principles and applications of microearthquake networks

150 6. Methods of Data Aizalysis
trial hypocenter (s?, yj*, zj*) of thejth earthquake, and rj* is the trial origin
time of thejth earthquake. Our objective is to adjust the trial hypocenter
and velocity parameters (e) simultaneously such that the sum of the
squares of the arrival time residuals is minimized. We now generalize
Geiger's method as given in Section 6.1.2 for our present problem.
The objective function for the least-squares minimization [Eq. (6.8)] as
applied to the simultaneous inversion is
(6.27)
n m
j=1 k=1
where yjk(6*) is given by Eq. (6.26). We may consider the set of arrival
time residuals yjk(tT), for k = 1, 2, . . - , rn, andj = 1, 2, . . . , iz, as
components of a vector r in a Euclidean space of mn dimensions, i.e.,
(6.28) r = (rI1, r12. . . . yll,t. rzl, rZ2, . . . , rZmr
... , rn1. rn2. . * . , G dT
The adjustment vector [Eq. (6.10)] now becomes
(6.29) 8e = @r1, 6~,, 6y,, SZ,, St,. SX,, Sy,, SZ,
. * - 7 6t,, Sxn, Sv,, 6~,, 6111, 6~2, . . . , ~ u L ) ~
and it is to be determined from a set of linear equations similar to Eq.
(6.19). We then replace the trial parameter vector e* by (5" + 66) and
repeat the iteration until some cutoff criteria are met.
To apply the Gauss-Newton method to the simultaneous inversion
problem, a set of linear equations is to be solved for the adjustment vector
8& at each iteration step. In this case, we may generalize Eq. (6.11) or
equivalently, Eq. (6.18), to read
(6.30) B SQ = -r
where B is the Jacobian matrix generalized to include a set of n earthquakes
and a set of L velocity parameters, i.e.,
i
I
0 I
I
I
I
.. 0 I
I
(6.31) B =
I
I C
I
\0 0 0 ... An I /
where the 0's are m X 4 matrices with zero elements, the A's are rn X 4
Jacobian matrices given in a manner similar to Eq. (6.13) as