# Seismic Multi-Parameter Full-Waveform Inversion Based on Rock Physical Constraints
## 3 Results  
### Data  
#### 3.1.1 The Layered Model  
A layered model was established to verify the applicability of the inversion framework proposed in this paper. In assuming that the model parameter information was obtained from drilling data and laboratory experiments, it is confirmed that there are four potential rock types in the target area. The average values and standard deviations of their physical property parameters are as follows:\[v_{n(1)}^{*} =1500\;\text{m/s},\;\rho_{n(1)}^{*}=1929.2\;\text{kg/m}^{3},\;\sigma_{n (1)}=200; \tag{12}\] \[v_{n(2)}^{*} =2000\;\text{m/s},\;\rho_{n(2)}^{*}=2073.1\;\text{kg/m}^{3},\;\sigma_ {n(2)}=200;\] \[v_{n(3)}^{*} =2500\;\text{m/s},\;\rho_{n(3)}^{*}=2192\;\text{kg/m}^{3},\;\sigma_ {n(3)}=200;\] \[v_{n(4)}^{*} =3000\;\text{m/s},\;\rho_{n(4)}^{*}=2294.3\;\text{kg/m}^{3},\;\sigma_ {n(4)}=200.\]  
The prior information represented by Equation (12) is graphically illustrated in Figure 2. White solid circles represent the average values of model parameters corresponding to different lithologies, the circular regions of different colors represent the range of the standard deviation of the corresponding lithology, and the blue solid line represents the fitted line of the velocity and density. It can be seen from Figure 2 that the lithological information based on the average value and standard deviation of physical parameters, as well as the petrophysical relationship information between velocity and density, can aptly describe the relationship between seismic wave velocity and medium density. Consequently, the prior information represented by Equation (12) can be employed as constraint terms in the multi-parameter FWI of seismic waves.  
Figure 2: The prior information of the layered model in the original domain and logarithmic domain. (**a**) Original domain with colored circles denoting distinct rock types: the red circle represents Type I rocks with statistical parameters (\(v_{n(1)}^{*}\)), \(\rho_{n(1)}^{*}\), \(\sigma_{n(1)}^{*}\)) corresponding to Equation (12); the blue circle represents Type II rocks with statistical parameters (\(v_{n(2)}^{*}\), \(\rho_{n(2)}^{*}\), \(\sigma_{n(2)}^{*}\)) corresponding to Equation (12); the orange circle represents Type III with statistical parameters (\(v_{n(3)}^{*}\), \(\rho_{n(3)}^{*}\), \(\sigma_{n(3)}^{*}\)) corresponding to Equation (12); and the green circle represents Type IV rocks with statistical parameters (\(v_{n(4)}^{*}\), \(\rho_{n(4)}^{*}\)) corresponding to Equation (12). (**b**) Logarithmic domain.  
The velocity model and density model constructed from the above data are shown in Figure 3a,b. The layered model comprises five strata with increasing velocity and density trends at greater depths as shown in Figure 3a,b. Two basal strata exhibit normal faults, inducing displacement and interface undulations between adjacent layers. When conducting multi-parameter inversion, homogeneous geological models were constructed as the initial model to validate the reliability of the proposed algorithm. Referencing the top-layer parameters of the layered model, the initial velocity and density were set to 2250 m/s and 2135 kg/m\({}^{3}\), respectively, as shown in Figure 3c,d.