# Seismic Multi-Parameter Full-Waveform Inversion Based on Rock Physical Constraints
## 3 Results
### Numerical Experiments of the Layered Model
#### 3.2.2 Numerical Experiments Based on Lithological Constraints  
In order to validate the influence of lithology information constraints on FWI, the model depicted in Figure 3 was chosen to perform inversion experiments. FWI experiments were conducted with Equation (5) as the objective function. It was assumed that the average values and standard deviations of the physical parameters of potential rock types within the target area were acquired, presented in Equation (12). The model penalty term based on lithology information can be obtained by substituting the above prior information into Equation (7). Another critical challenge in FWI is cycle skipping [28]. To reduce the dependence of inversion on the initial model, the inversion framework proposed in this paper employs the multiscale strategy with stage-wise data injection from low to high frequencies. Multiscale inversion was carried out using a 3 Hz starting frequency, 7.5 Hz band data, and 12.5 Hz original data following Wiener low-pass filtering. The assignment of regularization factors needs to balance the contributions of the data misfit term \(Q(\mathbf{m})\) and the constraint term \(\Phi_{L}(\mathbf{m})\). Considering the orders of magnitude of \(Q(\mathbf{m})\) and \(\Phi_{L}(\mathbf{m})\), lithological regularization factors \(\lambda_{L}\) were set to 0 and \(10^{6}\), respectively, representing the unconstrained and lithological constrained inversion strategies. The obtained inversion results are illustrated in Figure 8. It can be seen from Figure 8a,c that the interface information of the middle formation can be effectively recovered in the velocity inversion model with \(\lambda_{L}\) = 0 and \(\lambda_{L}\) = \(10^{6}\). However, in terms of the formation velocity parameter values, the inversion results with constraints show closer agreement with the true model. Specifically, compared to the unconstrained inversion results, the inverted values of the deep high-velocity formation after adding constraints are closer to 3000 m/s. As is shown in Figure 8b,d, for the density inversion in the deep strata, the inversion result obtained under lithological constrained inversion was closer to the true density model. Figure 8: Inversion results of multi-parameter FWI of the layered model based on lithological constrained under different lithological regularization factors. (**a**) Velocity and (**b**) density inversion results under the unconstrained strategy (\(\lambda_{L}\) = 0). (**c**) Velocity and (**d**) density inversion results under the lithological constrained strategy (\(\lambda_{L}\) = \(10^{6}\)). Section AA\({}^{\prime}\) was selected for the purpose of comparing the inversion parameters under lithological constraint with different lithological regularization factors. Figure 9a,b presents the velocity and density curve diagrams of section AA\({}^{\prime}\). Black solid lines represent the true model, blue dotted lines show the inversion results under constrained strategy, and red dashed lines represent the inversion results under the lithological constrained strategy. According to the comparison curves of the velocity inversion results, adding lithological constraints can make the inversion parameter values closer to the true model, especially in the deep zones of the layered model. Meanwhile, higher-precision density inversion results can be obtained by introducing lithological constraints, especially in the deep zones. In this numerical experiment, the introduction of lithological constraints could enhance the inversion accuracy, especially in the deep zones of the model, and produce inversion models that are more in line with the prior information. To further analyze the influence of lithological constraints on the inversion results, a scatter plot of the velocity and density is shown in Figure 10. White circles represent the true model's results, blue circles represent the inversion results under the unconstrained strategy, and red circles represent the inversion results under the lithological constrained strategy.