An observable consequence of the solar rotation is that frequencies of solar oscillation modes (n,l,m) are splitted in a m multiplet. for low rotation the frequency shifts are given by:

where is the unperturbed mode frequency, is the unknown rotation rate versus depth and latitude, , is the colatitude and r the normalized solar radius. The kernels depend on the model and the eigenfunctions of the mode which are assumed to be known exactly (e.g. Christensen-Dalsgaard & Berthomieu 1991).

We search for the unknown rotation as a linear combination of piecewise polynomials, projected on a tensorial product of B-spline basis (Corbard et al., 1995). We take into account the observational errors (with standard deviation ) and a regularization term to avoid the large spurious variations of the solution induced by the ill-conditioned inversion problem. Thus we search coefficients that represent the projection of the rotation rate on the chosen B-splines basis by minimizing:

The regularization parameters and and functions and are used to define the trade off between the resolution and the propagation of the input errors. We can choose any derivative of the rotation rate. This leads to different behavior of the rotation especially at locations where the observed modes are not enough to constrain the solution. For the results discussed here we choose , i=2 and with j=1,2.

The inversion depends on the number of piecewise polynomials, on the order of the spline functions and on the distribution of the break points of these polynomials. In what follows, the distribution of these points along the radius has been chosen according to the density of the turning points of the considered p-modes for a given dataset.

The inferred rotation at a point () is obtained as a linear combination of the data . Thus from Eq.(1) it follows that averaging kernels , which are a linear combination of rotational kernels , exist such that:

The radial and latitudinal full widths at mid height ( and ) of these kernels provide an estimation of the spatial resolution of the inversion at a point . It increases with lower regularizing term, at the expense of larger errors on the solution.

A guide for the choice of the trade-off parameters and could be a generalization of the so called L-curves currently used in one dimensional problems (Hansen, 1992). The aim is to find parameters that minimize both the value obtained for the fit of observed splittings (i.e. the first term in the sum Eq.(4)) and the two regularization terms and .

One can plot the value of each part of the regularization term against the value and this for different choices of and . An interesting result is that in a log-log plot, for large values of the trade off parameters the value gets large for little reduction of regularization terms and, in the opposite, for small values of lambdas the regularization terms get large for small reduction of the value. Reasonable choices of trade-off parameters are therefore those in the vicinity of the corner of the L-curve.

This figure shows an example of such L-curves obtained for dataset I and with second derivative (i=2) in the regularization term in radius () and first derivative (j=1) in .

The upper figure shows the logarithm of the value of the regularization term against where N is the total number of splittings, this for different values of and . One can see that all the points on this figure which are labeled by the same but different have the same location except for very large values of (i.e. ) . Thus it appears that the two terms ( and ) do not depend on the value of for a large domain of variation of this parameter ().

The full curves join points with the same ratio . According to the previous discussion, on the upper figure all these curves reduce to only one if we do not look at solutions obtained with that are too much regularized.

On the lower figure the value of the logarithm of the regularization term is plotted against for six values of the ratio . As decreases the position of the L-curve becomes lower showing that for a given value of , the value must be as large as possible (always with the limitation ) if one wants to reduce the value of the regularization term in latitude.

According to these two figures, a choice like point 1 on the figure (, , i.e. ) tends to minimize both the value and the two regularization terms.

When a regularization term in latitude is chosen with second derivative (j=2), the corresponding plots have similar behaviors but the domain of variation for , for which and depend only on , is smaller (.

L-curves reveal themself as being an useful tool to study the variation and mutual dependencies of each term in Eq.(4) for different choices of trade-off parameters and functions and . Nevertheless other methods for the optimal choice of trade-off parameters are possible. In particular, for a local estimation of the quality of the solution, we must have a look at the balance between the effect of propagating input errors and the resolution reached at a target location .

The inferred rotation rate is shown as a function of radius for ten latitudes from the equator up to the pole (). This solution is very smooth and has no latitudinal dependence below 0.6 solar radius.

Below the solution is not reliable because of the lack of modes able to describe these depths leading to averaging kernels that are large, not well localized and that consist in several peaks. The latitudinal independence found for these depths results essentially from the choice of j=1 in the regularization term . This choice could be judicious for an attempt to sound the very deep interior from such global inversion, without searching for a description of a latitudinal dependence in the core that should require very low errors in data. Nevertheless, our current investigations and the averaging kernels that we obtained do not allow us to draw an inference from the solution obtained below .

