Preconditioning a non-linear problem and its application to bidirectional deconvolution

Preconditioning a non-linear problem and its application to bidirectional deconvolution

Yi Shen, Qiang Fu and Jon Claerbout

Abstract:

Non-linear optimization problems suffer from local minima. When we use gradient-based iterative solvers on these problems, we often find the final solution to be highly dependent on the initial guess. Here we introduce preconditioning and show how it helps resolve these issues in our current problem--bidirectional deconvolution. Using three data examples, we show that results with preconditioning are more spiky than results without preconditioning. Additionally, field data results with preconditioning have fewer precursors, cleaner salt bodies, more symmetric wavelets, and faster convergence than those without preconditioning. In addition to the field data, we illustrate the theory and application of two methods of preconditioning: prediction-error filter (PEF) preconditioning and gapped anti-causal leaky integration followed by PEF (GALI-PEF) preconditioning. Unlike PEF preconditioning, GALI-PEF preconditioning helps constrain the spike to the central wavelet, or allows us to shift it to another position in the wavelet by manipulating the length of the gap.

Preconditioning a non-linear problem and its application to bidirectional deconvolution