# Seismic Multi-Parameter Full-Waveform Inversion Based on Rock Physical Constraints
## 3 Results
### Numerical Experiments of the Layered Model
#### 3.2.4 Numerical Experiment of Standard Geological Model  
In the development of energy resources, the acquisition of logging information is more convenient than that of laboratory analysis results, and the amount of logging data is larger. Therefore, prior information containing petrophysical relationships is of greater practical significance than lithological information in inversion. In this study, numerical experiments were carried out with the 1994 BP model to verify the applicability of the petrophysical relationship constrained inversion strategy proposed in this paper. Multiscale inversion was carried out using a 10 Hz starting frequency, 15 Hz band data, and 40 Hz original data following Wiener low-pass filtering. The inversion results obtained by setting the regularization factor \(\lambda_{R}\) to 0 and 500, respectively, are shown in Figure 14. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Inversion Parameters** & **TV Regularization** & **Lithological** & **Petrophysical Relationship** \\  & **Strategy** & **Constrained Strategy** & **Constrained Strategy** \\ \hline Number of iterations & 48 & 61 & 87 \\ Objective function & \(3.4017\times 10^{8}\) & \(1.5627\times 10^{8}\) & \(1.0651\times 10^{8}\) \\ Reconstruction error of velocity & 0.4208 & 0.3947 & 0.4026 \\ Reconstruction error of density & 0.3950 & 0.3923 & 0.3476 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The number of iterations, objective function, and reconstruction error of the layered model under different inversion strategies. Figure 13: Velocity–density scatter plot of the FWI results in the layered model based on petrophysical relationship constraints under different petrophysical relationship regularization factors. The velocity inversion results demonstrate that the constrained inversion strategy proposed in this paper improved the continuity of middle formations and achieved closer alignment with the true-velocity model in the middle and deep zones. In the density inversion results, under unconstrained inversion, there are many inversion artifacts in the middle zone of the model, especially in the low-density formations below the relatively high-density formation. After adding constraints, the artifact phenomenon of the middle formation improved, and the interfaces and parameter values of the deep high-density formation were closer to those of the true model. A horizontal cross section of the 1994 BP model (referring to the \(z=250\) m section indicated by the white dashed line AA\({}^{\prime}\) in Figure 4a) and a vertical cross section (referring to the \(x=800\) m section indicated by the white dashed line BB\({}^{\prime}\) in Figure 4a) were selected for the comparative analysis of velocity and density, and the inversion results are shown in Figure 15a,c and Figure 15b,d, respectively. Black solid lines represent the true model results, blue dotted lines show the inversion results under the unconstrained strategy, and red dashed lines represent the results under petrophysical relationship constraints. The accuracy of velocity inversion under these two inversion strategies was comparable. In the regions at both lateral sides and in the middle-deep part of the model, the results obtained after incorporating petrophysical relationship constraints were more congruent with the true model. After the addition of petrophysical relationship constraints, the density inversion accuracy significantly improved, with local oscillations mitigated, and the model values closer to those of the true model. Figure 14: Inversion results of the 1994 BP model based on petrophysical relationship constraints under different petrophysical relationship regularization factors. (**a**) Velocity and (**b**) density inversion results under unconstrained strategy (\(\lambda_{R}=0\)). (**c**) Velocity and (**d**) density inversion results under petrophysical relationship constrained strategy (\(\lambda_{R}=500\)).