Seismic reflectivity inversion with mixed L1-L2 norm regularization

Seismic reflectivity and the resulting elastic parameters can be used for mining engineering, mineral reservoir prediction, and other fields related to underground mineral resources. Seismic reflectivity is common obtained from post-stack seismic data. It is the so called seismic reflectivity inversion (SRI). Sparse regularization-based SRI (sparse-SRI) is a method capable of estimating reflectivity from seismic data, thereby enhancing the resolution of the seismic data. However, the current sparse-SRI method encounters two main challenges: (1) sparse regularization tends to cause a significant increase or numerical oscillation in the residual, which can negatively impact the inversion outcomes; (2) it provides inadequate preservation of small reflectivity components present in real seismic data. To address these limitations, this paper introduces a new mixed-norm regularization approach for SRI. The proposed method modifies the objective function by introducing the mixed-norm term that encompasses both the L1-norm and L2-norm of the model. Mixed-norm regularization is a well-established concept in statistics and machine learning for over a decade. Considered the gradual variation of underground geological features, we also add the L2-norm of the model gradients in the mixed-norm for further vertical smoothness. By tuning the weighting parameter that balances L1-norm and L2-norm regularization, one can control the sparsity of the solutions. Increasing the L1-norm's weight favors the creation of more sparse and concentrated models. Hence, this framework efficiently reconciles the sparsity of large reflectivity features with the smoothness of small reflectivity components. Synthetic tests show that the mixed L1-L2 norm inversion produces more compact and accurate reflectivity distributions compared to conventional single norm inversion, achieving better vertical resolution and minimizing the underestimation of model values. Field testing using actual seismic data from West China provides further validation of the algorithm's effectiveness. The application of mixed L1-L2 norm inversion results in reflectivity series distributions that are consistent with established geological structures, evidencing enhanced vertical confinement relative to conventional L1-norm or L2-norm inversion techniques.