principles and applications of microearthquake networks

5.3. Solving InL.erse Problems 121
independent columns. Such a definition is very difficult to apply in practice
for a general matrix. However, if the matrix is triangular, the rank is
simply the number of nonzero diagonal elements. Since an orthogonal
transformation, such as that in the singular value decomposition, does not
affect the rank, we see right away that the rank of A is equal to the rank of
S in its singular value decomposition. Hence a practical definition of the
rank of a matrix is the number of its nonzero singular values. Because of
finite precision in any computer, it may be difficult to distinguish a small
singular value and a zero one. Therefore, we define the effective rank as
the number of singular values greater than some prescribed tolerance
which reflects the accuracy of the machine and the data.
Reducing the rank of a matrix has the effect of improving the variance
of the solution. In Eq. (5.59), the singular value ui enters into the
covariance matrix as l/o:. Therefore, if oi is small, the covariance, and
hence the variance will be large. By discarding small singular values, we
decrease the variance. However, we pay the price of poorer resolution.
For a full-rank matrix, the resolution matrix R is simply the identity matrix
and we have full resolution. Decreasing the rank makes R further away
from being an identify matrix.
Since our model and data have uncertainties, we have to consider the
condition number of the matrix A. In practice, we are not solving Ax = b,
but Ax = b 2 Ab, assuming A to be exact at this moment. It can be shown
(Forsythe et al., 1977, pp. 41-43) that the error Ax in x resulting from the
error Ab in b is given by
(5.61)
II &I1 /II x II
Y [II Ab II /lib Ill
where y is the condition number of A, and 11 - 11 denotes the vector norm.
Forsythe et al. (1977, pp. 205-206) also show that we can define the
condition number of A by
(5.62)
i.e., y is the ratio of the largest and smallest singular values of A. We see
from Eq. (5.61) that the condition number y is an upper bound in magnifying
the relative error in our observations. In microearthquake studies, the
relative error in observations is about lop3. Therefore, if y is lo3, then the
error in our solution may be comparable to the solution itself.
In conclusion, the singular value decomposition (SVD) approach to
generalized inversion offers a powerful analysis of our problem. Numerical
computation of SVD has been worked out by Golub and Reinsch
(1970), and a SVD subroutine is generally available as a library subroutine
in most computer centers. Although SVD analysis requires more computational
time than, say, solving the normal equations by the Cholesky