# An Application of Elastic-Net Regularized Linear Inverse Problem in Seismic Data Inversion
## 3 Synthetic Data Tests  
We first use a synthetic seismic trace to test the effects of inversion with elastic-net regularization and its differences compared to conventional inversion methods. The synthetic trace is shown in Figure 2a, which is obtained from the convolution between a 40 Hz Ricker wavelet and the reflectivity series shown in Figure 1a. Then, three different regularized inversion methods are performed on this synthetic trace. The first one is an inversion with elastic-net regularization (ENRI), the second one is an inversion with L1-norm regularization (L1 RI), and the last one is an inversion with L2-norm regularization (L2 RI). The initial solution of inversion is obtained by smoothing the actual impedance well log through a high-cut filter with a threshold value of 15 Hz. In addition, this low-frequency trend log is served as the a priori model. The inverted reflectivity series traces and impedances from these three inversion methods are plotted in Figures 3-5, respectively. In Figures 3-5, the black curves are the actual reflectivity series or impedances well log, and the red curves are the inverted reflectivity series or impedances. Figure 3: The inverted reflectivity series and impedance from L2 RI performed on the noise-free synthetic trace. (**a**) Reflectivity series; (**b**) impedance. Figure 2: The synthetic seismic trace from the convolution between a 40 Hz Ricker wavelet and the reflectivity series shown in Figure 1a. (**a**) Noise-free synthetic trace; (**b**) noise-contaminated trace with SNR = 4 (SNR means the ratio of signal to noise); (**c**) noise-contaminated trace with SNR = 1. Figure 4: The inverted reflectivity series and impedances from L1R1 performed on the noise-free synthetic trace. (**a**) Reflectivity series; (**b**) impedance. Figure 5: The inverted reflectivity series and impedances from ENRI performed on the noise-free synthetic trace. (**a**) Reflectivity series; (**b**) impedance. From Figures 3-5, we can see that, generally speaking, all of the inverted reflectivity series can match with the actual reflectivity series, and all of the inverted impedances follow the relative trend of the actual well log very nicely. However, the inverted reflectivity series and impedances by the three inversion methods are very different in detail. First, the L2 RI swings some major large reflectivity series to make the inverted reflectivity series over oscillate. Hence, the inverted impedance by L2 RI is over oscillating, too. The formation boundary of the main stratigraphic sequence interface is not precise. Second, compared to the L2 RI, the L1 RI can obtain more blocky impedance. Additionally, the inverted reflectivity series is sparse with high resolution. However, many small reflectivity series are over suppressed. Hence, the inverted impedance corresponding to weak reflectors is underestimated. Furthermore, some interbeds are completely suppressed. Last, in the inverted reflectivity series by ENRI, both large and small reflectivity series are all recovered well. Hence, the inverted impedance by ENRI is the best match compared with the actual impedance well log. We calculate the relative errors (RE) of the inverted reflectivity series by different inversion methods through the following formula  
\[RE=\frac{||\mathbf{r}-\mathbf{r}_{\text{inverted}}||_{2}^{2}}{||\mathbf{r}||_{ 2}^{2}} \tag{9}\]  
where \(\mathbf{r}_{\text{inverted}}\) is the inverted reflectivity series, and \(\mathbf{r}\) is the actual reflectivity series. The three calculated REs of the inverted reflectivity series shown in Figures 3-5 are listed in Table 1. The RE for ENRI is much smaller than the other two inversion methods. Next, the synthetic seismic trace is added to different level zero-mean Gaussian random noise to test the anti-noise performance of ENRI. Figure 2b,c show the synthetic seismic trace with SNR = 4, and the synthetic seismic trace with SNR = 1, respectively. Then, ENRI is performed on these two noise-contaminated synthetic traces.