# Prestack Seismic Inversion via Nonconvex L\({}_{1-2}\) Regularization
## Appendix A Bayesian Derivation of the Objective Function  
According to [PERSON] theorem, the posterior probability distribution (PDF) of the estimated parameter \(\mathbf{r}\) may be expressed as  
\[p(\mathbf{r}|\mathbf{s})=\frac{\mathbf{p}(\mathbf{s}|\mathbf{r})\mathbf{p}( \mathbf{r})}{\mathbf{p}(\mathbf{s})} \tag{1}\]  
where \(p(\mathbf{r}|\mathbf{s})\) is the PDF of \(\mathbf{r}\). The likelihood function, \(p(\mathbf{r}|\mathbf{s})\), summarizes the strength of the fit between the observed data and \(\mathbf{r}\). \(p(\mathbf{r})\) represents the prior probability distribution of \(\mathbf{r}\) in the absence of observed data \(\mathbf{s}\). \(p(\mathbf{s})\) is the probability of the entire sample space, which is typically a constant. Therefore, Equation (1) is equivalent to  
\[p(\mathbf{r}|\mathbf{s})\propto\mathbf{p}(\mathbf{s}|\mathbf{r})\mathbf{p}( \mathbf{r}) \tag{2}\]  
In seismic exploration, the observed data are seismic data. Prior to performing the inversion, the seismic data must undergo preprocessing procedures such as dynamic correction, multiple wave suppression, and attenuation compensation. It is typically assumed that the likelihood function has a Gaussian distribution with a mean of zero and a variance of \(c_{d}^{2}\). Therefore,\(p(\mathbf{s}|\mathbf{r})\) may be expressed as  
\[p(\mathbf{s}|\mathbf{r})\propto\exp\biggl{[}-\frac{\mathbf{1}}{\mathbf{2}}( \mathbf{WCr}-\mathbf{s})^{\mathbf{T}}\mathbf{\Sigma}^{-1}(\mathbf{WCr}- \mathbf{s})\biggr{]} \tag{3}\]  
where \(\mathbf{\Sigma}\) is the covariance matrix that represents noise in the seismic data (\(\mathbf{\Sigma}=c_{d}^{2}\cdot\mathbf{I}\)) and \(\mathbf{I}\) is the identity matrix. Certain types of a priori information, such as structural and well-log data, significantly improve the stability of the inversion by providing useful constraints for the model results. According to [12; 30], there are two types of useful a priori information: (1) low-frequency data that adhere to a (0, \(c_{low}^{2}\)) Gaussian distribution and (2) constraints on the sparsity of the estimated parameter. The low-frequency information, which is independent of the seismic data and is typically obtained by interpolating and extrapolating the horizon and well-log data, is used to enhance the lateral continuity of the inversion. The sparsity information is used to characterize the strata boundaries. As such, \(p(\mathbf{r})\) may be defined as  
\[p(\mathbf{r})=p_{L}(\mathbf{r})p_{V}(\mathbf{r}) \tag{4}\]  
where \(p_{L}(\mathbf{r})\) is the lateral constraint for the low-frequency information and \(p_{V}(\mathbf{r})\) is the vertical sparsity constraint for the estimated parameter. The definition of \(p_{L}(\mathbf{r})\) is  
\[p_{L}(\mathbf{r})\propto\exp\biggl{[}-\frac{1}{2}(\mathbf{LBr}-\xi_{low})^{T} \Omega^{-1}(\mathbf{LBr}-\xi_{low})\biggr{]} \tag{5}\]  
where \(\mathbf{L}\) is the low-pass matrix, \(\xi_{low}\) is the low-frequency prior information (\(\xi_{low}(t)=\frac{1}{2}\ln\frac{I_{low}(t)}{I_{low}(t_{0})}\)), \(I_{low}\) is the low-frequency P impedance trend that was built from the well-log data, \(\mathbf{\Omega}\) is the covariance matrix of the model parameters (\(\mathbf{\Omega}=c_{low}^{2}\cdot\mathbf{I}\)). The L\({}_{1-2}\) regularized prior distribution function is defined as  
\[p(\mathbf{r})=\frac{1}{2\pi^{1-2}}\exp\biggl{[}-\frac{1}{2}((\|\mathbf{r}\|_{1 }-\|\mathbf{r}\|_{2})\biggr{]} \tag{6}\]which is equivalent to  
\[p(\mathbf{r})\propto\exp\Bigl{[}-\frac{1}{2}(\|\mathbf{r}\|_{1}-\|\mathbf{r}\|_{2} )\Bigr{]} \tag{10}\]  
By substituting the likelihood function (Equation (11)) and prior information equations (Equations (12) and (10)) into the Bayes theorem equation (Equation (10)), we can obtain the final expression for \(p(\mathbf{r}|\mathbf{s})\):  
\[