Theory

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

$$d = Lm \EQNLABEL{equ1}$$ (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:

$$m = (L^TL)^{-1}L^Td \EQNLABEL{equ2}$$ (2)

and, for the underdetermined case, as

$$m = L^T(LL^T)^{-1}d \EQNLABEL{equ3}$$ (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 L^T as

$$L_m^{\dagger} = (L^TL)^{-1}L^T \EQNLABEL{equ4}$$ (4)

whereas in equ3 the inverse for the underdetermined problem is

$$L_d^{\dagger} = L^T(LL^T)^{-1} \EQNLABEL{equ5}$$ (5)

Applying the pseudo-inverse of equ4 is equivalent to applying the adjoint operator L^T, followed by a spatial filtering of the model space by the inverse of L^TL. 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 LL^T 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 L^TL, alternatively, (LL^T).