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

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4. Data and inversion parameters used

4.1. The data: LOWL observations

The LOWL instrument is a Doppler imager based on a Potassium Magneto-Optical Filter that has been operating on Mauna Loa, Hawaii since 1994 (see Tomczyk et al. (1995a) for a detailed description). Both low- and intermediate-degree p-modes can be observed with this instrument.

The modes used in this paper are shown in Fig. 1. The estimations of frequency splittings result from a two year period of observation (2/26/94 - 2/25/96). The first year of observation has been analyzed and inverted by Tomczyk et al. (1996) and is referred as the one year dataset in the following. The second year of observations has been analyzed separately and an unweighted average of the two resulting datasets has been performed to produce the data that we use in this work. These data contain 1102 modes [FORMULA] with degrees up to [FORMULA] and frequencies lower than [FORMULA]. For each mode, individual splittings are given by, at best, five a-coefficients of their expansion on orthogonal polynomials [FORMULA] defined by Schou et al. (1994):


Estimations of standard deviations are given for each of these a-coefficients. To first order, the solar rotation contributes only odd j a-coefficients to the expansion Eq. (15). Even indexed a-coefficients arise from aspherical perturbations, centrifugal distortion and magnetic fields.

We have inverted both odd indexed a-coefficients and the set of splittings reconstructed from these coefficients. The errors assigned to these splittings are discussed in Appendix B. A [FORMULA] value can be calculated from the inversion of the a-coefficients by the first term in the sum Eq. (B7).

[FIGURE] Fig. 2. Basis of [FORMULA] B-splines functions of order 3 with a distribution of break points calculated from the density of turning points relative to the set of modes plotted in Fig. 1

4.2. The choice of inversion parameters

4.2.1. Splines basis

In all inversions shown in this paper the set of all B-splines [FORMULA] with [FORMULA], forms a basis for the linear space of the set of the piecewise polynomials of order 3 having their first derivatives continuous in [FORMULA] and with a distribution of break points, i.e. a partition [FORMULA] of [FORMULA] (see Appendix A), equidistant in [FORMULA]. Doing this we obtain a finer discretization near the equator than near the pole, in agreement with the fact that among all of the observed modes only a few of them have significant amplitude near the pole. The choice of only a few basis functions ([FORMULA]) to describe the latitudinal dependence of the rotation rate is related to the low number of odd indexed a-coefficients (3 maximum) given by observers to describe the azimuthal- or m -dependence of each splitting through Eq. (15).

B-splines in radius [FORMULA] with [FORMULA] will also be piecewise polynomials of order 3 with their first derivatives continuous at each break point, but the partition of [FORMULA] is chosen such that the number of basis functions used to describe an interval in radius is proportional to the number of modes having their turning points located in this interval (Fig. 4.2.1). Compared with an equally spaced partition with the same number of points, this distribution allows a better resolution in the layers which are well described by the data and acts as a regularization term in less well constrained zones.

4.2.2. The trade-off parameters [FORMULA] and [FORMULA]

Currently, most inverters who use this kind of regularization in helioseismic inversions take [FORMULA] in the regularization terms Eqs. (12) and (13) (Schou et al. 1994). Here the code allows constraining with the first derivative of the rotation in latitude (i.e. [FORMULA]). Using a high weight in the core (with a function [FORMULA] for example), this constraint is in better agreement with the regularity condition at the center given by Eq. (8). Both cases ([FORMULA]) have been performed and are discussed in the following with [FORMULA] and [FORMULA].

A generalization of the so-called L-curves currently used in one dimensional problems (Hansen 1992a, b) can be a guide for the choice of the trade-off parameters [FORMULA] and [FORMULA]. The aim is to find parameters that minimize both the [FORMULA] value obtained for the fit of data and the two regularization terms [FORMULA] and [FORMULA]. In the limit of strong regularization (large [FORMULA] and [FORMULA]) which aims to minimize [FORMULA] and [FORMULA], a small decrease in [FORMULA] and [FORMULA] can be obtained only at the expense of a rapidly increasing [FORMULA] value and the solution does not give a good fit of the data anymore. On the other hand, in the limit of low regularization which aims to minimize the [FORMULA] value, a little better fit of the data can be obtained only at the expanse of a strong increase of the terms [FORMULA] and [FORMULA] and the solution presents important oscillations. A good choice of trade-off parameters should be near the intersection of these two limit regimes.

