The vast size of the seismic imaging problems makes performing a direct inversion impossible with today's computer power, even if we are only dealing with a 2-D seismic line. Fortunately, we can closely approximate a direct inverse with iterative techniques. In particular, we can approximate a least-squares inversion with the conjugate-gradient minimization of this objective function:
where boldL is a linear modeling operator, boldd is the data, and boldm is the model. This minimization can be expressed more concisely as a fitting goal:
The first expression is the ``data fitting goal,'' meaning that it is responsible for making a model that is consistent with the data. The second expression is the ``model styling goal,'' meaning that it allows us to impose some idea of what the model should look like using the regularization operator . The strength of the regularization is controlled by the regularization parameter .
Unfortunately, the inversion process described by fitting goals (3) can take many iterations to produce a satisfactory result. We can reduce the necessary number of iterations by making the problem a preconditioned one. We use the preconditioning transformation Fomel et al. (1997); Fomel and Claerbout (2003) to give us these fitting goals:
is obtained by mapping the multi-dimensional regularization operator to helical space and applying polynomial division Claerbout (1998). This process is called Regularized Inversion with model Preconditioning (RIP).