Shaping regularization in geophysical estimation problems |

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

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

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 |

2013-07-26