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Appendix A: Demonstration of the error propagation formula for the non-linear extension of the acoustic radius inversion

Equation (26) is obtained with a little algebra. First, we treat the observed frequencies, ν_obs,i, and the inverted acoustic radius, τ_inv, as independent stochastic variables with ϵ_i being the individual noise realisations for each frequency, ϵ_τ the resultant deviation on τ_inv, and and the average of the stochastic variables ν_obs,i and τ_inv, respectively. Furthermore, we assume that: (A.3)Using the fact that τ_inv = q_optτ_ref, with the definition of q_opt given in Eq. (25), and the separation into stochastic and average contributions defined previously, we obtain: (A.4)where we assumed that ϵ_i is much smaller than 1, thereby allowing us to linearise the above equation. We now apply the formula for the variance of a linear combination of independent stochastic variables and obtain (A.5)where we used the following equivalences: (A.6)Equation (A.5)then leads directly to Eq. (26) when using the definition of q_opt given in Eq. (25).

Appendix B: Supplementary figures

The following figures illustrate the quality of the kernel fits for some of the test cases we presented in the article. Although these plots are redundant from the visual point of view, we wish here again to stress that they are crucial to the understanding of the quality of a SOLA inversion and justify the accuracy of the results presented in the previous sections.

	Fig. B.1 Least square fits of the kernels for model A′.
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	Fig. B.2 Same as Fig. 6 for the first test case with non-adiabatic frequencies.
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	Fig. B.3 Same as Fig. 6 for the test case with turbulent pressure.
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© ESO, 2015