Time-frequency analysis of seismic data using local attributes |

In this appendix, we review the theory of shaping regularization in inversion problems. Fomel (2007b) introduces shaping regularization, a general method of imposing regularization constraints. A shaping operator provides an explicit mapping of the model to the space of acceptable models.

Consider a system of linear equation given as , where is a forward-modeling operator, is the model, and is the data. By equation 8, we can find that the proposed time-frequency decomposition can be written as the form . The standard regularized least-squares approach to solving this equation seeks to minimize , where is the Tikhonov regularization operator (Tikhonov, 1963) and is scaling.

The formal solution, denoted by
, is given by

Substituting equation A-2 into equation A-1 yields a formal solution of the estimation problem regularization by shaping

Introducing scaling of by in equation A-3 , we obtain

The main advantage of shaping regularization is the relative ease
of controlling the selection of and in comparison with and
cite[]Fomel2009. In this paper, we choose to be the median value of
and the shaping operator to be a Gaussian smoothing operator.
The Gaussian smoothing operator is a convolution operator with
the Gaussian function that is used to smooth images and remove
detail and noise. In this sense it is similar to the mean filter, but
it uses a different kernel to represent the shape of a Gaussian
(bell-shaped) hump. The Gaussian function in 1D has the form,

The computational cost of generating a time-frequency representation with our method is , where is the number of time samples, is the number of frequencies, is the number of conjugate-gradient iterations, is the number of processors used to process different frequencies in parallel, and decreases with the increase of smoothing and is typically around 10.

Time-frequency analysis of seismic data using local attributes |

2013-07-26