3. OLA inversion of the transformed problem
The ultimate purpose of performing the discretization and SVD in Sect. 2 is to find out what suitable linear combinations of the original data will provide a smaller set of transformed data and yet retain the useful information in the original dataset. Guided by rotation inversions (Christensen-Dalsgaard & Thompson 1993), we take the elements from the matrix of left singular vectors as our coefficients in the transformation of Eq. (1) to Eq. (3). With the singular values ordered to be monotonic decreasing, we expect that we can just take the columns of U (i.e. , the index i) corresponding to the largest singular values. Applying this transformation yields transformed kernels
Corresponding to the singular values in Fig. 1, we show in Fig. 3 a few of the transformed kernels for squared sound speed and density. As in the case of rotation, the transformed kernels are increasingly oscillatory as i increases. Also, it is only for moderately large i that the transformed kernels are sensitive to structure in the deep interior, reflecting the fact that this region is relatively inaccessible to p modes. This is the reason why inversion at the core is often so difficult.
After deciding on the discretization used for the SVD, we still have considerable freedom in choosing the number of transformed kernels to retain for the subsequent inversion. To assess how large has to be to capture essentially all of the useful information in the full dataset, we compare inversions based upon the full and transformed sets of kernels, retaining different numbers of transformed kernels. We use a subtractive optimally localized averages (SOLA) inversion in all cases, since this is an example of an inversion method where it would be attractive to reduce the computational burden by means of the proposed transformation. As in the previous section, for now we do not consider the effect of a surface term.
The application of the SOLA method to structural inversions was described by Christensen-Dalsgaard & Thompson (1995). In brief, the aim is to choose coefficients such that (say) the first component of is localized about : then provided the other component is small, the corresponding linear combination of the data is a localized estimate of the first component of around . In SOLA, we choose the coefficients to minimize
where is a chosen target function (here a Gaussian), is a parameter which controls the influence of the second function on the solution obtained for the first function, µ is a trade-off parameter which controls the error magnification, and is the error covariance matrix for the data. (Although we are here assuming the errors to be uncorrelated and have uniform standard deviations, will usually have a more general form.) The minimization of Eq. (10) is constrained by the requirement that the averaging kernel
By way of illustration, we invert artificial data comprising frequency differences between our reference model and a test model. The physics of the test model is similar to that in the reference model, but it is younger, with an age of 4.52 Gyr. Fig. 4 shows the inferred difference in squared sound speed obtained with the original set of kernels (circles) and with various sets of transformed kernels, using 81 knots for the discretization. For , the resulting is somewhat different; for there are still noticeable differences in the core. With further increase in values of , the differences between the inversions rapidly become imperceptible on this scale. For , for example, the differences in the inferred values of sound speed are only a few parts in .
As a further diagnostic test, we may consider the averaging kernels. Panel (a) of Fig. 5 shows selected averaging kernels obtained in the complete inversion. If the corresponding averaging kernels obtained by inverting the first 120 transformed kernels were plotted on the same scale, the two sets of averaging kernels would be indistinguishable. Thus, for example, panel (b) shows the difference at a target radius of between the averaging kernel obtained by inverting the first 120 transformed kernels and the result of the complete inversion: the differences are at the level of . This provides further evidence that the set of 120 transformed kernels contains essentially the same information as the full set for this inversion.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998