principles and applications of microearthquake networks

116 5. Inversion and Optimization
Because of uncertainties in our model and data, an exact solution to Eq.
(5.44) may never be obtainable. Furthermore, there may be many sohtions
satisfying Ax = b. By Eq. (5.45), our estimate 5i is related to the true
solution by the product HA. Let us call this product the resolution matrix
R, i.e.,
(5.47) R = HA
and hence Eq. (5.45) becomes
(5.48) x = Rx
This equation tells us that the matrix R maps the entire set of solutions x
into a single vector x. Any component of vector x, say xk, is the product of
the kth row of matrix R with any vector x that satisfies Ax = b. Therefore,
R is a matrix whose rows are “windows” through which we may view the
general solution x and obtain a definite result. If R is an identity matrix,
each component of vector x is perfectly resolved and our solution ii is
unique. If R is a near diagonal matrix, each component, say &, is a
weighted sum of components of x with subscripts near k. Hence, the
degree to which R approximates the identity matrix is a measure of the
resolution obtainable from the data.
In a similar manner, we may call the product AH the information density
matrix D (Wiggins, 1972), i.e.,
(5.49) D = AH
It is a measure of the independence of the data. The theoretical datab that
satisfies exactly our solution X is given by
n
(5.50) b 3 AX = AHb = Db
In actual applications, the criteria (l), (2), and (3) mentioned previously
are not equally important and may be weighted for a specific problem. For
example, there is a trade-off between resolution and variance.
5.3. Computational Aspects of Solving Inverse Problems
There are many ways to solve the problem Ax = b. For example, one
learns Cramer’s rule and Gaussian elimination in high school. We also
know that by applying the least squares method, we can reduce any overor
underdetermined system of equations to an even-determined one. With
digital computers generally available, one wonders why the machines
cannot simply grind out the answers and let the scientists write up the
results. Unfortunately, this point of view has led many scientists astray.