Discussion: Error Distribution

Recall that in the earlier TLS formulation, the noise which contaminates both the operator and data is assumed to be white, with uniform variance. In practice, both the operator and data noise are likely to be correlated, with nonuniform variance. Björck (1996) notes that an appropriate change of variables can restore the validity of the assumptions. He defines a square matrix D which is applied, somewhat surprisingly, to the ``data matrix'' of equation (2).  

$$\left(\bf D [\bf L \;\; d ] + [\bf N \;\; n ]\right) \left[ \begin{array} {r} \bf m \\ -1 \end{array}\right] = \bold 0.$$ (6)

While it may seem intuitive to scale the noise, rather than the data, if the operator is diagonal (as it is in the fairytale world of uncorrelated noise), the inverse is trivial. Even if the noise is correlated, at SEP, we have considerable experience with the design of invertible decorrelation and balancing operators.

Are the restrictions (white, balanced) on the noise crippling? Zhu et al. (1997) claim that in scattering tomography experiments, correlated noise does not unduly harm the TLS result, and also that the TLS result in this case is still better than the normal LS result.