Astron. Astrophys. 321, 634-642 (1997)

5. The effect of realistic errors

Having 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 i, with the uniform-error singular values being higher than those for the non-uniform case. This probably reflects the fact that the most deeply penetrating (i.e. , low-degree) modes have higher than average uncertainties, so that it is more difficult to invert for the deep interior, relative to the outer envelope, if the data have realistic errors.

Fig. 9. Singular value spectra for realistic uncertainties in the data. The thick continuous line shows the spectrum when the variable combination is used, while the spectrum for the case is shown as a thick dotted line. For comparison we also show the spectra with uniform errors - thin continuous line for , and thin dotted line for . In all cases 81 radial knots and 11 surface terms have been used

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.

Fig. 10. Comparison of inversion results for the full modeset (circles) and the transformed set using 130 singular values (asterisks). The continuous lines are the exact model differences. The upper panel shows the inversion for squared sound speed, where the second variable is density. The lower panel shows the inversion for density (the second variable being helium abundance). Note that the results for full and transformed sets are virtually indistinguishable. The error bars show propagated errors for the full set; the errors for the transformed set are virtually the same

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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