Jökull - 01.12.1973, Side 46

rewrite the band-limited restriction of (34) in
terms of a system of linear algebraic equations
with a n x n matrix. This system is easily
derived from (34). The solution (42) can also
be replaced by a similar n x n system of linear
algebraic relations. Conversely, if v(x, h) is ob-
served in a discrete equidistant set of points
only, then only a band-limited solution can be
obtained. This underlines the connection be-
tween the present potential theoretical example
and the results of the previous section.
Another type of approximation is quite often
of help in solving interpretation problems of
the type (34). We can lump this magnetization
of 2 into a finite number of line-sources of
constant source density parallel to the y-axis.
The continuous sources density u(x) would
then be replaced by a finite number of 8-
functions, that is, by a so called Dirac comb.
The integral on the right of (34) reduces to a
finite sum. Let the number of lines be n and
the number of observation stations for v(x, h)
be m. The integral equation (34) then reduces
to a system of linear algebraic equations with
a m x n matrix such as (8). There is in this case
no compelling reason to make m =: n. One is
quite often interested in some degree of re-
dundancy by making m > n. This leads to an
overdetermined system of equations which can
be treated by the methods of the previous
section.
ACKNOWLEDGEMENT
This work was supported by the National
Science Foundation grant GA-25896.
REFEREN CES
Arsac, J. 1966: Fourier Transforms and the
Theory of Distributions, 318 pp., Prentice-
Hall, Englewood Cliffs, N.J.
Backus, G., and F. Gilbert. 1967: Numerical
applications of a formalism for geophysical
inverse problems, Geophys. 1. R. astr. Soc.,
13, 247-276.
— and F. Gilbert. 1968: The resolving power
of gross Earth data, Geophys. J. R. astr.
Soc., 16, 169-205.
Bodvarsson, G. 1971: Approximation methods
for equivalent strata, J. Geophys. Res., 16,
3932-3939.
— 1973: Downward continuation of con-
strained potential fields. J. Geophys. Res.,
78, 1288-1292.
Boullion, T. L., and P. L. Odell. 1971: Gener-
alized Inverse Matrices, 101 pp. Wiley-Inter-
science, New York, N.Y.
Carmichael, C. M. 1970: The mid-Atlantic ridge
near 45° N, 7, Magnetic properties and
opaque mineralogy of dredged samples,
Can. J. Earth Sci., 7 (2), 239-256.
Duff, G. F. D., and D. Naylor: 1966: Different-
ial Equations of Applied Mathematics, 423
pp., John Wiley and Sons, Inc., New York,
N.Y.
Grant, F. S., and G. F. West. 1965: Interpreta-
tion Theory in Applied Geophysics, 583
pp., McGraw-Hill, New York, N.Y.
Lanczos, C. 1961: Linear Differential Operators,
564 pp., D. Van Nostrand Co. Limited,
London.
Papoulis, A. 1962: The Fourier Integral and
its Applications, 318 pp. McGraw-Hill,
New York, N.Y.
Rao, C. R., and S. K. Mitra. 1971: Generalized
Inverses of Matrices and its Applicadons,
240 pp., John Wiley and Sons, Inc., New
York, N.Y.
Snow, C. 1923: Spectroradiometric analysis of
radio signals, Sci. Pap. NBS, 19 (477), 231 —
261.
ÁGRIP
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44 JÖKULL 23. ÁR