    Shaping regularization in geophysical estimation problems  Next: Smoothing by regularization Up: Fomel: Shaping regularization Previous: Introduction

# Review of Tikhonov's regularization

If the data are represented by vector , model parameters by vector , and their functional relationship is defined by the forward modeling operator , the least-squares optimization approach amounts to minimizing the least-squares norm of the residual difference . In Tikhonov's regularization approach, one additionally attempts to minimize the norm of , where is the regularization operator. Thus, we are looking for the model that minimizes the least-squares norm of the compound vector , where is a scalar scaling parameter. The formal solution has the well-known form (1)

where denotes the least-squares estimate of , and denotes the adjoint operator. One can carry out the optimization iteratively with the help of the conjugate-gradient method (Hestenes and Steifel, 1952) or its analogs. Iterative methods have computational advantages in large-scale problems when forward and adjoint operators are represented by sparse matrices and can be computed efficiently (Saad, 2003; van der Vorst, 2003).

In an alternative approach, one obtains the regularized estimate by minimizing the least-squares norm of the compound vector under the constraint (2)

Here is the model reparameterization operator that translates vector into the model vector , is the scaled residual vector, and has the same meaning as before. The formal solution of the preconditioned problem is given by (3)

where is the identity operator in the data space. Estimate 3 is mathematically equivalent to estimate 1 if is invertible and (4)

Statistical theory of least-squares estimation connects with the model covariance operator (Tarantola, 2004). In a more general case of reparameterization, the size of may be different from the size of , and may not have the full rank. In iterative methods, the preconditioned formulation often leads to faster convergence. Fomel and Claerbout (2003) suggest constructing preconditioning operators in multi-dimensional problems by recursive helical filtering.    Shaping regularization in geophysical estimation problems  Next: Smoothing by regularization Up: Fomel: Shaping regularization Previous: Introduction

2013-07-26