# Direct Inversion Method of Brittleness Parameters Based on Reweighted Lp-Norm
## 3 Experiments
### Theoretical Model Experiment  
The model used in the theoretical experiments was part of the Marmousi2 model. To carry controlled tests, we generated a synthetic dataset that was used as observational data in the present work. Such a dataset is generated by using the same mechanism to solve the forward problem. Figure 2 shows the angle stack data volume composed of reflection coefficient and wavelet convolution. The observed data made a synthesis according to Equation (7). Figure 2a-c show the angle superposition data volumes for the incident angles of 10\({}^{\circ}\), 20\({}^{\circ}\), and 30\({}^{\circ}\), respectively. The theoretical models are shown in Figure 3, which includes three parameters: Figure 3a shows the \(Ep\) theoretical model, Figure 3b shows the Poisson's ratio model, and Figure 3c shows the density theoretical model. The models have 800 traces in the cross-sections, and each trace includes 500 sampling points. We used the traditional method and the weighted Lp method to invert the theoretical models. Figures 4a, 5a and 6a show the inversion results based on the traditional method for the \(Ep\), Poisson's ratio, and density. Figures 4b, 5b and 6b show the inversion results based on the reweighted Lp method for the \(Ep\), Poisson's ratio, and density. Compared with the theoretical models, it was demonstrated that the inversion results of the reweighted Lp method had good continuity in the horizontal orientation and clear stratigraphic boundaries in the vertical orientation. The new method could adequately reflect the geological characteristics of the models and was highly consistent with the theoretical models. Compared with the traditional method, the reweighted Lp method showed approximately the same characteristics. However, through careful observation, it was found that the reweighted Lp method had higher accuracy and better stability. In the black rectangular boxes, the reweighted Lp method had better continuity and was closer to the theoretical models, which further proved the significance of the method's improvement. Figure 1: Inversion flow. Figure 3: Theoretical models: (**a**) _Ep_: (**b**) Poisson’s ratio; (**c**) density. Figure 2: Stacked data volume with different incident angles: (**a**) \(10^{\circ}\); (**b**) \(20^{\circ}\); (**c**) \(30^{\circ}\). In order to reflect the difference between the two methods more clearly, we calculated the absolute error values of inversion for the two methods. Figures 7-9 show the absolute error values of inversion for the two methods. Figures 7a, 8a and 9a show the absolute error values of inversion for the traditional approach for the \(E\rho\), Poisson's ratio, and density, respectively. Figures 7b, 8b and 9b show the absolute error values of inversion for the reweighted Lp method for the \(E\rho\), Poisson's ratio, and density, respectively. It can be seen from the three figures that the errors of the inversion results for the proposed method were significantly reduced, and the results were more stable, especially those in the black rectangular boxes. The above experiments show that this research method could improve inversion accuracy and had important significance for improvement. Figure 4: \(E\rho\) inversion results based on (**a**) traditional method and (**b**) reweighted Lp method. Figure 5: Poisson’s ratio inversion results based on (**a**) traditional method and (**b**) reweighted Lp method. Figure 6: Density inversion results based on (**a**) traditional method and (**b**) reweighted Lp method. Then, we analyzed the inversion results of the 150 th trace and the 500 th trace. Figure 10 shows the inversion results of the 150 th trace, and Figure 11 shows the inversion results of the 500 th trace, where the black, green, blue, and red curves represent the actual model, initial model, inversion outcome of the traditional approach, and inversion outcome of the reweighted Lp method, respectively.