Solution steering with space-variant filters

Preconditioning

Another important consideration is the speed of convergence of the problem. The size of most geophysical problems make direct matrix inversion methods impractical. An appealing alternative for linear problems is the family of conjugate gradient methods. Unfortunately, the operators used in seismic reflection problems are often computationally expensive. As a result it is important to minimize the number of steps it takes to get to a reasonable solution. One way that can reduce the number of iterations is by reformulating the problem in terms of some new variable ($\mathbf x$) with a preconditioning operator ($\mathbf B$). Changing a tradition inversion problem where the operator ($\mathbf C$) maps the model ($\mathbf m$) to the data ($\mathbf d$),

$$\mathbf d \approx \mathbf C \mathbf m$$ (2)

we can rewrite

$$\mathbf d \approx \mathbf C \mathbf B \mathbf x$$ (3)

where

$$\mathbf m = \mathbf B \mathbf x.$$ (4)

Solution steering with space-variant filters