# Seismic Multi-Parameter Full-Waveform Inversion Based on Rock Physical Constraints
## 4 Conclusions  
To mitigate the ill-posed nature of the multi-parameter FWI of seismic waves, this paper proposes a rock physical constraint inversion framework. The proposed framework synergistically combines the following: lithological constraints and petrophysical relationship constraints embedded into the FWI objective function through regularization terms, the multiscale inversion strategy to reduce the cycle skipping problem by progressively incorporating higher-frequency data components, and the L-BFGS optimization algorithm to enhance convergence efficiency. The inversion results of the simple layered model show that the proposed rock physical constraint inversion strategies based on lithological information and the relational expressions of physical property parameters can enhance the stability of inversion, reduce the parameter oscillation, and obtain multi-parameter inversion results with higher accuracy. Compared with unconstrained FWI, the constrained inversion framework proposed in this study can improve the density inversion accuracy without reducing the velocity inversion accuracy, thereby both the reconstruction error of velocity and density were reduced. The inversion results of the standard geological model 1994 BP indicate that the multi-parameter inversion strategy of seismic waves based on rock physical constraints has good applicability to complex models. While the proposed FWI framework demonstrates compatibility with diverse prior constraints, its current implementation is restricted to lithological and petrophysical rela \begin{table}
\begin{tabular}{c c c} \hline \hline
**Inversion Parameters** & **Unconstrained Strategy** & **Petrophysical Relationship Constrained Strategy** \\ \hline Number of iterations & 152 & 302 \\ Objective function & 7.0867 \(\times\) 10\({}^{9}\) & 1.2560 \(\times\) 10\({}^{9}\) \\ Reconstruction error of velocity & 0.4269 & 0.3575 \\ Reconstruction error of density & 0.6454 & 0.4869 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Number of iterations, and the objective function and reconstruction errors of the 1994 BP model under the unconstrained strategy and petrophysical relationship constraints. Figure 16: Velocity–density scatter plot based on the petrophysical relationship constrained inversion of the 1994 BP model. tionship constraints. Future work will focus on other types of constraints to improve the effect of inversion. Meanwhile, measured data will be used to verify the applicability of the proposed FWI framework. Conceptualization, C.C. and D.F. ; methodology, X.W. ; software, C.C. ; validation, C.C. ; formal analysis, X.W. ; investigation, X.W. ; resources, D.F. and J.T. ; data curation, X.W. ; writing--original draft preparation, C.C. ; writing--review and editing, D.F. ; visualization, C.C. ; supervision, D.F. ; funding acquisition, J.T. and D.F. All authors have read and agreed to the published version of the manuscript. This research was funded in part by the National Natural Science Foundation of China (NSFC) under Grants 42474196, 42104143, and 42474191 and the Natural Science Foundation of Hunan Province, China, under Grant 2025 JJ40033. Not applicable. The data presented in this study are available on request from the corresponding author. The data are not publicly available due to the data are part of an ongoing study. We would like to thank [PERSON] for his valuable advice on integrating geological exploration data with seismic data, and [PERSON] for his technical assistance in refining the layered model construction. We also acknowledge the support from the High Performance Computing Center of Central South University. Authors [PERSON] and [PERSON] were employed by the company Power China Zhongnan Engineering Corporation Limited and declare no conflicts of interest. [PERSON] received a research grant from the National Natural Science Foundation of China (NSFC) under Grant 42474191.