Summary

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).