# Seismic Multi-Parameter Full-Waveform Inversion Based on Rock Physical Constraints
## 3 Results
### Numerical Experiments of the Layered Model
#### 3.2.3 Numerical Experiment Based on Petrophysical Relationship Constraints  
To verify the effect of petrophysical relationship constraints on multi-parameter FWI, experiments were conducted using the model shown in Figure 3. The corresponding model penalty term can be formulated as follows:  
\[\Phi_{R}(\mathbf{m})=\frac{1}{2}\underset{x}{\sum}\left(\rho-310v^{\frac{1}{4 }}\right)^{2} \tag{13}\]  
Figure 9: Parameter sections of the FWI results in the layered model based on lithological constrained inversion under different lithological regularization factors. (**a**) Section of velocity at \(x=300\) m. (**b**) Section of density at \(x=300\) m.  
The constrained FWI objective function listed in Equation (8) can be expressed as:  
\[Q_{R}(\mathbf{m})=Q(\mathbf{m})+\lambda_{R}\frac{1}{2}\underset{x}{\sum}\left( \rho-310\upsilon\frac{1}{4}\right)^{2} \tag{14}\]  
The gradient of \(\Phi_{R}(\mathbf{m})\) with respect to \(\mathbf{m}(\upsilon,\rho)\) is  
\[ \begin{split}\
abla_{\upsilon}\Phi_{R}(\mathbf{m})& =\left(\rho-310\upsilon^{\frac{1}{4}}\right)\cdot\left(-\frac{16 5}{2}\upsilon^{-\frac{3}{4}}\right)\\ \
abla_{\rho}\Phi_{R}(\mathbf{m})&=\rho-310\upsilon ^{\frac{1}{4}}\end{split}  \tag{15}\]  
The multi-parameter FWI experiments based on the L-BFGS algorithm were carried out by setting the petrophysical relationship regularization factor \(\lambda_{R}=0\) and \(\lambda_{R}=3000\), respectively. Other inversion parameters were identical to those in the experiment of 3.2.2. The resultant inversion outcomes are graphically presented in Figure 11. Upon analyzing the velocity inversion results, it becomes evident that with the incorporation of constraints, the inversion image of the middle--layer formation interface attains a higher degree of clarity. Moreover, the parameter values of the deep high-velocity layer demonstrate a closer approximation to those of the true model. When examining the density inversion results, it is observable that after the addition of petrophysical relationship constraints, the formation interfaces become more distinct. The artifact issues within the middle low-density formation are notably ameliorated. The inversion parameters of the low-density formation with depths of 110 m to 170 m and deep-seated formation show a closer proximity to the true model. Section AA\({}^{\prime}\) was selected for the comparison of inversion parameters. Figure 12a,b displays the velocity and density curves, respectively, under section AA\({}^{\prime}\). 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 the petrophysical relationship constrained strategy. From the velocity inversion results, it can be inferred that the addition of petrophysical relationship constraints can reduce the inversion oscillations, bringing the inversion parameter values closer to those of the true model and resulting in higher inversion accuracy. The same conclusion can be drawn from the density inversion results. Figure 10: Velocity–density scatter plot of the FWI results in the layered model based on lithological constrained inversion under different lithological regularization factors. distributed around the fitted line without constraints. After adding petrophysical relationship constraints, the inversion results fit in the direction approaching the fitted line, thus obtaining inversion results that were more consistent with the geological prior information. The inversion results of the above three constraint strategies (corresponding to the numerical experimental results in Section 3.2.1, Section 3.2.2 and Section 3.2.3, respectively) are compared with each other (presented in Table 3). Compared with TV regularization, the inversion strategy constrained by rock physical information features a simpler process, eliminating the need for parameter selection, and the inversion results are closer to the true model.