principles and applications of microearthquake networks

126 5. Inversion criid Optimization
If the objective function is quadratic in x, the approximation used in
Newton’s method is exact, and the minimum can be reached from any x in
a single step. For more general objective functions, an iterative procedure
must be used with an initial estimate of x sufficiently near the minimum in
order for Newton’s method to converge. Since this can not be guaranteed,
modern computer codes for optimization usually include modifications to
ensure that (1) the 6x generated at each iteration is a descent or downhill
direction, and (2) an approximate minimum along this direction is found.
The general form of the iteration is6x = -aGg, where G is a positivedefinite
approximation to H -’ based on second-derivative information,
and a represents the distance along the -Gg direction to step. Alternatively,
as iterations proceed, an approximate Hessian matrix may be built
up without computing second derivatives. This is known as the Quasi-
Newton or variable metric method. For more details on the derivative
methods and their implementations, readers may refer, for example, to R.
Fletcher (1980), and Gill et al. (1981).
5.4.3. Nonlinear Least Squares
The application of optimization techniques to model fitting by least
squares may be considered as a method for minimizing errors of fit (or
residuals) at a set of rn data points where the coordinates of the kth data
point are (x)~, k = 1, 2‘, . . . , rn. The objective function for the present
problem is
(5.77)
where rk(x) denotes the evaluation of residual r(x) at the kth data point.
We may consider rk(x), k = 1,2, . . . , rn, as components of a vector in an
rn-dimensional Euclidean space and write
(5.78) r = M x), r2(x), . . . , r,WT
Therefore, Eq. (5.77) becomes
(5.79) F(x) = rTr
TO find the gradient vector g, we perform partial differentiation on Eq.
(5.77) with respect to xi, i = 1, 2, . . . , n, and obtain
(5.80) aF(x)/axi =
in
2rk(x)[ark(x)/dxi], i = 1, 2, . . . , n
k=1