When coherent noise is present in the data, residual variables are no longer IID. The coherent noise will add ``color'' to the spectrum of the residual. The key idea is to recognize that the goal of the inverse covariance matrix is to absorb this spectrum. Now, as Claerbout and Fomel (2002) assert:

Clearly, the noise spectrum is the same as the data covariance only if we accept the theoretician's definition thatThis statement is the basis of the filtering method. It says that the data residuals (squared) should be weighted by their inverse multivariate spectrum for optimal convergence. Because a prediction-error filter (PEF) whitens data from which it was estimated Burg (1975), it approximates the inverse power spectrum. Thus a PEF (squared) with the inverse spectrum of the coherent noise accomplishes the role of the inverse covariance matrix in equation (). The fitting goals in equation () become, omitting the regularization term,E(d)=Fm. There is no ambiguity and no argument if we drop the word ``variance'' and use the word ``spectrum''.

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5/5/2005