principles and applications of microearthquake networks

136 6. Methods oj Data Analysis
It is obvious from step 2 that a model of the seismic velocity structure
underneath the microearthquake network must be specified in order to
compute theoretical travel times and derivatives. Therefore, earthquakes
are located with respect to the assumed velocity model and not to the real
earth. Routine earthquake locations for microearthquake networks are
based mainly on a horizontally layered velocity model, and the problem of
lateral velocity variations usually is ignored. Such a practice is often due
to our poor knowledge of the velocity structure underneath the microearthquake
network and the difficulties in tracing seismic rays in a heterogeneous
medium (see Chapter 4). As a result, it is not easy to obtain accurate
earthquake locations, as shown, for example, by Engdahl and Lee
( 1976).
Although step 3 is straightforward, we must be careful in forming the
normal equations to avoid roundoff errors (see Section 5.3.1). Many numerical
methods are available to solve a set of four linear equations in step
4, but few will handle ill-conditioned cases (see Section 5.3.2). The matrix
G is often ill-conditioned if the station distribution is poor. Having data
from four or more stations is not always sufficient to locate an earthquake.
These stations should be distributed such that the earthquake hypocenter
is surrounded by stations. Since most seismic stations are located near the
earth’s surface and whereas earthquakes occur at some depth, it is difficult
to determine the focal depth in general. Similarly, if an earthquake
occurs outside a microearthquake network, it is also difficult to determine
the epicenter. To illustrate these difficulties, let us consider a simple case
with only four observations. In this case, we do not need to form the
normal equations, and our problem is to solve Eq. (6.18) form = 4. The
Jacobian matrix A becomes
Let us recall that the determinant of a matrix is zero if any column of it is a
multiple of another column (Noble, 1969, p. 204). Since the first column of
A is all l’s, it is easy for the other columns of A to be a multiple of it. For
example, if P-arrivals from the same refractor are observed in a layered
velocity model, then all elements in the aT/az column have the same
value, and the fourth column of A is thus a multiple of the first column.
Similarly, if an earthquake occurs outside the network, it is likely that the