principles and applications of microearthquake networks

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6. Methods of Data Analysis
(6.38)
fork = 1, 2, . . . ,rn, j = 1, 2, . . . ,n
Equation (6.38) is a generalization of Eq. (6.19) for the simultaneous inversion
problem.
6.3.2. Numerical Solution of the Simultaneous Inverswn Problem
In the previous subsection, we have shown that the simultaneous inversion
problem may be formulated in a manner similar to Geiger’s method of
determining origin time and hypocenter. The computations involve the
following steps:
Guess a trial parameter vector t* as given by Eq. (6.25).
Compute the theoretical travel time TJk and its spatial partial derivatives
dT,,/dx, dTik/dy, and dTj,/t3z evaluated at (x?, yJ, zj”)
for k = 1, 2, . . . , rn andj = 1, 2, . . . , n.
Compute matrix B as given by Eqs. (6.31), (6.32), and (6.37), and
compute vector r as given by Eqs. (6.28) and (6.26).
Solve the system of rnn linear equations as given by Eq. (6.30) for
the adjustment vector 65 with (4n + L) elements. This may be
accomplished via the normal equations approach, or the generalized
inversion approach as discussed in Chapter 5.
Replace the trial parameter vector (* by (e* + St).
Repeat steps 2-5 until some cutoff criteria are met. At this point,
we set 5 = 5” as our solution for the hypocenter and velocity
parameters .
Although numerical solution for the simultaneous inversion is computationally
straightforward, the pitfalls discussed in Section 6.1.3 apply here
also. The set of mn linear equations to be solved in step 4 can be very
large. Typically a few thousand equations and several hundred unknowns
are involved. Because of the positions of the zero elements in matrix B
[Eq. (6.31)], the arrival time residuals of one earthquake are coupled to
those of another earthquake only through the velocity coefficient matrix C
in Eq. (6.30). Therefore, it is possible to decouple mathematically the
hypocenter parameters from the velocity parameters in the inversion procedure
as shown by Pavlis and Booker (1980) and Spencer and Gubbins
(1980). Consequently, large amounts of arrival time data can be used in