In the whole convection zone the rotation rate presents the same latitudinal variation as at the surface with a positive gradient just beneath the surface for latitude from the equator up to . The transition to a latitudinal independent rotation in the radiative zone occurs between 0.8 and 0.6 solar radius for latitude up to .

Dotted curves represent the errors on the solution which are the result of propagating input errors on the splittings through the inversion. In this inversion these errors are very low because the regularization is very strong. In fact these errors represent the difference between the inferred rotation and the true rotation rate convolved with the averaging kernels as in Eq.(5).(see e.g. Schou et al 1994). A look at these kernels is needed to qualify the resolution of the inversion.

These figures show contour plots of the averaging kernels at solar pole, mid latitude and equator for . is the geometrical distance between position of the maximum value of the averaging kernel and the point () shown as a star on figures. This quantity is very high for averaging kernels calculated near the pole (), showing that for these latitudes we are not able to obtained well peaked kernels and inferred the solution is not reliable.

At the estimated rotation rate is latitudinal independent. Nevertheless the latitudinal resolution at is very low ( at mid latitude) and could produce a smoothing effect on a latitudinal variation if any exists.

The radial resolution at the base of the convection zone () is around at mid latitude and is almost as large as the transition observed on the solution (). Thus if the transition is sharper this inversion doesn't enable to resolve it. Inversions with less regularization have been performed to study this transition zone.

This inferred rotation rate has a sharper transition zone than the previous one (about ) and the averaging kernels at this location () are better peaked with an increasing radial resolution ( at mid latitude). Inversion performed with lower regularization doesn't enable to produce better radial resolution and increase the effect of input errors, leading to an increasing product . Thus, this inversion is the one that gives the best radial resolution at this target location without increasing to much the effect of input errors. The fact that the width of the transition zone decreases when the resolution increases allows us to say that the transition occurs in a zone between 0.66 and 0.76 solar radius but it might be sharper than this.

At averaging kernels at and are, for this inversion, sufficiently well localized in latitude so that the latitudinal independence of the inferred rotation rate at this depth can be considered as real.

At 0.4 the large extent in latitude of the averaging kernels should produce a smoothing effect i.e. almost the same averaged rotation rate for all latitudes. Nevertheless, the solution shows again a latitudinal dependence between 0.55 and 0.35 . This might argues in favor of the existance of a variation with the latitude at these depths but we are still studying how to read our results there: some contributions in averaging kernels generated by a lack of cancelation of rotational kernels near the surface are suspected and may explain this behavior of the inferred rotation rate in this kind of inversion with low regularization. Moreover this latitudinal dependence below is of lower importance with dataset II.

The possibility of introducing a discontinuous basis of B-splines might give a test of the reliability of the inferred rotation rate at a target radius. The idea is to allow the rotation to be discontinuous at a given depth and to look, after inversion, if the inferred rotation is discontinuous at this location or not. Solutions that are physically realistic should be continuous at each point but in zones where the solution is not well constrained such inversion could produce different values for the rotation rate on the right and on the left of the break point. Our preliminary results show that the gap between the right and left values is low where the solution is thought to be well constrained by the data and larger elsewhere.

The value obtained for this solution and the resolutions are almost the same than with the standard basis indicating that data are compatible with a very sharp transition zone.

Averaging kernels obtained at the equator and are shown for the solutions obtained with and without setting a discontinuous basis at (shown on the two transparencies). Although, without discontinuous basis the averaging kernel was equally distributed around , with a discontinuous basis the major part of the kernel is below 0.7. By this way the value of the rotation rate obtained at is less influenced by the true rotation above 0.7 and this could explain its lower value. This may argue in favor of a solid rotation up to but this conclusion would have been more reliable if the averaging kernel had strictly no part above . Furthermore the new averaging kernel is more oscillating between and 0.9 so that its interpretation is harder. A symmetrical example is shown by the averaging kernels calculated at the equator and with and without allowing the rotation to be discontinuous at . This could explain by similar way the higher value obtained just above when we perform the inversion with a discontinuous basis.

This work is very preliminary and we think that numerical experiments on the reconstruction of some arbitrary given test rotation rates should be useful to calibrate the effect of introducing a discontinuous basis.