principles and applications of microearthquake networks

5.2. Physical Consideration c$ Inverse Problems 115
eigenvalues rather loosely. Consequently, some works of geophysicists
who use Lanczos’ notation may be confusing to readers. Strictly speaking,
eigenvalues and eigenvectors are not defined for anm x n rectangular
matrix because eigenvalue analysis can be performed only for a square
matrix.
5.2. Physical Consideration of Inverse Problems
In the previous section, we considered the model matrix A and the data
vector b in the problem Ax = b to be exact. In actual applications, however,
our model A is usually an approximation, and our data contain
errors and have no more than a few significant figures. Consequently, we
should write our problem as
(5.43) (A 2 AA)x = b 5 Ab
where AA and Ab denote uncertainties in A and b, respectively. In this
section we investigate what constitutes a good solution, realizing that
there are uncertainties in both our model and data. The treatment here
follows an approach given by Jackson (1972).
Our task is to find a good estimate to the solution of the problem
15.44) Ax = b
where A is an rn X n matrix. Let us denote such an estimate by X; x may be
defined formally with the help of a matrix H of dimensions n X m operating
on both sides of Eq. (5.44)
(5.45) X=Hb=HAx
The matrix H will be a good inverse if it satisfies the following criteria:
(1)
(2)
(3)
(5.46)
AH = I, where I is an rn X n7 identity matrix. This is a measure of
how well the model fits the data, because b = AX = A(Hb) = (AH)b
= b, if AH = I.
HA == I, where I is an n X n identity matrix. This is a measure of
uniqueness of the solution, because X = x, if HA = I.
The uncertainties in X are not too large, i.e., the variance of x is
small. For statistically independent data, the variance of the kth
component of x is given by
var(ik) =
m
i=l
Hai var(bi)
where Hki is the (k, i)-element of H, and bf is the ith component
of b.