CGG with iteratively reweighted gradient

Another way to modify the gradient direction is to modify the gradient vector after the gradient is computed from a given residual. Since the gradient vector is in the model space, any modification of the gradient vector imposes some constraint in the model space. If we know some characteristics of the solution which can be expressed in terms of weighting in the solution space, we can use that a priori knowledge to redirect the gradient vector by applying a weight to it. This algorithm can be implemented as follows:

		 		 $\bold r \quad\longleftarrow\quad\bold L \bold m - \bold d$ 
		 iterate { 
		 		 $\bold W \quad\longleftarrow\quad{\bf diag}[f(\bold m)]$ 
		 		  $\Delta\bold m \quad\longleftarrow\quad\bold W\bold L^T\ \bold r$ 
		 		  $\Delta\bold r\ \quad\longleftarrow\quad\bold L \ \Delta \bold m$ 
		 		  $(\bold m,\bold r) \quad\longleftarrow\quad{\rm cgstep}
 (\bold m,\bold r, \Delta\bold m,\Delta\bold r )$ 
		 		 } . 

Even though weighting the gradient has different meaning from weighting the residual, the analysis is similar in both cases. As we redefined the contribution of each residual element by weighting it with the absolute value of itself to some power: we can do the same with each model element in the solution,

$$\bold W= \vert\bold m\vert^{p} ,$$ (9)

where p is a real number that depends on the problem we wish to. When we have a finite model space we are applying a uniform weight to the finite model space and zero weight to the outlying space. If the operator used in the inversion is close to unitary, the solution obtained after the first iteration already closely approximates the real solution. Therefore, weighting the gradient with some power of the absolute value of the previous iteration means that we down-weight the importance of small model values and improve the fit to the data by emphasizing model components that already have large values.