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

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1. Introduction

Global solar oscillation frequencies can be determined to very high precision. Hence comparison of computed and observed frequencies provides a stringent test of models of the solar interior. In fact, no physical model of the Sun reproduces the observed frequency spectrum at the level of accuracy provided by the observations. As a result, to infer the detailed structure of the Sun the helioseismic data must be inverted.

The observed global oscillations of the Sun are well described throughout most of the solar interior by the equations describing linear adiabatic oscillations. (cf. Unno et al. 1989). These equations, along with appropriate boundary conditions, constitute a self-adjoint eigenvalue problem, which leads to a variational principle connecting the eigenfrequency to the basic equilibrium state. This has been extensively used as the basis for inversions to determine the structure of the Sun (e.g. Gough & Kosovichev 1988, 1993; Dziembowski, Pamyatnykh & Sienkiewicz 1990; Däppen et al. 1991; Antia & Basu 1994; Dziembowski et al. 1994). Such inversions proceed by linearizing the equations of solar oscillations about a known reference model of the Sun. Schematically, we can write our inverse problem as


where [FORMULA] is the i th datum, [FORMULA] is a vector function, the kernel, which is a known function of the reference model, [FORMULA] is the unknown vector function for which we wish to invert, and [FORMULA] is the error in the i th datum. In our case, [FORMULA] is the relative differences between the observed frequency of the i th mode and the corresponding frequency of the reference model (more correctly, the mean multiplet frequency for given radial order and degree, averaged over azimuthal order); [FORMULA] is a 2-vector describing the relative structural differences between the Sun and the model relevant for adiabatic oscillations of a spherically symmetric, self-gravitating body in hydrostatic equilibrium; r is the radial coordinate describing position inside the Sun; and the integral runs from the centre of the Sun ([FORMULA]) to the surface (taken to be located at the chromospheric temperature minimum).

Various standard linear inversion techniques are available for inverting a set of constraints such as Eq. (1) - see the above references and Sect. 2 below. A characteristic feature of the inverse problem is the very large number of modes available for inversion and the large degree of redundancy in the information they contain (see Christensen-Dalsgaard, Hansen & Thompson 1993). This leads to an ill-posed problem, and unless precautions are taken, errors in the data cause large errors in the solution (e.g. Craig & Brown 1986). To control the errors, trade-off parameters have to be introduced in the inversion techniques, and these must be determined as a part of the solution. This and the large amount of data available in general makes inversion computationally expensive.

It was shown by Christensen-Dalsgaard & Thompson (1993) that in the case of inversion to determine solar rotation, it is possible to exploit the redundancy of the data to reduce the problem at hand. They demonstrated that, by using a singular value decomposition, one can form a new, smaller set of data that are linear combinations of the original data. This smaller set contains all the useful information of the original dataset, and because of its size can be inverted by some standard inversion techniques much more quickly than could the original set. Our purpose in this paper is to demonstrate that similar preprocessing is effective also in the case of the inversion for solar structure.

The essence of the preprocessing is therefore to find a smallish set of linear combinations of the data that contains essentially the same information about the solar interior as the full dataset. This will then be our new transformed dataset. The empirical test to show that all useful information has been retained is that the same inversion solutions and averaging kernels can be derived with the new dataset as with the original one. We can write the new data as


for some coefficients [FORMULA] ; and since this is just a linear operation it follows from Eq. (1) that


where [FORMULA] and [FORMULA] are the corresponding combinations of the kernels and the errors. The essential point is that the necessary number [FORMULA] of new data is smaller than the original number M of data, if the coefficients [FORMULA] are chosen judiciously.

How then are these coefficients [FORMULA] to be determined? The strategy we adopt is to form a matrix, each row of which is a discretized representation of the kernels [FORMULA], and to find the singular value decomposition (SVD) of that matrix. We then derive the required coefficients from the singular vectors corresponding to the largest, most significant, singular values of the kernel matrix. This is analogous to the approach Christensen-Dalsgaard & Thompson used for the rotation inversion, and is closely related also to methods that have been adopted in other areas of astrophysics for handling large datasets (e.g. Zaroubi et al. 1995; Tegmark et al. 1997).

One difference between inversion for solar rotation and that for solar structure is that in the case of rotation there is only one unknown scalar function, namely the rotation rate, whereas in structure inversion two unknown scalar functions determine the frequency differences, i.e. , the two components of vector function [FORMULA]. These two functions may be, for example, the relative differences in the adiabatic sound speed squared ([FORMULA]) and in density [FORMULA] ; or if, in addition to the assumption of hydrostatic equilibrium the equation of state and heavy element-abundance are assumed known, then the two functions may be e.g. the relative difference in density and the difference in helium abundance Y.

There is a second difference between inversions for rotation and structure. In the latter case, a significant contribution to the frequency differences that constitute the data arises from non-adiabatic effects and other inadequacies of the forward model in the near-surface layers. In the absence of any reliable formulation for these effects, they are taken into account in an ad hoc manner by including on the right-hand side of Eq. (1) an additive term:


Fortunately, the surface term cannot be a completely general function of mode parameters. Having factored out the inertia [FORMULA] of the mode, the function [FORMULA] can be expanded as (cf., Brodsky & Vorontsov 1993; Antia 1995; Gough & Vorontsov 1995)


where [FORMULA] is the angular frequency and [FORMULA]. Provided the degree l of the mode is not too large, all terms except the first can be ignored, i.e., the function [FORMULA] is a function of the frequency [FORMULA] but is independent of l. Moreover, it is a relatively slowly varying function of frequency. As a consequence, it is possible to perform a linear projection to remove the surface term and recast the inversion problem into the form (1). Alternatively, it is perhaps simpler (and therefore more commonly done) to work with the full Eq. (4), parametrizing [FORMULA] as a low-order polynomial with unknown coefficients which are themselves to be determined or else eliminated as part of the inversion procedure. We adopt the latter approach here, in Sect. 4.

The plan of the rest of the paper is as follows. In Sect. 2 we discuss the setting up and singular value decomposition of the matrix of discretized kernels. It should be borne in mind that the only purpose of discretizing the kernels is to obtain the coefficients for transforming problem (1) to problem (3): this transformation is performed in Sect. 3, and the results of inversions with transformed and full datasets are compared. For clarity, we exclude the possibility of a near-surface term [FORMULA] in Sects. 2 and 3. In Sect. 4 we indicate how things are modified in the presence of [FORMULA], and we consider the effect of more realistic errors in the data in Sect. 5.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998