## 2. Discretization and SVD of the matrix of kernelsIn this section and the next, we neglect the surface term , so that the full inverse problem is described by Eq. (1). Approximating each component of in terms of basis functions , the integral term on the right of Eq. (1) can be replaced by a
matrix product , where the elements of
are the and
. If there are where and are the two
components of the vector in Eq. (1). The
matrix where is a diagonal matrix containing the
singular values of We now show the results of applying the procedure above to a test
example. Following Antia & Basu (1994) we define a set of points
in the interval [0, ],
distributed equally in acoustic depth . Then our
basis functions are defined to be the cubic
B-spline basis functions over We expect the results of the SVD to be similar for kernels from any modern standard solar model. However, to be specific, let us note that for the following numerical results we have used model S of Christensen-Dalsgaard et al. (1996). The model was constructed with OPAL opacities (Iglesias, Rogers & Wilson 1992), and with the OPAL equation of state (Rogers, Swenson & Iglesias 1996). It incorporates gravitational settling of helium and other heavy elements, and has an age of 4.6 Gyr. This model is also the reference model of the inversion in subsequent sections. The results will depend more on the modeset used. We have used all modes from the first year of observations by the LOWL instrument (Tomczyk et al. 1995) except for 24 modes with unusual characteristics, such as much lower errors or linewidths than neighbouring modes. The set consists of 1130 modes of degrees 0 to 99 in the frequency range 1-3.5 mHz (Schou, Tomczyk & Thompson 1996; Schou & Tomczyk 1997). In this section and the next two we assume uniform uncertainties on the data: we relax this assumption in Sect. 5. First we consider the effect of the discretization on the spectrum
of singular values of matrix
To investigate further the correspondence between the structural
SVD and that for the rotation kernels, we have computed the singular
value spectra using just one component of the kernels, i.e.,
corresponding to just one of the unknown variables. Fig. 2 shows
that the spectrum for sound-speed kernels alone does have a flatter
plateau, terminated by a well defined shoulder, which is very similar
to the spectrum for rotation kernels. The spectrum for density kernels
alone is quite similar. Thus the less well-defined plateaux in
Fig. 1 are in some sense a consequence of having the two
variables at once. However, we note that at a given index
Other pairs of variables can be used, and Fig. 2 shows also the spectra for individual variables from another choice of variable pair: . Once again the spectra exhibit well-defined plateaux, of similar length. However, we note that the singular values for are much reduced relative to the others, except for the first few, reflecting the difficulty in obtaining detailed information about the helium abundance except in the dominant ionization zones of hydrogen and helium. © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |