3.5 Optimally localized averaging

In the Subtractive OLA (SOLA) approach (Backus and Gilbert, 19681970), the minimization is applied to the difference between the actual averaging kernels 𝒦 and a target kernel 𝒯, for example a 2-dimensional Gaussian or Lorentzian function. In this case (Pijpers and Thompson, 1992Jump To The Next Citation Point1994Jump To The Next Citation Point) the function minimized is
∫ R∫ π 2 0 0 [𝒯 (r0,𝜃0;r,𝜃) − 𝒦(r0,𝜃0;r,𝜃)] rdrd𝜃 (22 ) M + λ∑ [σ c(r ,𝜃 )]2. (23 ) i=1 i i 0 0
Both the tradeoff parameter λ and the radial and latitudinal resolution of the inversions must be chosen before running the inversion. If the choice of target kernel is poor – too narrow or too wide for the quantity and quality of the data – the reliability of the inversion will suffer. In OLA inversions, setting target locations outside the regions that can be resolved using the data will result in averaging kernels displaced from their targets, and this should be taken into account when interpreting the results. Figure 12View Image illustrates typical averaging kernels for a 2d SOLA inversion of an MDI data set.
View Image

Figure 12: As Figure 11View Image, for a SOLA inversion.

Another approach, older, and more computationally expensive, is the Multiplicative OLA (MOLA) described by Pijpers and Thompson (19921994). Here, no target form is imposed on the averaging kernel, but it is multiplied by a term which penalizes large values away from the target location.

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