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Equation (1) can be linearized for small perturbations around
the homogeneous solution as follows:
 

 
 (6) 
Using this approximation in equation (2) and minimizing it
with respect to all s leads to the following set of linear equations:
where each element is an
matrix, and and are column vectors
with L elements, L being the number of layers (or cells) of the model.
The elements of the matrix and vectors are given by the equations
 
(7) 
Michelena (1991) solved the linearized problem using a conjugate gradients
routine. He obtained a stable solution by stopping the process after
a prespecified number of iterations, thus avoiding the problems
arising from the illconditioning of matrix .
Another way to build a more stable algorithm is to introduce an
angledependent weighting factor into equation (2) and to
apply a weak constraint to the inversion:
 
(8) 
where is the angle of the ray as measured from the horizontal.
Next: WALSH FUNCTIONS
Up: INVERSION SCHEMES
Previous: Nonlinear schemes
Stanford Exploration Project
12/18/1997