Fig. 3 is a plot of the value of each regularization term against the [FORMULA] value for different choices of [FORMULA] and [FORMULA] ([FORMULA] being the number of degrees of freedom of the system i.e. the difference between the total number of a-coefficients [FORMULA] and the number [FORMULA] (see Appendix A) of searched coefficients [FORMULA]).

[FIGURE] Fig. 3. L-curves obtained by inverting the a-coefficients from the dataset shown in Fig. 1. a Regularization term in radius ([FORMULA]) (with second derivative ([FORMULA])) and b regularization term in latitude ([FORMULA]) (with first derivative ([FORMULA])) against the [FORMULA] value. Each graph marker corresponds to one value of [FORMULA]. Full curves join points with the same ratio [FORMULA] ([FORMULA] from the top to the bottom) and dotted curves have the same [FORMULA].
[FIGURE] Fig. 4. Inferred surface rotation rates against latitude from inverting a-coefficients. The solid curves show the observations of the plasma surface rotation Eq. (6). The dot-dashed curve shows the observations of the small magnetic feature surface rotation Eq. (7). The dashed curves show the inferred rotation rates: a inversion without surface constraint, b inversion with plasma surface constraints and [FORMULA] (This solution is the same as the one shown in Fig. 5.1).

An interesting result is that, on Fig. 3a, all the points which are labeled by the same [FORMULA] but different [FORMULA] have nearly the same location except for values of [FORMULA] and [FORMULA]. We define the corner of a curve that joins points with the same ratio [FORMULA] (full curves on Fig. 3) as the nearest point of the curve to the intersection of the two limit regimes asymptotes. For [FORMULA] the [FORMULA] value begins to increase rapidly for the largest values of [FORMULA] (the star graph marker goes on the right). For [FORMULA] (not shown on the figure) the corners of the L-curves give larger values of [FORMULA] and the corresponding values of parameters must be disregarded because they do not lead to the best compromise between the regularization and the fit of the data. Thus it appears that, near the corners of the L-curve, the [FORMULA] and [FORMULA] values do not depend on the value of [FORMULA] for a large domain of variation of the parameter [FORMULA] ([FORMULA]): they depend only on [FORMULA]. Consequently we minimize both the [FORMULA] and [FORMULA] values by choosing [FORMULA] at the corner of the L-curve i.e. [FORMULA]

The choice of [FORMULA] is then given by the analysis of Fig. 3b. On this figure, as [FORMULA] decreases the position of the L-curve becomes lower showing that for a given value of [FORMULA], the [FORMULA] value must be as large as possible (keeping in the previous interval for [FORMULA]) if one wants to reduce the value of the regularization term in latitude.

According to these two figures, a choice near [FORMULA] tends to minimize both the [FORMULA] value and the two regularization terms. When the regularization term in latitude [FORMULA] is chosen with second derivative ([FORMULA]), the corresponding plots have similar behaviors but the domain of variation of [FORMULA], for which [FORMULA] and [FORMULA] depend only on [FORMULA], is smaller ([FORMULA]). In this case the optimal choice for trade-off parameters becomes [FORMULA]. The inversion of individual splittings, instead of a-coefficients, leads to the same results for the choice of trade-off parameters and the L-curves analysis is not sensitive to the surface constraints parameter [FORMULA].

