# Prestack Seismic Inversion via Nonconvex L\({}_{1-2}\) Regularization
## 2 Method  
### The Forward Problem  
[PERSON]'s three-term formula, which approximates the relationship between the seismic reflectivity of the P/S impedance and the density for a given data set, is expressed as [2]:  
\[r(\theta)=A(\theta)\frac{\Delta I_{p}}{I_{p}}+B(\theta)\frac{\Delta I_{s}}{I_{s }}+C(\theta)\frac{\Delta\rho}{\rho} \tag{1}\]  
where, \(A(\theta)=\frac{1}{2}\big{(}1+\tan^{2}\theta\big{)}\), \(B(\theta)=-4\frac{V_{s}^{2}}{V_{p}^{2}}\sin^{2}\theta\), \(C(\theta)=-\frac{1}{2}\bigg{(}\tan^{2}\theta-4\frac{V_{s}^{2}}{V_{p}^{2}}\sin^ {2}\theta\bigg{)}\), \(\bar{I}_{p}=\frac{I_{p}^{t+1}+I_{p}^{t}}{2}\), \(\bar{I}_{s}=\frac{I_{p}^{t+1}+I_{p}^{t}}{2}\), \(\bar{\rho}=\frac{\rho^{t+1}+\rho^{t}}{2}\), \(\Delta I_{p}=I_{p}^{t+1}-I_{p}^{t}\), \(\Delta I_{s}=I_{s}^{t+1}-I_{s}^{t}\), \(\Delta\rho=\rho^{t+1}-\rho^{t}\), where \(t\) is sampling time point, \(\theta\) is the angle of incidence, \(I_{p}^{t}\), \(I_{s}^{t}\), \(\rho^{t}\) are the P impedance, \(S\) impedance, and density at \(t\), respectively. The matrix form of Equation (1) is  
\[r(\theta)=\big{[} \begin{array}{cc}A(\theta)&B(\theta)&C(\theta)\end{array}  \big{]}\left[ \begin{array}{c}\frac{\Delta I_{p}}{I_{p}}\\ \frac{\Delta I_{s}}{I_{s}}\\ \frac{\Delta\rho}{\rho}\end{array} \right] \tag{2}\]  
In the case of multiple sampling points, Equation (2) is extended as follows:  
\[\left[ \begin{array}{c}r_{1}(\theta)\\ r_{2}(\theta)\\ \vdots\\ r_{t}(\theta)\end{array} \right]=\left[ \begin{array}{ccc}A(\theta)&B(\theta)& C(\theta)\\ A(\theta)&B(\theta)&C(\theta)\\ \vdots&\vdots&\vdots\\ A(\theta)&B(\theta)&C(\theta)\end{array} \right]\times\left[ \begin{array}{c}r_ {p}\\ r_{s}\\ r_{p}\end{array} \right] \tag{3}\]where, \(r_{p}=\frac{\Delta I_{I}}{T_{p}},r_{s}=\frac{\Delta I_{s}}{I_{s}},r_{\rho}=\frac{ \Delta\rho}{\beta}\). With multiple sampling points and \(N\) incident angles, the matrix form of the approximate reflectivity equation is defined as:  
\[\underbrace{\left[ \begin{array}{c}\mathbf{R}(\theta_{1})\\ \mathbf{R}(\theta_{2})\\ \vdots\\ \mathbf{R}(\theta_{N})\end{array} \right]}_{\mathbf{R}}=\underbrace{\left[ \begin{array}{ccc}\mathbf{C}_{p}(\theta_{1})&\mathbf{C}_{s}(\theta_{1})& \mathbf{C}_{p}(\theta_{1})\\ \mathbf{C}_{p}(\theta_{2})&\mathbf{C}_{s}(\theta_{2})&\mathbf{C}_{p}(\theta_{2 })\\ \vdots&\vdots&\vdots\\ \mathbf{C}_{p}(\theta_{N})&\mathbf{C}_{s}(\theta_{N})&\mathbf{C}_{p}(\theta_ {N})\end{array} \right]}_{\mathbf{C}}\times\underbrace{\left[ \begin{array}{c} \mathbf{r}_{p}\\ \mathbf{r}_{s}\\ \mathbf{r}_{\rho}\end{array} \right]}_{\mathbf{r}} \tag{4}\]  
where \(\mathbf{R}(\theta_{N})=\left[ \begin{array}{cccc}r_{1}(\theta_{N})&r_{2}( \theta_{N})&\cdots&r_{t}(\theta_{N})\end{array} \right]^{T}\), \(\mathbf{r}_{p}\), \(\mathbf{r}_{s}\), and \(\mathbf{r}_{\rho}\) are the reflectivity series of the P impedance, S impedance, and density, respectively. \(\mathbf{C}_{p}(\theta_{N})\), \(\mathbf{C}_{s}(\theta_{N})\), and \(\mathbf{C}_{p}(\theta_{N})\) are the diagonal coefficient matrices composed of\(A(\theta)\), \(B(\theta)\), and \(C(\theta)\), respectively; for brevity, we have not included the forms of these matrices. According to the seismic convolution model, the relationship between the prestack angle gather seismic data and the reflectivity is expressed as  
\[\underbrace{\left[ \begin{array}{c}\mathbf{s}(\theta_{1})\\ \mathbf{s}(\theta_{2})\\ \vdots\\ \mathbf{s}(\theta_{N})\end{array} \right]}_{\mathbf{s}}\underbrace{\left[ \begin{array}{cccc}\mathbf{w}(\theta_{1})\\ \mathbf{w}(\theta_{2})\\ \ddots\\ \mathbf{w}(\theta_{N})\end{array} \right]}_{\mathbf{W}}\times\underbrace{ \left[ \begin{array}{c}\mathbf{R}(\theta_{1})\\ \mathbf{R}(\theta_{2})\\ \vdots\\ \mathbf{R}(\theta_{N})\end{array} \right]}_{\mathbf{R}} \tag{5}\]  
where \(\mathbf{w}(\theta_{N})\)is the wavelet matrix corresponding to the \(N\) th angle seismic data. Equations (4) and (5) can be written concisely in matrix form as  
\[\mathbf{s}=\mathbf{W}\mathbf{C}\mathbf{r} \tag{6}\]  
where \(\mathbf{s}=\left[