearly are linear; they are the linear terms of a Taylor expansion, neglect-
ing quadratic and higher terms. In the above notation
L
i
T
,thesymbol
L
i
denotes, for instance, the operation
1
γ
∂
∂x
L
i
=
(10-12)
applied to
T
at some point.
Putting
L
i
f
=
L
i
T
=
i
(10-13)
and substituting (10-2), we get
q
B
ik
b
k
=
i
with
B
ik
=
L
i
ϕ
k
,
(10-14)
i
=1
where
L
i
ϕ
k
denotes the number obtained by applying the operation
L
i
to the
base function
ϕ
k
; the coecient
B
ik
obtained in this way does not depend
on the measured values. Equation (10-14) is a linear system of
q
equations
for
q
unknowns, which is quite similar to (10-6). This method of fitting an
analytical approximating function to a number of given linear functionals is
called collocation and is frequently used in numerical mathematics.
It is clear that interpolation is a simple special case of collocation in
which
L
i
f
=
f
(
P
i
)
(10-15)
is the “evaluation functional”, giving the value of
f
at a point
P
i
.Thus
we see that in both interpolation and collocation the coecients
b
k
require
the solution of a linear system of equations (which in general will not be
symmetric).
