3.2 Averaging kernels

By substituting Equation (11View Equation) into the RHS of Equation (14View Equation) we obtain
∫ R⊙ ∫ π ¯Ω (r0,𝜃0) = 𝒦 (r0,𝜃0;r,𝜃)Ω(r,𝜃)drd 𝜃 + 𝜖i, (15 ) 0 0
∑M 𝒦 (r0,𝜃0;r,𝜃) ≡ ci(r0,𝜃0)Ki(r,𝜃) (16 ) i=1
is the averaging kernel for the location (r0,𝜃0). The averaging kernels are independent of the values of the data. However, because the uncertainties in the data are used to weight the inversion calculation that generates the coefficients c i, as described below in Sections 3.4 and 3.5, these do enter indirectly into the averaging kernels. The averaging kernels also depend on which modes are present in the input data set. They provide a useful tool for assessing the reliability of an inversion inference from a particular mode set (see, for example,  Schou et al., 1992Jump To The Next Citation Point1994Jump To The Next Citation Point).
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