Preconditioning, IRLS, and Ji's interpolation

The preconditioning operator in this example is a diagonal weighting matrix, but any operator may be used as a preconditioner, the choice depends on the application. Often a preconditioner is chosen to speed up an inversion Karrenbach (1994), though here the weighting matrix is chosen to impart a desired property to the model.

In a manner analogous to weighting the model, the residual can be weighted. This results in a problem formulated as Nichols (1994)

Wd = WHm.

Both sides of the equation are weighted in data space. This is the approach known as iteratively reweighted least squares.

Ji's 1994 approach is similar to preconditioning, in that he applies an iterative reweighting to the model space. However, it is not applied as a change of variables. That is, the weighting is not reapplied to the final solution. This approach is like defining a new operator

$$\tilde{H} = WH$$ (6)