# Prestack Seismic Inversion via Nonconvex L\({}_{1-2}\) Regularization
## 1 Introduction
derived the approximate reflectivity equations for Young's modulus, Poisson's ratio, and density based on the assumption of plane wave incidence, and obtained Young's modulus and Poisson's ratio, further enriching the prestack inversion method [16]. [PERSON] et al. performed an inversion to quantify the Gassmann fluid term and the shear modulus by applying the basis pursuit decomposition technique and the theory of dipole reflectivity decomposition to a prestack inversion. Using this inversion method, they also conducted a sensitivity analysis to determine how the elastic parameters changed in the presence of various reservoir fluids [17]. Furthermore, they made significant improvements to the prestack AVO inversion method; by refining the Zoeppritz approximate equation, they located and characterized hydrocarbon-bearing subsurface reservoirs using an inversion that was constrained by various elastic parameters such as the Young's modulus, Poisson's ratio, Gassmann fluid term, and brittleness index values [18; 19; 20]. [PERSON] et al. first inverted the reflectivity of seismic data with different angles and then extracted the brittleness index from those reflectivity values [21]. [PERSON] et al. used nonconvex L\({}_{1-2}\) regularization to extract stable elastic impedance values from seismic data and then employed a Bayesian inversion framework to back out the P/S velocities and the densities from the elastic impedance values [22]. However, this method requires the use of logging data to establish the relationship between the elastic impedance and the inverted parameters. Therefore, the reliability, accuracy, and precision of this inversion method must be determined in future studies. Meanwhile, the high dimensionality of multi-angle seismic data increases the ill-posedness and decreases the stability of a prestack AVA inversion. Typically, this issue is resolved using regularization methods such as the Tikhonov and many variations of the L\({}_{1}\) regularization methods. However the Tikhonov regularization method often results in over-smoothed solutions, it is not an ideal technique to use with applications related to identifying and characterizing formation boundaries. More importantly, some existing AVO inversion methods that use L\({}_{1}\) regularization techniques fail to generate the optimal sparse solutions because they exacerbate the existing prestack AVA high dimensionality issue by requiring additional model constraints. With the potential to generate inversion results that are both optimally sparse and highly accurate, nonconvex optimization algorithms have become quite popular in recent years. [PERSON] et al. proposed an L\({}_{p}\)-norm minimization algorithm that is solved by iteratively reweighting the least-squares solution [23]. [PERSON] et al. proposed L\({}_{1}\)/L\({}_{2}\) and L\({}_{1-2}\) minimization algorithms and confirmed the superiority of nonconvex optimization algorithms in terms of solution accuracy and efficiency using both theoretical examples and actual case studies [24]. Because of its good performance in promoting sparsity, nonconvex optimization algorithms have been widely applied to geophysical problems. [PERSON] et al. proposed a L\({}_{1/2}\) norm regularization to reconstruct highly incomplete seismic data and obtained obviously better results [25]. Recently, [PERSON] et al. used L\({}_{1-2}\) norm regularized logarithmic absolute misfit function to improve the stability and fidelity of the prestack seismic inversion [26]. In the same year, they combined with the logarithmic absolute-criterion-based misfit function with L\({}_{1-2}\) norm-based penalty for P-P and P-SV waves joint inversion and achieved good performance in both resolution and accuracy [27]. With this motivation, we employ nonconvex L\({}_{1-2}\) regularization to simultaneous P/S-impedance and density inversion from pre-stack seismic data. In this study, we derive the multi-angle, multi-reflection interface prestack AVA forward model using the approximate reflectivity equation given by [PERSON] [2].