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

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4. The effect of the surface term

Thus far, we have neglected the surface term [FORMULA] which is certainly present when comparing observed frequencies with the frequencies of present-day solar models. As explained in Sect. 1, this term arises from near-surface errors in the model and mode physics, and introduces an extra term in Eq. (4) compared with Eq. (1). To the level of approximation we consider here, the function [FORMULA] depends on frequency only.

There are two differences in the way in which we transform and invert data in the presence of the surface term. First, the SVD is based on a discretization that includes the surface term as well as the integral contribution from the kernels. We assume that [FORMULA] can be expanded in terms of a set of basis functions [FORMULA] ([FORMULA]). Then the new matrix A is of order [FORMULA], where [FORMULA], and A has elements

[EQUATION]

Here we have replaced the inertia [FORMULA] in Eq. (4) by the inertia ratio [FORMULA], where [FORMULA] is the inertia of a radial mode at frequency [FORMULA] ; thus [FORMULA] is of order unity, at least for modes of low or moderate degree (e.g. Christensen-Dalsgaard & Berthomieu 1991). We take the [FORMULA] to be proportional to Legendre polynomials, such that [FORMULA] is between -1 and 1, with suitably scaled argument (cf. Däppen et al. 1991). In Fig. 6, the singular values of this matrix are compared with those of the matrix given by Eq. (7), where the surface term was not taken into account.

[FIGURE] Fig. 6. The singular value spectrum for matrix A as defined in Eq. (12) (thick continuous line) when the variable combination [FORMULA] is used. Also shown is the spectrum for the case [FORMULA] (thick dotted). For comparison we also show the spectra without the surface terms - thin continuous line for [FORMULA], and thin dotted line for [FORMULA] In all cases 81 radial knots are used. We have expanded the surface term in Legendre polynomials of degree between 0 and 10

The change in the singular-value spectra due to the inclusion of the surface term is quite striking. The main difference seems to be some comparatively large singular values in the case with the surface term. These are undoubtedly due to the extra basis functions of frequency in Eq. (12). One way to gauge the nature of the extra singular values is to look at the corresponding columns of the matrix U. For the singular values that arise due to the surface terms, [FORMULA] is expected to be a function of the frequency of the j th mode. Singular vectors associated with the interior of the model, on the other hand, are predominantly functions of the turning point of the mode, which can be represented by w (see also Christensen-Dalsgaard et al. 1993). In the present case we find that the first 11 columns of U are indeed essentially functions of frequency and hence correspond to the basis functions of frequency. The rest are largely functions of w. A few such columns are shown in Fig. 7. Note the qualitatively different behaviour for [FORMULA] and for [FORMULA]. We remark also that the addition of the singular values associated with the surface term causes a shift in the remaining singular values, as seen in Fig. 6.

[FIGURE] Fig. 7. Some columns of the matrix U, obtained by the decomposition of the matrix A defined in Eq. (12), plotted against cyclic frequency [FORMULA]

The second difference in approach in the presence of a surface term is in the OLA inversion itself. Evidently, the linear combination of data of the form (4) will produce an extra term in Eq. (3), which becomes

[EQUATION]

To find the coefficients [FORMULA] for the inversion problem in Eq. (4), we minimize Eq. (10) subject to the condition in Eq. (11), and apply the additional constraints

[EQUATION]

In the transformed case we do something completely analogous, imposing the constraints

[EQUATION]

The inversion results are shown in Fig. 8, where we again compare the results obtained by inverting the full modeset and those found by inverting the preprocessed modeset. Whereas without a surface term transformation with 120 singular values was sufficient to reproduce the original solution (cf. Fig. 4), we find that 130 singular values are required once the surface term is included. This increase evidently corresponds approximately to the number of singular values associated with the surface term.

[FIGURE] Fig. 8. A comparison of the inversion results of the full modeset (circles) and the transformed set with 81 knots and using 130 singular values (asterisks). The top 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
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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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