# An Application of Elastic-Net Regularized Linear Inverse Problem in Seismic Data Inversion
## 2 Methods
### Sesimic Inversion with Elastic-Net Regularization  
The basic assumption behind sparse regularization in seismic inversion is that the reflectivity series of underground formation is sparsely distributed. Hence, the final impedance from sparse reflectivity series is blocky through the exponential integration transformation [2]. In fact, the reflectivity series of underground formation is not strictly sparse from the observation of actual well log. For example, Figure 1a shows the reflectivity series calculated from an actual impedance well log shown in Figure 1b. The well log has been transformed into time domain and scaled into the frequency band of seismic data. We can see that the major large reflectivity is sparse, which corresponds to main stratigraphic sequence interface of underground formation, such as stratigraphic uncontomority, lithology boundary, and so forth. However, there are also many small reflectivity series, which represent weak reflectors of thin interbed between major reflectivity. The large reflectivity is sparse, but small reflectivity is dense. In addition, small reflectivity is highly correlated to the major reflectivity right above it. Figure 1: The reflectivity series and the corresponding impedance from an actual well log. (**a**) Reflectivity series; (**b**) impedance. To combat this issue, we adopt elastic-net regularization as an alternative to sparse regularization. The objective function with elastic-net regularization is  
\[\min f(\mathbf{r})=||\mathbf{Wr}-\mathbf{d}||_{2}^{2}+\lambda_{1}||\mathbf{r}||_ {1}+\lambda_{2}||\mathbf{r}||_{2}^{2} \tag{6}\]  
The elastic net was first proposed by [PERSON] and [PERSON] to solve LASSO (Least Angle Selection and Shrinkage Operator) problem in statistics [36]. In elastic-net regularization, the L1-norm and L2-norm have opposing effects. L1-norm imposes a sparse solution, whereas L2-norm promotes density of solution. Hence, it is called elastic net. The combined effect of elastic-net regularization in seismic inversion is that it yields sparse large reflectivity, but the dense effect of L2-norm yields only correlated small reflectivity [36; 37]. Usually, seismic inversion can greatly improve the identification capability of formations compared with the original seismic data by compensating for the lacking low- and high-frequency components in the original seismic data [16; 32]. However, it may be unreasonable just through L1-norm or L2-norm regularization. This is because the low-frequency components compensated by these regularizations may not coincide with the true geological background [38]. The a priori model with effective low-frequency components can be used to serve as a regularization term to compensate for the lacked low-frequency components in the original seismic data [4]. This a priori model can be obtained from some credible prior information, such as well-log data, seismic velocity analysis, and geological knowledge. The a priori model regularization can make the low-frequency information compensated by the inversion results more consistent with the actual work area [32]. Hence, objective function (6) is updated as  
\[\min f(\mathbf{r})=||\mathbf{Wr}-\mathbf{d}||_{2}^{2}+\lambda_{1}||\mathbf{r}|| _{1}+\lambda_{2}||\mathbf{r}||_{2}^{2}+\mu||\mathbf{Cr}-\mathbf{\eta}||_{2}^{2} \tag{7}\]  
where \(\mu\) is the regularization parameter of a priori model, \(\mathbf{\eta}\) is the logarithmic impedance from a priori impedance model, \(\mathbf{C}\) is the integral operator matrix. The a priori model can be obtained from some credible prior information, such as well-log data, seismic velocity analysis information, and prior geological knowledge.