principles and applications of microearthquake networks

5. Generalized Inversion and
Nonlinear Optimization
Many scientific problems involve estimating parameters from nonlinear
models or solving nonlinear differential equations that describe the physical
processes. In the previous chapter, the problem of finding the
minimum-time ray path between the earthquake source and a receiving
station involves solving a set of first-order differential equations. These
equations are approximated by a set of nonlinear algebraic equations,
which are then solved by an iterative procedure involving the solution of a
set of linear equations. In the next chapter, calculation of earthquake
hypocenters will be treated as a nonlinear optimization problem which
also involves solving a set of linear equations. Therefore, solving linear
equations is frequently required in microearthquake research as it is for
other sciences. In fact, 75% of scientific problems deal with solving a set
of rn simultaneous linear equations for n unknowns, as estimated by
Dahlquist and Bjorck (1974, p. 137).
In solving linear equations, it is convenient to use matrix notation as
commonly developed in linear algebra or matrix calculus. Readers are
assumed to be familiar with vectors and matrices and their properties.
Among numerous texts on these topics, we recommend the following:
Lanczos (1956, 1961), Noble (1969), G. W. Stewart (1973), and Strang
(1976). In general, components of a matrix can be either real or complex
numbers. However, in most microearthquake applications, we deal with
real numbers. Hence, we will be concerned here only with real matrices.
In matrix notation, the problem of solving a set of linear equations may
be written as
(5.1) AX = b
where A is a real matrix of rn rows by n columns, x is an n-dimensional
vector, and b is an m-dimensional vector. Our problem is to solve for the
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