### 3.3 Inversion errors

If the errors on the input data are uncorrelated and properly described by a normal distribution whose
width corresponds to the quoted uncertainty , the formal uncertainty on the inferred profile is given by
In the (usually unrealistic) case where the errors on the input data are all equal, we can write
where the “error magnification” is given by
As discussed, for example, by Christensen-Dalsgaard et al. (1990), a quantitative choice of
regularization parameters can then be made by finding the “knee” of a tradeoff curve where the error
magnification is plotted against the width of the averaging kernel. However, in the two-dimensional case
this does not always give a clear result, and this formulation of the error magnification is not
very useful for modern data sets where the uncertainties on the parameters are anything but
uniform. Instead, one can consider the uncertainty on the inferred quantity at a particular
location.
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

this can easily be generalized to the two-dimensional case. Howe and Thompson (1996) found that the
spatial scale over which the inversion errors are significantly correlated is usually similar to that for the
averaging kernels, though for some cases where the inversion parameters have been badly chosen
the results can be correlated over long distances even when the averaging kernels appear well
formed.
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).