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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).
| |
(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.

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Stanford Exploration Project

11/11/2002