Seismic data interpolation using nonlinear shaping regularization |

Linear shaping regularization was proposed by Fomel (2007b) for regularizing under-determined geophysical inverse problems. Compared with the commonly used Tikhonov regularization (Tikhonov, 1963), shaping regularization enjoys a number of advantages, including easier control on properties of the estimated model or in some cases significantly faster convergence. In the past five years, the linear form of shaping regularization has been utilized broadly. Fomel (2007a) used it to define the local correlation, which was then utilized by Liu et al. (2011a,2009) for optimal stacking and by Liu and Fomel (2012a); Liu et al. (2011b) for local time-frequency analysis. Du et al. (2010) proposed to use shaping regularized inversion to estimate the Q factor of seismic attenuation by treating a spectral division as a non-stationary inversion problem. Liu et al. (2012) applied shaping regularization to random noise attenuation, and achieved a better result than predictive filter in the case of non-stationary seismic signal. Chen et al. (2012) found an application of smooth shaping in Gabor deconvolution. All these methods use a linear operator to constrain the model when iteratively solving the inverse problem.

Fomel (2007b) extended linear shaping regularization to its nonlinear form. In the nonlinear form, the shaping operator is not limited to be linear, and thus produce more convenience in implementing the iterative framework. However, the properties and applications of nonlinear form of shaping regularization have been barely mentioned since that. In this abstract, we build a bridge among the well known iterative shrinkage thresholding (IST), projection onto convex sets (POCS) algorithms and shaping regularization. We derive the two well-known inversion formulations (IST and POCS) in the basis of shaping regularization. We also propose a faster version of shaping regularization, where a linear combination operator is used. The faster version utilizes the information of both the current and previous shaping regularized model to form a new model, which demonstrates an obviously faster convergence.

Seismic data interpolation using nonlinear shaping regularization |

2015-11-24