Geoscience Reference
In-Depth Information
solves the third boundary-value problem for the exterior of the sphere
r
=
R
.
The straightforward verification is analogous to the case of (1-127).
In the determination of the geoidal undulations, the constants
h, k
have
the values
2
R
,
h
=
−
k
=
−
1
,
(1-132)
so that
R
r
n
+1
Y
n
(
ϑ, λ
)
n
V
e
(
r, ϑ, λ
)=
R
∞
(1-133)
−
1
n
=0
solves the
boundary-value problem of physical geodesy
.
As we have seen in the preceding section, the first boundary-value prob-
lem can also be solved directly by
Poisson's integral
. Similar integral formulas
also exist for the second and the third problem. The integral formula that
corresponds to (1-133) for the boundary-value problem of physical geodesy
is
Stokes' integral
, which will be considered in detail in Chap. 2.
Remark on inverse problems
Boundary-value problems give the potential
outside
the earth, where there
are no masses and where the potential, satisfying Laplace's equation, is har-
monic. The determination of the potential
inside
the earth is of a quite
different character since the earth is filled by masses, and the interior po-
tential satisfies Poisson's rather than Laplace's equation, as we have seen in
Sect. 1.2. Unfortunately, the density
inside the earth is generally unknown.
To see the diculties of the problem, let us consider Newton's integral
(1-12). If the interior masses were known, we could easily use this formula
to compute the potential inside (and outside) the earth, in a direct and
straightforward way. The determination of the potential from the masses is
a “direct” problem. The “inverse” problem is to determine the masses from
the potential, finding a solution of Newton's integral for the density
,which
is essentially more dicult.
In fact, it is impossible to determine uniquely the generating masses
from the external potential. This
inverse problem of potential theory
has no
unique solution. Such inverse problems occur in geophysical prospecting by
gravity measurements: underground masses are inferred from disturbances
of the gravity field. To determine the problem more completely, additional
information is necessary, which is furnished, for example, by geology or by
seismic measurements.
Generally, nowadays we know that many problems in geophysics and
other sciences including medicine (e.g., seismic and medical tomography) are
inverse problems. We cannot pursue this interesting problem here and refer
