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

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2. Discretization and SVD of the matrix of kernels

In this section and the next, we neglect the surface term [FORMULA], so that the full inverse problem is described by Eq. (1). Approximating each component of [FORMULA] in terms of basis functions [FORMULA],

[EQUATION]

the integral term on the right of Eq. (1) can be replaced by a matrix product [FORMULA], where the elements of [FORMULA] are the [FORMULA] and [FORMULA]. If there are M modes in the set (i.e. , M data), and there are N basis functions [FORMULA], then A is an [FORMULA] matrix where [FORMULA]. The elements of matrix A are:

[EQUATION]

where [FORMULA] and [FORMULA] are the two components of the vector [FORMULA] in Eq. (1). The matrix A can be written in terms of its singular value decomposition (SVD)

[EQUATION]

where [FORMULA] is a diagonal matrix containing the singular values [FORMULA] of A ([FORMULA]), U (of order [FORMULA]) and V (of order [FORMULA]) are orthonormal matrices, with the property that [FORMULA] and [FORMULA] are equal to an identity matrix.

We now show the results of applying the procedure above to a test example. Following Antia & Basu (1994) we define a set of points [FORMULA] in the interval [0, [FORMULA] ], distributed equally in acoustic depth [FORMULA]. Then our basis functions [FORMULA] are defined to be the cubic B-spline basis functions over r centred at the knots [FORMULA] ; these are normalized to a maximum value of unity so that, roughly, [FORMULA] and [FORMULA] are of the magnitude of the corresponding component of [FORMULA].

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 A. We choose [FORMULA] as our unknown variables. Fig. 1 shows the singular value spectrum for different numbers N of radial knots. Except when very few knots are used, there is a plateau of about one hundred singular values, beyond which the singular values rapidly become much smaller. A similar behaviour was seen in the case of rotation (Christensen-Dalsgaard et al. 1993), though there the plateau of high singular values was rather flatter than in the present case. The behaviour as the number of knots is increased is qualitatively similar in the two cases. Provided a sufficient number of knots is used, the large singular values are essentially independent of N. These SVD components represent the significant information about solar structure contained in the modeset. Indeed, the minimum-norm regularized least-squares solution to the inverse problem can essentially be obtained from the components corresponding to the largest singular values, truncating the expansion at a level which depends on the errors in the data (e.g. Hanson 1971; Christensen-Dalsgaard et al. 1993). Potential information from components corresponding to small singular values will in practice be inaccessible because of data errors. The smallest singular values are probably dominated by round-off errors. With our discretization and modeset, 61 knots seem just about adequate to capture the behaviour of the plateau of large singular values correctly. For 71 or more knots the spectrum appears to reach the round-off error limit. Thus we believe that a minimum of about 70 knots in radius are required. For the rest of the paper, we use 81 knots in r.

[FIGURE] Fig. 1. The singular value spectrum of the matrix A as defined in Eq. (7), when the variable combination of [FORMULA] is used. The points are joined together for the sake of clarity. Shown are the spectra for 41 knots in r (continuous line), 51 knots (dotted line), 61 knots (short-dashed line), 71 knots (long-dashed line), 81 knots (dot-short dashed line), 91 knots (dot-long dashed line) and 101 knots (short dash-long dashed line)

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 i, [FORMULA] for the combined case is similar to, but slightly larger than, the value obtained for [FORMULA] alone. It is perhaps not surprising that inversion for two variables allows extraction of a larger number of significant components of the solution, and hence leads to a more extended spectrum of comparatively large singular values, although at the expense of the less sharply defined transition to small singular values. It is interesting also that the largest few singular values in the combined case are virtually identical to those for [FORMULA] ; the same is true of the corresponding components of the U and V matrices. This indicates that the most well-determined components of the solution in the combined case are associated with sound speed, as might indeed have been anticipated.

[FIGURE] Fig. 2. The singular value spectrum of just the [FORMULA] kernels at constant [FORMULA] (continuous line), the [FORMULA] kernels at constant [FORMULA] (dotted), the [FORMULA] kernels at constant helium abundance Y (dashed), and the Y kernels at constant [FORMULA] (dot-dashed). In all cases, 81 knots were used. For comparison, the long-dashed line shows the corresponding singular values obtained with the variable combination [FORMULA]

Other pairs of variables can be used, and Fig. 2 shows also the spectra for individual variables from another choice of variable pair: [FORMULA]. Once again the spectra exhibit well-defined plateaux, of similar length. However, we note that the singular values for [FORMULA] 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.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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