principles and applications of microearthquake networks

112 5. Inversion and Optimization
number of unknowns is infinite. But because geophysical data are limited,
the number of observations is finite. In other words, we write a finite
number of equations corresponding to our observations, but our unknown
vector x is of infinite dimension. In order to obtain a unique answer, the
solution chosen is the one which maximizes or minimizes a subsidiary
integral. In actual practice, we minimize a sum of squares to produce a
smooth solution. A typical mathematical formulation may look like
(5.32)
where the top block A represents the underdetermined constraint equations
with the observed data vector b, the vector x contains the unknowns,
and the bottom block is a band matrix which specifies that some filtered
version ofx should vanish. As pointed out by Claerbout (1976, p. 120), the
choice of a filter is highly subjective, and the solution is often very sensitive
to the filter chosen. The Backus-Gilbert inversion is intended for
analysis of systems in which the unknowns are functions, i.e., they are
infinite-dimensional, as opposed to, say, hypocenter parameters in the
earthquake location problem, which are four-dimensional. For an application
of the Backus-Gilbert inversion to travel time data, readers may
refer, for example, to Chou and Booker (1979).
5.1.3.
Analysis of an Overdetermined System
Most scientists are modest in their data modeling and would have more
observations than unknowns in their model. On the surface it may seem
very safe, and one should not expect difficulties in obtaining a reasonable
solution for an overdetermined system. However, an overdetermined system
may in fact be underdetermined because some of the equations may
be superfluous and do not add anything new to the system. For example, if
we use only first P-arrivals to locate an earthquake which is coniiderably
outside a microearthquake network, the problem Is underdetermined no
matter how many observations we have. In other words, in many physical
situations we do not have sufficient information to solve our problem
uniquely. Unfortunately, many scientists overlook this difficulty. Therefore,
it is instructive to quote the general principle given by Lanczos
(1961, p. 132) that “a lack of information cannot be remedied by any
mathematical trickery. ”
Usually a given overdetermined system is mathematically incompatible,
i.e., some equations are contradictory because of errors in the observations.
This means that we cannot make all the components of the re-