# Seismic Multi-Parameter Full-Waveform Inversion Based on Rock Physical Constraints
## 3 Results
### Numerical Experiments of the Layered Model
#### 3.2.1 Numerical Experiments Based on Total Variation Regularization
Figure 6a,b shows the velocity and density curves under section AA\({}^{\prime}\), respectively. Black solid lines represent the true model value, blue dashed lines represent the inversion results with \(\lambda_{TV}\) = 0, green dotted lines represent the inversion results with \(\lambda_{TV}\) = 1000, and red dashed lines represent the inversion results with \(\lambda_{TV}\) = 500. From the comparison curves of the velocity inversion results in Figure 6a, it can be seen that the addition of TV regularization can improve the accuracy of FWL especially for the low-velocity formation with a depth ranging from 100 m to 180 m. It can be observed from the enlarged view in Figure 6a that the inversion results under \(\lambda_{TV}\) = 500 were closest to the true model; the velocity parameter curve does not have as strong oscillations as under unconstrained conditions. However, when the regularization factor was not appropriately selected, the inversion model deviated significantly from the true values. The same conclusion can be drawn from the comparison curves of the density inversion results shown in Figure 6b. The addition of TV regularization can effectively improve the inversion accuracy, especially for the middle and shallow formation, the oscillation of the density parameter curve was the least when \(\lambda_{TV}\) = 500, and the model density value was closer to the true model. Figure 7a shows the objective function curves of the multi-parameter FWI based on TV regularization under different selections of TV regularization factors. Black solid lines represent the inversion results with \(\lambda_{TV}\) = 0, blue dashed lines represent the inversion results with \(\lambda_{TV}\) = 1000, and red dotted lines represent the inversion results with \(\lambda_{TV}\) = 500. Compared with the inversion results without TV regularization, the number of inversion iterations was larger, the objective function was smaller, and the convergence during the inversion process was relatively stable when \(\lambda_{TV}\) = 500. The value of the inversion objective function was relatively large when \(\lambda_{TV}\) = 1000. Figure 7b shows the velocity and density reconstruction error curves corresponding to the selection of TV regularization factors. The number of inversion iterations was larger, the reconstruction error of the model parameters was smaller, and the inversion was more accurate when \(\lambda_{TV}\) = 500. Figure 6: Parameter sections of the FWI results in the layered model based on TV regularization under different TV regularization factors. (**a**) Section of velocity at \(x\) = 300 m. (**b**) Section of density at \(x\) = 300 m.  
Table 2 lists the number of inversion iterations, the objective function values, and the parameter reconstruction errors of the layered model under different selections of TV regularization factors. In the three sets of numerical experiments conducted, the number of inversion iterations was larger, the inversion objective function was smaller, and the reconstruction errors of velocity and density were smaller when \(\lambda_{TV}\) = 500. \begin{table}
\begin{tabular}{c c c c} \hline
**Inversion Parameters** & \(\lambda_{TV}\) **= 0** & \(\lambda_{TV}\) **= 500** & \(\lambda_{TV}\) **= 1000** \\ \hline Number of iterations & 39 & 48 & 38 \\ Objective function & 3.7335 \(\times\) 10\({}^{8}\) & 3.4017 \(\times\) 10\({}^{8}\) & 6.8019 \(\times\) 10\({}^{8}\) \\ Reconstruction error of velocity & 0.4339 & 0.4208 & 0.4356 \\ Reconstruction error of density & 0.4066 & 0.3950 & 0.4084 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of number of iterations, objective function, and parameter reconstruction errors of the layered model based on TV regularization with different TV regularization factors. Figure 7: (**a**) Objective function curves, (**b**) reconstruction error curves of the velocity, and (**c**) reconstruction error curves of the density of the layered model based on TV regularization.