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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 *L*^{T} 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 *L*^{T}, followed by a spatial filtering of the model
space by the inverse of *L*^{T}*L*. 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*^{T}*L*, alternatively, (*LL*^{T}).

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Stanford Exploration Project

10/9/1997