Jökull - 01.12.1973, Side 44

purpose. We can select a matrix B with a
stronger negating power than A'. This would
reduce equations (10) to an underdetermined
problem
BAx0 = Bb (33)
which would then have to be treated by the
methods described above. As will be discussed
below there are cases where such an overnega-
tion is convenient. The adjoint A' is optimal
in the sense that it negates those and only
those components of b which have to be negat-
ed for the inversion of A.
AN EXAMPLE
FROM POTENTIAL THEORY
In the following we will discuss an example
from potential theory which is of interest in
the interpretation of marine magnetic field
anomalies. These anomalis are believed to re-
sult from the magnetization of thin layers of
lava at the ocean floor (Carmichael, 1970).
Consider a horizontal magnetic stratum X
carrying a unidirectional vertically oriented
magnetization of density u(S) where S are the
points on X- We will place the (x, y) plane in
X with the z-axis vertical. Moreover, for the
purpose of simplification let u(S) depend on
the x-coordinate only and let u = 0 for x out-
side the interval (0, L). We have then a one-
dimensional case where u(x) has a bounded
support.
Using MKS units we find (Grant and West,
1965) that the scalar magnetic potential v(x, z)
due to X a field point (x, z) is
v(x, z) - (z/2tr) j
u(x')dx'
Z2+ (x-x')2
(34)
The vertical magnetic field is
K (x, z) = - fi0 (3v/3z) (35)
which can also be expressed
v(x,z) = - J bz(x, z')dz' (36)
AC z
Assuming that the magnetic field can be ob-
served in a horizontal plane located at a dist-
ance h above X> the fundamental problem of
magnetic field interpretation consists in deriv-
ing u(x) on the basis of the observed bz(x, h).
This is equivalent to solving the integral equa-
tion (34) for u(x) when z = h and v(x, h) is
obtained on the basis of (36). The latter step
is a simple integration obtained on the basis
of an outward continuation of bz(x, h) which
presents no formal difficulties.
Equation (34) is an integral equation of the
first kind where the operation on the right is
a simple convolution. Hence, the most con-
venient method of solution consists in applying
the Fourier transformation with respect to x
to (34). Let U(k) and V(k, h) be the transforms
of u(x) and v(x, h), respectively, where k is the
transform variable, that is, the wave-number in
the transformed space. Using ■ transform tables
(.Duff and Naylor, 1966) we find that the trans-
formed version of (34) is
V(k,h) = (i/2)exP(-|h|k)U(k) (37)
and hence
U(k) = 2 exp(|h|k)V(k, h) (38)
which formally represents the Fourier trans-
form of the solution of (34).
Unfortunately, the exponential factor on the
right of (38) is unbounded as k-^=° and does not
possess an inverse transform. We are therefore
unable to retrieve u(x) from U(k) unless re-
strictions are imposed on V(k,h) so that the
product on the right of (38) can be inverted.
The integral equation (34) for u(x) is simply an
improperly posed mathematical problem. This
is evidenced by the fact that as k-»<® the ex-
ponential factor in (38) causes an unbounded
magnification of the high-frequency compon-
ents of v(x, h). This situation is typical for
many problems in interpretation theory. Equa-
tion (34) is one of the most elementary ex-
amples of an improperly posed interpretation
problem.
The simplest and most obvious way out of
this difficulty is to reject any high-frequency
components of v(x, h) beyond a certain cutoff
wavenumber k0. Adopting this procedure, we
42 JÖKULL 23. ÁR