principles and applications of microearthquake networks

6.3. Simulta ti eou s Inversion 151
(6.32)
and C is a rnn X L matrix. Each row of the velocity coefficient matrix C
has L elements and describes the sampling of the velocity blocks of a
particular ray path. If rjkis the ray path connecting the kth station and the
jth earthquake, then the elements of the ith row [where i = k + ( j - l)m]
of C are given by
(6.33) c. 11 = -fl jkl(aTjk/dVt)lc for = 1, 2, , . . , L
where njkl is defined by
(6.34) njkl =
1 if the Ith block is penetrated by
the ray path rjk
0 otherwise
If ii2 is the slowness in the Ith block defined by
(6.35) LIl = I/c,
then we have
To approximate dTjk/dZ+ in Eq. (6.33) to first-order accuracy, we note
that d Tjk/duz = d Tjkl/du,, where Tjkl is the travel time spent by the ray r jk
in the Ith block. In addition, we note that Tjkl = UlSjkl, where sjkl is the
length of the ray path rjk in the Ith block. Assuming that the dependence
Of Sjkl on Ul is Of second order, dTjkl/dl4, --- Sjk, = 2,1Tjkl. Hence Eq. (6.33)
becomes
(6.37) Cil = IIjk/Tjkl((*)/cy for I = 1. 2, . . . , L
because (dTjk/du,) Svl = (dTjk/8ul) 6uI. and using Eq. (6.36).
Since each ray path samples only a small number of blocks, most elements
of matrix C are zero. Therefore, the matrix B as given by Eq. (6.31)
is sparse, with most of its elements being zero.
Equation (6.30) is a set of rnn equations written in matrix form, and by
Eqs. (6.28),(6.29), (6.31), (6.32), and (6.37), it may be rewritten as