Regularisation parameter adaptive selection and its application in the prestack AVO inversion

The numerical solution of the inverse problem is usually obtained by solving a set of linear algebraic equations, while the system of equations may suffer from ill-posedness due to insufficient data. Regularisation is a technique for making the estimation problems well posed by adding indirect constraints on the estimated model, but the regularisation parameter selection is difficult. In geophysics, without explicit calculation methods and quantitative evaluation criteria, it is usually based on the experience of the inversion engineers to try to achieve the best inversion results by continuously modifying the regularisation parameter. For prestack amplitude variation with offset (AVO) inversion for real seismic data, fixed regularisation parameters cannot satisfy the optimisation conditions in the seismic data with different signal-to-noise (SNR) in one area. Besides, fixed regularisation parameter may cause that the model constraint misfit is too large or too small compared to data misfit, which may guide the inversion to generate undesirable results. Therefore, adaptive selection of regularisation parameter according to the seismic data can help guarantee a good inversion result. Based on the traditional L-curve criterion, we derive the theoretical formula of the adaptive computation of the regularisation parameter, which can be applied to any norm constraint. We proposed the application of this selection scheme in prestack AVO inversion. Model tests show that the improved L-curve method has better stability than its main competitor, the generalised cross-validation (GCV) method. Prestack AVO inversion on logging data and real seismic data demonstrate that the proposed method can improve the accuracy of the inversion, and it is more immune to strong noise.