principles and applications of microearthquake networks

134 6. Methods of Data Analysis
x* + 6x, and repeat the same procedure until some cutoff criteria are
satisfied to stop the iteration. In other words, the nonlinear earthquake
location problem is solved by an iterative procedure involving only solutions
of a set of four linear equations.
Instead of solving an even-determined system of four linear equations
as given by Eq. (6.14), we may derive an equivalent overdetermined system
of rn linear equations from Eq. (6.11) as
(6.18) A6x = -r
where the Jacobian matrix A is given by Eq. (6.13). Equation (6.18) is a set
of rn equations written in matrix form, and by Eqs. (6.9), (6.10), and
(6.13), it may be rewritten as
By using generalized inversion as discussed in Chapter 5, the system of rn
equations as given by Eq. (6.19) may be solved directly for the adjustments
St, Sx, Sy, and Sz. This way, we may avoid some of the computational
difficulties associated with solving the normal equations as discussed
in Section 5.3.2.
6.1.3. Implementation of Geiger’s Method
Geiger’s method as derived in the previous subsection is computationally
straightforward, but the method is too tedious for hand computation
and hence was ignored by seismologists for nearly 50 years. In the late
1950s digital computers became generally available, and several seismologists
quickly seized this opportunity to implement Geiger’s method
for determining origin time and hypocenter (e.g., Bolt, 1960; Flinn, 1960;
Nordquist, 1962; Bolt and Turcotte, 1964; Engdahl and Gunst, 1966).
Since then, many additional computer programs have been written for this
problem, especially for locating local earthquakes (e.g., Eaton, 1969;
Crampin, 1970; Lee and Lahr, 1975; Shapira and Bath, 1977; Klein, 1978;
Herrmann, 1979; Johnson, 1979; Lahr, 1979; Lee ef al., 1981).
In deriving Geiger’s method, we have not specified any particular seismic
arrival times. Normally, the method is applied only to first P-arrival
times because first P-arrivals are easier to identify on seismograms and the
P-velocity structure of the earth is better known. However, if arrival
times of later phases (such as S, P,, etc.) are available, we can set up