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Astron. Astrophys. 324, 298-310 (1997)

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Appendix A: the space of solutions

In this work, we search the rotation rate as a piecewise polynomial of arbitrary order in two dimensions. Let us define more precisely what piecewise polynomials are: a piecewise polynomial [FORMULA] of order m on a given partition [FORMULA] of [FORMULA] [FORMULA] is defined as a function which coincides, on each sub-interval [FORMULA] [FORMULA] with a polynomial of degree [FORMULA]. We can define at each break point the kind of connection which is required between the right and the left pieces of polynomials. Formally, the rules of connection can differ from one break point to the next: at some of them, [FORMULA] can be discontinuous, at some others the left and right pieces can be tied to fulfilling the continuity of their first derivatives or only of [FORMULA], ...etc.

It can be shown that a basis of such a space of piecewise polynomials can be obtained from B-splines in 1 dimension and a tensorial product of B-splines in two dimensions (Schumaker 1981). B-splines basis are a local basis. Moreover, at a given q ([FORMULA]), only m B-splines of order m are not identically zero and their sum is equal to 1. These properties have two principal useful consequences in our case. First they are easy to compute and the evaluation of the rotation at a given target location needs only a few calculations. Second, they allow us to easily study the boundary conditions in the core and at the solar surface.

Using [FORMULA] and [FORMULA] we obtain respectively at the surface and the center:

[EQUATION]

From Eq. (A1), the knowledge of the surface rotation at [FORMULA] different and well chosen latitudes allows us to fix in theory the [FORMULA] coefficients [FORMULA], [FORMULA] which form a vector named [FORMULA] in the following. Nevertheless, the observations of the surface motions Eq. (6) are given with some error bars. Moreover, the depth which defines the solar surface depends on the choice of indicator and may differ from the surface of the solar model which gives the upper boundary for the p-modes. Consequently, we prefer to include these observations in the minimization procedure rather than to calculate directly the vector [FORMULA] only from data concerning the motion of the surface.

The relation Eq. (A2) allows us to search less coefficients to describe the core than for the rest of the solar interior. This is reasonable because of the lack of observed modes able to describe this zone even if we invert data including both low and intermediate degrees. This introduces a scalar value [FORMULA] which is the value of the rotation at the center of the Sun and that is used only to describe the rotation rate in depth where the first B-spline [FORMULA] is not identically zero i.e. in most practical cases under 0.2 solar radii (see Sect. 4.1, Fig. 4.2.1)

The relation Eq. (9) becomes:

[EQUATION]

Appendix B: the functions [FORMULA]

We apply a least-squares method on values of both observed splittings and observed surface rotation. More precisely, we search the vector

[EQUATION]

by minimizing the quantity:

[EQUATION]

where:

  • [FORMULA], [FORMULA] are the diagonal matrix of the inverse of errors given on splittings and surface rotation values. These errors are therefore used as weights in the whole minimization procedure.
  • [FORMULA] is the vector of observed splittings [FORMULA],
  • - [FORMULA] is a matrix computed by the discretization of Eq. (2) using Eq. (A3) and Gaussian integrations.
  • [FORMULA] is the vector of the values of the surface rotation [FORMULA] according to Eq. (6),
  • [FORMULA] is a matrix defined by: [FORMULA] [FORMULA] according to Eq. (A1)and
  • - [FORMULA] is a parameter used to define the weight assigned to the fit of surface observations. If [FORMULA] is the solution of the problem:

    [EQUATION]

    and [FORMULA] the solution of the equality constrained least-squares problem:

    [EQUATION]

    then [FORMULA] (Golub & Van Loan 1989). Therefore a high value of the parameter [FORMULA] tends to give a good fit of these observations but one can take small values or even [FORMULA] if the observed p-modes are thought to be adequate to describe the surface rotation.

Unfortunately, up to now most of the observers do not give individual splittings [FORMULA] but rather few coefficients (typically [FORMULA] or 9) of their expansion on chosen polynomials [FORMULA] (Eq. (15)). This latter equation can be rewritten in matrix form:

[EQUATION]

by building the vector [FORMULA] of odd indexed a-coefficients for all modes [FORMULA] and the appropriate rectangular matrix [FORMULA] of polynomials [FORMULA]. Therefore, there are two ways for performing the inversion: we can build all individual splittings from Eq. (B5) and minimize [FORMULA] ; or we can express a-coefficients as a linear combination of individual splittings:

[EQUATION]

where [FORMULA] is the pseudo-inverse of [FORMULA] (assuming that this one exists for the chosen polynomials), and minimize:

[EQUATION]

[EQUATION]

In this case we can use directly the quoted errors on the a-coefficients (matrix [FORMULA]).

When we invert splittings, we must take care of weights that we assign to individual splittings through matrix [FORMULA]: Eq. (15) implies that individual splittings calculated from a-coefficients are correlated and thus there is no evident diagonal matrix [FORMULA]. One possibility is to calculate the true covariance matrix [FORMULA] on individual splittings using Eq. (B5):

[EQUATION]

and to take only the diagonal part of this matrix as matrix [FORMULA] (Sekii 1991; Corbard et al. 1995). This leads to individual errors that depend on m. In this work, however, we assume that individual splittings are uncorrelated and independent of m for each [FORMULA] (Schou et al. 1992) and we calculate their errors such that they lead at best in a least-squares sense to the errors given on the a-coefficients if these ones were calculated by a least-squares fit to individual splittings. By this way, we obtain individual errors that are higher than in the previous case especially for low m.

In order to have a more immediate interpretation of the result found by inverting individual splittings it will be of much interest to have accurate observations for individual splittings along with their associated errors. This is already the case with the ground based GONG experiment and should probably be possible with the SOHO space mission instruments.

In this paper [FORMULA] denote both [FORMULA] and [FORMULA] depending on the kind of inversion we perform.

Appendix C: averaging kernels

For all linear inversion techniques, the inferred rotation rate at a target location [FORMULA] can be expressed as a linear combination of the data. Namely, in our implementation these data are the splittings [FORMULA] and, if [FORMULA], the observed rotation rates at the surface [FORMULA]:

[EQUATION]

Averaging kernels [FORMULA] are defined by:

[EQUATION]

From Eqs. (2) and (C1) we get:

[EQUATION]

Here [FORMULA] denote a Dirac distribution in two dimensions. Each surface constraint induces a term proportional to a [FORMULA] function, localized at the corresponding point of the surface, in the averaging kernel. The same relations exist when a-coefficients are inverted instead of individual splittings.

From Eq. (C2) the value of the inferred rotation rate [FORMULA] can be regarded as a weighted average of the true rotation rate where the averaging kernel [FORMULA] is the weighting function. Ideal averaging kernels would be close to a [FORMULA] function leading to [FORMULA]. In practice averaging kernels have a peak near [FORMULA] and we can evaluate the latitudinal and radial full width at mid height (FWMH) of this peak, [FORMULA] and [FORMULA], respectively. These quantities provide a measure of the resolution of the inversion in the sense that it gives a limit for the finest details that the inversion is able to resolve for a given depth and latitude. It should be noted that averaging kernels formally do not depend on the data but only on errors on these data. Nevertheless they depend on the regularization used (high regularization decreases the resolution) and the regularization used itself depends on the set of data that we want to invert. A complete description of averaging kernels and their properties can be found in Christensen-Dalsgaard et al. (1990) and Schou et al. (1994).

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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