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An important assumption in least-squares inversion is that the
residual has independent and identically distributed components (IID). If
the data are contaminated with noise that the
operator does not model properly, the residual will not be IID, thus
affecting the estimated model. Theoretically, the data covariance
matrix mitigates the effects of modeling uncertainties by
filtering the bad data points. In practice, the data covariance operator is
often approximated with diagonal operators with similar variance for
each point, which can be a valid assumption in the presence of white
noise only. For more complicated noise sources with multidimensional
spectral components, more elaborated covariance matrices are needed to
obtain IID residuals.
In this Chapter, two methods that lead to IID residuals are introduced. One method
replaces the data covariance matrix with prediction-error filters that
absorb the noise spectrum in the data residual. The other method improves the
operator by incorporating a noise-modeling component in it. This
method can be reduced to a weighting of the data residual with
projection filters. On a synthetic and field data examples, while both
methods achieve similar results in terms of estimated models, the
modeling approach produces better residual panels (i.e., smallest
error and smallest coherent energy left).

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Stanford Exploration Project

5/5/2005