## 5. The effect of realistic errorsHaving investigated the uniform-error case, we turn our attention to a more realistic case in which the errors are still assumed to be independent and Gaussian-distributed, but with individual standard deviations as estimated for LOWL observational 1-year data (Schou et al. 1996; Schou & Tomczyk 1997). As in Sect. 4 we include the effects of a surface term. As a first step, for each datum we renormalize by dividing through Eq. (1) by , where is some typical magnitude for the errors. This renormalizes the data and the kernels in such a way as to give once again a problem in which the errors have identical standard deviations. The factor is included simply to avoid rescaling the kernels by a large factor. After renormalization, the problem looks formally identical to the one we have already considered. We proceed in exactly the same way with the SVD, constructing the transformed kernels and data and performing the SOLA inversion. The singular value spectrum for the case with realistic errors is
shown in Fig. 9. For comparison, we also show the spectrum for
the uniform-error case. The factor was chosen
so as to make the value of the largest singular value the same in the
case of non-uniform errors as in the uniform-error case. In this
example, the spectra of singular values in the case of realistic
errors are very similar to those for uniform errors. The reason for
this is that for this modeset, which only extends up to 3.5mHz, there
is not a great range in the quoted uncertainties on the frequencies.
There would probably be more substantial differences if
higher-frequency data, which are generally less certain, were
included. There are differences between the two spectra for
intermediate
The inversion results are shown in Fig. 10, where we again compare the results obtained by inverting the full modeset, and those by inverting the preprocessed modeset. As we can see again, and a discretization with 81 knots gives very good results. The propagated errors in the two inversions are essentially identical at each point. Moreover, as we have seen earlier (cf. Fig. 5), the averaging kernels (and hence the resolution) are the same in both inversions. Fig. 10 also shows the exact differences between the test and reference models, thus demonstrating that the inversions do indeed do a good job of estimating the true differences.
© European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |