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## Theory

The linear inverse problem we solve for each frequency component can be written as a matrix equation:
 (1)

where the vector d represents the irregular input data, L represents the modeling operator, and m stands for the regularly sampled model. The least-squares solution to equation equ1 can be expressed, for the overdetermined case, as:
 (2)

and, for the underdetermined case, as
 (3)

These solutions define a least-squares inverse or pseudo-inverse to the operator L. From equation equ2, we write this inverse in terms of L and its adjoint LT as
 (4)

whereas in equ3 the inverse for the underdetermined problem is
 (5)
Applying the pseudo-inverse of equ4 is equivalent to applying the adjoint operator LT, followed by a spatial filtering of the model space by the inverse of LTL. Therefore, we refer to this inverse as model-space inverse. In equation equ5 the adjoint operator is applied after the data have been filtered with the inverse of LLT and, consequently, we refer to this inverse as data-space inverse. To compute any of these two inverses, the problem reduces to estimating an inverse for the cross-product matrix LTL, alternatively, (LLT).

Next: The cross-product filter Up: Problem formulation Previous: Problem formulation
Stanford Exploration Project
10/9/1997