### 3.5 Optimally localized averaging

In the Subtractive OLA (SOLA) approach (Backus and Gilbert, 1968, 1970), the minimization is
applied to the difference between the actual averaging kernels and a target kernel , for example a
2-dimensional Gaussian or Lorentzian function. In this case (Pijpers and Thompson, 1992, 1994) the
function minimized is
Both the tradeoff parameter 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 12 illustrates typical averaging kernels for a 2d SOLA inversion of an MDI data
set.
Another approach, older, and more computationally expensive, is the Multiplicative OLA (MOLA)
described by Pijpers and Thompson (1992, 1994). 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.