principles and applications of microearthquake networks

106 5. Inversion and Optimization
unknown vector x, given a model matrix A and a set of observations b.
One is very tempted to write down
(5.2) x = A-'b
where A-' denotes the inverse of A, and hand the problem to a programmer
to solve by a computer. In due time, the programmer brings
back the answers and everyone lives happily thereafter. However, it may
happen that the scientist is very cautious and notices that a certain unknown
that from experience should be positive turns out to be negative as
given by the computer. But the programmer assures the scientist that he
has used the best matrix inversion subroutine in the computer program
library and has checked the program very carefully. The scientist may be
at a loss about what to do next. He may look up his old high-school notes
on Cramer's rule, sit down to program the problem himself, and discover
many surprises. Alternatively, he may consult his colleagues in computer
science, and take advantage of recent techniques in computational mathematics
to solve his problem.
In order to help the reader become better acquainted with some recent
advances in computational mathematics, we briefly review the mathematical,
physical, and computational aspects of the inversion problem in the
first part of this chapter. In the second part, we briefly discuss nonlinear
optimization problems that arise in scientific research and how to reduce
these problems to solving linear equations. The material in this chapter
includes some fundamental mathematical tools for analyzing scientific
data. We recognize that such material is not normally expected to be given
in a work such as this one. However, by presenting these mathematical
tools, some methods for analyzing microearthquake data may be developed
more systematically.
Throughout this chapter, we will refer to many publications treating the
inversion problem from the mathematical point of view. There are also
numerous papers presenting the inversion problem from the geophysical
point of view. Readers may refer, for example, to Backus and Gilbert
(1967, 1968, 1970), Jackson (1972), Wiggins (1972), Jupp and Vozoff
(1975), Parker (1977), and Aki and Richards (1980).
5.1. Mathematical Treatment of Linear Systems
If one is asked to solve a set of linear equations in the form of Ax = b, it
is reasonable to expect that there be as many equations as unknowns. If
there are fewer equations than unknowns, we know right away that we are
in trouble. If there are more equations than unknowns, we may transform