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

Conclusion

In this paper, we illustrate the importance of preconditioning in non-linear problems, and we apply preconditioning to bidirectional deconvolution. The results of three data examples show that wavelets are more spiky in the results with preconditioning than in those without preconditioning. However, the results with preconditioning in the 1D and 2D synthetic sections show slight ringing around the spike, which may encourage us to use more prior knowledge in the preconditioning. For field data, the results with preconditioning have fewer precursors, a cleaner salt body, and a more symmetric wavelet than those without preconditioning. This proves that preconditioning can guide the gradient along sensible pathways, thus avoiding potential local minima, making the results more reliable, and speeding convergence.

In addition, we introduce two methods of preconditioning --PEF and GALI-PEF--and apply them to the field data. Both approaches improve the bidirectional deconvolution result and improve the convergence speed. But unlike PEF preconditioning, GALI-PEF preconditioning helps constrain the spike to the center of the wavelet (or other positions in the wavelet if we change the length of gap). However, we have tested these two methods on only one set of field data. More experiments on other datasets are needed to illustrate the effectiveness and limitations of these two methods of preconditioning in our future work.

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