# Prestack Seismic Inversion via Nonconvex L\({}_{1-2}\) Regularization
## 1 Introduction  
Prestack amplitude variation angle (AVA) inversion methods can be used to estimate multiple elastic parameters in a given study area by including observational logging and petrophysical data [1]. Those methods are of great interest to those in the field of reservoir fluid identification and evaluation. There are two main types of prestack seismic inversions: prestack direct inversion and elastic impedance inversion. Prestack direct inversion, also known as AVA/AVO inversion, is based on the P-wave approximate reflectivity formulas in the Zoeprpritz equation and uses angle gather data as constraints in the prestack elastic parameter inversion [2, 3]. For example, [PERSON] improved the accuracy of the AVO inversion results by using the covariance matrix to remove the correlation between the initial elastic parameter constraints [4]. To improve the identification of thin interpbeds, [PERSON] et al. derived a detailed multi-angle prestack forward model by extending the theory of poststack reflectivity decomposition(the dipole decomposition principle)to the inversion of prestack P/S velocities and densities [5, 6]. However, due to the lack of far-angle gather seismic data, the density parameters were used as the initial inversion constraints in those specific application examples. To highlight the \"_blocky_\" inversion effect, [PERSON] et al. performed a three-parameter prestack inversion using a weighted L\({}_{1,2}\) mixed norm [7]. In this inversion method, both the correlation of theinversion parameters and the correlation of the model parameters are considered; using the fast iterative shrinkage-thresholding algorithm (FISTA), they were able to solve the inversion objective function and achieve high-resolution inversion results [8]. In a later study, [PERSON] et al. quantified the [PERSON]'s modulus, [PERSON]'s ratio, and density values using a combination of L\({}_{1}\), L\({}_{2}\), and total variation (TV) norm regularization methods [9]. Of these three types of methods, L\({}_{1}\) norm regularization reflects the sparsity of the inversion results, L\({}_{2}\) norm regularization forces the inversion results to conform to the initial constraints, and TV norm regularization enhances the lateral continuity of the inversion results. To eliminate the P/S velocity ratio as a parameter in the traditional approximate reflectivity equation, [PERSON] et al. first re-derived the reflectivity equation to define the P/S impedance and then developed the P/S impedance inversion method [10]. Using high-order TV regularization, [PERSON] et al. ran multiple three-parameter prestack inversions that accounted for various topographical characteristics in multiple study areas [11]. Those methods are sensitive to noise and the amount and quality of the observational data used in the inversion. To improve the stability of the inversion method, [PERSON] et al. proposed a novel P/S impedance inversion used in conjunction with L\({}_{1-2}\) regularization, and they demonstrated the applicability and feasibility of their method, however, which method also faces difficulties in producing accurate density values [12]. The other types of prestack seismic inversion methods are elastic impedance inversion. Those methods are undertaken in two parts. In the first part, the elastic impedance is inverted from partially angularly stacked seismic data. In the second part, the to-be-inverted prestack elastic parameters are extracted from the elastic impedance. The concept of elastic impedance was introduced by [PERSON] and later extended by [PERSON] to provide a theoretical basis for prestack inversion [13; 14]. [PERSON] et al. derived the Zoeppritz approximate equation for P/S-wave moduli and performed an inversion to determine the prestack P/S-wave moduli [15]. In a later study, [PERSON] et al.