# 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  
To validate the influence of TV regularization on FWI, the model shown in Figure 3 was selected to carry out TV regularization inversion experiments. In Figure 3a,b, the black crosses denote the positions of the source points, while the red circles signify the positions of the receiver points. Both the source and receiver points were situated at a depth of \(z=50\) m. Spanning from \(x=0\) m to 600 m, a source point was established every 20 m, amounting to a total of 30 shot points. A receiver point was set every 10 m, and a total of 60 channels of data were collected. The forward simulation was carried out with a 12.5 Hz Ricker wavelet, a time interval of 0.5 ms, and a total simulation duration of 0.6 s. The initial model was a homogeneous medium model with a velocity of 2250 m/s and a density of 2135 kg/m\({}^{3}\), as illustrated in Figure 3c,d. There were three termination criteria for the inversion iteration: first, the total number of iterations is 300; second, the ratio of the relative objective function \(Q_{k}/Q_{0}\) is less than \(10^{-3}\); third, the step size is smaller than \(10^{-5}\). Equation (5) was used as the objective function to conduct the FWI experiment based on L-BFGS (details are provided in Appendix A). When conducting the TV regularization experiments, regularization factor \(\lambda_{TV}\) was determined through manual adjustment. To identify an optimal regularization factor, this study carried out model experiments under different TV regularization factors while maintaining other inversion parameters constant. The model experiments were the following: without loading TV regularization (\(\lambda_{TV}=0\)), with \(\lambda_{TV}=500\), and with \(\lambda_{TV}=1000\). The inversion results obtained are shown in Figure 5.  
Figure 4: True models and initial models of 1994 BP model. (**a**) True model of velocity of 1994 BP model. (**b**) True model of density of 1994 BP model. (**c**) Initial model of velocity of 1994 BP model. (**d**) Initial model of density of 1994 BP model. From the velocity inversion results without regularization, the interfaces of middle and deep formations were indistinct, particularly the top boundary of the bottom high-velocity formation, which is difficult to identify in the inversion result. In the density inversion result, the inversion outcomes on both sides of the middle low-density formation exhibit significant deviation from the true model, and the top boundary of the deep high-density formation is also obscure. After the implementation of TV regularization, the interface of the low-velocity formation was clear under \(\lambda_{TV}\) = 500, and the model parameters showed a closer resemblance to those of the true model. The density inversion results also yield analogous conclusions. In the density inversion result, formation boundaries can be distinguished more distinctly, and the density value is closer to that of the true model. Figure 5: Inversion results of multi-parameter FWI of layered model based on TV regularization under different TV regularization factors. (**a**) Velocity and (**b**) density inversion results with \(\lambda_{TV}\) = 0. (**c**) Velocity and (**d**) density inversion results with \(\lambda_{TV}\) = 500. (**e**) Velocity and (**f**) density inversion results with \(\lambda_{TV}\) = 1000.  
model. When \(\lambda_{TV}\) was set to 1000, the contour of the high-velocity body and the inversion accuracy of parameters were relatively unsatisfactory in the velocity inversion results. The reconstructed parameter values of the high-density formation in the middle and deep parts were remarkably close to those of the true model, though the shallow formation density inversion result still shows deviations from the true model. A cross section (the black dashed line AA\({}^{\prime}\) in Figure 3a) was selected for the comparative analysis of model parameters.