Jökull - 01.12.1973, Page 39

Remarks on Generalized Solutions of Improperly
Posed Problems in the Exploration Sciences
GUNNAR BÖÐVARSSON
DEPARTMENT OF MATHEMATICS AND SCHOOL OF OCEANOGRAPHY,
OREGON STATE UNIVERSITY, CORVALLIS, OREGON 97331
abstract
Mathematically improperly posed problems
are of considerable interest in the exploration
sciences. The interpretatio?i of incomplete and
noisy field data leads to underdetennined or
°verdeter?nined problems. The principal char-
actenstics of such proble?ns are discussed on
the basis of examples fro?n linear algebra. Both
two dimensional and more general cases are
discussed. Generalized solutions are defined and
derived in a convenient matrix language. The
discussion is extended to a case in potential
theory involving the interpretation of one-
dimensional marine magnetic data. An integral
ef]uation of the first kind is solved in the space
°f band-limited functions. A generalized solu-
tion is defined and it is shown that the case is
fuite similar to the algebraic examples discuss-
ed earlier.
introduction
The processing and interpretation of observa-
tional geophysical data very frequently require
the solving of numerical problems which are
trnproperly posd from the mathematical point
°f view. Many of thes problems may not possess
any well. defined solutions at all, others may
have many, or more frequently infinitely many,
s°lutions. Although this fact is quite well
known, the role of the improperly posed pro-
blems in interpretation theory has not been
appreciated as much as it really deserves. The
problem setting is frequently quite different
from the standard textbook type of mathema-
tical problems.
The present paper gives an elementary re-
view of two of the simplest types of improperly
posed problems encountered in geophysical
interpretation work. These two cases are (1)
systems of linear algebraic equations where the
number of equations differs from the number
of the unknowns and (2) linear integral equa-
tions of the first kind which are improperly
posed for similar but not quite as obvious
reasons. The concept of a generalized solution
is introduced and applied to these two cases.
For a much more elaborate discussion of gen-
eralized inverse matrices and solutions of al-
gebraic equations, the reader is referred to two
rather recent monographs by Boullion and
Odell (1971) and Rao and Mitra (1971). Some
of the more complex problems arising in the
inversion of global earth data and the well
known methods derived for this purpose by
Backus and Gilbert (1967, 1968) will not re-
ceive attention.
GENERALIZATION
OF THE SOLUTION CONCEPT
Cases with two unknown
Systems of linear algebraic equations with
only a finite number of unknowns pose some
of the simplest mathematical problems of solv-
ing for unknown quantities. It is customary to
restrict the concept of a solution of such syst-
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