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 σi, the formal uncertainty on the inferred profile is given by
σ2[Ω(r ,𝜃 )] = ∑ [c (r ,𝜃 )σ ]2. (17 ) 0 0 i i 0 0 i
In the (usually unrealistic) case where the errors on the input data are all equal, we can write
σ2 [Ω (r0,𝜃0)] = Λ (r0,𝜃0)σ, (18 )
where the “error magnification” is given by
Λ (r ,𝜃 ) = ∑ [c(r ,𝜃 )2]1∕2. (19 ) 0 0 i i 0 0
As discussed, for example, by Christensen-Dalsgaard et al. (1990Jump To The Next Citation Point), 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 (1996Jump To The Next Citation Point). (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 ci for that datum is non-zero.) In the one-dimensional case, the correlation between the errors for two points r0 and r1 is given by

∑ ci(r0)ci(r1)σ2 C(r0,r1) = -∑--2-----2-1∕2-∑--2--i--2-1∕2; (20 ) [ ci(r0)σi] [ ci(r1)σi]
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 (17View Equation), 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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