Let C denote the approximate Hessian matrix:
C is a symmetric, positive semidefinite matrix that can be decomposed into a product of three matrices:
where L is a diagonal matrix of non-negative eigenvalues and
A is a matrix of eigenvectors columns. Given the decomposition
of C, its inverse 
which is needed for solving the
linear system in (2.26) can be calculated as:
where 
is a diagonal matrix whose elements are the inverse of the
eigenvalues.
An analysis of the eigenvalues spectrum of C highlights important information about the modeling problem characteristics. The ratio between the highest and lowest eigenvalues defines the condition number of the extraction. A large condition number denotes an ill-conditionned problem typical of models with correlated parameters or with a weak response to certain parameters.
A typical remedy in such cases is to fix the values of parameters
associated with small eigenvalues to a constant.
In some cases, this solution is not adequate and a multi-stage
update of the model parameters to resolve the uncertainty is more appropriate.
The model parameters are divided into groups that contain uncorrelated
parameters with eigenvalues of similar magnitude.
During every iteration of the Levenberg-Marquardt algorithm,
the parameters in the same subgroup are updated while keeping the parameters
in the other subgroups constant. This divides the parameter spaces into
orthogonal subspaces where the modeling problem is well conditioned.
Correlated parameters, and parameters with varying sensitivity level can
then be extracted separately. The current implementation provides the
capability of dividing model parameters into distinct subgroups.
In addition, a granular control of the optimization allows a great
flexibility in devising the appropriate extraction strategy
.