L-curves are a useful tool to study variations and mutual dependencies of each term in Eq. (10) for different choices of trade-off parameters and functions [FORMULA] and [FORMULA]. Nevertheless other criteria for the optimal choice of trade-off parameters are possible (Craig & Brown 1986; Thompson & Craig 1992). In particular, for the solar rotation problem, the method of generalized cross validation (GCV) (Golub & Van Loan 1989) has been applied to the 1D RLS inversion method by Thompson (1992) and Barrett (1993). For a local estimation of the quality of the solution, we must look at the balance between the effect of propagating input errors and the resolution (as defined in Appendix C) reached at a target location [FORMULA]. Thus a local optimal choice of trade-off parameters could be based on plots showing resolution against the error on the inferred rotation rate for different choices of trade-off parameters. Such curves have been plotted by Christensen-Dalsgaard et al. (1990) for different 1D inversion techniques and by Schou et al. (1994) for a 2D inversion.

4.2.3. Surface constraints and the [FORMULA] parameter

Different surface constraints can be used and introduced in Eq. (B2) with the parameter [FORMULA] which defines the weight assigned to the fit of surface observations. The choice of surface constraints is suggested by the behavior of [FORMULA] obtained if we do not impose any surface constraint ([FORMULA]) and with the previous choice of trade-off parameters (Fig. 4a).

This figure clearly points out that the surface rotation estimated from the helioseismic data alone is closer to the plasma observations than to the small magnetic feature observations. However, the estimated surface rotation has no latitudinal dependence in the region covering [FORMULA] around the equator which is in evident contradiction with all surface observations and can be a consequence of the lack of high degree modes in the data. Thus we have to fix the value of [FORMULA] in such a way that the rotation rate obtained at the surface becomes close to the imposed surface values [FORMULA] at the [FORMULA] points.

Let us define the [FORMULA] value by:


and the relative contribution [FORMULA] of the surface term to the estimated rotation rate at [FORMULA] by (according to Eq. (C1)):


where [FORMULA] is a function of [FORMULA].

The Fig. 5a shows the variations against [FORMULA] of the relative contribution of the surface term to the estimated rotation rate at the solar surface ([FORMULA]) and at the equator ([FORMULA]).

The Fig. 5b and c show the variations of [FORMULA] and [FORMULA] against [FORMULA] for the two kinds of observations Eq. (6) and Eq. (7) used as surface constraints. With the use of plasma rotation observations, we can obtain a small [FORMULA] value for [FORMULA] and the helioseismic data still contributes more than 30 percent in the computation of the surface rotation rate. If we want to obtain roughly the same value of the [FORMULA] with the use of the small magnetic feature observations, we have to set [FORMULA] and then the helioseismic data contributes less than 10 percent in the computation of the surface rotation rate.

Furthermore, as [FORMULA] decreases, the [FORMULA] value increases greatly for the small magnetic feature constraint (from 2.0 for [FORMULA] up to 2.5 for [FORMULA]) but not so much for the plasma constraints (Fig. 5b, 5c). This behavior and the very large value of [FORMULA] in Fig. 5b for small magnetic feature clearly indicates that the use of these observations for surface constraints is not compatible with helioseismic data, probably because these observations correspond to the rotation not of the solar surface but of deeper layers.

[FIGURE] Fig. 5. Variation of, a the relative contribution of the surface term in the estimation of the rotation rate at the surface and the equator (i.e. [FORMULA] Eq. (17)), b the [FORMULA] value and c the [FORMULA] value, against [FORMULA] and by inverting a-coefficients. Dot-dashed curves are for the small magnetic feature observations and full curves for the plasma observations used as surface constraints.
[FIGURE] Fig. 6. Variation with depth of the relative contribution of the surface term in the estimation of the rotation rate at the equator and for [FORMULA]. The line styles are the same as in Fig. 5a-c.

The Fig. 4.2.3 shows the variation of the surface contribution with depth (at fixed [FORMULA] and at the equator), showing that the major contribution of the surface constraints occurs above [FORMULA]. Nevertheless some residual (negative) contributions exist below this depth and are more important for the small magnetic feature observations than for the plasma observations.

For these reasons we choose in the following to use plasma observations as surface constraints with [FORMULA]. With this choice, the inferred rotation rate, shown in Fig. 4b as a function of latitude at the surface, is close to the observed one and remains compatible with LOWL data (Fig. 5c).

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998