## 4. The effect of the surface termThus far, we have neglected the surface term 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 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
can be expanded in terms of a set of basis
functions (). Then the
new matrix Here we have replaced the inertia in Eq. (4) by the inertia ratio , where is the inertia of a radial mode at frequency ; thus is of order unity, at least for modes of low or moderate degree (e.g. Christensen-Dalsgaard & Berthomieu 1991). We take the to be proportional to Legendre polynomials, such that 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.
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
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 To find the coefficients for the inversion problem in Eq. (4), we minimize Eq. (10) subject to the condition in Eq. (11), and apply the additional constraints In the transformed case we do something completely analogous, imposing the constraints 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.
© European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |