principles and applications of microearthquake networks

6.3. Simul ta 11 eoirs Iii v ers ioti 149
and the P-velocity structure underneath the stations. Let us denote the
first P-arrival time observed at the kth station for thejth earthquake as T~~
along the ray path rjk.
Since there are n earthquakes, the total number of hypocenter parameters
to be determined is 4n. Let us denote the hypocenter parameters for
thejth earthquake by (tf,xf,yf,z~), where tf is the origin time and xy,yf, and
zf are the hypocenter coordinates. Thus, all the hypocenter parameters for
iz earthquakes may be considered as components of a vector in a Euclidean
space of 412 dimensions, i.e., ct~,s?..\.?.z~,f20,~vY20,YZO.zZO, . . . , t~,,vO,,yO,,z~)T,
where the superscript T denotes the transpose.
The earth underneath the stations may be divided into a total of L
rectangular blocks with sides parallel to the x, Y, and z axes in a Cartesian
coordinate system. Let each block be characterized by a P-velocity denoted
as u. If I is the index for the blocks, then the velocity of the Ith block
is ul, I = I, 2, . . . . L. We further assume that the set of n earthquakes and
the set of rn stations are contained within the block model. Thus, the
velocity structure underneath the stations is represented by the L parameters
of block velocity. These L velocity parameters may be considered as
components of a vector in a Euclidean space of L dimensions, i.e., (ul, u2,
... 7 VJT.
Assuming that every block of the velocity model is penetrated by at
least one ray path, the total number of parameters to be determined in the
simultaneous inversion problem is (4n + L). These hypocenter and velocity
parameters may also be considered as components of a vector & in a
Euclidean space of (4n + L) dimensions, i.e.,
In order to formulate the simultaneous inversion problem as a nonlinear
optimization problem, we assume a trial parameter vector e* given by
(6.25) e* = (t" 1, x;, y;, z;, f3, x3. y$, z:,
. . . , t:, x;, y;, z;. o;, ?I$, . . . , ot)T
The arrival time residual at the kth station for the jth earthquake may
be defined in a manner similar to Eq. (6.7) as
(6.26)
rjk(e*) = T jk - Tj,(g*) - tf
k = l , 2
,..., rn, j = l , 2 ,..., n
where 7jk is the first Farrival time observed at the kth station for the jth
earthquake, Tjk(( 9 is the corresponding theoretical travel time from the