Geoscience Reference
In-Depth Information
Thus, we use simultaneously the earth model EM for the longer wave-
lengths and the topographic-isostatic geological model TI for the shorter
wavelengths. Since the spherical-harmonic expansions are generalizations,
for the sphere, of Fourier series for the circle, we can speak of wavelengths.
Denoting the maximum degree of the spherical-harmonic expansion with
N
,
this can be associated with a shortest resolvable wavelength
λ
according to
λ
=
2
π
N
=
360
◦
N
.
(11-8)
For an expansion to degree
N
= 180 (say), we have
λ
= 360
◦
/
180 = 2
◦
,
which roughly corresponds to 200 km on a meridian or on the equator. In
many cases, the half wavelength
λ/
2 is considered (see Seeber 2003: p. 469).
Since EM (approximately) takes care of the long waves up to a certain
maximum degree
N
, it is resonable to represent the remaining short waves
from
N
to infinity. This sequence
N
+1
,N
+2
,...,
will be denoted by
CN
,where
CN
is the abbreviation of the “complement” of the sequence
from 2 to
N
.
Thus, we may write for the residuals
∞
T
EM
−
T
CN
δT
=
T
−
,
TI
(11-9)
EM
−
CN
δ
=
−
.
TI
The collocation procedure will be applied to these residuals.
Remark
As we have noted at the beginning of Sect. 10.2, the remove-restore process
aims at removing all known major trends:
•
the local topography produces
Bouguer anomalies
,
•
the regional features (i.e., their isostatic compensation), in addition to
the Bouguer effect, produce
topographic-isostatic anomalies
,
•
the global irregularities are expressed by an earth model and lead to
what is modestly called the
“residual anomalies”
.
It is clear that what is “removed” before the computation, must be fully
“restored” after the computation.
11.2
Geoid in Austria by collocation
Austria is a nice country, and in spite of being small, it has all types of
topography: flat, hilly, and alpine, up to an elevation of 3800 m. Thus, beyond
being a pleasant place to live, it is an interesting geodetic test area.
