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Applying a mask in equation (6) eliminates the contribution
of the empty traces in the model space, making them invisible to the
inversion. Therefore, by simply remodeling a data panel from
the estimated model after inversion without the mask,
the missing traces are reconstructed. Then, the interpolated data
vector can be estimated as follows:
| |
(9) |

where **I** is the identity matrix. Now, for the noise removal, we
simply (1) apply a mute in the radon domain that isolates and
preserves the signal, and (2) transform the muted panel in the data
space as follows:
| |
(10) |

where is the estimated signal (specular reflections and impinging source).
The estimated noise (diffracted energy and ambient noise) is
obtained by subtracting the estimated signal from the input data:
| |
(11) |

Note that the estimated noise and signal in equations (10) and
(11) are for the non-interpolated data. To compute the
estimated noise and signal for the interpolated data, **M** must be removed
in equations (10) and (11) and **d** must be
replaced by in equation (11).

** Next:** Synthetic Test: Data Interpolation
** Up:** Theory of noise attenuation
** Previous:** Sparse inversion
Stanford Exploration Project

5/3/2005