principles and applications of microearthquake networks

120 5. Iwersioti arid Optirnizntiori
(5.58) x = A-b
is always unique. If matrix A is of full rank (i.e., Y = n), then R = I, and D
= I,. Thus, fr: corresponds to the usual least squares solution. If matrix A
is rank deficient (i.e., r < n), then there is no unique solution to the least
squares problem. In this case, we must choose a solution by some criteria.
A simple criterion is that the solution k has the least vector length, and Eq.
(5.58) satisfies this requirement. Finally, we introduce the unscaled
covariance matrix C (Lawson and Hanson, 1974, pp. 67-68)
(5.59) c = V( ST)2VT
The Moore-Penrose generalized inverse is one of many generalized
inverses (Ben-Israel and Greville, 1974). In other words, there are other
choices for the matrix H, which can serve as an inverse, and thus one can
obtain different approximate solutions to the problem Ax = b. For example,
in the method of damped least squares (or ridge regression), the
matrix H is chosen to be
(5.60a)
H = VFSTUT
In this equation, F is an n X n diagonal filter matrix whose components are
(5.60b)
F- 22 = u:/(uf + 02) for i = I, 2, . . . , n
where 8 is an adjustable parameter usually much less than the largest
singular value ul.
The effect of this H as the inverse is to produce an estimate Er whose
components along the singular vectors corresponding to small singular
values are damped in comparison with their values obtained from the
Moore-Penrose generalized inverse solution. This Er can be shown (Lawson
and Hanson, 1974) to solve the problem
(5.60~) minimize {IIAx - bI(2 + O z ~ ~ ~ ~ z }
and thus represents a compromise between fitting the data and limiting the
size of the solution. Such an idea is also used in the context of nonlinear
least squares where it is known as the Levenberg-Marquardt method (see
Section 5.4.3).
Unlike the ordinary inverse, the Moore-Penrose generalized inverse
always exists, even when the matrix is singular. In constructing this generalized
inverse, we are careful to handle the zero singular values. In Eq.
(539, we define u: = l/at only for ui > 0, and ui = 0, for ui = 0. This
avoids the problem of the ordinary inverse, where (T;' = l/ui. Furthermore,
singular values give insight to the rank and condition of a matrix.
The usual definition of rank of a matrix is the maximum number of