Geoscience Reference
In-Depth Information
11 Computational methods
11.1
The remove-restore principle
Let us start with gravity reduction according to the modern view of mea-
suring and calculating the gravity field in principle always
at the earth's
surface,
or briefly,
on the ground
,orequivalently,
at point level
.Thisisused
in the sense of Sects. 8.9 and 8.14. More precisely, it is
topographic-isostatic
reduction at ground level
.
The most practical way to realize this idea is least-squares collocation,
because it automatically works in three-dimensional space, by simply putting
the desired topographic height
h
as parameters for input (measurements:
gravity anomalies, deflections of the vertical, etc.) and output (potential
T
or its functionals to be computed). Symbolically, this means
T
=
L
(
)
(11-1)
or
output =
L
(input)
,
(11-2)
where
denotes the linear operation of least-squares collocation (not to be
confused with a linear functional
L
as used, e.g., in Eq. (10-13)).
In Sect. 8.9 we have introduced gravity reduction from the point of view
of the modern theory. To repeat, immediately specializing to topographic-
isostatic reduction, we have
L
•
measured gravity anomalies ∆
g
at ground level,
reduced topographic-isostatic anomalies ∆
g
c
obtained by
removing
the
attraction of the topographic-isostatic masses
δg
TI
,
•
“co-potential”
T
c
=
L
(∆
g
c
) computed by collocation, and
•
•
“real potential”
T
by
restoring
the “indirect effect” of the topographic-
isostatic masses
δT
TI
.
Mathematically this may be written
T
=
L
(∆
g
−
δg
TI
)+
δT
TI
.
(11-3)
This is a reinterpretation of the gravity reduction of Sect. 8.9. It must
be correct since if
δT
TI
=
L
(
δg
TI
)
(11-4)
