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If the system of equations, ,is underdetermined, then a standard approach is to find the solution with
the minimum norm. For the *L*2 norm, this is the solution to
| |
(8) |

As a corollary to the the methodology outlined above for creating
model-space weighting functions, Claerbout (1998) suggests constructing
diagonal approximations to by probing the
operator with a reference data vector, .
This gives data-space weighting functions of the form,

| |
(9) |

which can be used to provide a direct approximation to the
solution in equation (8),
| |
(10) |

Alternatively, we could use as a data-space
preconditioning operator to help speed up
the convergence of an iterative solver:

| |
(11) |

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** Up:** Rickett: Normalized migration
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Stanford Exploration Project

4/29/2001