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Astron. Astrophys. 321, 634-642 (1997)

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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 [FORMULA] from the matrix of left singular vectors as our coefficients [FORMULA] 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

[EQUATION]

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.

[FIGURE] Fig. 3. A sample of the transformed kernels, obtained using 81 knots in r. The panel on the left are the transformed sound-speed kernels, while those on the right are the transformed kernels for density

After deciding on the discretization used for the SVD, we still have considerable freedom in choosing the number [FORMULA] of transformed kernels to retain for the subsequent inversion. To assess how large [FORMULA] 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 [FORMULA] such that (say) the first component of [FORMULA] is localized about [FORMULA]: then provided the other component is small, the corresponding linear combination of the data is a localized estimate of the first component of [FORMULA] around [FORMULA]. In SOLA, we choose the coefficients to minimize

[EQUATION]

where [FORMULA] is a chosen target function (here a Gaussian), [FORMULA] 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 [FORMULA] is the error covariance matrix for the data. (Although we are here assuming the errors to be uncorrelated and have uniform standard deviations, [FORMULA] will usually have a more general form.) The minimization of Eq. (10) is constrained by the requirement that the averaging kernel

[EQUATION]

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 [FORMULA], the resulting [FORMULA] is somewhat different; for [FORMULA] there are still noticeable differences in the core. With further increase in values of [FORMULA], the differences between the inversions rapidly become imperceptible on this scale. For [FORMULA], for example, the differences in the inferred values of sound speed are only a few parts in [FORMULA].

[FIGURE] Fig. 4. a The inversion results obtained from the equations transformed with 81 knots in r compared with those obtained from the full set of kernels; results for the latter case are shown with circles. The crosses are for a transformation including 60 singular values, the asterisks for are 80 singular values, the triangles are for 100 singular values and the squares are for 120 singular values. b The difference in the inversion results between the solution with the full modeset and those obtained using 120 singular values

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 [FORMULA] 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 [FORMULA]. This provides further evidence that the set of 120 transformed kernels contains essentially the same information as the full set for this inversion.

[FIGURE] Fig. 5. a Averaging kernels produced by inversion of the full modeset. b The difference, for a target radius of [FORMULA], between the averaging kernel shown in panel a and the corresponding kernel for the transformed set with 81 knots and [FORMULA]
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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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