Even when the errors on the input data are uncorrelated, the errors on the inferred profile will not be, as discussed by Howe and Thompson (1996). (As a simple way to understand this, consider the case where one measurement is significantly “off”; this will affect the inferred profile at every location where the inversion coefficient for that datum is non-zero.) In the one-dimensional case, the correlation between the errors for two points and is given by
Error correlations by definition should not distort the inferred profile beyond the distribution predicted by the formal uncertainty on the inferences, provided always that the input uncertainties are correct. However, the finite width of averaging kernels also gives rise to a systematic error that can be much larger. Consider, for example, the case where a thin shear layer is not resolved; then all the estimated rotation rates on one side of the shear could be underestimated, and those on the other side overestimated, by several times the formal uncertainty. Such systematic errors and their relationship to the averaging kernels have been discussed, for example, by Christensen-Dalsgaard et al. (1990).
Gough et al. (1996) pointed out that it is not sufficient for the rotation rates at two locations to have non-overlapping errors as calculated in Equation (17), and described a method for increasing the error estimates on inversions to allow truly significant differences between the inferred rotation rate at different locations to be determined. This method, however, has not been widely used.
Because the input data are noisy and of finite resolution, the inversion problem does not have a unique solution; there will always be a tradeoff between noise and good localization. Two widely-used approaches to balancing these criteria are “regularized least squares” (RLS) and “optimally localized averaging” (OLA).
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