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Most inversion problems start by formulating the forward problem,
which describes the forward operator, ,
that transforms the model vector to the data vector :

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(1) |

In general, the measured data may be inexact, and
the forward operator may be ill-conditioned.
In that case, instead of solving the above equation directly,
different approaches are used to find an optimum solution
for a given data .The most popular method is finding a solution that minimizes the misfit between
the data and the modeled data .The misfit, or the residual vector, , is described as follows:

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(2) |

In least-squares inversion, the solution is the one that minimizes
the squares of the residual vector as follows:

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(3) |

Iterative solvers for the LS problem search the solution space
for a better solution in each iteration step,
along the gradient direction (in steepest-descent algorithms),
or on the plane made by the current gradient vector and
the previous descent-step vector (in conjugate-gradient algorithms).
Following Claerbout 1992,
a conjugate-gradient algorithm for the LS solution
can be summarized as follows:

`
iterate {
} ,
`

where the subroutine cgstep() remembers
the previous iteration descent vector,
, where *i* is the iteration step,
and determines the step size by minimizing
the quadrature function composed from
(the conjugate gradient)
and (the previous iteration descent vector),
as follows Claerbout (1992):
Notice that
the gradient vector () in the CG method for LS solution
is the gradient of the squared residual
and is determined by taking the derivative of the squared residual
(i.e. the *L*^{2}-norm of the residual, ,
with respect to the model ):

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(4) |