## 3. OLA inversion of the transformed problemThe 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 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
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, be unimodular. 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 |