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Model-space regularization implies adding equations to system

| |
(15) |

to obtain a fully constrained (well-posed) inverse problem. The
additional equations take the form
| |
(16) |

The full system of equations ()-() can be
written in a short notation as

| |
(17) |

where is the effective data vector:
| |
(18) |

and is a *column* operator:
| |
(19) |

The estimation problem () is fully constrained. We can
solve it by means of unconstrained least-squares optimization,
minimizing the squared power of the
compound residual vector

| |
(20) |

The formal solution of the regularized optimization problem has a
known form, which coincides with formula (). One can
carry out the optimization iteratively with the help of the
conjugate-gradient method Hestenes and Steifel (1952) or its analogs
Paige and Saunders (1982).
The next subsection introduces an alternative formulation of the
optimization problem.

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Stanford Exploration Project

12/28/2000