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

Non-linear optimization problems suffer from local minima. When we use gradient based iterative solvers on these problems, we often find the final solution highly dependent on the initial guess. Here we introduce preconditioning and show how it helps resolve in our current problem—bidirectional deconvolution. The results in three data examples show that preconditioning helps us get a more spiky result when compared with the results without preconditioning. Additionally field data results with preconditioning have fewer precursors, cleaner salt body, more symmetric wavelet, and faster convergence rate than those without preconditioning. In addition to the field data, we theoretically illustrate and practically apply two ways of preconditioning: Prediction-error filter (PEF) preconditioning and Gapped anti-causal leaky integration followed by PEF (GALIP) preconditioning. Unlike PEF preconditioning, GALIP helps produce the result in the central wavelet or other position of the wavelet if we change the length of gap.