Partial stacking the data recorded with irregular geometries within offset and azimuth ranges yields uniformly sampled common offset/azimuth cubes. In order to enhance the signal and reduce the noise, the reflections should be coherent among the traces to be stacked. Normal Moveout (NMO) is a common method to create this coherency among the traces.

Let's define a simple linear model that links the recorded traces (at arbitrary midpoint locations) to the stacked volume (defined on a regular grid). Each data trace is the result of interpolating the stacked traces and equal to the weighted sum of the neighboring stacked traces. In matrix notation, this transforms to:

(1) |

(2) |

This simple operation does not yield satisfactory results for an uneven fold distribution. To compensate for this uneveness, it is common practice to normalize the stacked traces by the inverse of the fold (), thus:

(3) |

Alternatively, it is possible to apply the general theory of inverse least-squares to the stacking normalization problem. The formal solution of the inverse least-squares problem takes the form:

(4) |

With the knowledge of model regularization in the least-squares inversion theory, it is possible to introduce smoothing along offset/azimuth in the model space. The simple least-squares problem becomes:

(5) |

The fold, which normalizes the data based on the traces distribution, is introduced by a diagonal scaling factor. The weights, for the regularized and preconditioned problem, are thus computed as:

(6) |

(7) |

11/11/2002